experimental investigation of the aerodynamics of a wing in a slipstream . an experimental study of a wing in a propeller slipstream was made in order to determine the spanwise distribution of the lift increase due to slipstream at different angles of attack of the wing and at different free stream to slipstream velocity ratios . the results were intended in part as an evaluation basis for different theoretical treatments of this problem . the comparative span loading curves, together with supporting evidence, showed that a substantial part of the lift increment produced by the slipstream was due to a /destalling/ or boundary-layer-control effect . the integrated remaining lift increment, after subtracting this destalling lift, was found to agree well with a potential flow theory . an empirical evaluation of the destalling effects was made for the specific configuration of the experiment .
simple shear flow past a flat plate in an incompressible fluid of small viscosity . in the study of high-speed viscous flow past a two-dimensional body it is usually necessary to consider a curved shock wave emitting from the nose or leading edge of the body . consequently, there exists an inviscid rotational flow region between the shock wave and the boundary layer . such a situation arises, for instance, in the study of the hypersonic viscous flow past a flat plate . the situation is somewhat different from prandtl’s classical boundary-layer problem . in prandtl’s original problem the inviscid free stream outside the boundary layer is irrotational while in a hypersonic boundary-layer problem the inviscid free stream must be considered as rotational . the possible effects of vorticity have been recently discussed by ferri and libby . in the present paper, the simple shear flow past a flat plate in a fluid of small viscosity is investigated . it can be shown that this problem can again be treated by the boundary-layer approximation, the only novel feature being that the free stream has a constant vorticity . the discussion here is restricted to two-dimensional incompressible steady flow .
the boundary layer in simple shear flow past a flat plate . the boundary-layer equations are presented for steady incompressible flow with no pressure gradient .
approximate solutions of the incompressible laminar boundary layer equations for a plate in shear flow . the two-dimensional steady boundary-layer problem for a flat plate in a shear flow of incompressible fluid is considered . solutions for the boundary- layer thickness, skin friction, and the velocity distribution in the boundary layer are obtained by the karman-pohlhausen technique . comparison with the boundary layer of a uniform flow has also been made to show the effect of vorticity .
one-dimensional transient heat conduction into a double-layer slab subjected to a linear heat input for a small time internal . analytic solutions are presented for the transient heat conduction in composite slabs exposed at one surface to a triangular heat rate . this type of heating rate may occur, for example, during aerodynamic heating .
one-dimensional transient heat flow in a multilayer slab . in a recent contribution to the readers’ forum wassermann gave analytic solutions for the temperature in a double layer slab, with a triangular heat rate input at one face, insulated at the other, and with no thermal resistance at the interface . his solutions were for the three particular cases.. i propose here to give the general solution to this problem, to indicate briefly how it is obtained using the method of reference 2, and to point out that the solutions given by wassermann are incomplete for times longer than the duration of the heat input .
the effect of controlled three-dimensional roughness on boundary layer transition at supersonic speeds . experiments were performed in the 12-in. supersonic wind tunnel of the jet propulsion laboratory of the california institute of technology to investigate the effect of three-dimensional roughness elements (spheres) on boundary-layer transition on a tained at local mach numbers of 1.90, 2.71, and 3.67 by varying trip size, position, spacing, and reynolds number per inch . the results indicate that (1) transition from laminar to turbulent flow induced by three-dimensional roughness elements begins when the double row of spiral vortices trailing each element contaminates and breaks down the surrounding field of vorticity, (2) transition appears rather suddenly, becoming more violent with increasing roughness height relative to the boundary-layer thickness, (3) after the breakdown of the vorticity field, the strength of the spiral vortices may still persist in the sublayer of the ensuing turbulent flow, (4) lateral spacing of roughness elements has little effect upon the initial breakdown (contamination) of the laminar flow, and (5) the trip reynolds number where u and v are the velocity and kinematic viscosity at the outer edge of the boundary layer and k is roughness height, such that transition occurs at the roughness position, varies as the position reynolds number to the one-fourth power, viz., where x is trip position .
measurements of the effect of two-dimensional and three-dimensional roughness elements on boundary layer transition . in his study of the effect of roughness on transition, h. l. dryden found, on the basis of available data, that the effect of a two-dimensional roughness element such as a /trip wire/ could be represented reasonably well in terms of a functional relation between and, where is the reynolds number of transition based on distance from the leading edge, is the height of the roughness element, and is the boundary-layer displacement thickness at the position of the element . at his suggestion some additional data were obtained, primarily to extend the range to higher values of, during the course of an investigation of transition on a flat plate conducted at the national bureau of standards . after the results on the two- dimensional roughness elements were obtained, it appeared to be desirable to see whether a row of three-dimensional roughness elements would behave in the same way .
transition studies and skin friction measurements on an insulated flat plate at a mach number of 5.8 . an investigation of transition and skin friction on an insulated flat plate, 5 by 26 in., was made in the galcit 5 by 5 in. hypersonic wind tunnel at a nominal mach number of 5.8 . the phosphorescent lacquer technique was used for transition detection and was found to be in good agreement with total-head rake measurements along the plate surface and pitot boundary- layer surveys . it was found that the boundary layer was laminar at reynolds numbers of at least 5 x 10 . transverse contamination caused by the turbulent boundary layer on the tunnel sidewall originated far downstream of the flat plate leading edge at reynolds numbers of 1.5 to 2 x 10, and spread at a uniform angle of 5 compared to 9 degree in low-speed flow . the effect of two-dimensional and local disturbances was investigated . the technique of air injection into the boundary layer as a means of hastening transition was extensively used . although the onset of transition occurred at reynolds numbers as low as 10, a fully developed turbulent boundary layer was not obtained at reynolds numbers much below 2 x 10 regardless of the amount of air injected . a qualitative discussion of these results is given with emphasis on the possibility of a greater stability of the laminar boundary layer in hypersonic flow than at lower speeds . direct skin-friction measurements were made by means of the floating element technique, over a range of reynolds numbers verified as being laminar over the complete range . with air injection, turbulent shear was obtained only for reynolds numbers greater than 2 x 10, this value being in good agreement with earlier results of this investigation . the turbulent skin-friction coefficient was found to be approximately 0.40 of that for incompressible flow for a constant value of r, and 0.46 for an effective reynolds number between 5 and 6 x 10 .
the theory of the impact tube at low pressure . a theoretical analysis has been made for an impact tube of the relation between free-stream mach number and the impact and free-stream pressures and densities for extremely low pressures . it is shown that the results differ appreciably from the corresponding continuum relations .
similar solutions in compressible laminar free mixing problems . there are in supersonic aerodynamics many situations of practical interest wherein streams of different velocities and, in general, different stagnation pressures mix with one another . in the majority of these problems the interaction between the two streams takes place in the presence of an axial pressure gradient . its effect on the characteristics of the mixing may influence significantly the performances of the devices wherein the phenomena cited above occur . a theoretical and experimental program of research to study mixing in the presence of axial pressure gradients is being carried on at the polytechnic institute of brooklyn .
some structural and aerelastic considerations of high speed flight . the dominating factors in structural design of high-speed aircraft are thermal and aeroelastic in origin . the subject matter is concerned largely with a discussion of these factors and their interrelation with one another . a summary is presented of some of the analytical and experimental tools available to aeronautical engineers to meet the demands of high-speed flight upon aircraft structures . the state of the art with respect to heat transfer from the boundary layer into the structure, modes of failure under combined load as well as thermal inputs and acrothermoelasticity is discussed . methods of attacking and alleviating structural and aeroelastic problems of high-speed flight are summarized . finally, some avenues of fundamental research are suggested .
similarity laws for stressing heated wings . it will be shown that the differential equations for a heated plate with large temperature gradient and for a similar plate at constant temperature can be made the same by a proper modification of the thickness and the loading for the isothermal plate . this fact leads to the result that the stresses in the heated plate can be calculated from measured strains on the unheated plate by a series of relations, called the /similarity laws ./ the application of this analog theory to solid wings under aerodynamic heating is discussed in detail . the loading on the unheated analog wing is, however, complicated and involves the novel concept of feedback and /body force/ loading . the problem of stressing a heated box-wing structure can be solved by the same analog method and is briefly discussed .
piston theory – a new aerodynamic tool for the aeroelastician . representative applications are described which illustrate the extent to which simplifications in the solutions of high-speed unsteady aeroelastic problems can be achieved through the use of certain aerodynamic techniques known collectively as /piston theory ./ based on a physical model originally proposed by hayes and lighthill, piston theory for airfoils and finite wings has been systematically developed by landahl, utilizing expansions in powers of the thickness ratio and the inverse of the flight mach number m . when contributions of orders and are negligible, the theory predicts a point-function relationship between the local pressure on the surface of a wing and the normal component of fluid velocity produced by the wing’s motion . the computation of generalized forces in aeroelastic equations, such as the flutter determinant, is then always reduced to elementary integrations of the assumed modes of motion . essentially closed-form solutions are given for the bending- torsion and control-surface flutter properties of typical section airfoils at high mach numbers . these agree well with results of more exact theories wherever comparisons can be fairly made . moreover, they demonstrate the increasingly important influence of thickness and profile shape as m grows larger, a discovery that would be almost impossible using other available aerodynamic tools . the complexity of more practical flutter analyses-e.g., on three-dimensional wings and panels-is shown to be substantially reduced by piston theory . an iterative procedure is outlined, by which improved flutter eigenvalues can be found through the successive introduction of higher-order terms in and . other applications to unsteady supersonic problems are reviewed, including gust response and rapid maneuvers of elastic aircraft . steady-state aeroelastic calculations are also discussed, but for them piston theory amounts only to a slight modification of ackeret’s formulas . suggestions are made regarding future research based on the new aerodynamic method, with particular emphasis on areas where computational labor can be reduced with a minimum loss of precision . it is pointed out that a mach number zone exists where thermal effects are appreciable but nonlinear viscous interactions may be neglected, and that in this zone piston theory is the logical way of estimating air loads when analyzing aerodynamic- thermoelastic interaction problems .
on two-dimensional panel flutter . theory and experiments of the flutter of a buckled plate are discussed . it is shown that an increase in the initial deviation from flatness or a static pressure differential across the plate raises the critical value of the /reduced velocity ./ the applicability of the galerkin method to the linearized problem of flutter of an unbuckled plate has been questioned by several authors . in this paper the flutter condition was formulated in the form of an integral equation and solved numerically by the method of iteration and the method of matrix approximations, thus avoiding the constraint of assumed modes . for a plate (with finite bending rigidity) the results confirm those given by the galerkin method . an approximate analysis of the limiting form and amplitude of the flutter motion for a buckled plate is presented .
transformation of the compressible turbulent boundary layer . the transformation of the compressible turbulent boundary- layer equations to their incompressible equivalent is demonstrated analytically . the transformation is essentially the same as that for the laminar layer, first given by stewartson, except that the explicit relation between the viscosity and temperature is not required . a key point in the analysis is the modification of the stream function to include a mean of the fluctuating components and the postulate that the apparent turbulent shear, associated with an elemental mass, remains invariant in the transformation . the values of the incompressible friction coefficients and of pressure rise causing separation thus transformed show good agreement with the experimentally measured and independently reported results . an application of the transformation to the self-preserving boundary layers and to the computations of general boundary-layer flow is shown .
remarks on the eddy viscosity in compressible mixing flows . in connection with a study of the wakes behind bodies in hypersonic flow carried out for the missile and space vehicle division of the general electric company, it was desired to estimate the eddy viscosity in axisymmetric, compressible wakes . because of the lack of applicable experimental data, it was found necessary to make such an estimate by rationally extending the few available data for incompressible flows to the compressible case . this suggested the application and extension of the transformations applied to turbulent boundary layers in reference infinitesimal mass are invariant with transformation, mager showed that the partial differential equations for the compressible turbulent boundary layer can be transformed to incompressible form . the validity of this assumption and of the transformations was established for several boundary-layer flows by comparison with experiment .
the flow field in the diffuser of a radial compressor . this note discusses the two-dimensional diffuser flow field in a radial compressor outside the impeller wheel . it is assumed that the diffuser has guide vanes arranged in a circular row at a radius . the impeller wheel has the radius (see fig. 1) . the flow in the diffuser starts at the circle with the radius . the velocity components, and in the r and directions of the velocity vector on this circle are prescribed together with the thermal state of the gas . the flow so prescribed on the radius will, if no disturbances are present (i.e., no boundary conditions in the flow other than zero velocity at infinity are to be fulfilled), develop in a spiral flow .
an investigation of the pressure distribution on conical bodies in hypersonic flows . a large amount of work on conical flow fields without axial symmetry at supersonic speed is presently available . however, no apparent hypersonic approximation has yet been derived . in this note, experimental data on two elliptical cones at m = 6 are presented and a hypersonic approach obtained from physical considerations is suggested .
generalised-newtonian theory . author generalizes lees’s (amr 10(1957), rev. 2601) modification of newtonian theory for blunt-nose bodies to apply to pointed- nose bodies as well . the result is expressed by sin where is the local inclination of the body surface and the subscript /max/ refers to the maximum local inclination and pressure coefficient . for blunt-nose bodies and the generalized theory reverts to lees’s blunt-nose modification with given by normal shock relations . author shows, by comparison of newtonian and generalized-newtonian theory with exact solutions, the superiority of generalized-newtonian theory . he also shows that both two-dimensional and axisymmetric shapes are correlated by this generalization . results are presented in two figures that support author’s generalization and indicate the independence of the correlation from variations in both the hypersonic similarity parameter k = m(d1) and the ratio of specific heats y . reviewer believes this generalization should be of interest to those engaged in development of hypersonic hardware as well as theory .
on heat transfer in slip flow . a number of authors have considered the effect of slip on the heat transfer and skin friction in a laminar boundary layer over a flat plate . reference 1 considers this by a perturbation on the usual laminar boundary-layer analysis while some other studies.dash e.g., reference the impulsive motion of an infinite plate .
on slip-flow heat transfer to a flat plate . assuming that continuum flow energy equation in a boundary layer remains valid well into slip region and taking account of the temperature jump in a moving rarefied gas and for influence of large mean free path through appropriate boundary conditions, a solution is found for the temperature gradient in the slip region . then from maslen expression (j. aero. sci. 25, 6, 400-401, june slipping fluid to a flat plate, and behavior confirms results for small values of knudsen number .
skin-friction and heat transfer characteristics of a laminar boundary layer on a cylinder in axial incompressible flow . a solution is given for the case of the laminar boundary layer of an incompressible fluid of constant properties on the exterior of a cylinder with flow parallel to the cylinder axis . this case differs from the blasius solution for flow along a flat plate by considering the effect of the curvature in a plane transverse to the flow direction . the local skin-friction and heat-transfer coefficients for a prandtl number of 0.715 are evaluated and compared to the similar magnitudes for flat plate flow, and the effect of the curvature is shown to be significant in some practical cases . recovery factors are evaluated, and this quantity is found to be insensitive to the effect of curvature of the boundary .
theory of stagnation point heat transfer in dissociated air . the boundary-layer equations are developed in general for the case of very high speed flight where the external flow is in a dissociated state . in particular the effects of diffusion and of atom recombination in the boundary layer are included . it is shown that at the stagnation point the equations can be reduced exactly to a set of nonlinear ordinary differential equations even when the chemical reactions proceed so slowly that the boundary layer is not in thermochemical equilibrium . two methods of numerical solution of these stagnation point equations are presented, one for the equilibrium case and the other for the nonequilibrium case . numerical results are correlated in terms of the parameters entering the numerical formulation so as not to depend critically on the physical assumptions made . for the nonequilibrium boundary layer, both catalytic (to atom recombination) and noncatalytic wall surfaces are considered . a solution is represented which shows the transition from the /frozen/ boundary layer (very slow recombination rates) to the equilibrium boundary layer (fast recombination rates) . a recombination rate parameter is introduced to interpret the nonequilibrium results, and it is shown that a scale factor is involved in relating the equilibrium state of a boundary layer on bodies of different sizes . it is concluded that the heat transfer through the equilibrium stagnation point boundary layer can be computed accurately by a simple correlation formula and that the heat transfer is almost unaffected by a nonequilibrium state of the boundary layer provided the wall is catalytic and the lewis number near unity .
inviscid hypersonic flow over blunt-nosed slender bodies . at hypersonic speeds the drag area of a blunt nose is much larger than the drag area of a slender afterbody, and the energy contained in the flow field in a plane at right angles to the flight direction is nearly constant over a downstream distance many times greater than the characteristic nose dimension . the transverse flow field exhibits certain similarity properties directly analogous to the flow similarity behind an intense blast wave found by g. i. taylor, s. c. lin, and a. sakurai . a comparison with the experiments of hammitt, vas, and bogdonoff on a flat plate with a blunt leading edge at in helium shows that the shock-wave shape is predicted very accurately by this similarity analysis . the predicted surface pressure distribution is somewhat less satisfactory . experimental results on a hemisphere-cylinder obtained at in the galcit air tunnel indicate that not only the shock-wave shape but also the surface pressures for this body are given very closely by the similarity theory, except near the hemisphere-cylinder junction . energy considerations combined with a detailed study of the equations of motion show that flow similarity is also possible for a class of bodies of the form, provided that, where for a two-dimensional body and for a body of revolution . when the shock shape is not similar to the body shape, and the entire flow field some distance from the nose must depend to some extent on the details of the nose geometry . by again utilizing energy and drag considerations one finds that at hypersonic speeds the inviscid surface pressures generated by a blunt leading edge are larger than the pressures induced by boundary-layer growth on an insulated flat surface for an insulated blunt-nosed slender body of revolution the corresponding distance is given by . (here is free-stream reynolds number based on leading-edge thickness, or nose diameter .) in free flight these constants are replaced by 1,700 and 20, respectively, so that viscous interaction effects are important over the forward portion of a blunt- nosed slender body only for relatively low values of . however, /far downstream/ of the nose the inviscid over-pressure is small and viscous interaction phenomena will have to be taken into account .
inviscid leading-edge effect in hypersonic flow . current interest in the problem of inviscid-viscous interaction has led to the realization of the significant effect of the leading-edge thickness in hypersonic flow . the purpose of this note is to give an account of the downstream influence of the blunt leading edge on the basis of the hypersonic small perturbation theory .
newtonian flow theory for slender bodies . as an aid to the aerodynamieist in the design of air frames for hypersonic speeds (speeds faster than about mach 5), newtonian flow theory is examined from the point of view of gas dynamics and hypersonic small-disturbance theory . the usual theory is shown to result as the first approximation of an expansion valid for small . a basic similarity parameter is introduced . a general solution of the first approximation for the flow past slender bodies (bodies which cause only a small disturbance to the stream) at zero angle of attack is given . an important condition which limits the application of the theory is noted-namely, that the pressure coefficient on the surface not fall to zero . the theory is then applied to cones and to bodies whose shape is .
a note on the explosion solution of sedov with application to the newtonian theory of unsteady hypersonic flow . an exact analytical solution of the equations of inviscid compressible unsteady flow has been given by sedov (reference to the solution may be made through hayes and probstein) . this solution is the similarity solution for a constant-energy point explosion . in view of the recent work on problems of hypersonic flow in the limiting form of the ratio of specific heats near 1 solution in this limit and inquire what form such a solution would take . einbinder, in a recent note, has examined the solution for various but does not mention the interesting case of . it may be shown that the convergence to the limit is nonuniform over the flow field . it is also not difficult to show that the non- uniform behavior exhibited here is that which one would expect from the newtonian formulation as derived in reference 3 .
a simple model study of transient temperature and thermal stress distribution due to aerodynamic heating . the present work is concerned with the determination of transient temperatures and thermal stresses in simple models intended to simulate parts or the whole of an aircraft structure of the built- up variety subjected to aerodynamic heating . the first case considered is that of convective heat transfer into one side of a flat plate, representing a thick skin, and the effect of the resulting temperature distribution in inducing thermal stresses associated with bending restraint at the plate edges . numerical results are presented for the transient temperature differentials in the plate when the environment temperature first increases linearly with time and then remains constant, the period of linear increase representing the time of acceleration of the aircraft . corresponding thermal stress information is presented . the second case is that of the wide-flanged i-beam with convective heat transfer into the outer faces of the flanges . numerical results are presented for transient temperature differentials for a wide range of values of the applicable parameters and for an environment temperature variation as described above . corresponding thermal stresses in a beam of infinite length are determined . a theoretical analysis of the stress distribution in a beam of finite length is carried out and numerical results obtained for one case . an experimental investigation of temperatures and stresses in such a beam is described, and results are presented which indicate good agreement with corresponding theoretical results .
photo-thermoelastic investigation of transient thermal stresses in a multiweb wing structure . photothermoelastic experiments were performed on a long multiweb wing model for which a theoretical analysis is available in the literature . the experimental procedures utilized to simulate the conditions prescribed in the theory are fully described . correlation of theory and experiment in terms of dimensionless temperature, stress, time, and biot number revealed that the theory predicted values higher than the experimentally observed maximum thermal stresses at the center of the web . detailed temperature measurements in the flange suggested that the major source of this discrepancy can be traced to the one-dimensional heat conduction analysis of the flange employed in the theory .
thermal buckling of supersonic wing panels . the temperature and thermal stress distributions are analyzed in multicellular supersonic wing structures . a buckling criterion is established for the panels of cover plates subjected to thermal stresses .
the dynamic motion of a missile descending through the atmosphere . a method is presented for computing rapidly, yet accurately, the dynamic motion of a ballistic-type missile descending through the atmosphere . the equations of motion are separated into a set of /static/ trajectory equations (zero angle of attack) and a set of /rotational/ equations describing the oscillatory motion of the missile about its center of gravity . a transformation allows the rotational equations to be written in a manner analogous to the equation for an undamped oscillating spring mass system with the mass equal to unity and a time variable spring constant . for given initial conditions this equation can be solved to obtain the envelope of maximum angle of attack . an additional transformation allows the calculation of the complete oscillatory motion at any time during the trajectory as a function of the maximum angle of attack at that time . this solution shows that the maximum angle of attack of a missile descending through the atmosphere at relatively constant speed is reduced even when the aerodynamic damping is neglected .
the prospects for magneto-aerodynamics . the equations describing the flow of an electrically conducting fluid in the presence of electric and magnetic fields are written down with the aid of certain simplifications appropriate to aeronautical applications . in order to estimate the probable significance of magneto-aerodynamic effects, some data on conductivity of pure and /seeded/ air are first examined . dimensionless quantities representing the ratios of forces and of currents are then formed and their values studied for conditions of flight in the atmosphere . some examples of magneto-hydrodynamic and magneto- gasdynamic effects in simple flows are given . these include two cases of poiscuille flow of conducting liquids with applied magnetic fields and the case of quasi-one-dimensional gas flow with applied electrical and magnetic fields . in the last case, attractive possibilities are found for controlled acceleration or deceleration of gas at subsonic and supersonic speeds, even in constant-area channels . the behavior of the flow is characteristically different in different regimes of mach number and flow speed relative to certain /significant speeds/ that are dependent on the ratio of electrical to magnetic field strengths . these are studied, and a chart is constructed to relate the length to the speed ratio of a maximum-acceleration constant-area channel . it is concluded that the advantages that may accrue from magneto-aerodynamic methods are sufficiently attractive to justify the considerable research and engineering development that will be required . among the unsolved engineering problems are the reduction of surface resistance of electrodes in contact with a conducting gas, development of techniques for seeding, and provision of the required magnetic fields in flight .
constant-temperature magneto-gasdynamic channel flow . in the course of investigating boundary-layer flow in continuous plasma accelerators with crossed electric and magnetic fields, it was found advantageous to have at hand simple closed-form solutions for the magneto-gasdynamic flow in the duct which could serve as free-stream conditions for the boundary layers . nontrivial solutions of this sort are not available at present, and in fact, as in the work of resler and sears, the variation of conditions along the flow axis must be obtained through numerical integration . consequently, some simple solutions of magneto-gasdynamic channel flow were sought, possessing sufficient algebraic simplicity to serve as free-stream boundary conditions for analytic investigations of the boundary layer in a physically reasonable accelerator . in particular, since the cooling of the accelerator tube is likely to be an important physical problem because of the high gas temperatures required to provide sufficient gaseous conductivity, channel flow with constant temperature appears interesting . some simple algebraic solutions for the case of a constant temperature plasma are developed in the following paragraphs .
stagnation point of a blunt body in hypersonic flow . the purpose of this paper is to present a method of calculation devised to yield all the important information on the symmetric inviscid hypersonic flow in the stagnation point region of a blunt body . the problem is the same as that considered by hayes who used a slightly different approach . it is demonstrated that hayes’ results are valid in the stagnation point region and can hence be considered a basis for constructing less restricted solutions . equations are presented giving velocity, pressure, detachment distance, and vorticity . the values of shock detachment distance and body pressure coefficient are compared with experimental data for spheres . the pressure comparison shows that the results of hayes and the theory presented herein represent a better approximation than the newtonian impact theory for hypersonic mach numbers . in conclusion, the possibility of refinements to this analysis is discussed .
supersonic flow around blunt bodies . the newtonian theory of impact has been shown to be useful for pressure calculations on the forward facing part of bodies moving at high speed . it is now a familiar practice to use this information to calculate nonviscous velocities at the wall and then to estimate rates of heat transfer . this procedure is perhaps open to question,. heat-transfer rates depend on velocity gradients which are not given by the newtonian analysis . nor can one obtain information on boundary-layer stability or all the body stability derivatives . it seems, therefore, inevitable that, as design proceeds with these hypersonic missiles, there will be a greater need for more accurate aerodynamic theories either to predict what will happen in unfamiliar flight conditions or to effect an extrapolation from a known test result to the design condition .
a new technique for investigating heat transfer and surface phenomena under hypersonic flow conditions . on the forebody of many practically interesting hypersonic vehicles, there is little interaction between the inviscid flow field and the boundary layer . therefore, inviscid flow theory can be used to determine, independent of surface phenomena, the physically interesting quantities such as shock shape, shock detachment distance, sonic line shape, and pressure distribution . furthermore, the pressure distribution so determined can then be used for the study of heat transfer, materials behavior, and other surface phenomena . thus, for these bodies, the prandtl boundary-layer concept can be utilized for the calculation of both the inviscid flow and the boundary-layer behavior . it is the purpose of this note to point out that this concept can also be applied experimentally in order to provide, in conjunction with a conventional hypersonic wind-tunnel air supply, a means for investigating hypersonic heat transfer and surface phenomena under conditions of flight reynolds numbers .
on the prediction of mixed subsonic/supersonic pressure distributions . high-speed wind-tunnel results are analyzed to derive a semiempirical scheme for the prediction of transonic pressure distributions . the supersonic and subsonic parts of the flow are treated separately, and then linked by an empirical shock pressure rise relation . the significance of the empirical results is considered in relation to the physical mechanism of transonic flows . it is also shown that theoretical solutions can be improved by introducing the empirical shock relation .
on the flow of a sonic stream past an airfoil surface . this study of the flow about an airfoil in a near-sonic stream indicates the important factors determining the pressure distribution on the airfoil . analysis of the mach wave pattern suggests that the supersonic domain of the flow can be derived from two simple-wave flows, one arising from the mach waves reflected at the sonic line and the other from the changes in airfoil surface slope . the compressive effect of the reflected mach waves is determined quantitatively as a function of airfoil leading-edge geometry from an analysis of measured pressure distributions for uncambered airfoils,. and it is shown how this can be superimposed on the wave system from the curved surface to give an equivalent simple-wave flow over the airfoil . an application of this scheme to the calculation of the pressure distribution over an airfoil in a sonic stream gives results in good agreement with experiment .
experiments on boundary layer transition at supersonic speeds . tests were conducted in the 12-in. continuous supersonic wind tunnel of the jet propulsion laboratory, california institute of technology, to determine the effects of surface cooling on boundary-layer transition at supersonic speeds . the effects of cooling were investigated at test section mach numbers of 1.97, smooth cone in the presence of three levels of supply-stream turbulence (0.4, 2, and 9 per cent) and several single-element roughnesses at fixed axial location . transition data were obtained optically by means of a magnified-schlieren system . the results, for the range of mach number investigated, indicate that (1) transition on a smooth cone can definitely be delayed by surface cooling, (2) transition promoted by either supply-stream turbulence or surface roughness can also be delayed by surface cooling depending upon degree of turbulence or relative roughness respectively, and (3) the adverse effects of increased turbulence and roughness decrease with increasing mach number .
on transition experiments at moderate supersonic speeds . studies of transition over a flat plate at mach number 1.76 were carried out using a hot-wire anemometer as one of the principal tools . the nature and measurements of free-stream disturbances at supersonic speeds are analyzed . the experimental results are interpreted in the light of present overall information on transition at supersonic speeds and conclusions as to further fruitful experiments are drawn .
the gyroscopic effect of a rigid rotating propeller on engine and wing vibration modes . in many wing vibration analyses it is found necessary to take into account the effect of flexibly mounted engines . hence, it is reasonable to ask what vibratory gyroscopic effect this flexibility may give rise to when propellers are whirling . an engine mount may be thought of as a horizontal beam cantilevered from the wing, having both horizontal and vertical flexibility . if this beam were infinitely rigid horizontally, then, when it vibrated, the gyroscopic moments induced in the propeller due to the resultant pitching motion of its axis would not produce propeller axis yaw . however, engine-mount lateral stiffness tical stiffness, so that gyroscopic effects will play a role as the propeller axis undergoes pitching vibrations at the tip of the cantilever engine mount . the purpose of this paper is to investigate this role under the assumption that the propeller itself is a rigid disc . the paper is divided into four parts . part (1) deals briefly with classical gyroscope theory . part (2) presents engine vibration mode studies-experimental photographic techniques on a model gyroscope mounted at the ends of two different cantilever beams . part (3) presents the theory of the coupled motion of an elastic wing upon which a gyroscope is mounted to simulate an engine-propeller system on an airplane . part (4) consists of an example of the theory of part (3), in which, by taking what are thought to be reasonable parameters, results are obtained showing how the whirling of a rigid propeller may materially affect wing normal mode shapes and frequencies .
the relation between wall temperature and the effect of roughness on boundary layer transition . the experimentally demonstrated rise and subsequent fall of transition reynolds number with decreasing wall- to-ambient temperature ratio has been the subject of two recent notes . in both cases it was argued that the increased effectiveness of roughness due to wall cooling was not sufficient to explain the transition-reversal phenomenon on nominally smooth bodies . in one case, the criterion for transition reversal was taken to be and in the other values of as low as eter is a reynolds number formed from velocity and kinematic viscosity based on calculated conditions at the height of roughness element k in the undisturbed, laminar boundary layer at the station of roughness location . the present note is submitted to show that another method for evaluating the effect of roughness on transition leads to an opposite conclusion .
tip-bluntness effects on cone pressures at m=6.85 . there is, at present, considerable interest in the characteristies of blunted bodies from both an aerodynamic and a heat-transfer standpoint . the use of blunt shapes is contemplated to reduce the heat-transfer problem at body noses, but there are also applications for blunt noses which occur from mainly aerodynamic considerations . an actual reduction in drag may be the beneficial result of blunting the nose of a cone or a similar slender shape under certain conditions . although the sphere has received considerable treatment, the nose shapes are not necessarily tangent spheres . in the case, let us say, of a total head tube situated in the nose of a given body, the blunting may be quite flat, and nose sections blunter than spherical shape may conceivably be desirable, in some cases, from the heat- transfer standpoint . the purpose of the present investigation is to examine the aerodynamic effect of a simple type of nose blunting on a basic body . the incompressible flow of an electrically conducting fluid past a porous plate y = 0 with constant suction velocity in the presence of a transverse uniform strength has recently been investigated by gupta . in this note, the problem is generalized to take into account the effect of free convection, when a body force g per unit mass is acting in the negative x-direction parallel to the wall . the fluid is assumed to be semi- incompressible as usual . in addition to the obvious practical significance, this problem is also interesting in the sense that it provides another exact solution of the magnetohydrodynamic equations, since the only electromagnetic assumptions involved are constant properties and freedom from excessive charges .
an investigation of separated flows, part ii: flow in the cavity and heat transfer . the first portion of this paper describes studies of the internal structure of the separated flow in a notch at a free-stream mach number of 3 . observations include.. flow visualization, spark- schlieren pictures of the fluctuations of the free shear layer, and studies of the diffusion of heat from sources placed in the separated region . the second part describes measurements of local heat transfer to the wall . the external mach number, the length-to-depth ratio of the cavity, the ratio of the oncoming boundary layer thickness to the notch depth (in the turbulent flow region), the thermal to-momentum thickness ratio of the boundary layer and, finally, the geometry of the internal boundary of the separated region are varied as systematically as possible . on the basis of these observations, a simple model of the flow in and the heat transfer across the separated region is formulated .
some comments on the inversion of certain large matrices . the subject of matric structural analysis has been treated in two recently published papers in the journal . the authors of these papers have made a number of statements about the inversion of certain large matrices . it is the purpose of this note to bring to the attention of the reader certain facts that shed new light on this important problem . it is shown here that the situation is not as hopeless as the above- mentioned authors intimate .
analysis of low-aspect-ratio aircraft structures . two methods are presented for the analysis of complex low- aspect-ratio aircraft structures . both methods provide for arbitrary external loading, are general with respect to the orientation of structural members, and permit arbitrary boundary conditions . for purposes of analysis a structure is idealized as a network of flexural members with interconnected torsion boxes . in the first method, sets of linear equations are obtained by expressing boundary conditions, member deflection equations, equilibrium requirements, and slope-compatibility relationships in terms of deflections and internal forces . the solution for deflections and internal forces is then formed as the product of an inverse structural matrix and a column matrix of load functions . in the second method, the conditions at a given boundary are assembled as a column matrix and are transferred in a step by-step fashion over the entire structure to an opposite boundary . the transfer is accomplished by successive multiplications of square matrices composed independently for the different transfer ranges . the final operation is the inversion of a relatively small matrix and provides the solution for the unknown boundary conditions . comparisons of theoretical results with experimental data and electric-analog solutions are favorable .
supersonic flow at the surface of a circular cone at angle of attack . formulas for the inviscid flow properties on the surface of a cone at angle of attack are derived for use in conjunction with the m.i.t. cone tables . these formulas are based upon an entropy distribution on the cone surface which is uniform and equal to that of the shocked fluid in the windward meridian plane . they predict values for the flow variables which may differ significantly from the corresponding values obtained directly from the cone tables . the differences in the magnitudes of the flow variables computed by the two methods tend to increase with increasing free-stream mach number, cone angle and angle of attack .
temperature and velocity profiles in the compressible laminar boundary layer with arbitrary distribution of surface temperature . an analysis is presented which enables the temperature profiles, veiocity profiles, heat transfer, and skin friction to be calculated for laminar flow over a two-dimensional or axially symmetric surface without pressure gradient but with an arbitrary analytic distribution of surface temperature . the general theory is applicable to a gas of any prandtl number, although the numerical results given herein have been computed for air . the predictions of the theory for the special case of constant surface temperature are compared with the calculations of crocco . on the basis of this comparison, it is inferred that the present theory enables heat-transfer and skin-friction calculations accurate to within about 5 per cent to be made for flight conditions up to mach numbers near 5 and to within about 1 or 2 per cent for supersonic wind-tunnel conditions up to considerably higher mach numbers . a particular effort has been made to present the results, which are simple considering their generality, in a form that can be used readily in practical applications . from the mathematical point of view, the theory is applicable to an arbitrary analytic distribution of surface temperature, but in any given practical case it is necessary that the surface-temperature distribution be approximated by a polynomial . the only unknowns in the final equations developed are the coefficients of this polynomial, so that the work involved in applying the theory in any given case depends entirely on the work involved in approximating a given surface- temperature distribution by a polynomial . an example is worked out in detail which illustrates some of the principal effects of variable surface temperature . it is shown that both positively infinite and negatively infinite heat-transfer coefficients can occur . the anomaly of infinite and negative heat-transfer coefficients is discussed and attributed to the customary definition of the heat-transfer coefficient, which is shown to be fundamentally inappropriate for flows with variable surface temperature . in the particular example considered, a conventional method for calculating the net heat transferred yields completely incorrect results . a brief qualitative discussion of the possible effects of the heat transfer on flow separation is given . in order to facilitate the use of the results, all of the principal equations developed are collected and summarized in the section entitled /practical use of results ./
investigation of laminar boundary layer in compressible fluids using the crocco method . in the present investigation of the flow of air in a thin laminar boundary layer on a flat plate, the crocco method has been used to solve the simultaneous differential equations of momentum and energy involved in such flow . the crocco method was used because it gave accurate results for arbitrary prandtl number near unity . the prandtl number was taken at 0.75, the specific heat was held constant, and the sutherland law of viscosity-temperature variation was assumed to represent the viscosity data starting with an initial ambient temperature of -67.6 f . the main results presented here are the skin-friction and heat-transfer coefficients as functions of reynolds number, mach number, and wall-to-free-stream temperature ratio . variations of shear, velocity, temperature, and mach number across the boundary layer are included . the crocco method is discussed in detail .
theory of aircraft structural models subjected to aerodynamic heating and external loads . the problem of investigating the simultaneous effects of transient aerodynamic heating and external loads on aircraft structures for the purpose of determining the ability of the structure to withstand flight to supersonic speeds is studied . by dimensional analyses it is shown that .. constructed of the same materials as the aircraft will be thermally similar to the aircraft with respect to the flow of heat through the structure will be similar to those of the aircraft when the structural model is constructed at the same temperature as the aircraft . external loads will be similar to those of the aircraft . subjected to heating and cooling that correctly simulate the aerodynamic heating of the aircraft, except with respect to angular velocities and angular accelerations, without requiring determination of the heat flux at each point on the surface and its variation with time . acting on the aerodynamically heated structural model to those acting on the aircraft is determined for the case of zero angular velocity and zero angular acceleration, so that the structural model may be subjected to the external loads required for simultaneous simulation of stresses and deformations due to external loads .
procedure for calculating flutter at high supersonic speed including camber deflections, and comparison with experimental results . a method which may be used at high supersonic mach numbers is described for calculating the flutter speed of wings having camber in their deflection modes . the normal coupled vibration modes of the wing are used to derive the equations of motion . chord deflections of the vibration modes are approximated by polynomials . the wing may have a control surface and may carry external stores although no aerodynamic forces on the stores are presented . the aerodynamic forces that are assumed to be acting on the wing are obtained from piston theory and also from a quasi-steady form of a theory for two-dimensional steady flow . airfoil shape and thickness effects are taken account of in the analysis . the method is used to calculate the flutter speed of some wings which had been previously tested at mach numbers of 1.3 to 3.0 . comparison of the calculations and experiment is made for flat-plate 60 and 45 delta wings and also for an untapered 45 sweptback wing .
transition reynolds numbers of separated flows at supersonic speeds . experimental research has been conducted on the effects of wall cooling, mach number, and unit reynolds number on the transition reynolds number of cylindrical separated boundary layers on an ogive-cylinder model . results were obtained from pressure and temperature measurements and shadowgraph observations . the maximum scope of measurements encompassed mach numbers between 2.06 and 4.24, reynolds numbers (based on length of separation) between 60,000 and 400,000, and ratios of wall temperature to adiabatic wall temperature between 0.35 and 1.0 . within the range of the present tests, the transition reynolds number was observed to decrease with increasing wall cooling, increase with increasing mach number, and increase with increasing unit reynolds number . the wall-cooling effect was found to be four times as great when the attached boundary layer upstream of separation was cooled in conjunction with cooling of the separated boundary layer as when only the separated boundary layer was cooled . wall cooling of both the attached and separated flow regions also caused, in some cases, reattachment in the otherwise separated region . cavity resonance present in the separated region for some model configurations was accompanied by a large decrease in transition reynolds number at the lower test mach numbers .
method for calculation of compressible laminar boundary layer characteristics in axial pressure gradient with zero heat transfer . the karman-pohlhausen method is extended primarily to sixth-degree velocity profiles for determining the characteristics of the compressible laminar boundary layer over an adiabatic wall in the presence of an axial pressure gradient . it is assumed that the prandtl number is unity and that the coefficient of viscosity varies linearly with the temperature . a general approximate solution which permits a rapid determination of the boundary-layer characteristics for any given free-stream mach number and given velocity distribution at the outer edge of the boundary layer is obtained . numerical examples indicate that this solution will in practice lead to results of satisfactory accuracy, including the critical reynolds number for stability . for the special purpose of calculating the location of the separation point in an adverse pressure gradient, a short and simple method, based on the use of a seventh-degree velocity profile, is derived . the numerical example given here indicates that this method should in practice lead to sufficiently accurate results . for the special case of flow near a forward stagnation point it is shown that the karman-pohlhausen method with the usual fourth-degree profiles leads to results of adequate accuracy, even for the critical reynolds number .
separation, stability and other properties of compressible laminar boundary layer with pressure gradient and heat transfer . a theoretical study is made of the effect of pressure gradient, wall temperature, and mach number on laminar boundary-layer characteristics and, in particular, on the skin-friction and heat-transfer coefficients, on the separation point in an adverse pressure gradient, on the wall temperature required for complete stabilization of the laminar boundary layer, and on the minimum critical reynolds number for laminar stability . the prandtl number is assumed to be unity and the coefficient of viscosity is assumed to be proportional to the temperature, with a factor arising from the sutherland relation . a simple and accurate method of locating the separation point in a compressible flow with heat transfer is developed . numerical examples to illustrate the results in detail are given throughout .
an analysis of the applicability of the hypersonic similarity law to the study of the flow about bodies of revolution at zero angle of attack . the hypersonic similarity law as derived by tsien has been investigated by comparing the pressure distributions along bodies of revolution at zero angle of attack . in making these comparisons, particular attention was given to determining the limits of mach number and fineness ratio for which the similarity law applies . for the purpose of this investigation, pressure distributions determined by the method of characteristics for ogive cylinders for values of mach numbers and fineness ratios varying from 1.5 to 12 were compared . pressures on various cones and on cone cylinders were also compared in this study . the pressure distributions presented demonstrate that the hypersonic similarity law is applicable over a wider range of values of mach numbers and fineness ratios than might be expected from the assumptions made in the derivation . this is significant since within the range of applicability of the law a single pressure distribution exists for all similarly shaped bodies for which the ratio of free-stream mach number to fineness ratio is constant . charts are presented for rapid determination of pressure distributions over ogive cylinders for any combination of mach number and fineness ratio within defined limits .
applicability of the hypersonic similarity rule to pressure distributions which include the effects of rotation for bodies of revolution at zero angle of attack . the analysis of technical note 2250, 1950, is extended to include the effects of flow rotation . it is found that the theoretical pressure distributions over ogive cylinders can be related by the hypersonic similarity rule with sufficient accuracy for most engineering purposes . the error introduced into pressure distributions and drag of ogive cylinders by ignoring the rotation term in the characteristic equations is investigated . it is found that the influence of the rotation term on pressure distribution and drag depends only upon the similarity parameter k (mach number divided by fineness ratio) . although the error in drag, due to neglect of the rotation term, is negligible at k=0.5, the error is about 30 percent at k=2.0 . charts are presented for the rapid determination of pressure distributions for rotational flow over ogive cylinders for all values of the similarity parameter between 0.5 and of mach number and fineness ratio .
pressure measurements on sharp and blunt 5 and 15 half-angle cones at mach number 3.86 and angles of attack to 100 . measured pressure distributions on cones are compared with modified newtonian theory . deviations as large as 14 percent of the stagnation pressure behind a normal shock are found . by combining empirical results for cylinders normal to the flow with newtonian concepts, a method of calculating pressures on cones at high angles of attack is developed . calculations by this method differ from the experimental results on sharp cones by only 2 percent of the stagnation pressure behind a normal shock . for blunted cones, additional deviations up to 8 percent are noted near the nose . schlieren pictures of the flow show an attached shock on the sharp of attack . detachment of the shock appears to be associated with the attainment of sonic speed immediately behind the shock . an orifice size effect is found which can increase the indicated pressure above the true value, if the orifice width is greater than one-tenth the local radius of curvature .
tables of exact laminar-boundary layer solutions when the wall is porous and fluid properties are variable . the three partial differential equations of the laminar boundary layer for two-dimensional steady-state compressible flow have been transformed into two ordinary differential equations by the method of pohlhausen, falkner, and skan . the ordinary equations include parameters for expressing the simultaneous effects of pressure gradient in the main-stream flow through a porous wall and property changes in the fluid due to large temperature differences between the wall and the free stream . a total of 58 cases have been solved numerically by the method of picard . the euler number (nondimensional pressure-gradient parameter) ranges in value from 1 (stagnation-point value) to the negative values found at the laminar separation points . three rates of flow through the porous wall were considered (including the impermeable case where the flow rate is 0) . five temperature ratios (stream temperature divided by wall temperature) were used .. the uncooled and unheated case (temperature ratio of 1), two cooled cases (temperature ratios of ture ratios of and ) . velocity, weight-flow, and temperature distributions are tabulated as are the dimensionless stream function of falkner and skan and its derivatives and the dimensionless temperature function of pohlhausen and its derivatives . for each case, displacement, momentum, and convection thicknesses, as well as nusselt number and coefficient of friction at the wall, were computed .
estimation forces and moments due to rolling for several slender tail configurations at supersonic speeds . the velocity potentials, span loadings, and corresponding force and moment derivatives have been theoretically evaluated for a number of slender-tail arrangements performing a steady rolling motion at supersonic speeds . the method of analysis is based upon an application of conformal-transformation techniques . the utilization of these techniques allows the simple determination of the complex potentials for various types of two-dimensional boundary-value problems . in addition, two simple and often-used approximations to the rolling derivatives have been compared with the corresponding exact values determined by the method presented in this report . in order to show the importance of wing-tail interference, the effect of the flow field behind a rolling wing on the tail characteristics has been illustrated for a simple wing-tail arrangement .
on flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic field . the use of a magnetic field to control the motion of electrically conducting fluids is studied . the boundary-layer solutions are found for flow over a flat plate when the magnetic field is fixed relative to the plate or to the fluid . the equations are integrated numerically for the effect of the transverse magnetic field on the velocity and temperature profiles, and hence, the skin friction and rate of heat transfer . it is concluded that the skin friction and the heat-transfer rate are reduced when the transverse magnetic field is fixed relative to the plate and increased when fixed relative to the fluid . the total drag is increased in all the cases studied .
similar solutions for the compressible laminar boundary layer with heat transfer and pressure gradient . stewartson’s transformation is applied to the laminar compressible boundary-layer equations and the requirement of similarity is introduced, resulting in a set of ordinary nonlinear differential equations previously quoted by stewartson, but unsolved . the requirements of the system are .. prandtl number of 1.0, linear viscosity-temperature relation across the boundary layer, an isothermal surface, and the particular distributions of free-stream velocity consistent with similar solutions . this system admits axial pressure gradients of arbitrary magnitude, heat flux normal to the surface, and arbitrary mach numbers . the system of differential equations is transformed to an integral system, with the velocity ratio as the independent variable . for this system, solutions are found for pressure gradients varying from that causing separation to the infinitely favorable gradient and for wall temperatures from absolute zero to twice the free-stream stagnation temperature . some solutions for separated flows are also presented . for favorable pressure gradients, the solutions are unique . for adverse pressure gradients, where the solutions are not unique, two solutions of the infinite family of possible solutions are identified as essentially viscid at the outer edge of the boundary layer and the remainder essentially inviscid . for the case of favorable pressure gradients with heated walls, the velocity within a portion of the boundary layer is shown to exceed the local external velocity . the variation of a reynolds analogy parameter, which indicates the ratio of skin friction to heat transfer, is from zero to 7.4 for a surface of temperature twice the free-stream stagnation temperature, and from zero to 2.8 for a surface held at absolute zero where the value 2 applies to a flat plate .
hypersonic viscous flow over slender cones . viscous self-induced pressures on 3 -semivertex-angle cones were measured over the range 3.7 free-stream mach number 5.8 and 0.5 viscous-interaction parameter 2.3 . the data were found to be in good agreement with results obtained by talbot on 5 cones in the range rameter 3.5 . all these data were correlated reasonably well by the viscous-interaction parameter, which is defined as where and are the mach number and reynolds number based on ideal taylor-maccoll flow conditions and c is the chapman-rubesin factor . a new method for calculating self-induced pressures is presented which takes into account the interaction between boundary-layer growth and the inviscid-flow field at the outer edge of the boundary layer . pressures calculated by this method were only 10 to 20 percent higher than the measured values .
unsteady oblique interaction of a shock wave with plane disturbances . analysis is made of the flow field produced by oblique impingement of weak plane disturbances of arbitrary profile on a plane normal shock . three types of disturbance are considered .. moves . the sound wave refracts either as a simple isentropic sound wave or as an attenuating isentropic pressure wave, depending on the angle between the shock and the incident sound wave . a stationary vorticity wave of constant pressure appears behind the shock . reflects as a sound wave, and a stationary vorticity wave is produced . the shock . the incident wave refracts as a stationary vorticity wave, and either a sound wave or attenuating pressure wave is also produced . computations are presented for the first two types of incident wave, over the range of incidence angles, for shock mach numbers of 1, 1.5, and .
convection of a pattern of vorticity through a shock wave . an arbitrary weak spatial distribution of vorticity can be represented in terms of plane sinusoidal shear waves of all orientations and wave lengths (fourier integral) . the analysis treats the passage of a single representative weak shear wave through a plane shock and shows refraction and modification of the shear wave with simultaneous generation of an acoustically intense sound wave . applications to turbulence and to noise in supersonic wind tunnels are indicated .
some effects of joint conductivity on the temperature and thermal stresses in aerodynamically heated skin-stiffener combinations . temperatures and thermal stresses in typical skin-stiffener combinations of winglike structures subjected to aerodynamic heating have been obtained with the aid of an electronic differential analyzer . variations were made in an aerodynamic heat-transfer parameter, in a joint conductivity parameter, and in the ratio of skin width to skin thickness . the results, which are presented in nondimensional form, indicate that decreasing the joint conductivity parameter lowers both the interior and the average temperature ratios, increases the peak thermal stress ratios in the skin, and may considerably increase the peak stiffener stress ratios,. increasing the aerodynamic heat-transfer parameter decreases the interior and average temperature ratios, increases the peak skin stress ratios somewhat, but greatly increases the peak stiffener stress ratios,. and increasing the ratio of skin width to skin thickness produces only moderate decreases in the peak skin stress ratios while moderately increasing the peak stiffener stress ratios .
dynamic stability of vehicles traversing ascending or descending paths through the atmosphere . an analysis is given of the oscillatory motions of vehicles which traverse ascending and descending paths through the atmosphere at high speed . the specific case of a skip path is examined in detail, and this leads to a form of solution for the oscillatory motion which should recur over any trajectory . the distinguishing feature of this form is the appearance of the bessel rather than the trigonometric function as the characteristic mode of oscillation .
some aspects of air-helium simulation and hypersonic approximations . some illustrations of the differences that may be expected between results obtained in hypersonic wind tunnels that employ air and results obtained in those that employ helium as the test medium (imperfect-gas effects are not considered) are compiled and presented herein . simple expressions are presented that demonstrate the possibility of simulating air results in helium tests and of transforming helium data to equivalent air data . nonviscous and viscous simulations are considered . in most cases, the methods and the general forms of the expressions for simulation that are derived are applicable to any two ideal gases having different ratios of specific heats .
predicted shock envelopes about two types of vehicles at large angles of attack . methods based on oblique- and normal-shock relationships and the continuity of mass flow through suitably chosen volume elements between the shock and body were developed to predict shock envelopes about two types of vehicles being considered for atmosphere entry . one type is a high-drag capsule shape . the other type is essentially a slender triangular wing capable of providing high lift or high drag, depending on the angle of attack . predicted and measured shock envelopes were compared for a mach number range of 3 to 15 for vehicles at high angles of attack,. good agreement was found . most of the available experimental data were in a speed and temperature range in which no important real-gas effects occurred .
a study of flow changes associated with airfoil section drag rise at supercritical speeds . a study of experimental pressure distributions and section characteristics for several moderately thick airfoil sections was made . a correlation appears to exist between the drag-divergence mach number and the free-stream mach number for which sonic velocity occurs at the airfoil crest, the chordwise station at which the airfoil surface is tangent to the free-stream direction . it was found that, since the mach number for which sonic velocity occurs at the airfoil crest can be estimated satisfactorily by means of the prandtl-glauert rule, a method is provided whereby the drag-divergence mach number of an airfoil section at a given angle of attack can be estimated from the low-speed pressure distribution and the airfoil profile . this method was used to predict with a reasonable degree of accuracy the drag-divergence mach number of a considerable number of airfoil sections having diverse shapes and a wide range of thickness-chord ratios . the pressure distributions and section force characteristics of several moderately thick airfoil sections at mach numbers above the drag-divergence mach number were analyzed . some of the characteristics of the flow over these airfoils at supercritical mach numbers are discussed .
laminar boundary layer behind shock advancing into stationary fluid . a study was made of the laminar compressible boundary layer induced by a shock wave advancing into a stationary fluid bounded by a wall . for weak shock waves, the boundary layer is identical with that which occurs when an infinite wall is impulsively set into uniform motion shocks . velocity and temperature profiles, recovery factors, and skin-friction and heat-transfer coefficients are tabulated for a wide range of shock strengths .
boundary layer behind shock or thin expansion wave moving into stationary fluid . the boundary layer behind a shock or thin expansion wave advancing into a stationary fluid has been determined . laminar and turbulent boundary layers were considered . the wall surface temperature behind the wave was also investigated . the assumption of a thin expansion wave is valid for weak expansions but becomes progressively less accurate for strong expansion waves . the laminar-boundary-layer problem was solved by numerical integration except for the weak wave case, which can be solved analytically . integral (karman-pohlhausen type) solutions were also obtained to provide a guide for determining expressions which accurately represent the numerical data . analytical expressions for various boundary-layer parameters are presented which agree with the numerical integrations within 1 percent . the turbulent-boundary-layer problem was solved using integral methods similar to those employed for the solution of turbulent compressible flow over a semi-infinite flat plate . the fluid velocity, relative to the wall, was assumed to have a seventh-power profile . the blasius equation, relating turbulent skin friction and boundary-layer thickness, was utilized in a form which accounted for compressibility . consideration of the heat transfer to the wall permitted the wall surface temperature, behind the wave, to be determined . the wall thickness was assumed to be greater than the wall thermal-boundary-layer thickness . it was found that the wall temperature was uniform (as a function of distance behind the wave) for the laminar-boundary-layer case but varied with distance for the turbulent-boundary-layer case .
investigation of the stability of the laminar boundary layer in a compressible fluid . in the present report the stability of two-dimensional laminar flows of a gas is investigated by the method of small perturbations . the chief emphasis is placed on the case of the laminar boundary layer . part 1 of the present report deals with the general mathematical theory . the general equations governing one normal mode of the small velocity and temperature disturbances are derived and studied in great detail . it is found that for reynolds numbers of the order of those encountered in most aerodynamic problems, the temperature disturbances have only a negligible effect on those particular velocity solutions which depend primarily on the viscosity coefficient (/viscous solutions/) . indeed, the latter are actually of the same form in the compressible fluid as in the incompressible fluid, at least to the first approximation . because of this fact, the mathematical analysis is greatly simplified . the final equation determining the characteristic values of the stability problem depends on the /inviscid solutions/ and the function of tietjens in a manner very similar to the case of the incompressible fluid . the second viscosity coefficient and the coefficient of heat conductivity do not enter the problem,. only the ordinary coefficient of viscosity near the solid surface is involved . part 2 deals with the limiting case of infinite reynolds numbers . the study of energy relations is very much emphasized . it is shown that the disturbance will gain energy from the main flow if the gradient of the product of mean density and mean vorticity near the solid surface has a sign opposite to that near the outer edge of the boundary layer . a general stability criterion has been obtained in terms of the gradient of the product of density and vorticity, analogous to the rayleigh-tollmien criterion for the case of an incompressible fluid . if this gradient vanishes for some value of the velocity ratio of the main flow exceeding 1-1/m (where m is the free stream mach number) .
an experimental study of the turbulen coundary layer on a shock tube wall . interferometric measurements were made of the density profiles of an unsteady turbulent boundary layer on the flat wall of a shock tube . the investigation included both subsonic and supersonic flow (mach numbers of 0.50 and 1.77) with no pressure gradient and with heat transfer to a cold wall . velocity profiles and average skin-friction coefficients were calculated . effects on the velocity profile of surface roughness and flow length are examined .
studies of structural failure due to acoustic loading . some discussion of the acoustic fatigue problem of aircraft structures is given along with data pertaining to the acoustic inputs from some powerplants in common use . comparisons are given for results of some fatigue tests of flat panels and cantilever beams exposed to both random- and discrete-type inputs . in this regard it appears that both the stress level of the test and the type of model are significant,. hence, no generalization can be made at this time . with regard to increasing the fatigue life, it was noted that increased stiffening of a panel due to curvature and pressure differential is particularly beneficial .
flight measurement of wall pressure fluctuations and boundary-layer turbulence . the results are presented for a flight test program using a fighter type jet aircraft flying at pressure altitudes of 10,000, 20,000, and apparatus was used to measure and record the output of microphones and hot-wire anemometers mounted on the forward-fuselage section and wing of the airplane . mean-velocity profiles in the boundary layers were obtained from total-pressure measurements . the ratio of the root-mean-square fluctuating wall pressure to the free-stream dynamic pressure is presented as a function of reynolds number and mach number . the longitudinal component of the turbulent-velocity fluctuations was measured, and the turbulence-intensity profiles are presented for the wing and forward-fuselage section . in general, the results are in agreement with wind-tunnel measurements which have been reported in the literature . for example, the variation of (is the root mean square of the wall-pressure fluctuation, and q is the free-stream dynamic pressure) with reynolds number was found to be essentially constant for the forward fuselage-section boundary layer, while variations at the wing station were probably unduly affected by the microphone diameter, which was large compared with the boundary-layer thickness .
a comparative analysis of the performance of long range hypervelocity vehicles . long-range hypervelocity vehicles are studied in terms of their motion in powered flight, and their motion and aerodynamic heating in unpowered flight . powered flight is analyzed for an idealized propulsion system which rather closely approaches present-day rocket motors . unpowered flight is characterized by a return to earth along a ballistic, skip, or glide trajectory . only those trajectories are treated which yield the maximum range for a given velocity at the end of powered flight . aerodynamic heating is treated in a manner similar to that employed previously by the senior authors in studying ballistic missiles (naca tn 4047), with the exception that radiant as well as convective heat transfer is considered in connection with glide and skip vehicles . the ballistic vehicle is found to be the least efficient of the several types studied in the sense that it generally requires the highest velocity at the end of powered flight in order to attain a given range . this disadvantage may be offset, however, by reducing convective heat transfer to the re-entry body through the artifice of increasing pressure drag in relation to friction drag – that is, by using a blunt body . thus the kinetic energy required by the vehicle at the end of powered flight may be reduced by minimizing the mass of coolant material involved . the glide vehicle developing lift-drag ratios in the neighborhood of and greater than 4 is far superior to the ballistic vehicle in ability to convert velocity into range . it has the disadvantage of having far more heat convected to it,. however, it has the compensating advantage that this heat can in the main be radiated back to the atmosphere . consequently, the mass of coolant material may be kept relatively low . the skip vehicle developing lift-drag ratios from about 1 to 4 is found to be superior to comparable ballistic and glide vehicles in converting velocity into range . at lift-drag ratios below 1 it is found to be about equal to comparable ballistic vehicles while at lift-drag ratios
an analytical treatment of aircraft propeller precession instability . an analytical investigation is made of a precession-type instability which can occur in a flexibly supported aircraft-engine-propeller combination . by means of an idealized mathematical model which is comprised of a rigid power-plant system flexibly mounted in pitch and yaw to a fixed backup structure, the conditions required for neutral stability are determined . the paper also examines the sensitivity of the stability boundaries to changes in such parameters as stiffness, damping, and asymmetries in the engine mount, propeller speed, airspeed, mach number, propeller thrust, and location of pitch and yaw axes . stability is found to depend strongly on the damping and stiffness in the system . with the use of nondimensional charts theoretical stability boundaries are compared with experimental results obtained in wind-tunnel tests of an aeroelastic airplane model . in general, the theoretical results, which do not account for wing response, show the same trends as observed experimentally,. however, for a given set of conditions calculated airspeeds for neutral stability are consistently lower than the measured values . evidently, this result is due to the fact that wing response tends to add damping to the system .
effects of extreme surface cooling on boundary layer transition . an investigation was made to determine the combined effects of surface cooling, pressure gradients, nose blunting, and surface finish on boundary-layer transition . data were obtained for various body shapes at a mach number of 3.12 and reynolds numbers per foot as high as 15×10 . previous transition studies, with moderate cooling, have shown agreement with the predictions of stability theory . for surface roughnesses ranging from 4 to 1250 microinches the location of transition was unaffected with moderate cooling . with extreme cooling, an adverse effect was observed for each of the parameters investigated . in general, the transition reynolds number decreased with decreasing surface temperature . in particular, the beneficial effects of a favorable pressure gradient obtained with moderate cooling disappear with extreme cooling, and a transition reynolds number lower than that observed on a cone is obtained . further, an increase in the nose bluntness decreased the transition reynolds number under conditions of extreme cooling .
effect of distributed three-dimensional roughness and surface cooling on boundary layer transition and lateral spread of turbulence at supersonic speeds . an investigation was made in the langley 4 by 4-foot supersonic pressure tunnel at mach numbers of 1.61 and 2.01 to determine (1) the effect of distributed roughness on boundary-layer transition with the model surface at adiabatic wall temperature and cooled and (2) the effect of surface cooling on the lateral spread of turbulence . both distributed granular-type and single spherical roughness particles were used, and transition of the boundary layer was determined by hot-wire anemometers . the transition-triggering mechanism of the three-dimensional roughness at supersonic speeds appeared to be the same as that previously observed at subsonic speeds . in fact, the critical value of the roughness reynolds number parameter (that is, the value at which turbulent spots are initiated by the roughness) was found to be approximately the same at supersonic and subsonic speeds when complete local conditions at the top of the roughness, including density and viscosity, were considered in the formulation of the roughness reynolds number . for three-dimensional roughness at a reynolds number less than its critical value, the roughness introduced no disturbances of sufficient magnitude to influence transition . surface cooling, although providing a theoretical increase in stability to small disturbances, did not increase to any important extent the value of the critical roughness reynolds number for three-dimensional roughness particles . cooling, therefore, because of its effect on the boundary-layer thickness, density, and viscosity actually promoted transition due to existing three-dimensional surface roughness for given mach and reynolds numbers . the measured lateral spread of turbulence in the boundary layer appeared to be unaffected by the increased laminar stability derived from the surface cooling .
compressible laminar flow and heat transfer about a rotating isothermal disk . the flow and heat transfer about a rotating isothermal disk are re-examined to include the effects of compressibility and property variations . if viscous dissipation is neglected, the compressible problem is correlated to the incompressible problem by assuming linear variations of viscosity and thermal conductivity with temperature . certain inaccuracies in several previous incompressible solutions are noted and corrected herein . the effect of compressibility appears as a distortion of the normal coordinate and normal velocity component and as a multiplicative factor in the heat-transfer coefficient, the nusselt number, and in the expressions for the skin-friction components and torque required to rotate the disk .
theoretical investigation of the ablation of a glass-type heat protection shield of varied material properties at the stagnation point of a re-entering irbm . the melting-type heat protection at the stagnation point of a re-entering irbm is treated by employing homogeneous, opaque, and nondecomposing glass shields which do not exceed a temperature of some effects due to variations of the glass properties . the ballistic re-entry vehicle has a nose diameter of 0.635 m, a ballistic factor of 3.5 x 10, a re-entry angle of 124.9 (from the vertical) at an altitude of 100 km, and a re-entry speed of 4.5 . the performance of 36 different glass shields with assumed combinations of material properties is investigated by employing a calculation method which yields practically exact, transient solutions for the problem . as a corollary, results for a certain steady flight state are also given . the discussions made it possible to derive under realistic flight conditions some thermal characteristics for the employment of thin, or light-weight, glass shields . investigation of these hypothetical glass shields leads to the conclusion that a low thermal conductivity and a high specific heat, and thus, a small thermal diffusivity are most desirable . a small thermal diffusivity yields high surface temperatures, causing a high radiative heat transfer out of the shield,. and steep temperature profiles normal to the surface, causing a small thermal penetration across the shield with little total ablation of the shield . results show that for the assumed irbm re-entry, the necessary thickness of the employed glass shields increases monotonically with thermal diffusivity which is the only material parameter affecting this thickness . a high viscosity level and a high emissivity constant of the surface of the supposedly opaque shield are also desirable,. although, these two properties exert a comparatively small influence on the overall performance when disregarding glass shields with an extremely low viscosity level .
discussion of solar proton events and manned space flights . as a result of studies made during the international geophysical year (igy) and the international geophysical cooperation (igc), it is known that a considerable fraction of large solar flares give rise to almost pure streams of protons which reach the earth and continue to arrive for as long as 11 days . the energies of these particles lie within a very steep spectrum extending from 20 to least 500 mev . because of the frequency of large flares during times of high solar activity, and owing to the long duration of each solar proton emission, these particles were present in detectable intensity near the top of the earth’s atmosphere for about 15 percent of the time from 1957 to 1960 . the number of large flares that accelerated and released these particles during this three-year period was about 30 . the event that began on august 22, 1958 contributed greatly toward the understanding of the solar and terrestrial sequence of events, and in addition provided the first identification of the emitted particles . a flare on may of protons in the neighborhood of the earth that this phenomenon was recognized as an additional radiation hazard to manned vehicles in the high atmosphere and in most parts of the solar system . the three very intense events that occurred in july, 1959 further supported this conclusion, and the possibility of predicting such events became an important consideration . in addition to its value in the protection of human beings, effective forecasting clearly would be of great value in the detailed scientific study of this phenomenon . this paper presents a preliminary discussion of some aspects of predicting the arrival of protons at the earth following the appearance of solar activity features and, equally important, of forecasting the periods when this penetrating radiation is unlikely to occur .
experimental investigation of the downstream influence of stagnation point mass transfer . this report presents the results of an experimental investigation of the downstream influence of localized mass transfer in the stagnation region of a blunt body under hypersonic flow conditions . the coolant is injected through a porous plug coaxial with the centerline of symmetry of the model . the tests were carried out in a wind tunnel with a mach number of 6.0, stagnation temperatures of approximately 1,600 r., and a stagnation pressure of approximately 600 psia . four different gases were injected over a range of mass flows . the heat transfer on the impermeable section was measured under isothermal wall conditions,. for the higher rates of mass flow, adiabatic surface temperatures were also determined . the theoretical analysis of the boundary-layer flow is investigated in order to establish the similarity parameters for the flow system . these parameters permit the extrapolation of the test results to other flow conditions, provided that laminar flow prevails . helium is found to be the most efficacious coolant .
on trails of axisymmetric hypersonic blunt bodies flying through the atmosphere . the trail left in the atmosphere by a body moving at hypersonic speeds is the subject of theoretical treatment . the times required for ionization and dissociation (and their inverse processes) to go to completion, when compared to the flow times of a gas particle, are important in determining the observable effects of hypersonic trails-i.e., emitted thermal radiation and reflection of electromagnetic waves from the trail . in order to simplify the theoretical treatment, the trail is divided into two regions .. (1) the expansion-controlled trail, which treats the behavior of the wake behind the body up to a point, along the direction of flight, where the pressure decays to the free-stream value and cooling is controlled principally by the expansion of the flow, and (2) the conduction-controlled trail, where the trail cools mainly by diffusion of heat away from the high-temperature core . the influence of the details of the body shape on the observables are discussed and a simple computational procedure for the behavior of the conduction-controlled trail is developed based on integral methods . results of calculations that assume thermodynamic equilibrium of the flow field give the values of the thermodynamic variables in the trail of a sphere, axial distributions of emitted thermal radiation, and maps of electron density distribution . it is shown that the cooling of the conduction-controlled trail is essentially due to conduction of heat and that viscous effects are not important . it is found that this portion of the trail does not widen as one proceeds downstream . flight velocities considered vary between 15,000 and 35,000 ft sec and altitudes range between 100,000 and 250,000 ft .
inviscid-incompressible flow theory of static peripheral jets in proximity to the ground . an /exact/ flow theory of peripheral jets issuing symmetrically from a hovering aerial-ground vehicle is presented . the theory is exact insofar as no simplifying assumptions have been made in obtaining a solution of the governing inviscid, two-dimensional hydrodynamical flow equations . the results are valid for all jet thickness vehicle height ratios . the limit of applicability of existing theories (very low thickness height ratios) are defined . jet reaction, lift, and power coefficients for static conditions are introduced and computed . lift augmentation and lift power ratios are also calculated . applications to three-dimensional vehicles with rotational symmetry are indicated .
free-convection magnetohydrodynamic flow past a porous flat plate . the incompressible flow of an electrically conducting fluid past a porous plate with constant suction velocity in the presence of a transverse uniform strength has recently been investigated by gupta . in this note, the problem is generalized to take into account the effect of free convection, when a body force is acting parallel to the wall . the fluid is assumed to be semi-incompressible as usual . in addition to the obvious practical significance, this problem is also interesting in the sense that it provides another exact solution of the magnetohydrodynamic equations, since the only electromagnetic assumptions involved are constant properties and freedom from excessive charges .
magnetohydrodynamic free-convection pipe flow . it has been shown that transverse magnetic fields of practical strengths exert considerable influence on liquid-metal, free-convection, vertical, flat-plate and parallel-plate flow fields . the extent of influence was determined by the magnitude of a nondimensional parameter a which is the ratio of the hartmann number to the fourth root of the grashof number, and is a measure of the relative influence of the magnetic and buoyant forces . in this note the steady, fully developed, laminar, free-convection flow of a fluid of electrical conductivity through a fully submerged, open-ended, constant-temperature, vertical pipe located in a transverse magnetic field of strength is analyzed in terms of the same parameter . the magnitude of its influence on the velocity and temperature profiles, the surface shear and heat transfer, and the volumetric flow rate is determined .
an investigation of separated flows, part i: the pressure field . the present article describes an investigation of several types of separated regions such as blunt-base wakes and cavities formed in cutouts in the boundaries and ahead of or behind two dimensional steps in supersonic (mach numbers 2 to 4) and subsonic flow . the conditions for the existence, the geometry, and the pressure field are described in this paper . a second article (to be published) will describe investigations of the internal flow and the heat transfer across such separated regions . it is found that there is a maximum (critical) ratio of the length of the separated free-shear layer to the depth of the depression in the boundary beyond which the cavity collapses, leaving mutually independent separated regions at each protrusion . this critical length changes greatly upon laminar-turbulent transition in the oncoming boundary layer,. in either laminar or turbulent flow it is approximately independent of mach and reynolds numbers . a semiempirical correlation predicting the conditions under which the flow will span a depression of arbitrary depth is proposed . detailed pressure distributions along the boundaries of a cavity (in turbulent flow) are presented as a function of the ratio of the cavity length to the critical length, which is found to be the pertinent similarity parameter . for short notches the impact pressure due to the reversal of the inner portion of the shear layer at recompression tends to thicken the shear layer and a type of boundary layer-free stream interaction governs the pressure field . the pressure in the cavity is nearly constant and can be higher than free-stream . in long notches the shear layer bends inward at separation and curves back gradually ahead of the recompression point . the floor-pressure variation is pronounced and the recovery pressure at reattachment is small . the variation of the drag coefficient with mach number reflects the change from one to the other mechanism of recompression . detailed surveys of the mach-number distributions in a blunt-body wake and the mixing region behind its throat, as well as in the shear layer spanning a cutout in a wall, are presented and analyzed . it is found that, in general, the assumptions of the simple supersonic-wake models which rely on a principle of steady flow with mass conservation in the cavity are not adequate for cavities in which there is recompression against a boundary . results showing the influence of the thickness of the initial boundary layer (in the range of 0.3 to 3 times the notch depth) and of the geometry of the notch are also presented .
periodic temperature distributions in a two-layer composite slab . an investigation to determine the feasibility of using an insulating thermal barrier to protect exposed solid propellant motors from atmospheric or environmental temperature variations has recently been completed . in one portion of this study, a solution was developed for the periodic temperature distribution in a two-layer composite slab . one exposed surface of this composite slab was adiabatic, and the other exposed surface was subjected to a sinusoidal temperature variation . the technique used in the analysis was similar to that of grober . in this note, pertinent features of the development of the solution are given .
periodic temperature distribution in a two-layer composite slab . in a recent contribution to the reader’s forum, under the above title, stonecypher outlined a method for finding the periodic temperature distribution in a two-layer composite slab, one exposed surface of the slab being insulated and the other subject to a sinusoidal temperature variation . perfect thermal contact between the two layers, and constant thermal properties were assumed . two years ago i drew attention in these pages to a method for determining the transient temperature in such a two-layer slab resulting from a triangular heat-input pulse . i should like to point out that this same method also is applicable to the case where one external face is given a sinusoidal temperature variation with time . the method is based on the analogy between one-dimensional heat flow and the flow of an electric current in a simple transmission line having only series resistance and parallel capacitance .
the analysis of redundant structures by the use of high-speed digital computers . large-scale redundant structure analyses are currently feasible by the use of modern high-speed digital computers . this capability opportunely meets the urgent need to solve complex problems which otherwise would be hopelessly beyond the capacity of the hand desk computer . however, the difficulties have now shifted from tedious hand computations to the problems of adequately representing the structure by a model and of the peculiarities of irregular geometrical configurations . a wide scope of problem types can be handled by a generalized program approach . matrix formulation is used for the organization of input data and for handling data transfer in the large complex of subroutines, including the formation of equilibrium and continuity conditions to the final loads and deflections . simultaneous treatment of thermal expansions and plasticity is included . the use of minimum-size redundant systems is emphasized, starting from the philosophy of cutting members to provide a statically determinate structure . improved numerical accuracy and problem size capacity is gained for a given computer . examples are discussed ranging from simple plane-load diffusion problems to pressurized fuselage cutouts and complex wing-fuselage-shell intersection-type problems .
the supersonic blunt body problem – review and extensions . a survey of existing analytical treatments of the supersonic or hypersonic blunt-body problem indicates that none is adequate for predicting the details of the flow field . reasons are given for the failure of various plausible approximations . a numerical method, which is simpler than others proposed, is set forth for solving the full inviscid equations using a medium-sized electronic computer . results are shown from a number of solutions for bodies that support detached shock waves described by conic sections .
the transverse curvature effect in compressible axially symmetric laminar boundary layer flow . the viscous transverse curvature effect in compressible axially symmetric laminar boundary-layer flow has been investigated, and it is found that the effect is characterized by the parameter which is essentially the ratio of the boundary-layer thickness to body radius . it is shown that the busemann and crocco integrals of the two-dimensional energy equation for are still valid for axially symmetric flow in which the transverse curvature effects are considered . by a generalization of mangler’s transformation it is then shown that the boundary-layer equations are reducible to an almost two-dimensional form, making the analysis simpler for two asymptotic flow regions characterized by and less than or of the order of unity . it is with the latter region that the present paper is primarily concerned, and for this case it is shown that the additional term in the momentum and energy equations, which differentiates them from the two-dimensional form, behaves like an external favorable pressure gradient . except for certain special cases it is necessary to obtain the of the order of unity by means of asymptotic expansions in ascending powers of a parameter that is small compared to unity but proportional to . it is shown how the asymptotic solutions can be found for (1) the velocity and temperature distributions for the compressible zero pressure gradient case when the body shapes are given by and and (2) the velocity distribution for incompressible flow with an external velocity of the form past a body given by . the zeroth approximation is the mangler result . for the cases of a linear external velocity distribution, similar profiles can be found for all values of . more generally it is shown that similar profiles exist if the exponents n and m satisfy the condition that . here, similar is used in the restricted meaning that the distributions are derivable from ordinary differential equations . in the case of the cone and cylinder with zero pressure gradient where the equations have been numerically integrated for, the first-order correction to the mangler formulation shows that the effect on both the skin-friction coefficient and heat-transfer rate can become appreciable in the range where is less than or of the order of unity . at a constant, the effects are increased in magnitude when either the ratio of wall to free-stream temperature, or mach number, is increased . also, all other conditions being equal, for the same value of the skin-friction coefficient and heat-transfer increase on the cylinder is greater than that on the cone . for flows with pressure gradient, the transverse curvature term behaves again like a favorable pressure gradient and tends to delay both separation and transition when compared with axially symmetric flows in which the transverse curvature effect is neglected .
temperature distribution and thermal stresses in a model of a supersonic wing . the transient temperature distribution and the thermal stresses in an idealized wing structure considered by hoff and torda in reference 1 are determined . only the effects of aerodynamic heating and of heat conduction are included,. radiation and convection effects are neglected . the present work differs from that of reference 1 in that the conduction from the cap to the web is considered when the temperature of the cap is calculated, and the spar cap temperature is assumed to be a function of both space and time . graphs of temperature and thermal stress distributions are presented, and the results are compared with those of reference 1 .
review of published data on the effect of roughness on transition from laminar to turbulent flow . a review is presented of the published data on the effect of roughness, especially single roughness elements, on transition from laminar to turbulent flow, in which an attempt is made to reanalyze and correlate the available information . the reanalysis shows that the transition reynolds number of a flat plate with zero pressure gradient is a function of the ratio of the height of the roughness element to the displacement thickness of the boundary layer at the element, this functional relation being a better representation of the data than a constant critical reynolds number of the roughness element . other data show that the effects of roghness are similar in streams of different initial turbulence and that a plot of the ratio of transition reynolds number of the rough plate to that for the smooth plate against the ratio of the height of the roughness element to displacement thickness of the boundary layer at the element gives good correlation of all the data for a given shape when transition occurs downstream from the roughness element . at a certain value of the height-thickness ratio dependent on the stream speed, location of roughness element, and airstream turbulence, the transition position reaches the element and remains there as the height or the stream speed is further increased . the paper also discusses available data on the effect of distributed roughness on transition on a flat plate, as well as some of the published data on roughness effects on transition on air-foils .
a mixing theory for the interaction between dissipative flows and nearly isentropic streams . by means of a simplified theoretical /model,/ the present paper treats the general class of flow problems characterized by the interaction between a viscous or dissipative flow near the surface of a solid body, or in its wake, and an /outer/ nearly isentropic stream . for the present, the external flow is taken to be a plane, steady, supersonic flow, which makes a small angle with a plane surface or plane of symmetry, although the methods used can be extended to curved surfaces, to axially symmetric supersonic flows, and also to subsonic flows . the internal dissipative flow is regarded as quasi-one-dimensional and parallel to the surface on the average, with a properly defined mean velocity and mean temperature . the nonuniformity of the actual velocity distribution is taken into account only approximately by means of a relation between mean temperature and mean velocity . mixing, or the transport of momentum from outer stream to dissipative flow, is considered to be the fundamental physical process determining the pressure rise that can be supported by the flow . with the aid of this concept, a large number of flow problems is shown to be basically similar, such as boundary-layer shockwave interaction, wake flow behind blunt-based bodies (base pressure problem), flow separation in overexpanded supersonic nozzles, separation on wings and bodies, etc .
heat transfer by laminar flow to a rotating plate . an exact solution of the heat-transfer problem for the von karman example of the laminar flow of a viscous fluid over a rotating plate is given in dimensionless form and physically discussed . the solution is explicitly given for a constant temperature on the plate with viscous dissipation included . the numerical results are given for prandtl numbers from 0.5 to 10 .
the fundamentals of the statistical theory of turbulence . statistical theory in general considers mean values of certain quantities . in the case of the turbulent motion one is interested in mean values of velocities and of their derivatives, and in mean values of squares and products of velocities and their derivatives . it was o. reynolds who first expressed the so-called apparent or turbulent stresses by the mean values of the products of the velocity components . the different theories suggested so far have as their common objective the establishment of relations between certain mean values, e.g. between the turbulent shear stresses given by the mean products of velocity fluctuations and the derivatives of the mean velocities, i.e. the measured mean velocity gradients . in this sort of investigations the conception of the /correlation/ is of paramount importance . the late a. friedman tried to introduce the correlations as unknown variables in the hydrodynamic equations., however, he could not carry his investigations to practical results, i.e., to results which can be compared with the experimental evidence . recently, g. i. taylor had success in his analysis of /isotropic/ turbulence by means of correlation calculations, and was able to discuss, theoretically, the problem of the decay of turbulence in a windstream behind a turbulence producing device . his theory raised considerable interest because it is concerned with the important problem of wind-tunnel turbulence and its results could be compared directly with experimental work done by dryden in this country and by fage, townend and simmons in england . the present paper is concerned with two fundamental problems.. with uniform isotropic turbulence and with the turbulent friction in a parallel stream . first, the general theory of isotropic turbulence is developed . this general theory includes taylor’s consideration as a special case . however, it
vibration isolation of aircraft power plants . vibration in aircraft structure can almost always be traced to vibratory forces originating from the power plant . these forces are transmitted to the aircraft in two ways .. (1) by the action of air forces upon the surfaces of the aircraft in, or adjacent to, the slip stream of the propeller, and (2) by direct transmission of unbalanced forces from the power plant through the engine mounting . the latter has always caused the preponderance of disturbance . vibratory stresses induced in the engine mounting structure occasionally produce fatigue failures in the associated parts, and always shorten the useful life of the entire aircraft structure . more important, however, are the psychological and physiological effects of continuous vibration and its attendant noise on the passengers and crew . this may very likely be the major source of the rapid fatigue which is so intimately associated with flying . the importance and desirability of drastically reducing vibration can hardly be questioned . this paper is limited to a consideration of the directly transmitted forces and, further, considers the power plants as rigid bodies attached by flexible means to the aircraft which is also considered as a rigid body of relatively large mass . it is also limited to the case of engines and engine supporting structures having axial symmetry (radial engines), although the methods employed could easily be extended to other cases .
laminar heat transfer over blunt-nosed bodies at hypersonic flight speeds . this paper deals with two limiting cases of laminar heat transfer over blunt-nosed bodies at hypersonic flight speeds, or high stagnation temperatures.. (a) thermodynamic equilibrium, in which the chemical reaction rates are regarded as /very fast/ compared to the rates of diffusion across streamlines., (b) diffusion as rate-governing, in which the volume recombination rates within the boundary layer are /very slow/ compared to diffusion across streamlines . in either case the gas density near the surface of a blunt-nosed body is much higher than the density just outside the boundary layer, and the velocity and stagnation enthalpy profiles are much less sensitive to pressure gradient than in the more familiar case of moderate temperature differences . in fact, in case (a), the nondimensionalized enthalpy gradient at the surface is represented very accurately by the /classical/ zero pressure gradient value, and the surface heat-transfer rate distribution is obtained directly in terms of the surface pressure distribution . in order to illustrate the method, this solution is applied to the special cases of an unyawed hemisphere and an unyawed, blunt cone capped by a spherical segment . in the opposite limiting case where diffusion is rate-controlling the diffusion equation for each species is reduced to the same form as the low-speed energy equation, except that the prandtl number is replaced by the schmidt number . the simplifications introduced in case (a) are also applicable here, and the expression for surface heat transfer rate is similar., the maximum value of the ratio between the rate of heat transfer by diffusion alone and by heat conduction alone in the case of thermodynamic equilibrium is given by.. (prandtl no./schmidt no .) when the diffusion coefficient is estimated by taking a reasonable value of atom-molecule collision cross section this ratio is 1.30 . additional theoretical and (especially) experimental studies are clearly required before these simple results are accepted .
advantages and limitations of models . summary .. the use of models for structural test investigations in the presence of kinetic heating effects is examined . the principal features of the complex process to be represented are discussed under the classifications external air flow, internal heat transfer, elastic response . of these the second is found to influence most model design, and an analysis of a typical structure is included to illustrate the various contributions to internal heat transfer .
theory of mixing and chemical reaction in the opposed jet diffusion flame . an idealization of the flow system used by potter and butler is analyzed . the differential equation of mixing is solved exactly, to give the location of, and burning rate in, the flame . the solutions to the chemical kinetic differential equation are discussed, relations being derived between the jet flow rate at extinction, the chemical kinetic constants and the laminar flame speed in premixed gases . it is shown that the jet flow rate at extinction is independent of the transport properties . comparison is made with the experimental data of potter, heimel and butler . it is argued that experiments must be carried out at higher reynolds numbers if the measurements are to be quantitatively analyzable .
similar solutions of a free convection boundary layer equation for an electrically conducting fluid . author investigates the existence of a class of similar solutions for free convection from a vertical flat plate, such as are known for free convection in a nonconducting fluid . the magnetic field acts transversely to the fluid motion and is assumed to remain constant in the direction perpendicular to the plate . this introduces into the momentum equation a retarding force which is a function only of x, the distance along the plate length . for similarity it is found that the magnetic inductance must vary as . if the plate temperature is constant . if n = 0, the magnetic inductance is constant while the plate temperature increases linearly with x .
the asymptotic boundary layer on a circular cylinder in axial incompressible flow . in this paper the incompressible boundary layer over a circular cylinder in an axial flow is investigated far from the leading edge . if u and v are the velocity components in the x and r direction respectively and a stream function is introduced by and, then for a constant free-stream velocity has the following asymptotic form .. where the p’s are determined successively, first for s=1 and all t, then s=2 and all t, etc., from ordinary differential equations . here and log c=euler’s constant . it is shown that the effect of the curvature of the body (in planes perpendicular to the flow) is to increase the skin friction . also the case in which the free-stream velocity is proportional to (at the method breaks down), is studied . it is concluded that the effect of the curvature of the cylinder, when the boundary layer has a thickness comparable with its radius of curvature, is to delay separation .
the transverse potential flow past a body of revolution . it is shown that in the potential flow of an incompressible inviscid fluid past a body of revolution set with its axis at right angles to the stream, the velocity components at the surface along and perpendicular to the meridians vary with azimuthal angle round the body in a simple manner . this is shown by entirely elementary considerations .
on the mixing of two parallel streams . using the techniques of boundary-layer theory, the proper third boundary condition for the mixing of two parallel streams is derived from the compatibility condition of the higher order approximation . it is shown that the commonly adopted third boundary condition of balancing of transverse momentum is correct only for the mixing problem of two semi-infinite incompressible streams . for the fulfillment of the proper third boundary condition, the possibility of introducing the similar solution of blasius type is examined for various cases .
properties of the confluent hypergeometric function . the confluent hypergeometric functions have proved useful in many branches of physics . they have been used in such problems involving diffusion and sedimentation, as isotope separation and protein molecular weight determinations in the ultracentrifuge . the solution of the equation for the velocity distribution of electrons in high frequency gas discharges may frequently be expressed in terms of these functions . the high frequency breakdown electric field may then be predicted theoretically for gases by the use of such solutions together with kinetic theory . this report presents some of the properties of the confluent hypergeometric functions together with six-figure tables of the functions .
the production of uniform shear flow in a wind tunnel . a nearly uniform shear flow was obtained in the working section of a wind tunnel by inserting a grid of parallel rods with varying spacing . the function of such a grid is to impose a resistance to the flow, so graded across the working section as to produce a linear variation in the total pressure at large distances downstream without introducing an appreciable gradient in static pressure near the grid . a method of calculating a suitable arrangement of the rods is described . although this method is strictly applicable only to weakly sheared flows, an experiment made with a grid designed for a shear parameter as large as 0.45 gave results in close agreement with the theory . there was no evidence from the experiment of any large-scale secondary flow accompanying the shear–a danger inherent in an empirical attempt to grade the resistance of the grid–nor was any tendency observed for the shear to decay with increasing distance from the grid .
dynamics of a dissociating gas . this is a lucid introduction to the effects of dissociation in gas dynamics . the problem in view is that of air flow past a bluff body at speeds somewhat above 2 km sec . thermodynamic equilibrium is assumed,. theories of near equilibrium for transport properties and of large departures from equilibrium being promised in parts 2 and 3 . following a survey of the equilibrium statistical thermodynamics of a pure dissociating diatomic gas, a new model is introduced . this /ideal dissociating gas/ is characterized by only three constants, the characteristic temperature, density and internal energy for dissociation . physically, it may be regarded as having its vibrational modes always just half excited (so that at low temperatures the ratio of specific heats approaches 4 3 rather then 7 5) . thermodynamic properties of the ideal gas are derived, and the oblique shock wave relations deduced in the /strong-shock/ approximation (including an elegant relation between the principal curvatures of any bow shock and the subsequent vorticity) . useful relations are given for the isentropic changes that take place along streamlines between shocks . various of these results are applied to the problem typified by a sphere flying at high mach number . the newtonian impact theory and its empirical modification are dismissed as lacking theoretical basis, in favor of the limit for large values of both mach number and density ratio across the shock . it is suggested that the zero surface pressure sometimes predicted by the latter theory corresponds to separation not of the flow but of the shock wave from the surface . an estimate is given for the subsequent shape of the shock . finally, another approximation is applied to the region near the stagnation streamline . the fluid is assumed incompressible, but rotational in accord with the shock relations,. and it is shown that a spherical shock corresponds to a concentric spherical body . the resulting surface pressure is within 1 per cent of that predicted by freeman’s second approximation based on the newtonian-plus-centrifugal solution (same j. 1 (1956),
the laminar boundary layer equation: a method of solution by means of an automatic computer . a method, very suitable for use with an automatic computer, of solving the hartree-womersley approximation to the incompressible boundary-layer equation is developed . it is based on an iterative process and the choleski method of solving a simultaneous set of linear algebraic equations . the programming of this method for an automatic computer is discussed . tables of a solution of the boundary-layer equation in a region upstream of the separation point are given . in the upstream neighbourhood of separation this solution is compared with goldstein’s asymptotic solution and the agreement is good .
steady motion of conducting fluids in pipes under transverse magnetic fields . this paper studies the steady motion of an electrically conducting, viscous fluid along channels in the presence of an imposed transverse magnetic field when the walls do not conduct currents . the equations which determine the velocity profile, induced currents and field are derived and solved exactly in the case of a rectangular channel . when the imposed field is sufficiently strong the velocity profile is found to degenerate into a core of uniform flow surrounded by boundary layers on each wall . the layers on the walls parallel to the imposed field are of a novel character . an analogous degenerate solution for channels of any symmetrical shape is developed . the predicted pressure gradients for given volumes of flow at various field strengths are finally compared with experimental results for square and circular pipes .
acoustical signal detection in turbulent airflow . improvement in detected signal-to-noise ratio is obtained for a periodic signal masked by additive noise and turbulent noise backgrounds . comparisons are made between autocorrelation, crosscorrelation, and a combination of frequency filtering and crosscorrelation . although the latter method provided the greatest improvement, the crosscorrelation technique was the most successful single method . it turned out that the maximum improvement obtainable was limited by the dynamic range of the correlator computer and not by errors due to finite averaging time and scanning the delay . the improvement for signals masked by turbulent noise was found to be about 5 db less than that obtained for additive noise .
response of plates to a decaying and convecting randon pressure field . following the methods of lyon, an analysis of the vibratory response of a plate to a random pressure field is given . the pressure correlation of the random field is assumed to have a scale small compared to the plate size, to decay exponentially, and to convect with constant speed over the plate . two cases are considered, one in which the convection speed is much less than the speed of free flexural waves in the plate, the other in which the convection speed is the same order as the flexural wave speed . the mean square plate displacement is shown to be relatively independent of convection for speeds much less than the flexural wave speed, and to increase significantly for speeds in the order of the flexural wave speed . it is shown that damping is usually, but not always, an effective means of vibration reduction . in the case of convection speeds much smaller than the flexural speed, the use of hysteretic damping for reduction of the displacement response is shown to be limited by the decay of the assumed random pressure field .
on turbulent lubrication . the paper concerns the hydrodynamic turbulent motion in the lubricant layer . proceeding from the reynolds equations and introducing the approximations currently used in lubrication problems, owing to the lubricant film thickness, the general motion equations for turbulent lubrication are written . using the prandtl mixing length hypothesis, exact and approximate solutions are obtained for the velocity distribution into the lubricant layer . the results are discussed by pointing out the pressure gradient and the reynolds number influence on the velocity distributions, as well as the differences with respect to the laminar flow . in order to obtain simple formulae, the exact dependence of the rate of flow on the pressure gradient into a dimensionless form is replaced by a linear relation, the slope of which depends on the reynolds number . this approximation allows the obtainment of the pressure differential equation under a simple form . the pressure equation is integrated in case of journal bearings, by assuming a constant or a variable viscosity of the lubricant . the results are compared to the experimental data obtained by m. i. smith and d. d. fuller and the good qualitative agreement is pointed out .
the elliptic cylinder in a shear flow with hyperbolic velocity profile . the stream function for the shear flow with hyperbolic velocity profile past an elliptic cylinder has been determined as an infinite series of mathieu functions . it is found that the stagnation streamline of the flow is displaced towards a region of higher velocity, this displacement increasing the main stream, (2) as the stream becomes progressively non-uniform, (3) with increase of minor axis length when the major axis length remains invariant . in each case the displacement reaches a limiting value as the cylinder moves away from the axis of symmetry of the stream . these limiting values are reached at critical distances from the axis of symmetry, which decrease as the stream becomes progressively non-uniform, but these distances are approximately independent of incidence . the pressure coefficients and the resultant force and moment coefficients associated with the cylinder have also been obtained, and investigated numerically for the flat plate type of cylinder .
the motion of a viscous liquid past a paraboloid . an approximate solution for the steady flow of incompressible viscous liquid past a paraboloid of revolution is described . an assumption is made for the form of the stokes stream function and substituted into the navier-stokes equations using paraboloidal coordinates . after making suitable approximations, a non-linear differential equation for a function f is deduced . the solutions of this equation depend on the reynolds number of the flow considered . examples found by numerical integration are given to illustrate the properties of the function f for reynolds numbers varying from 0.0001 to is found, and it is shown that this approximate solution tends to the perfect fluid flow away from the boundary, allowance being made for the displacement effect of what may be called the boundary layer .
the transonic flow of a compressible fluid through an axially symmetrical nozzle . by a method similar to that developed by s. tomotika and k. tamada (quart. appl. math. 7, 381-397 (1950),. these rev. 11, 275) for computing two-dimensional mixed isentropic flows in the sonic region, the flow in the vicinity of the throat of an axially symmetrical nozzle is studied . several exact solutions to von karman’s equation for axially symmetrical transonic flows are obtained and the one that gives flows through a converging and diverging nozzle is considered in detail . this solution consists of four branches of which two are rejected because of singularities . of the remaining two branches, one gives pure supersonic flow and the other gives taylor’s type of flow with a local supersonic region in the throat . by varying a parameter, the latter branch approaches two asymptotes which yield meyer’s type of asymmetrical flows .
conduction of fluctuating heat flow in a wall consisting of many layers . van gorcum has pointed to interesting and important analogies between the theory of a passive four-pole and the conduction of heat waves through stratiform bodies . this paper generalizes in certain regards van gorcum’s ideas and draws their consequences for the case of a solid, bounded by two infinite parallel planes and consisting of any number of layers made from different materials .
measurement of convective heat transfer by means of the reynolds analogy . preston’s method for measuring skin friction in pipes has been extended to include non-uniform flow, with and without pressure gradients, over flat surfaces . by means of a modified form of the reynolds analogy, the local convective heat transfer coefficient can be related to the skin friction, and it is proposed that the method be used in aerodynamic models of furnaces and in heat transfer plant of simple geometry . more investigations are required of the effects of fluid turbulence, surface roughness and surface curvature on convective heat transfer and skin friction .
a theory for base pressures in transonic and supersonic flow . a physical flow model is devised based on the concepts of interaction between the dissipative shear flow and the adjacent free stream and the conservation of mass in the wake . four flow components are integrated in the model,. namely, the flow approaching the trailing edge, the expansion around the trailing edge, the mixing within the free-jet boundary, and the recompression at the end of the wake . a unique and stable solution results for the base pressure . theoretical results obtained for thin approaching boundary layer do not require empirical information and are, therefore, best suited to evaluate the merits of the theory . here emphasized is the case of isoenergetic constant-pressure mixing in the turbulent free-jet boundary and agreement is found between theory and experimental data .
a simplified approximate method for the calculation of the pressure around conical bodies of arbitrary shape in supersonic and hypersonic flow . exact conical-flow solutions are available only for circular cones at zero angle of attack . for nonaxisymmetric cones or cones at angle of attack, only approximate methods exist . these methods are generally quite complicated and further limited to certain body shapes or certain mach-number ranges . a great need was therefore felt for a simple approximate method applicable to any arbitrarily shaped conical body at zero incidence as well as at angle of attack . such a method has been developed recently at lockheed and is presented here in abbreviated form . the method is based on the /equivalent-cone/ theory . this theory determines the pressure on a conical body utilizing information for a symmetric cone at zero angle of attack with the same normal component of the free stream with respect to the surface as the local element of the body considered . this method works relatively well at high mach numbers . however, it is quite inconsistent at lower mach numbers, especially for bodies which deviate considerably from circular cones . the equivalent-cone method does not give satisfactory results, mainly due to the fact that it considers only the local surface element on the body independent of the other body elements in the newtonian-theory manner .
the downstream influence of mass transfer at the nose of a slender cone . the influence of localized mass transfer at the nose of a slender cone under hypersonic flow conditions has been studied by experimental and theoretical means . two gaseous coolants, nitrogen and helium, are injected through a porous plug subtending a half angle of 30 . the effect of the mass transfer on the shock shape, pressure distribution, heat transfer, and transition are investigated . the experimental work involved tests in the mach-number-8.0 tunnel at pibal . the theoretical analysis involved a study of the effect of mass transfer on the shock stand-off distance and leads to an inviscid-flow parameter permitting the experimentally determined shock shape and pressure distribution to be extrapolated to other than test conditions and to other coolant gases . there is obtained the maximum value of this parameter resulting in no significant alteration of the pressure distribution on the cone and thus defining the flows in which boundary-layer-type similarity applies . significant reductions in heat transfer are obtained with injection . indeed, with small amounts of helium injection the peak heating is found to occur downstream on the cone and to be an order of magnitude less than would occur at the stagnation point without mass transfer . with nitrogen early transition is found to occur, so that local heating rates are actually increased over those prevailing at the same reynolds number without injection .
a summary of the supersonic pressure drag of bodies of revolution . a number of approximate theories for supersonic and hypersonic flow over bodies of revolution at zero angle of attack are appraised by a critical comparison with characteristics and second-order results, with the use of hypersonic similarity as a basis for the comparison . most of the approximate theories are inadequate except over very limited ranges of fineness ratio and mach number . the combination of second-order supersonic theory and second-order shock-expansion theory provides consistently good results throughout the supersonic speed range . on the basis of exact (or nearly exact) supersonic solutions and a limited amount of test data and theory in the transonic region, summary design curves are developed that give the pressure drag of conical and ogive noses and conical and ogive boattails over the complete range of transonic, supersonic, and hypersonic mach numbers . other shapes can be analyzed in the same manner, provided that an equivalent amount of data is available . the analysis is made with the assumption of inviscid flow, so that the effects of boundary-layer growth, shock boundary-layer interaction, and flow separation are not included . the present correlations provide a sound basis of inviscid-flow results from which these additional viscous effects can be evaluated .
measurements of skin friction of the compressible turbulent boundary layer on a cone with foreign gas injection . measurements of average skin friction of the turbulent boundary layer have been made on a 15 total included angle cone with foreign gas injection . measurements of total skin-friction drag were obtained at free-stream mach numbers of 0.3, 0.7, 3.5, and x 10 with injection of helium, air, and freon-12 through the porous wall . substantial reductions in skin friction are realized with gas injection within the range of mach numbers of this test . the relative reduction in skin friction is in accordance with theory–that is, the light gases are most effective when compared on a mass flow basis . there is a marked effect of mach number on the reduction of average skin friction,. this effect is not shown by the available theories . limited transition location measurements indicate that the boundary layer does not fully trip with gas injection but that the transition point approaches a forward limit with increasing injection . the variation of the skin-friction coefficient, for the lower injection rates with natural transition, is dependent on the flow reynolds number and type of injected gas,. and at the high injection rates the skin friction is in fair agreement with the turbulent boundary-layer results .
an investigation of two-dimensional supersonic base pressures . an investigation of the base pressure behind wedges at mach numbers 2 and 3 in the laminar and the transitional regime is reported . temperature and velocity traverses through the mixing zone are shown and exploratory investigations of the wake vortex by use of hot wires and flow-visualization techniques are described . it is found that the laminar two-dimensional base pressure agrees well with chapman’s theoretical predictions . the shear layer exhibits gross velocity distributions characteristic of the free jet mixing zone, but also shows disturbances that originate in the expansion-turning of the oncoming boundary layer . an interesting trailing vortex is observed, which is explained in terms of nonuniform mixing rate in the wake .
supersonic axially symmetric nozzles . at each of twenty-one exit mach numbers, ranging from 1.008 to 8.238, ten supersonic axially symmetric nozzle shapes with plane sonic surfaces have been computed on the eniac by the method of characteristics . the boundary of the shortest of each group of ten has a sharp edge at the sonic plane, while the others have smooth boundaries . this report describes the computational procedures and presents a sample of the results for twenty nozzles . more extensive and elaborate tables of the results of the entire computations are available at the ballistic research laboratories . nozzle contours can be obtained accurately from them by interpolation for exit mach numbers between 1.479 and 8.238 for a wide range of ratios of nozzle length to throat diameter .
effects of free stream vorticity on the behaviour of a viscous boundary layer . theoretical investigation is considered of the two-dimensional steady flow field at large distance from a finite object set in a viscous incompressible fluid . study is made of coordinate-type expansions for pressure and velocity for large r, uniformly in, for fixed reynolds number, assuming exact boundary conditions at infinity and regularity of flow with zero net mass flow across a simple curve enclosing the object . mathematical nature of the distinction between parameter and coordinate-type expansions is discussed with description of inner and outer expansions and matching techniques . a feature of the expansion procedure is the introduction of an artificial parameter . inner and outer expansions are matched with the aid of known solutions of the navier-stokes equations . analysis requires simple consideration of the heat and laplace equations without resort to special methods . paper is worth studying by those interested in asymptotic expansion procedures .
an investigation of the noise produced by a subsonic air jet . to investigate the theoretical predictions of lighthill on aerodynamic sound, measurements have been made of the sound field of a 1 in. air jet issuing from a long pipe . the measurements have been made over a wide frequency band (30 to 10,000 cycles/sec.) and in one-third octave bands in this frequency range . the mean mach number at the pipe orifice was varied from 0.3 to 1.0 . the dependence of the apparent position of the noise sources on frequency and jet speed was investigated . at a given frequency a source is situated farther from the jet orifice the higher the jet speed . lower frequency sources appear farther downstream than ones of higher frequency, consistent with their association with larger eddies . the directional characteristics of the sound field at different frequencies and jet speeds are illustrated by means of scale diagrams showing lines of constant sound intensity . these sound fields are analyzed in terms of the moving quadrupole sources of lighthill’s theory and good agreement obtained . it is shown that the apparent spread of the sources at low frequencies is due to the doppler effect . at low frequency relative to the frequency of maximum power output) the radiation is predominantly that of three mutually orthogonal longitudinal quadrupoles which, except for the effect of convection upon it, has a sound field like a monopole source . at higher frequencies the sound fields of lateral and longitudinal quadrupoles predominate .
the behaviour of non-linear systems . many of the phenomena that occur in the world around us are governed by nonlinear relationships . in the development of the mathematical sciences, the difficulties of nonlinear analysis have hindered the formulation of nonlinear concepts that would permit us to understand such phenomena . in the present article, our progress in understanding the behavior of nonlinear systems is reviewed and an attempt is made to present the resulting concepts in such a way that they may be applied with some generality to other problems .
two-dimensional jet mixing of a compressible fluid . the mixing and divergence of a supersonic jet exhausting into a supersonic stream are investigated theoretically . in the first part of this paper, the flow is assumed to be laminar . when the velocity and temperature in the jet are different slightly from those of the surrounding stream, by the method of small perturbations and under ordinary boundary layer assumptions, the equation of motion of two-dimensional flow will be reduced to a form of the well-known equation of heat conduction, whose solution is known for any given boundary conditions . it has also been shown that the exact solution of the two dimensional jet mixing of viscous compressible fluids can be obtained by successive approximations starting with the solution of small perturbations . velocity and temperature distributions for two cases–one is the mixing of two-uniform flows and the other is the mixing of a jet of compressible fluid from a two-dimensional nozzle with full expansion exhausting into a supersonic stream–have been calculated . the properties of the jet mixing depend mainly on the momentum of the jet regardless of whether the change of momentum is due to the change of velocity or the change of temperature–i.e., the change of density . compressibility has a considerable effect on the properties of the jet . in the second part, the cases of turbulent flow are investigated . by means of reichardt’s theory of free turbulence, the turbulent shearing stress may be expressed as it has been shown in this paper that where is a constant that can be determined experimentally . the value of n lies between 0 and 1 . the exact value of n depends on the condition of mixing . when the expression of turbulent shearing stress given above is used instead of the viscous stress in the equation of motion, by suitable transformation of variables, it has been shown that the equation of two-dimensional turbulent jet mixing is identical to that of the laminar case . hence, the solution of the first part of this paper can be applied to the turbulent case, provided that the characteristic constants and n have been properly chosen .
viscosity effects in sound waves of finite amplitude: in survey in mechanics . this article has as its subject /the conflicting influence on sound propagation of convection on the one hand, and of diffusion and relaxation on the other/, whose importance in the determination of the structure of shock waves was first appreciated clearly by sir geoffrey taylor . as an essential introduction to the main topics, author gives an exceptionally clear and valuable account of the physical mechanisms of viscosity, thermal conductivity, and other diffusion effects, including relaxation . the classical theory of shock-wave formation is then discussed, and some extensions are made . the remainder of the article is based on the demonstration that the nonlinear equation for plane progressive sound waves, in which convection and diffusion are taken into account to a first approximation, can be transformed into burgers’s equation, the general solution of which was given by hopf and cole . this approach, in which all flows are continuous (they become discontinuous at shock waves in the limit as viscosity, etc., tend to zero), allows the author to re-derive and extend whitham’s theory of the formation and decay of weak plane shock waves, and to derive many new results, such as the velocity distributions during the union of two shock waves and during the formation of a shock wave . the application of the same idea to non-plane shock waves is also discussed, but more briefly,. in these cases, burgers’s equation is not quite such a good approximation as before . the article concludes with sections on sound waves whose reynolds numbers based on the length scale of the flow and the velocity amplitude are comparable with unity, and on the effects of relaxation on the properties of shock waves . the whole is much more than a survey, and represents a very substantial advance in the theory of sound waves . it is the finest possible tribute to sir geoffrey taylor that he should be able to inspire articles such as this and the others in this volume .
some effects of surface curvature on laminar boundary layer flow . the laminar flow of a viscous incompressible fluid over a two-dimensional curved surface is investigated for two cases, one in which the curvature is /large/ and the other in which it is cases are obtained as approximations from the exact equations of motion by an order-of-magnitude analysis . these equations are solved for flow over a particular surface with zero surface pressure gradient . in this analysis, the pressure gradient normal to the surface is included, and the outer boundary conditions are modified in accordance with the requirements of flow over a curved surface . the results indicate that for equal reynolds numbers, the stress on convex surfaces is less than the flat-plate value, while the stress on concave surfaces is greater than for a flat plate . the most important effect of surface curvature, for the cases considered, is the modification of the shape of the velocity profile near the /outer edge/ of the boundary layer . the requirement that a smooth transition exist between the viscous flow and the potential flow at the outer edge of the layer causes the profile to have a negative slope near the outer edge for convex surface curvature and a positive slope for concave surface curvature .
note on an interaction between the boundary layer and the inviscid flow . according to the classical boundary-layer theory the flow about bodies at reynolds numbers of aeronautical interest can be considered as composed of two regimes.. an outside inviscid flow and a thin boundary-layer region adjacent to the body . this point of view leads to the approximation that, on a slightly curved surface, throughout the layer is negligibly small . the additional assumption that the inviscid flow is irrotational leads to the requirement that is zero at the outer edge of the boundary layer . in this theory any interaction between the two regimes is accountable by a simple correction to the body shape based on the boundary-layer displacement thickness . recently, in connection with hypersonic laminar boundary layers, this classical point of view has been modified., an interaction between the two flow regimes leading to a self-induced axial pressure gradient has been considered . it is the purpose of the present note to point out another type of interaction which may be of practical importance and of fundamental interest even at mach numbers below those considered in the hypersonic boundary-layer theory and which may have to be considered in that theory .
the calculation of wall shearing stress from heat-transfer measurements in compressible flows . it has been shown by ludwieg that the wall shearing stress of a laminar or turbulent boundary layer in an incompressible flow can be determined from a heat-transfer measurement at the surface . the instrument used in that investigation was essentially a small, locally insulated, heating element embedded in the test surface . the size of the instrument was restricted by the condition that the thermal boundary layer generated by the heating element be contained locally within the laminar sublayer . in the present analysis ludweig’s theory for such an instrument is extended to compressible flow over an insulated flat plate . with the same limitations on the design and operation of the instrument as mentioned above, it can also be assumed for compressible laminar and turbulent boundary layers that only the flow in the immediate vicinity of the wall or the laminar sublayer will be affected in the region of the heated element . this assumption then permits the use of the laminar boundary-layer equations as the governing equations for this analysis for both laminar and turbulent boundary layers .
recent developments in rocket nozzle configurations . existing configurations of supersonic portion of rocket nozzles are described and compared . survey covers bell-type conical and contoured nozzles, annular nozzles, plug nozzles, and the author’s own /e-d/ (expansion-deflection) nozzle . the latter is a bell-type nozzle in which the gases are first deflected radially outward by a small central plug, then expanded radially inward around the base of the plug, and finally deflected back to a nearly axial direction by the nozzle wall, in compressive turning .
the generation of sound by aerodynamic means . a summary is given of some of the more important experimental results relating to the noise radiated from a cold subsonic turbulent jet . these are then related to the predictions of lighthill’s general theory of aerodynamic noise .
wakes in axial compressors . the tendency in the past has been to assume that when wakes or non-uniform total head profiles are fed into an axial compressor then substantially constant static pressure prevails at the entry, the variations in total head appearing as variations in velocity . this variation in velocity causes variation in incidence on the early stage blade rows and thus can give rise to excitation of blade vibration . this assumption is implicit, for instance, in references 1 and 2, but we think has been a common assumption by most of the people working in this field . where the compressor is fed by a duct of substantially parallel walls for a reasonable length ahead, such an assumption appeared justifiable . such a duct when given an air flow test with its outlet discharging, for instance, to atmosphere instead of to the compressor, then the distribution assumed would normally be obtained and in fact many surveys of such ducts have been represented in this fashion . the object of this note is to show that, in fact, this distribution will not normally occur when the compressor is present and we may normally expect much more nearly a constant velocity into the compressor with attendant static pressure distributions to match with the total head variations ahead of the intake, with of course, the attendant curved flow to support the static pressure gradients .
viscous effects on pitot tubes at low speeds . measurements were made of the pressure in a blunt-nosed pitot tube, in an air stream at reynolds numbers from about 15 to 1000 . the results are expressed in terms of a pressure coefficient density of the fluid, and p and v are the static pressure and velocity in the undisturbed stream . as found in previous investigations, becomes greater than 1 at low reynolds numbers, the increase being about at a reynolds number of 50 (based on external tube radius) . in disagreement with the work of hurd, chesky, and shapiro, no decrease of below 1 was found at any reynolds number . when the values of found by various experiments are plotted against reynolds numbers based on internal tube radius, it is found that the curves are in closer agreement than when the external radius is used .
the determination of turbulent skin friction by means of pitot tubes . a simple method of determining local turbulent skin friction on a smooth surface has been developed which utilises a round pitot tube resting on the surface . assuming the existence of a region near the surface in which conditions are functions only of the skin friction, the relevant physical constants of the fluid and a suitable length, a universal non-dimensional relation is obtained for the difference between the total pressure recorded by the tube and the static pressure at the wall, in terms of the skin friction . this relation, on this assumption, is independent of the pressure gradient . the truth and form of the relation were first established, to a considerable degree of accuracy, in a pipe using four geometrically similar round pitot tubes–the diameter being taken as representative length . these four pitot tubes were then used to determine the local skin friction coefficient at three stations on a wind tunnel wall, under varying conditions of pressure gradient . at each station, within the limits of experimental accuracy, the deduced skin friction coefficient was found to be the same for each pitot tube, thus confirming the basic assumption and leaving little doubt as to the correctness of the skin friction so found . pitot traverses were then made in the pipe and in the boundary layer on the wind tunnel wall . the results were plotted in two non-dimensional forms on the basis already suggested and they fell close together in a region whose outer limit represented the breakdown of the basic assumption, but close to the wall the results spread out, due to the unknown displacement of the effective centre of a pitot tube near a wall . this again provides further evidence of the existence of a region of local dynamical similarity and of the correctness of the skin friction deduced from measurements with round pitot tubes on the wind tunnel wall . the extent of the region in which the local dynamical similarity may be expected to hold appears to vary from about to of the boundary-layer thickness for conditions remote from, and close to, separation respectively .
free-flight techniques for high speed aerodynamic research . the development rocket-borne and rocket-launched high-speed airplane model test is described . details of airborne components, telemetering units, tracking, and their calibration are also discussed . tests on controls, drag measurements, longitudinal stability evaluations, lift measurements, pressure measurements, aeroelastic estimations, and sonic bang recordings are effected . the reynolds numbers involved are much higher than are usual in the wind tunnel, and extensions of mach numbers are obtained beyond the tunnel limits, both free of the tunnel wall interference .
the problem of aerodynamic heating . paper is a good review of knowledge to date on convective heat transfer to objects moving through air at low and high speeds . theoretical and experimental information is given on recovery factors and heat-transfer coefficients for isothermal surfaces of unswept flat plates, wedges and cones with attached shock waves, and stagnation points of blunt bodies of revolution, for both laminar and turbulent boundary layers . a convenient nomograph for calculating flat plate turbulent boundary-layer heat-transfer coefficients is given . effects of surface cooling, surface roughness, and supply stream turbulence on transition are discussed and shown graphically .
interplanetary orbits . the basic equations under simplified conditions for interplanetary flight are derived . for a voyage from planet to planet an unlimited number of orbits is possible . in order to give a clear survey of these possible orbits a diagram is developed from which the approximate energy-requirement, the duration, and other particulars of a voyage can be easily found .
heat flow in composite slabs . this paper presents the solution of the heat flow problem in composite walls under heat transfer conditions which are typical of uncooled rocket engine walls . analytic expressions in the form of fourier sums are obtained for the temperature distribution in a composite wall consisting of an inner (refractory) medium and an outer metallic) medium under newtonian heat transfer into the first medium with negligible heat transfer from the second medium to the exterior . the expressions obtained are based on a plane parallel composite slab as a representative model for relatively thin cylindrical walls, with thickness-to-radius ratio not exceeding 0.2 . the general results for the composite slab are simplified for the limiting cases of a thin refractory shield with a thick shielded medium and a thick refractory shield with a thin shielded medium .
skin friction in the laminar boundary layer in compressible flow . from an analysis of the work of crocco and others, semi-empirical formulae are derived for the skin friction on a flat plate at zero incidence with a laminar boundary layer . these formulae are for the general case of heat transfer, and when there is no heat transfer . the problem of heat transfer and the effect of radiation are discussed in the light of these formulae . the second formula is then utilised in the development of an approximate method for solving the momentum equation of the boundary layer on a cylinder without heat transfer . the method indicates that with increase of mach number there is a marked forward movement of separation from a flat plate in the presence of a constant adverse velocity gradient .
supersonic flow past slender bodies with discontinuous profile slope . ward’s slender-body theory is extended to derive first approximations to the external forces on slender bodies of general cross section with discontinuous profile slope . two classes of body are considered .. bodies whose profile (typified by the local radius) is continuous between the nose and base, and certain bodies whose profile is discontinuous, such as bodies with annular or side air intakes and wing-bodies on which the wing has an unswept leading edge . (where air intakes are concerned, it is assumed that they are sharp-edged and that there is no /spillage/ of the internal flow) . the following conclusions apply to the former class of bodies . the variation of drag with mach number is found to depend only on the discontinuities in the longitudinal rate of change of the cross-sectional area, and is thus independent of cross-sectional shape . the drag itself is unchanged if the direction of the flow is reversed . the expressions for lift and moment assume the same forms as for smooth pointed bodies, the lift depending only on conditions at the base of the body . the general theory is applied to winged bodies of revolution with an unswept wing leading edge .. the results bear a marked resemblance to those obtained by ward . the results for wings alone are seen to be applicable, with one modification, to subsonic as well as to supersonic speeds .
supersonic flow past slender pointed wings with ?similar? cross sections at zero lift . some recent theoretical work on slender pointed wings at zero lift is co-ordinated and extended . the wings considered may have any pointed plan form shape, provided that the trailing edge is straight and unswept . the root section profile and cross-section shapes are arbitrary, provided that, on any one wing, the latter are /descriptively similar/ (diamond or parabolic biconvex for instance), though not necessarily geometrically similar . the chief aim of the work is to find wings with simple geometry, low wave drag and pressure distributions which are unlikely to be seriously affected by viscous effects . wave drag and pressure distributions are calculated by slender-wing theory . general formulae, which are both simple and instructive, are given for the wave drag and the overall pressure distribution, with particular emphasis on the root pressure distribution . results for a number of wings of special interest are presented and discussed .
on displacement thickness . four alternative theoretical treatments of ‘displacement thickness’, and, generally, of the influence of boundary layers and wakes on the flow outside them, are set out, first for two-dimensional, and then for three-dimensional, laminar or turbulent, incompressible flow . they may be called the methods of ‘flow reduction’, ‘equivalent sources’, ‘velocity comparison’ and the principal expression obtained for the displacement thickness in three-dimensional flow may be written if, as orthogonal coordinates (x,y) specifying position on the surface, we choose x as the velocity potential of the external flow, and y as a coordinate, constant along the external-flow streamlines, such that h dy is the distance between (x,y) and z is the distance from the surface, u and v are the x and y components of velocity, and u takes the value u just outside the boundary layer .
expansions at small reynolds number for the flow past a sphere and a circular cylinder . this paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of stokes and oseen . since the perturbation theory arising from the consideration of small non-zero reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small reynolds numbers . the technique adopted is as follows . separate, locally valid the regions close to, and far from, the obstacle . reasons are presented for believing that these ‘stokes’ and ‘oseen’ expansions are, respectively, of the forms where are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, r is the reynolds number, and and vanish with r . substitution of these expansions in the navier-stokes equation then yields a set of differential equations for the coefficients and, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the stokes expansion, and the uniform-stream condition for the oseen expansion) so that unique solutions cannot be derived immediately . however, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion . it is thus possible to determine alternately successive terms in each expansion . the leading terms of the expansions are shown to be closely related to the original solutions of stokes and oseen, and detailed results for some further terms are obtained .
integration of the boundary layer equations . the equations of the boundary layer are integrated by an expression of the form where f(x) is a positive function with x=0 as the stationary point,. (x) is slowly varying,. the integral contains an unknown parameter which is found from the condition . the integral is evaluated by the method of steepest descent . the expressions obtained are usually divergent, except in few cases which include blasius’s equation,. the divergent expressions are summed by euler’s transformation . to check the procedure it is applied to falkner and skan’s equation . the results obtained are very striking,. few terms in the expansions are sufficient to obtain close agreement with hartree’s laborious numerical computations . the method is also applied to the general boundary-layer equation for the case of flow past an elliptic cylinder, measured by schubauer . the results obtained are in close agreement with schubauer’s measurements for the velocities, almost up to separation, for the position of the separation point,. and in satisfactory agreement downstream of separation .
the generation of noise by isotropic turbulence . a finite region, with fixed boundaries, of an infinite expanse of compressible fluid is in turbulent motion . this motion generates noise and radiates it into the surrounding fluid . the acoustic properties of the system are studied in the special case in which the turbulent region consists of decaying isotropic turbulence . it is assumed that the reynolds number of the turbulence is large, and that the mach number is small . the noise appears to be generated mainly by those eddies of the turbulence whose contribution to the rate of dissipation of kinetic energy by viscosity is negligible . it is shown that the intensity of sound at large distances from the turbulence is the same as that due to a volume distribution of simple acoustic sources occupying the turbulent region . in this analogy, the whole fluid is to be regarded as a stationary and uniform acoustic medium . the local value of the acoustic power output p per mass of turbulent fluid is given approximately by the formula where a is a numerical constant, u is the mean-square velocity fluctuation, is the time, and c is the velocity of sound in the fluid . the constant a is expressed in terms of the well-known velocity correlation function f(r) by assuming the joint probability distribution of the turbulent velocities and their first two time-derivatives at two points in space to be gaussian . the numerical value is then obtained by substituting the form of f(r) corresponding to heisenberg’s theoretical spectrum of isotropic turbulence . it is found that the effects of decay make only a small contribution to the value of a, and that the order of magnitude of a is not changed when widely differing forms of the function f(r) are used .
on the flow of compressible fluid past an obstacle . it is well known that according to classical hydrodynamics a steady stream of frictionless incompressible fluid exercises no resultant force upon an obstacle, such as a rigid sphere, immersed in it . the development of a /resistance/ is usually attributed to viscosity, or when there is a sharp edge to the negative pressure which may accompany it (helmholtz) . in either case it would seem that resistance involves something of the nature of a wake, extending behind the obstacle to an infinite distance . when the system of disturbed velocities, although it may mathematically extend to infinity, remains as it were attached to the obstacle, there can be no resistance . the absence of resistance is asserted for an incompressible fluid., but it can hardly be supposed that a small degree of compressibility, as in water, would affect the conclusion . on the other hand, high relative velocities, exceeding that of sound in the fluid, must entirely alter the conditions . it seems worth while to examine this question more closely, especially as the first effects of compressibility are amenable to mathematical treatment .
on the steady motion of viscous, incompressible fluids, with particular reference to a variation principle . except in exceptional cases, it is not possible to represent the motion of a viscous incompressible liquid by means of a variation principle, but all cases of such motion that have yet been discovered belong to this class of /exceptional cases ./ the appropriate functions are given .
velocity and temperature distributions in the turbulent wake behind a heated body of revolution . recently (see abstract 954 (1938)) goldstein made calculations based on theories of vorticity transfer, of the distributions of velocity and temperature in the turbulent wake behind a heated body of revolution, and the present authors now record an experimental determination of these distributions in a low-turbulence wind tunnel . difficulty was experienced in obtaining a truly symmetrical wake and observations have been reduced to mean values, curves of which are given .
on the solution of the laminar boundary layer equations . the problem of the flow along a flat plate placed edgewise to a steady stream, when a retarding pressure gradient varying linearly as the distance x from the leading edge of the plate is superposed is discussed . if y denotes distance measured perpendicular to the plate, a solution is obtained in the form of a power series in x where coefficients are functions of . differential equations are obtained for these coefficients . seven of the coefficients have been obtained with reasonable accuracy, and the eighth and ninth roughly . unfortunately it appears that about eight more terms are required to carry the solution to the point of separation,. the work involved in their determination is prohibitive . two approximate methods have been developed for determining the error when the first seven terms of the series are used as an approximation . these methods lead to the determination of the point of separation and are in agreement as to its position . if is the velocity at the edge of the boundary layer at the leading edge of the plate and is the velocity gradient, separation is found when . a method is developed for the solution of the boundary layer equations in any retarded region . it is obtained by replacing the velocity distribution at the edge of the boundary layer by a circumscribing polygon of infinitesimal sides and applying the preceding solution to each of these sides, making the momentum integral continuous at each vortex . the problem is thereby reduced to the solution of a first order differential equation .
the effect of shallow water on wave resistance . the general character of experimental results dealing with the effect of shallow water on ship resistance may be stated briefly as follows ..–at low velocities the resistance in shallow water is greater than in deep water, the speed at which the excess is first appreciable varying with the type of vessel . as the speed increases, the excess resistance increases up to a maximum at a certain critical velocity, and then diminishes . with still further increase of speed, the resistance in shallow water ultimately becomes, and remains, less than that in deep water at the same speed . the maximum effect is the more pronounced the shallower the water . for further details and references one may refer to standard treatises, but one quotation may be made in regard to the critical velocity .. /this maximum appears to be at about a speed such that a trochoidal wave travelling at this speed in water of the same depth is about times as long as the vessel . it was at one time supposed that the speed for maximum increase in resistance was that of the wave of translation . this, however, holds only for water whose depth is less than for greater depths the speed of the wave of translation rapidly becomes greater than the speed of maximum increase of resistance ./ in a recent analysis of the data, h. m. weitbrecht expresses a similar conclusion by stating that for each depth of water there is a critical velocity, but that the critical velocity does not vary as the square root of the corresponding depth .
the hodographic transformation in transonic flow . the author studies the problem of finding the shape of a symmetrical nozzle with the velocity along the axis (x-axis) specified . the velocity along each streamline is assumed to increase steadily . the singularity at the sonic velocity and to the axis of the nozzle) is first studied in the physical plane by using a power series in . in the hodograph plane, the two characteristics of the hodograph differential equation passing through the sonic point and are lines of branch points . the region between these lines is a region of triple-valuedness for the stream function . outside this region is single-valued . there are also singularities at the sonic point and the point corresponding to the specified condition at the exit of the nozzle . the author then proposes to construct in the hodograph plane by at the exit velocity and (3) a finite sum regular throughout .. sin, where r is the square of the velocity and the are hypergeometric functions . the a’s are fixed by the required approximation to the specified velocity distribution along the axis . this solution is single-valued, convergent and represents except a region near the sonic point in the nozzle . for this excluded region, the author inverts the solution to obtain a power series in for 0 . this is shown to be convergent for the region of interest . the type of solution considered by the author gives a nozzle having an infinitely long supersonic part .
temperature charts for induction and constant temperature heating . charts are presented for determining complete temperature historics in spheres, cylinders, and plates . it is shown that for values of the dimensionless time ratio x greater than 0.2 the heating equations reduce to such a simple form that for each shape two charts which give temperatures at any position within the heated or cooled bodies can be plotted . it is also shown that the usual simple heating and cooling charts can also be used for the determination of temperatures and heating times in bodies heated by a constant rate of heat generation at the surface (induction heating) . finally, a two dimensional chart is given for finding heating times in short cylinders, thereby eliminating the trial-and-error solution that is necessary when heating times are found from the present one-dimensional charts .
numerical methods for transient heat flow . this paper deals with the application of numerical methods for the solution of heat-conduction problems, their generality being extended in the following ways .. may proceed most rapidly to a solution, or may proceed more slowly and with greater precision,. (b) criteria are developed for the choice of modulus to insure convergence . this is most important at a convective surface,. (c) a method is developed for handling k and c when these properties vary independently with temperature . a comprehensive appendix gives the derivations, and the use of equations and charts is demonstrated by typical examples .
approximate analytical solutions for hypersonic flow past slender power-law bodies . approximate analytical solutions are presented for two-dimensional and axisymmetric hypersonic flow over blunt-nosed slender bodies whose shapes follow a power law variation . in particular, the body shape is given by where is the transverse body ordinate, is the streamwise distance from the nose, and m is a constant in the range . both zero-order solutions and first-order (small but nonvanishing values of solutions are presented, where m is the free-stream mach number and is a characteristic body or streamline slope . the zero-order shock shape is similar to the body shape for these flows . the solutions are found within the framework of hypersonic-slender-body theory . the limiting case m=1 corresponds to a wedge or cone flow . the limiting case corresponds to a constant-energy flow . the latter cases are included so that the present study may be applied to all flows wherein the zero-order shock shape is given by with m in the range . flow fields associated with shock shapes having values of m outside this range are also discussed . for all values of, except m=1, certain portions of the flow field riolate the hypersonic-slender-body approximations, while other portions are consistent with these approximations . for m=1, all portions of the flow field are consistent with the approximations . the approximate solutions are found as follows . the asymptotic form of the flow in the vicinity of the body surface is used as a guide to write approximate expressions for the dependent variables . these expressions exactly satisfy the continuity and energy equations and contain arbitrary constants which are evaluated so as to satisfy boundary conditions at the shock . the approximate solutions do not satisfy the lateral momentum equation except at the shock and (for the first-order problem) at the body surface . the results of the approximate solutions are compared with numerical integrations of the equations of motion for various values of m and (ratio of specific heats) . good agreement is noted, particularly when m and are both near one . the shock is relatively close to the body for the latter cases . sufficient results are presented to evaluate the accuracy of the approximate method for various values of m and .
supersonic flow past a family of blunt symmetric bodies . some 100 numerical computations have been carried out for unyawed bodies of revolution with detached bow waves . the gas is assumed perfect with . free-stream mach numbers are taken as 1.2, 1.5, 2, 3, 4, 6, 10, and . the results are summarized with emphasis on the sphere and paraboloid .
nearly circular transfer trajectories for descending satellites . simplified expressions describing the transfer from a satellite orbit to the point of atmospheric entry are derived . the expressions are limited to altitude changes that are small compared with the earth’s radius, and velocity changes small compared with satellite velocity . they are further restricted to motion about a spherical, nonrotating earth . the transfer orbit resulting from the application of thrust in any direction at any point in an elliptic orbit is considered . expressions for the errors in distance (miss distance) and entry angle due to an initial misalinement and magnitude error of the deflecting thrust are presented . the largest potential contributing factor towards a miss distance stems from the misalinement of the retrovelocity increment . if this velocity increment is pointed in direct opposition to the flight path, a 1 misalinement leads to a miss distance of 34.5 miles . however, it is shown that this error can be avoided by applying the velocity increment at an angle between 120 and 150 below the flight-path direction . the guidance and accuracy requirements to establish a circular orbit, in addition to the corrections applied to transform elliptic orbits into circular ones, are also discussed .
an analysis of the corridor and guidance requirements for supercircular entry planetary atmospheres . an analysis is presented of supercircular entry into a planet’s atmosphere giving particular attention to the corridor through which spacecraft must be guided in order to accomplish various maneuvers . a dimensionless parameter based on conditions at the conic perigee altitude is introduced for characterizing supercircular entries and conveniently prescribing corridor widths associated with elliptic, parabolic, or hyperbolic approach trajectories . the analysis applies to vehicles of arbitrary weight, shape, and size . illustrative calculations are made for venus, earth, mars, jupiter, and titan . for nonlifting vehicles having fixed aerodynamic coefficients, curves are presented of dimensionless parameters from which can be calculated the maximum deceleration, maximum rate of laminar convective heating, and total laminar heat absorbed during single-pass entry at velocities up to twice circular velocity . for lifting vehicles, curves are presented of the maximum deceleration and overshoot boundary of an entry corridor,. equations are presented for estimating laminar aerodynamic heating from the maximum deceleration . it is shown that the corridor width is independent of vehicle weight, dimensions, and drag coefficient, provided these are the same at the overshoot boundary as at undershoot . the corridors of certain planets can be broadened markedly by the application of aerodynamic lift,. for example, the 10-earth-g corridor width for single-pass, nonlifting, parabolic entry is increased from to 52, 51, and 52 miles, respectively, by employing a lift-drag ratio of 1 . the use of aerodynamic lift does not increase appreciably the corridors of mars and titan . all corridor widths decrease rapidly as the entry velocity is increased . terminal guidance requirements on accuracy of velocity and flight path angle for successfully entering various corridors are compared with analogous requirements for putting a satellite into orbit, for hitting the moon from the earth, and for achieving icbm accuracy . consideration is given to the terminal guidance problem involved in using a planet’s atmosphere–rather than rocket fuel–to effect orbital transfers from heliocentric to planeto-centric motion, thereby converting a hyperbolic approach trajectory to an elliptic orbit about the target planet . this fuel saving maneuver appears technologically feasible for certain planetary voyages, and implies the possibility of achieving a large reduction in required earth lift-off weight of chemical propulsion systems .
an approximate analytical method for studying entry into planetary atmospheres . the pair of motion equations for entry into a planetary atmosphere is reduced to a single, ordinary, nonlinear differential equation of second order by disregarding two relatively small terms and by introduring a certain mathematical transformation . the reduced equation includes various terms, certain of which represent the gravity force, the centrifugal acceleration, and the lift force . if these particular terms are disregarded, the differential equation is linear and yields precisely the solution of allen and eggers applicable to ballistic entry at relatively steep angles of descent . if all the other terms in the basic equation are disregarded (corresponding to negligible vertical acceleration and negligible vertical component of drag force), the resulting truncated differential equation yields the solution of sanger for equilibrium flight of glide vehicles with relatively large lift-drag ratios . a number of solutions for lifting and nonlifting vehicles entering at various initial angles also have been obtained from the complete nonlinear equation . these solutions are universal in the sense that a single solution determines the motion and heating of a vehicle of arbitrary weight, dimensions, and shape entering an arbitrary planetary atmosphere . one solution is required for each lift-drag ratio . these solutions are used to study the deceleration, heating rate, and total heat absorbed for entry into venus, earth, mars, and jupiter . from the equations developed for heating rates, and from available information on human tolerance limits to acceleration stress, approximate conditions for minimizing the aerodynamic heating of a trimmed vehicle with constant lift-drag ratio are established for several types of manned entry . a brief study is included of the process of atmosphere braking for slowing a vehicle from near escape velocity to near satellite velocity .
skin-friction measurements in incompressible flow . experiments have been conducted to measure in incompressible flow the local surface-shear stress and the average skin-friction coefficient for a turbulent boundary-layer on a smooth flat plate having zero pressure gradient . the local surface-shear stress was measured by a floating-element skin-friction balance and also by a calibrated total head tube located on the surface of the test wall . the average skin-friction coefficient was obtained from boundary-layer velocity profiles . the boundary-layer profiles were also used to determine the location of the virtual origin of the turbulent boundary layer . data were obtainec for a range of reynolds numbers from 1 million to about 45 million with an attendant change in mach number from 0.11 to 0.32 . the measured local skin-friction coefficients obtained with the floating-element balance agree well with those of schultz-grunow and kempf for reynolds numbers up to 45 million . the measured average skin-friction coefficients agree with those given by the schoenherr curve in the ranges of reynolds numbers from 1 to 3 million and 30 to 45 million . in the range of reynolds numbers from 3 to 30 million the measured values are less than those predicted by the schoenherr curve . the results show that the /univeral skin-friction constants/ proposed by coles appraoch asymptotically a constant value at reynolds numbers exceeding mentioned constants and the limited reynolds number range of the present investigation, there is some doubt as to the validity of any turbulent skin-friction law written on the basis of the present results . hence, no new friction law is proposed . the frictional resistance of a flat plate was calculated by means of the momentum method and also the integrated measured local surface shear . for reynolds numbers from 14 million to 45 million both methods give about the same result,. whereas at lower values of reynolds number the momentum method based on velocity profiles uncorrected for the effects of turbulence results in a frictional resistance as much as 4 percent higher than that of the integrated shear . the measurement of local surface shear by a calibrated preston tube appears to be accurate and inexpensive . the calibration as given by preston must be modified slighlty, however, to yield the results obtained from the floating-element skin-friction balance .
flow of chemically reacting gas mixtures . suitable forms of the equations for the flow of an inviscid, non-heat-conducting gas in which chemical reactions are occurring are derived . the effects of mass diffusion and non-equilibrium amongst the internal modes of the molecules are neglected . special attention is given to the speeds of sound in such a gas mixture and a general expression for the ratio of frozen to equilibrium sound speeds is deduced . an example is given for the ideal dissociating gas . the significance of the velocity defined by the ratio of the convective derivatives of pressure and density is explained . it is the velocity which exists at the throat of a convergent-divergent duct under maximum mass flow conditions, and it is shown that this velocity depends on the nozzle geometry as well as on the ‘reservoir’ conditions . as an illustration the phenomena of sound absorption and dispersion are discussed for the ideal dissociating gas . the results can be concisely expressed in terms of the frozen and equilibrium sound speeds, the frequency of the (harmonic) sound vibration and a characteristic time for the rate of progress of the reaction .
linearized flow of a dissociating gas . the equations for planar two-dimensional steady flow of an ideal dissociating gas are linearized, assuming small disturbances to a free stream in chemical equilibrium . as an example of their solution, the flow past a sharp corner in a supersonic stream is evaluated and the variations of flow properties in the relaxation zone are found . numerical illustrations are provided using an ‘oxygen-like’ ideal gas and comparisons made with a characteristics solution . the flow past a sharp corner can be studied in a conventional shock tube and it may be possible to verify the present theory experimentally . in particular it may prove feasible to use the results to obtain a measure of the reaction rates in the gas mixture .
heat conduction through a gas with one inert internal model . the rate of energy transfer between parallel flat plates is evaluated when the (stagnant) gas between them is polyatomic with one inert internal mode . deviations of the thermal conductivity from the complete equilibrium of the inert mode relaxation time and the effectiveness of the walls in exciting or de-exciting this mode . the results are obtained via a linear theory consistent with small temperature differences between the plates . it is found that the eucken-value of conductivity could be exceeded if the relaxation times are non-zero and the plates very effective in exciting the inert mode . when relaxation times are very short the effect of the walls on the energy transfer rate is small, but the walls make their presence felt by distorting the temperature profiles in /boundary layers/ adjacent to the walls which are of order in thickness time) . this result is analogous to hirschfelder’s (1956) for the case of chemical reactions . for experimental measurement of conductivity in a hot wire cell type of apparatus it is shown that extrapolation of measured reciprocal conductivities to zero reciprocal pressure should load to the full eucken value . it is also shown that the slope of reciprocal apparent (measured) conductivity versus reciprocal pressure curves is a function of relaxation time as well as of the accommodation coefficients . it is quite possible that the relaxation effect here is comparable with the temperature jump effects, even for rotation in diatomic molecules .
on the sudden contact between a hot gas and a cold solid . the flow induced by the sudden contact between a semi-infinite expanse of gas and a solid, initially at different temperatures, is examined on the basis of a linear continuum theory . for times large compared with the mean time between molecular collisions in the gas, the velocity and pressure disturbances are found to be concentrated around a wave front propagating out from the interface at the ambient isentropic sound speed, whilst, near to the interface, these disturbances are small and the gas temperatures are nearly equal to those predicted by the classical constant pressure heat conduction theory . the possible significance of these results in connection with reflected shock wave techniques to measure high temperature gas properties is commented upon .
the interaction of a reflected shock wave with the boundary layer in a shock tube . ideally, the reflection of a shock from the closed end of a shock tube provides, for laboratory study, a quantity of stationary gas at extremely high temperature . because of the action of viscosity, however, the flow in the real case is not one-dimensional, and a boundary layer grows in the fluid following the initial shock wave . in this paper simplifying assumptions are made to allow an analysis of the interaction of the shock reflected from the closed end with the boundary layer of the initial shock afterflow . the analysis predicts that interactions of several different types will exist in different ranges of initial shock mach number . it is shown that the cooling effect of the wall on the afterflow boundary layer accounts for the change in interaction type . an experiment is carried out which verifies the existence of the several interaction regions and shows that they are satisfactorily predicted by the theory . along with these results, sufficient information is obtained from the experiments to make possible a model for the interaction in the most complicated case . this model is further verified by measurements made during the experiment . the case of interaction with a turbulent boundary layer is also considered . identifying the type of interaction with the state of turbulence of the interacting boundary layer allows for an estimate of the state of turbulence of the boundary layer based on an experimental investigation of the type of interaction .
a low density wind tunnel study of shock wave structure and relaxation phenomena in gases . the profiles and thicknesses of normal shock waves of moderate strength have been determined experimentally in terms of the variation of the equilibrium temperature of an insulated transverse cylinder in free-molecule flow . the shock waves were produced in a steady state in the jet of a low-density wind tunnel, at initial mach numbers of 1.72 and 1.82 in helium and 1.78, the shock thickness, determined from the maximum slope of the cylinder temperature profile, varied from mean free path in the supersonic stream . a comparison between the experimental shock profiles and various theoretical predictions leads to the tentative conclusions that .. (1) the navier-stokes equations are adequate for the description of the shock transition for initial mach numbers up to 2, and (2) the effects of rotational relaxation times in air can be accounted for by the introduction of a /second/ or /bulk/ viscosity coefficient equal to about two-thirds of the ordinary shear viscosity .
some aerodynamic considerations of nozzle afterbody combination . the aerodynamic problems associated with propulsion-system installations have assumed a role of vital importance in the development of supersonic aircraft . although air-induction systems have received moderate attention in the literature, considerably less information can be found on the design and installation of turbojet exit nozzles . this condition should not be interpreted to indicate a lack of problems in jet-exit design . as flight speeds reach supersonic levels, it becomes increasingly difficult to achieve nozzle installations which are efficient over the entire speed range . the difficulties largely stem from the fact that the goals of high jet thrust and low afterbody drag are not always compatible . in many of the compromise solutions, it is generally unsatisfactory to examine isolated nozzle and afterbody performance . rather they must be treated as a unit, and the complex effects of jet interaction with the external stream must be taken into account . to accomplish this, the nozzle and air-frame designers must closely coordinate their efforts . some of the aerodynamic problems of nozzle afterbody combinations are outlined in this report . particular attention is devoted to the influence of the jet-stream interaction on both nozzle thrust and after-body drag . for this purpose, use is made of shock- boundary-layer-interaction concepts . this approach, although not precise, correctly predicts many trends and is generally enlightening .
the effect of a central jet on the base pressure of a cylindrical afterbody in a supersonic stream . this report describes an experimental investigation of the factors affecting the base flow and jet structure behind a cylindrical after-body with a central nozzle . seven interchangeable nozzles were tested . six of these were convergent-divergent, with a design mach number of 2.0, jet base diameter ratios ranging from 0.2 to 0.8 and nozzle divergence angles ranging from convergent with a jet base diameter ratio of 0.6 . in the main experimental programme the free-stream mach number was 2.0 and the boundary layer was turbulent both on the after-body and in the nozzle . measurements were made of the base pressure, the surface pressure distribution inside the nozzle, the overall thrust and the nozzle mass flow, over a range of jet pressures . this programme was supplemented by comparative tests with the jet exhausting into still air (static tests) . readings were taken of the internal nozzle pressures and the jet thrust at different jet pressures . schlieren photography was used extensively throughout . the results of the tests with external flow are presented in the form of curves showing the separate effects of jet pressure ratio, jet base diameter ratio, nozzle design mach number and nozzle divergence angle on the base pressure and overall thrust . the special case of base bleed is discussed separately . similar curves are included for the static tests . these show the effect of jet pressure ratio and nozzle geometry on the jet thrust . a general method of correlating data on annular base pressures is proposed and discussed . essentially, this method compares the pressure on an annular base with the calculated pressure on the corresponding two-dimensional base . it correlates the present results reasonably well, but is less successful when applied to more extensive data .
investigation at supersonic speeds of the effects of jet mach number and divergence angle of the nozzle upon the pressure of the base annulus of a body of revolution . an investigation has been conducted in the langley 9-inch supersonic tunnel to determine the jet effects for varying jet mach number and nozzle divergence angle upon the pressure on the base annulus of a model with a cylindrical afterbody . the tests were conducted over a wide range of jet static pressure ratios and at a reynolds number of approximately free-stream mach numbers of 1.62, 1.94, and 2.41 . all testing was conducted with an artificially induced turbulent boundary layer along the model . in the lower range of jet static pressure ratios, jet flow from a sonic or supersonic nozzle affected the pressure acting on the base annulus in essentially the same manner as shown in naca rm e53h25 which covers jet static pressure ratios up to about present results showed that the base pressure tends to level off with increasing jet static pressure ratio, and at the extreme static pressure ratios reached in tests with sonic nozzles the base pressure began to decrease . except in the lower range of jet static pressure ratios, nozzle divergence angle generally had a larger effect on the base pressures than nozzle mach number,. the increase in base pressure for a change in divergence angle from 0 to 10 was small compared to the increase when the divergence angle was changed from and other data indicates that the effects of divergence angle were reduced when the ratio of jet exit diameter to base diameter was decreased . jet mach number effects increased with increase in stream mach number .
experiments with static tubes in a supersonic airstream . systematic tests have been made at a mach number of 1.6 on a family of static tubes . the variables which have been investigated are the shape of the nose, the distance of the holes downstream, and the inclination of the tube to the flow . pressure measurements have also been made in the vicinity of a shock wave and close to a wall .
base pressure at subsonic speeds in the presence of a supersonic jet . this paper presents the results of an experimental investigation into the effect of supersonic jets upon the base pressure of a bluff cylinder in a uniform subsonic flow . the ratio of jet diameter to base diameter was 0.1875 . jet stagnation pressures giving slight under-expansion of the jet cause an increase in the base pressure but for larger jet stagnation pressures the base pressure is again reduced . a simple theory, based on a momentum integral, shows the dependence of the base drag upon the jet and free stream speeds and upon the dimensions of the jet and the base .
the mixing of free axially-symmetrical jets of mach number 1.40 . axially-symmetrical, supersonic, fully-expanded jets of diameter about 0.75 in. and of mach number 1.40 issuing into an atmosphere at rest were investigated by schlieren and shadow photography and by pressure traversing . the development of the jets was found to depend critically on the strength of the shock waves in the core of the jet at the nozzle exit . with strong shock waves present the jet spread very rapidly and was very unsteady . the jet did in some cases break up into large eddies of the same size as the diameter of the jet . when no disturbances were present in the core of the jet the spreading was far more gradual and the jet showed only slight unsteadiness . the turbulent mixing region of the first part of the jet with strong shock waves was investigated in detail by pitot tubes . the first inch was found to correspond to a two-dimensional half-jet . the velocity profiles were similar and well represented by the error integral . the rate of spreading was only half the value for low-speed flow . by integrations across the mixing region the entrainment and the loss of kinetic energy were determined . these quantities were found to agree well with the values estimated by assuming an error-integral velocity profile .
on full dispersed shock waves in carbon dioxide . it is pointed out that, for shock mach numbers between 1 and that the adjustments in the energy in all the degrees of freedom proceed slowly and in parallel and occur over a distance large compared with the mean free path . theoretical velocity profiles for such shock waves are given and found to be in excellent agreement with interferometric shock-tube observations .
an analysis of base pressure at supersonic speeds and comparison with experiment . in the first part of the investigation an analysis is made of base pressure in an inviscid fluid, both for two-dimensional and axially-symmetric flow . it is shown that for two-dimensional flow, and also for the flow over a body of revolution with a cylindrical sting attached to the base, there are an infinite number of possible solutions satisfying all necessary boundary conditions at any given free-stream mach number . for the particular case of a body having no sting attached only one solution is possible in an inviscid flow, but it corresponds to zero base drag . accordingly, it is concluded that a strictly inviscid-fluid theory cannot be satisfactory for practical applications . since the exact inviscid-fluid theory does not adequately describe the conditions of a real fluid flow, an approximate semi-empirical theory for base pressure in a viscous fluid is developed in a second part of the investigation . the semi-empirical theory is based partly on inviscid-flow calculations, and is restricted to airfoils and bodies without boat-tailing . in this theory an attempt is made to allow for the effects of mach number, reynolds number, profile shape, and type of boundary-layer flow . the results of some recent experimental measurements of base pressure in two-dimensional and axially-symmetric flow are presented for purposes of comparison . some experimental results also are presented concerning the support interference effect of a cylindrical sting, and the interference effect of a reflected bow wave on measurements of base pressure in a supersonic wind tunnel .
boundary layer over a flat plate in presence of shear flow . the governing equations of an incompressible boundary layer over a flat plate in the presence of a shear flow with finite vorticity are derived . for large vorticity, a similarity solution is obtained . for moderate vorticity, one of the governing equations is replaced by an approximate one for which similarity solutions exist .
some problems on heat conduction in stratiform bodies . problems on heat conduction in multilayer bodies lead usually to complicated calculations . the present paper gives an idea of specific difficulties arising in the case of infinite composite solides . general deductions are applied to a special class of questions .
effect of roughness on transition in supersonic flow . further experiments carried out in the 12-inch supersonic wind tunnel of the jet propulsion laboratory of the california institute of technology to investigate the effect of three-dimensional roughness elements (spheres) on boundary-layer transition on a 10 transfer are reported herein . the local mach number for these tests was minimum (effective) size of trip required to bring transition to its lowest reynolds number varies as the one-fourth power of the distance from the apex of the cone to the trip . use of available data at other mach numbers indicates that the mach number influence for effective tripping is taken into account by the simple expression .
properties of impact pressure probes in free molecule flow . an expression has been derived for the mass flow through a circular tube in free molecule flow when the tube and gas are in relative motion . the gas entering the tube is assumed to have a maxwellian distribution function and the molecular reflection process at the wall is assumed to be diffuse . the theory has been used to determine the pressure read by an impact probe in free molecule flow . although the general expressions derived apply to any value of gas velocity and tube size, the detailed calculations for the pressure probe are difficult except for the case of low speeds and long tubes . an experimental check of the theory has been carried out using impact probes in a whirling arm apparatus and in the utia low density wind tunnel . agreement between theory and experiment is quite satisfactory .
scale models for thermo-aeroelastic research . an investigation is made of the parameters to be satisfied for thermo-aeroelastic similarity . it is concluded that complete similarity obtains only when aircraft and model are identical in all respects, including size . by limiting consideration to conduction effects, by assuming the major load carrying parts of the structure are in regions where the flow is either entirely laminar, or entirely turbulent, and by assuming a specific relationship between reynolds number and nusselt number, an approach to similarity can be achieved for small scale models . experimental and analytical work is required to check on the validity of these assumptions . it appears that existing hot wind tunnels will not be completely adequate for thermo-aeroelastic work, and accordingly a possible layout for the type of tunnel required is described . automatic programmed control of the tunnel would appear to be necessary .
some possibilities of using gas mixtures other than in aerodynamic research . a study is made of the advantages that can be realized in compressible-flow research by employing a substitute heavy gas in place of air . most heavy gases considered in previous investigations are either toxic, chemically active, or (as in the case of the freons) have a ratio of specific heats greatly different from air . the present report is based on the idea that by properly mixing a heavy monatomic gas with a suitable heavy polyatomic gas, it is possible to obtain a heavy gas mixture which has the correct ratio of specific heats and which is nontoxic, nonflammable, thermally stable, chemically inert, and comprised of commercially available components . calculations were made of wind-tunnel characteristics for 63 gas pairs comprising 21 different polyatomic gases properly mixed with each of three monatomic gases (argon, krypton, and xenon) . for a given mach number, reynolds number, and tunnel pressure, a gas-mixture wind tunnel having the same specific-heat ratio as air would be appreciably smaller and would require much less power than a corresponding air wind tunnel . analogous though different advantages can be realized in compressor research and in firing-range research . the most significant applications, perhaps, arise through selecting and proportioning a gas mixture so as to have at ordinary wind-tunnel temperatures certain dimensionless characteristics which air at flight temperatures possesses but which air at ordinary wind-tunnel temperatures does not possess . characteristics which involve the relaxation time (or bulk viscosity), the variation of viscosity with temperature, and the variation of specific heat with temperature fall within this category . other applications arise in heat-transfer research since certain gas mixtures can be concocted to have any prandtl number in the range at least between 0.2 and 0.8 .
base pressure in supersonic flow . the problem of accurately predicting the pressure and wake configuration at the base of bodies in supersonic flow is an extremely important one inasmuch as a sizeable portion of the total drag of a given body may be attributable to the low pressure in this region . although a great deal of theoretical and experimental work has been done in this field, there does not yet exist a satisfactory method for accurate predictions . this paper represents an excellent effort to experimentally confirm analytically deduced concepts . a large amount of experimental data on body shapes such as wedges, cones, and cone-cylinders has been obtained over a range of mach numbers up to 4 . the data are thoroughly discussed with respect to analytical deductions . on the basis of the evidence accumulated it is concluded that the boundary-layer thickness has only a small effect on the base pressure for axisymmetric bodies and for two-dimensional bodies when the base height-to-chord ratios are of the order . reviewer believes this report is a significant contribution in the field of base pressure and wake flow phenomena .
investigation of separated flows in supersonic and subsonic streams with emphasis on the effect of transition . experimental and theoretical research has been conducted on flow separation associated with steps, bases, compression corners, curved surfaces, shock-wave boundary-layer reflections, and configurations producing leading-edge separation . results were obtained from pressure-distribution measurements, shadow-graph observations, high-speed motion pictures, and oil-film optics . the maximum scope of measurement encompassed mach numbers between 0.4 and 3.6, and length reynolds numbers between 4000 and 5000000 . the principal variable controlling pressure distribution in the separated flows was found to be the location of transition relative to the reattachment and separation positions . classification is made of each separated flow into one of three regimes .. and /turbulent/ with transition upstream of separation . by this means of classificaiton it is possible to state rather literal results regarding the steadiness of flow and the influence of reynolds number within each regime . for certain pure laminar separations a theory for calculating dead-air pressure is advanced which agrees well with subsonic and supersonic experiments . this theory involves no empirical information and provides an explanation of why transition location relative to reattachment is important . a simple analysis of the equations for interaction of boundary-layer and external flow near either laminar or turbulent separation indicates the pressure rise to vary as the square root of the wall shear stress at the beginning of interaction . various experiments substantiate tnis variation for most test conditions . an incidental observation is that the stability of a separated laminar mixing layer increases markedly with an increase in mach number . the possible significance of this observation is discussed .
an analysis of base pressure at supersonic velocities and comparison with experiment . in the first part of the investigation an analysis is made of base pressure in an inviscid fluid, both for two-dimensional and axially symmetric flow . it is shown that for two-dimensional flow, and also for the flow over a body of revolution with a cylindrical sting attached to the base, there are an infinite number of possible solutions satisfying all necessary boundary conditions at anh given free-stream mach numger . for the particular case of a body having no sting attached only one solution is possible in an inviscid flow, but it corresponds to zero base drag . accordingly, it is concluded that a strictly inviscid-flow theory cannot be satisfactory for practical applications . an approximate semi-empirical analysis for base pressure in a viscous fluid is developed in a second part of the investigation . the semi-empirical analysis is based partly on inviscid-flow calculations . in this theory an attempt is made to allow for the effects of mach number, reynolds number, profile shape, and type of boundary-layer flow . some measurements of base pressure in two-dimensional and axially symmetric flow are presented for purposes of comparison . experimental results then are presented concerning the support interference effect of a cylindrical sting, and the interference effect of a reflected air wave on measurements of base pressure in a supersonic wind tunnel .
experimental investigation of base pressure on blunt-trailing-edge wings of supersonic velocities . measurements of base pressure are presented for 29 blunt-trailing-edge wings having an aspect ratio of 3.0 and various airfoil profiles . the different profiles comprised thickness ratios between 0.05 and 0.10, boattail angles between –2.9 and 20, and ratios of trailing-edge thickness to airfoil thickness between 0.2 and 1.0 . the tests were conducted at mach numbers of 1.25, 1.5, 2.0, and 3.1 . for each mach number, the reynolds number and angle of attack were varied . the lowest reynolds number investigated was 0.2 x 10 and the highest was 3.5 x 10 . measurements on each wing were obtained separately with turbulent flow and laminar flow in the boundary layer . span-wise surveys of the base pressure were conducted on several wings . the results with turbulent boundary-layer flow showed only small effects on base pressure of variations in reynolds number, airfoil profile shape, boattail angle, and angle of attack . the principal variable affecting the base pressure for turbulent flow was the mach number . at the highest mach number investigated (3.1), the ratio of boundary-layer thickness to trailing-edge thickness also affected the base pressure significantly . the results obtained with laminar boundary-layer flow to the trailing edge showed that the effect of reynolds number on base pressure was large . in all but a few exceptional cases the effects on base pressure of variations in angle of attack and in profile shape upstream of the base were appreciable though not large . the principal variable affecting the base pressure for laminar flow was the ratio of boundary-layer thickness to trailing-edge thickness . for a few exceptional cases involving laminar flow to the trailing edge, the effects on base pressure of variations in profile shape, boattail angle, and angle of attack were found to be unusually large . in such cases the variation of base pressure with angle of attack was discontinuous and exhibited a hysteresis . stroboscopic schlieren observations at a mach number of 1.5 indicated that these apparently special phenomena were associated with a vortex trail of relatively high frequency .
on magnetohydrodynamic shock waves . in the earlier attempts at finding the jump conditions across a hydromagnetic shock wave (1, 2, 3) various simplifying assumptions regarding the shape of the shock and the dimensions and the character of the motion are made . from that analysis it is possible to write down the jump conditions in a higher degree of generality (4) . the shock conditions for magnetohydrodynamic flows can, however, be derived in their full generality with the help of the transport equation as used by thomas (5) in the derivation of shock conditions in conventional gas dynamics . the purposes of this paper are .. cover the present more general case . that every flow and field quantity downstream from the shock wave is expressible separately in terms of the known values of these quantities upstream from the shock wave . in this rearranged form of the equations, various effects of the shock wave can be easily read off . the shock conditions along the same lines as in conventional gas dynamics .
a theory for the core of a leading edge vortex . in the flow past a slender delta wing at incidence can be observed a roughly axially symmetric core of spiralling fluid, formed by the rolling up of the shear layer that separates from a leading edge . the aim in this report is to predict the flow field within this vortex core, given appropriate conditions at its outside edge . the basic assumptions are core . in addition it is assumed that the flow is axially symmetric and incompressible . together, these admit outer and inner solutions for the core from the equations of motion . for the outer solution the sub-core is ignored, and the flow is taken to be inviscid (but rotational) and conical . the resulting solution consists of simple expressions for the velocity components and pressure . for the inner solution, which applies to the diffusive sub-core, the flow is taken to be laminar, and approximations, some based on the boundary conditions and some analogous to those of boundary layer theory, are made . the solution obtained in this case is a first approximation, and is presented in tabular form . a sample calculation yields results which are in good qualitative and fair quantitative agreement with experimental measurements .
on the hypersonic viscous flow past slender bodies of revolution . a similar solution of the hypersonic viscous flow past slender bodies of revolution is deduced for a special case when the radial coordinate of the body surface at section x is proportional to x, where the radial coordinate have the comparable order value with the thickness of the boundary layer . here, /similar/ is used in the direct meaning that distributions in the boundary layer keep the similar form lengthwise . calculations are accomplished for the region of strong interaction between the boundary layer and the shock wave . from several calculations it may be expected that if the thickness of the body becomes small, the thickness of the layer in which the longitudinal velocity component u is rapidly decreased also becomes small, and in the major part of the boundary layer, only the normal component v is increased . further if the thickness of the body is increased, then, the height of the shock wave, the pressure on the wall, and the shear stress at the wall are also increased while the boundary layer thickness is decreased . the nose region is excluded by the reason that the ordinary boundary layer theory will be invalid there .
a study of inviscid flow about air foils at high supersonic speeds . steady flow about curved airfoils at high supersonic speeds is investigated analyticially . with the assumption that air behaves as a diatomic gas, it is found the the shock-expansion method may be used to predict the flow about curved airfoils up to extremely high mach numbers, provided the flow deflection angles are not too close to those corresponding to shock detachment . this result applies not only to the determination of the surface pressure distribution, but also to the determination of the whole flow field about an airfoil . verification of this observation is obtained with the aid of the method of characteristics by extensive calculations of the pressure gradient and shock-wave curvature at the leading edge, and by calculations of the pressure distribution on a 10-percent-thick biconvex airfoil at 0 angle of attack . an approximation to the shock-expansion method for thin airfoils at high mach numbers is also investigated and is found to yield pressures in error by less than 10 percent at mach numbers above three and flow deflection angles up to 25 . this slender-airfoil method is relatively simple in form and thus may prove useful for some engineering purposes . effects of caloric imperfections of air manifest in disturbed flow fields at high mach numbers are investigated, particular attention being given to the reduction of the ratio of specific heats . so long as this ratio does not decrease appreciably below to include the effects of these imperfections, should be substantially as accurate as for ideal-gas flows . this observation is verfied with the aid of a generalized shock-expansion method and a generalized method of characteristics employed in forms applicable for local air temperatures up to about 5000 rankine . the slender-airfoil method is modified to employ an average value of the ratio of specific heats for a particular flow field . this simplified method has essentially the same accuracy for imperfect-gas flows as its counterpart has for ideal-gas flows . an approximate flow analysis is made at extremely high mach numbers where it is indicated that the ratio of specific heats may approach close to 1 . in this case, it is found that the shock-expansion method may be in considerable error,. however, the busemann method for the limit of infinite free-stream mach number and specific-heat ratio of 1 appears to apply with reasonable accuracy .
general theory of airfoil sections having arbitrary shape or pressure distribution . in this report a theory of thin airfoils of small camber is developed which permits either the velocity distribution corresponding to a given airfoil shape, or the airfoil shape corresponding to a given velocity distribution to be calculated . the procedures to be employed in these calculations are outlined and illustrated with suitable examples .
correlation of theoretical and photo-thermoelastic results on thermal stresses in idealized wing structure . after a rather complete exploratory program described in previous papers, the photo-thermoelastic method was applied to the experimental evaluation of the thermal-stress theories . the new technique was correlated with several theories which analyzed the transient thermal stresses in idealized wing structures of high-speed aircraft . various theories were investigated which represented the same idealized wing models and differed from each other only in the simplifying assumptions regarding the temperature distributions in skin and webs . the theories were evaluated by duplicating the boundary and initial conditions on plastic models and then by correlating the theories with the observed fringe orders in nondimensional form . a significant general conclusion was reached after correlating the available theories and experimental results . owing to simplifying assumptions concerning the thermal behavior in the flanges, thermal stresses predicted by the available theories are all higher than the experimental observation . in some cases the discrepancy is as great as 30 per cent .
pressure distributions . axially symmetric bodies in oblique flow . a simple picture, known from the work of i. lotz, of the flow over the forward part of a body of revolution in oblique flow is derived here from entirely elementary considerations . the pressure at any point of the (forward part of the) body at any angle of incidence depends on three parameters whose values vary along the body . the variation of these parameters along the body can be determined from a relatively small number of wind tunnel or water tunnel measurements . the necessary water tunnel measurements have been made for four axially symmetric head shapes . additional measurements have been made to illustrate the theoretical conclusions . the data for each head shape are adequate for a determination of the pressure coefficient at any point on the head shapes at any angle of incidence (up to 6, say) . in particular they can be used to determine the peak suction at any angle of incidence and so the conditions for the onset of cavitation on the head .
pressure distributions on three bodies of revolution to determine the effect of reynolds number up to and including the transonic speed range . this paper presents the results of an investigation conducted in the langley 16-foot transonic tunnel to determine the effects of varying reynolds number on the pressure distribution on a transonic body of revolution at angles of attack through the transonic speed range . the effect of a change in sting cone angle on the pressure distributions and a comparison of experimental incremental pressures with theory is also included . the models were tested through a mach number range from 0.60 to 1.09 . the reynolds number range based on body length was from 9 x 10 to 39 x diameter was 1.3 x 10 to 4.53 x 10 for the model at 8 angle of attack . an increase in reynolds number from 9 x 10 to 39 x 10 affected the longitudinal pressure distributions very slightly . these effects were of such a nature as to cause an increase of 0.05 in the normal-force coefficient of the body when tested in the subcritical cross-flow reynolds number range . this increase is in agreement with theoretical approximations . a comparison between experimental and theoretical values of the incremental pressure coefficient due to angle of attack indicated good agreement except at angles where separated flow areas existed over the body . the effect of a change in sting-cone angle from 5 to 9 on the pressure distribution of the 120-inch model was negligible up to a mach number of 1.05 . at this mach number the effect was to cause a small increase in the velocity over the rear of the body .
investigation of a systematic group of naca 1 – series cowlings with and without spinners . an investigation has been conducted in the langley propeller-research tunnel to study cowling-spinner combinations based on the naca 1-series nose inlets and to obtain systematic design data for one family of approximately ellipsoidal spinners . in the main part of the investigation, 11 of the related spinners were tested in various combinations with 9 naca open-nose cowlings, which were also tested without spinners . the effects of location and shape of the spinner, shape of the inner surface of the cowling lip, and operation of a propeller having approximately oval shanks were investigated briefly . in addition, a study was conducted to determine the correct procedure for extrapolating design conditions determined from the low-speed test data to the design conditions at the actual flight mach number . the design conditions for the naca 1-series cowlings and cowling-spinner combinations are presented in the form of charts from which, for wide ranges of spinner proportions and rates of internal flow, cowlings with near-maximum pressure recovery can be selected for critical mach numbers ranging from spinners and the effects of the spinners and the propeller on the cowling design conditions are presented separately to provide initial quantitative data for use in a general design procedure through which naca 1-series cowlings can be selected for use with spinners of other shapes . by use of this general design procedure, correlation curves established from the test data, and derived compressible-flow equations relating the inlet-velocity ratio to the surface pressures on the cowling and spinner, naca 1-series cowlings and cowling-spinner combinations can be designed for critical mach numbers as high as 0.90 .
measurement of two dimensional derivatives on a wing-aileron-tab system . measurements have been made of the direct two-dimensional damping and stiffness derivatives for a in incompressible flow . corrections arising from the apparatus are discussed and reference is made to an attempt to measure the direct tab derivatives . the effects are shown of frequency parameter, amplitude of oscillation, reynolds number, aileron angle and position of transition on the wing . variation with frequency parameter is substantially the same as for vortex-sheet theory and variation of amplitude produces little change in both derivatives . at the lowest reynolds number there is little change in both derivatives with variation of aileron angle for the condition of natural transition, but at higher reynolds numbers the stiffness derivatives increase at . a forward movement of transition reduces the stiffness derivatives at the smaller aileron angles, but at, at the lowest reynolds number, an increase results . similar trends are observed for the damping derivatives above . comparison with vortex-sheet theory shows that the measured values of the stiffness and damping derivatives are approximately 0.6 of the theoretical values . measurements have been made of the direct tab derivatives and cross aileron-tab derivatives for a per cent aileron and 4 per cent (approx.) tab . in addition some measurements of the direct aileron derivatives have been made for comparison with earlier results together with a number of static derivatives for the wing and controls . the influence is shown of frequency parameter, reynolds number, position of transition, mean tab angle and sealing of the control hinge gaps . some tests have been made with the ailcron set at minus 8 deg and the tab at plus 12 deg for which condition the hinge moment on the aileron was zero . reasonable agreement with the values given by the /equivalent profile/ theory is shown for both direct damping derivatives and for the direct tab stiffness derivative . the direct aileron stiffness derivative shows some departure from the theoretical value when . at and the natural transition, comparison with the values given by flat-plate theory gives the following approximate factors, where suffix denotes the theoretical values ..
calculation of derivatives for a cropped delta wing with subsonic leading edges oscillating in a supersonic airstream . the lift, pitching moment and full-span constant-chord control hinge-moment are derived for a cropped delta wing describing harmonic plunging and pitching oscillations of small amplitude and low-frequency parameter in a supersonic air stream . it is assumed that (a) the wing has subsonic leading edges, (b) the wing is sufficiently thin and the mach number sufficiently supersonic to permit the use of linearised theory . expressions for the various derivative coefficients are obtained for a particular delta wing of aspect ratio 1.8 and taper ratio these are avaluated and tabulated for mach numbers 1.1, 1.15, 1.2, 1.3, 1.4, 1.5, 1.6 and 1.944 .
supersonic flow past oscillating airfoils including nonlinear thickness effects . a solution to second order in thickness is derived for harmonically oscillating two-dimensional airfoils in supersonic flow . for slow oscillations of an arbitrary profile, the result is found as a series including the third power of frequency . for arbitrary frequencies, the method of solution for any specific profile is indicated, and the explicit solution derived for a single wedge . nonlinear thickness effects are found generally to reduce the torsional damping, and so to enlarge the range of mach numbers within which torsional instability is possible . this destabilizing effect varies only slightly with frequency in the range involved in dynamic stability analysis, but may reverse to a stabilizing effect at high flutter frequencies . comparison with a previous solution exact in thickness suggests that nonlinear effects of higher than second order are practically negligible . the analysis utilizes a smoothing technique that replaces the actural problem by one involving no kinked streamlines . this stratagem eliminates all consideration of shock waves from the analysis, yet yields the correct solution for problems that actually contain shock waves .
aircraft flutter . the term flutter is used here to denote maintained or violent oscillations of a structure due to aerodynamic forces acting in conjunction with both elastic and inertial forces . attention is restricted to this particular branch of the more general field of aeroelasticity, which embraces buffeting, divergence, and reversal of control, as well as flutter,. airscrew flutter is not specifically considered . the monograph is divided into three main parts, each of which has been made self-contained for the convenience of readers . in the first part, general methods for the investigation of aircraft flutter, by theoretical analysis and by experiments on flutter models, are set out and discussed . a detailed account of the aerodynamic theory of wings in non-uniform motion is not included, since this has already been provided elsewhere, but methods for the evaluation of the aerodynamic forces required in a theoretical flutter analysis are logically developed, and a bibliography of researches on the aerodynamic theory is given in the appendix . investigations on specific types of aircraft flutter–namely wing flutter, control surface flutter, and tab flutter–are discussed in part these various types of flutter are considered, but the practical details of flutter-prevention devices are omitted . finally, in part 3, methods for the experimental determination of airloads on oscillating aerofoil systems are described, and available airload measurements are analysed and compared with theoretical results . an attempt has been made to refer in the text to all relevant british work reported by the early part of 1947 . foreign work has been mentioned in parts 1 and 2 only where necessary for the sake of completeness, but in part 3 and the appendix all relevant foreign references known to the author have been included . matrix notation has been used for the theoretical treatment in part 1, but otherwise its use has been avoided .
calculated velocity distributions and force derivatives for a series of high-speed aerofoils . the polygon method of woods is used to calculate the velocity distribution over a number of two-dimensional aerofoils at low incidence, subcritical flows only being considered . lift slopes and aerodynamic centres at zero lift are also calculated . some comparisons with experimental results are made, and these show good agreement at zero incidence .
a study of the application of airfoil section data to the estimation of the high subsonic speed characteristics of swept wings . estimates of the variation with mach number of the aerodynamic characteristics of swept wings are made on the basis of airfoil section data combined with span-loading theory . the analysis deals with examinations of some 26 wings and wing-body combinations ranging in sweep angle from 30 to 60 and for mach numbers between 0.6 and 1.0 . results of the study indicate that the two-dimensional section data afford good qualitative information for such high-speed aerodynamic characteristics as the variation with mach number of drag, zero-lift pitching-moment coefficient, and lift coefficient for flow separation . quantitative estimates of the force and moment divergence mach numbers could not be made with any degree of certainty from the airfoil data alone . somewhat improved quantitative estimates for a given configuration were obtainable by basing the estimates on the measured characteristics for a wing of similar plan form but different section, and adjusting for the effects of differences in section on the basis of section data .
a correlation of airfoil section data with the aerodynamic loads measured on a 45 sweptback wing at subsonic mach numbers . an investigation has been made of the possibility of correlating airfoil section data with measured pressure distributions over a 45 sweptback wing in the mach number range from 0.50 to 0.95 at a free-stream reynolds number of approximately 2 million . the wing had an aspect ratio of 5.5, a taper ratio of 0.53, naca 64a010 sections normal to the quarterchord line, and was mounted on a slender body of revolution . at mach numbers of 0.85 and below, and for wing normal-force coefficients below the maximum normal-force coefficient for an infinite-aspect-ratio wing yawed 45 to the flow (derived from airfoil section data by simple sweep relations), good correlation was obtained over most of the wing between wing-section and two-dimensional-airfoil pressure distributions . for greater normal-force coefficients lateral boundary-layer flow permitted the inboard wing sections to rise to high maximum section normal-force coefficients . the effectiveness of this lateral boundary-layer flow disappeared towards the tip . for all mach numbers, the influence of plan-form effects on the pressure distributions limited the quality of the correlation at the 20- and 95-percent-semispan stations . above a mach number of about 0.85 the shock waves originating at the juncture of the body and the wing trailing edge spread over the span, preventing further application of two-dimensional data . the spanwise load distributions at moderate normal-force coefficients could be predicted from span-loading theory for the entire mach number range of the tests .
the applications of the polygon method to the calculation of the compressible subsonic flow round two-dimensional profiles . this paper sets out the method now used by the author of applying the polygon method to the calculation of the compressible subsonic flow round two-dimensional aerofoils . tables have been constructed which can be used for all aerofoil shapes, putting the polygon method on the same footing numerically as goldstein’s method has the advantage over approximation 3 that it can be applied in the following cases which are beyond the scope of goldstein’s method .. conventional aerofoils, (b) the low-speed flow about very thick aerofoils, e.g., in reference 3 it is applied to circular cylinders, (c) the flow about symmetric aerofoils between either straight or constant pressure walls, (d) flow in asymmetric channels, and (e) more difficult problems of the flow about aerofoils in the presence of one or two constraining walls (to be published) . a method of calculating lift and moment coefficients, and their rates of change with incidence (a) is also given in the paper . as an example the velocity distribution and the rates of change of the lift and moment coefficients with a are calculated for the aerofoil r.a.e.104 at values of m (mach number at infinity) of 0, and 0.7, for various values of the incidence, a . the velocity distributions for zero incidence are found to be in fair agreement with the corresponding experimental results . the results at incidence are in satisfactory agreement with the experimental results, not for the same incidence, but for the same lift coefficient . it is found, for example, that at m = 0.7 the theory for a = 0.8 agrees best with experiment for a = 1.0, when the lift coefficients are approximately the same .
laminar boundary layer oscillations and transition on a flat plate . this is an account of an investigation in which oscillations were discovered in the laminar boundary layer along a flat plate . these oscillations were found during the course of an experiment in which transition from laminar to turbulent flow was being studied on the plate as the turbulence in the wind stream was being reduced to unusually low values by means of damping screens . the first part of the paper deals with experimental methods and apparatus, measurements of turbulence and sound, and studies of transition . a description is then given of the manner in which oscillations were discovered and how they were found to be related to transition, and then how controlled oscillations were produced and studied in detail . the oscillations are shown to be the velocity variations accompanying a wave motion in the boundary layer, this wave motion having all the characteristics predicted by a stability theory based on the exponential growth of small disturbances . a review of this theory is given . the work is thus experimental confirmation of a mathematical theory of stability which had been in the process of development for a period of approximately 40 years, mainly by german investigators .
the hall effect in the viscous flow of ionized gas between parallel plates under transverse magnetic field . the electrical conductivity of an ionized gas is anisotropic in the presence of magnetic field (hall effect) . the conductivity is expressed by a tensor in the same form for both fully and partially ionized gases . by the use of modified ohm’s law and conventional magnetohydrodynamical equations the incompressible viscous flow between parallel plates under the transverse magnetic field is analyzed and an exact solution is obtained when the magnetic reynolds number is small . the numerical results reveal a remarkable effect of anisotropy of conductivity . the acceleration and deceleration of viscous ionized gas under combined electric and magnetic fields are also calculated .
boundary layer induced noise in the interior of aircraft . at high speeds the turbulent boundary layer washing the airplane fuselage excites appreciable skin vibration, promoting strong noise in the interior . the fluctuating exciting pressure distribution can be represented as a pattern of moving waves (fourier integral) . a running ripple in the skin follows underneath each wave, and the noise is ultimately due to these ripples . the acoustic effects of the running ripples are calculated for an infinite sheet,. this is considered the main result of the paper . supersonically moving ripples radiate strong sound in the form of mach waves,. subsonically moving ripples radiate no sound . formulas for the mean square surface pressure and the energy flux are obtained for an assumed idealized turbulent pressure spectrum . the results are adapted to provide a tentative estimate of the noise generated at subsonic speeds in a practical fuselage . the running ripples are almost noise-free, but multiple reflections at the frames and stringers promote standing waves . an assumption is used to link the two kinds of waves, and this leads to provisional calculations of noise level .. on this basis the noise level is predicted to vary as for thin boundary layers, changing progressively to for thick layers ( = external air density, = speed, = layer thickness, = panel thickness) . some comparisons are made with experiment . finally, an idea for minimizing the noise is presented .
propeller in yaw . it was realized as early as 1909 that a propeller in yaw develops a side force like that of a fin . in 1917, r. g. harris expressed this force in terms of the torque coefficient for the unyawed propeller . of several attempts to express the side force directly in terms of the shape of the blades, however, none has been completely satisfactory . an analysis that incorporates induction effects not adequately covered in previous work and that gives good agreement with experiment over a wide range of operating conditions is presented herein . the present analysis shows that the fin analogy may be extended to the form of the side-force expression and that the effective fin area may be taken as the projected side area of the propeller . the effective aspect ratio is of the order of 8 and the appropriate dynamic pressure is roughly that at the propeller disk as augmented by the inflow . the variation of the inflow velocity, for a fixed-pitch propeller, accounts for most of the variation of side force with advance-diameter ratio v nd . the propeller forces due to an angular velocity of pitch are also analyzed and are shown to be very small for the pitching velocities that may actually be realized in maneuvers, with the exception of the spin . further conclusions are .. a dual-rotating propeller in yaw develops up to one-third more side force than a single-rotating propeller . a yawed single-rotating propeller experiences a pitching moment in addition to the side force . the pitching moment is of the order of the moment produced by a force equal to the side force, acting at the end of a lever arm equal to the propeller radius . this cross-coupling between pitch and yaw is small but possibly not negligible . the formulas for propellers in yaw derived herein (with the exception of the compressibility correction) and a series of charts of the side-force derivative calculated therefrom have been presented without derivation in an earlier report .
effect of slight blunting of leading edge of an immersed body on the flow around it at hypersonic speed . manufacturing and maintainance of ideally sharp leading edges and noses is practically impossible, hence a discrepancy arises between the theory established for sharp edges and actual flow around slightly blunted edges, where a detached shock is formed with a subsonic adjacent region . semi-empirical method is worked out showing that the pressure distribution in the vicinity of the leading edge is the same for different thin profiles having the same shape of bluntness on their edges or noses . the data for a flat plate can be used for all of them . for moderate supersonic speed the pressure on the remaining body is practically unaffected by the nose bluntness, and can be computed from a sharp-edge theory . for high supersonic speed a slight blunting of the edge can considerably alter the pattern of flow over a large region . the method consists in replacing blunted edge by action of concentrated forces on the flow,. it is applied to blunted wedge where it shows doubling of the drag computed by classic theory, and to cones, where the drag of a blunted cone may become smaller than that of a sharp one .
theory and tunnel tests of rotor blade for supersonic turbines . in special circumstances where a large work output is required from a turbine in a single stage it is necessary to use high pressure ratios across the nozzle blades, thus producing supersonic velocities at inlet to the rotor . as part of an investigation into such turbines, several designs for the inter-blade passages of the rotor have been tested in a two-dimensional tunnel, a design theory being developed concurrently . the first design, featuring constant passage width and curvature as in steam-turbine practice, but having thin leading and trailing edges, was found to suffer from focusing of the compression waves from the concave surface, with consequent flow separation from the opposite convex surface . it gave a velocity coefficient of measured at an inlet mach number of 1.90 and turning angle of 140 deg . the measured value compares favourably with values from previous steam tests, where the results have been in the range from 0.65 to 0.92 . from theoretical reasoning, and from additional test observations, a subsequent passage was designed having an inlet transition length of small curvature, leading to a free-vortex passage of double the transition curvature,. a small amount of contraction was incorporated . schlieren photographs showed the flow in this passage to be almost shock free . a thin region of low-energy air existed close to the convex surface, but liquid-injection tests located only one small bubble of reversed flow . pressure traverses at exit indicated a velocity coefficient of 0.952, based on the area-mean total pressure . when allowance is made for turning angle and reynolds number this result appears to compare quite favourably with previous work . it would seem that the optimum blade pitching in a turbine would be about 20 to 30 per cent closer than in a two-dimensional cascade . however, the resultant pitching tends to become very close, except at very large turning angles, with the result that in some applications difficulties could arise in the practical design and manufacture . several uncertainties remain and the present design must be regarded as still experimental .
the performance of supersonic turbine nozzles . an investigation has been conducted at the national gas turbine establishment into the performance of turbines having high pressure ratios per stage . the present report discusses the mode of operation of supersonic nozzles for such turbines, and describes a cascade experiment . both theory and experiment demonstrate that the conditions imposed upon the supersonic flow immediately downstream of the nozzles (e.g., by a following row of rotor blades) exert an overriding influence upon the nozzle outlet flow angle, and hence upon the maximum pressure ratio obtainable across the nozzle–providing that the axial component of velocity is subsonic . this is an important difference from the more familiar flow of subsonic turbine nozzles, where, for example, the downstream gas angle is controlled predominantly by the nozzle blade shape and spacing . a suitable test technique using a closed-jet tunnel is demonstrated . the particular nozzles tested, of convergent-divergent form, had a straight-sided divergent portion of to axial direction) and a design mach number of 2 . the flow was found to be well behaved as regards shock pattern, losses, and starting over the range of pressure ratios tested–between 9 1 and 19 1 . in particular the efficiency at the design pressure ratio of 16.6 1 was high, the velocity coefficient calculated from traverses of pitot and static tubes being 0.98 . for the conversion of pitot to total pressure at a mach number of 2.5 a high accuracy is important in the measurement of the static pressure,. nevertheless readings from a conventional four-hole instrument appear to be reliable .
on the testing of supersonic compressor cascades . to facilitate the development of high speed axial-flow compressors, an investigation was made into the possibility of measuring blade performance in a stationary cascade at supersonic speeds . a suitable technique was developed and the losses in a variety of cascades were measured, but these losses were too high for the blading to have any possible application . it was concluded that if a useful compressor is to result, it is essential to test the cascades at mach numbers close to the existing technique was suitable only for zero incidence tests, and thus a new approach is necessary . some of the fundamentals of this cascade testing at low supersonic speeds are discussed in the light of the current understanding of the mode of operation of supersonic compressors at transonic speeds .
the test performance of highly loaded turbine stages designed for high pressure ratio . a blade design for a highly loaded two-stage turbine is described and the test performance of the turbine is presented . some of the factors affecting the performance and matching of turbine blade rows operating at supersonic gas velocity are discussed and investigated by means of tests on a three-dimensional nozzle cascade tunnel and on a variety of single-stage turbine builds .
the supersonic axial flow compressor . an investigation has been made to explore the possibilities of axial-flow compressors operating with supersonic velocities into the blade rows . preliminary calculations showed that very high pressure ratios across a stage, together with somewhat increased mass flows, were apparently possible with compressors which decelerated air through the speed of sound in their blading . the first phase of this investigation, which has been reported in naca acr l5d20, was the development of efficient supersonic diffusers to decelerate air through the speed of sound . the present report is largely a general discussion of some of the essential aerodynamics of single-stage supersonic axial-flow compressors . in the supersonic flow about isolated bodies, large energy losses usually occur due to wave systems which extend far from the bodies . supersonic flow entering a cascade is considered and, in this case, the possibility of entirely eliminating this extended wave system is demonstrated,. thus, no reason for supersonic compressors to be necessarily inefficient is apparent . the conditions that occur as the flow through the compressor is being started are discussed and a hypothesis as to the type of transonic flow which will be encountered is proposed . as an approach to the study of supersonic compressors, three possible velocity diagrams are discussed briefly . because of the encouraging results of this study, an experimental single-stage supersonic compressor has been constructed and tested in freon-12 . in this compressor, air decelerates through the speed of sound in the rotor blading and enters the stators at subsonic speeds . a pressure ratio of about 1.8 at an efficiency of about 80 percent has been obtained .
flow pattern in a converging-diverging nozzle . the present report describes a new method for the prediction of the flow pattern of a gas in the two-dimensional and axially symmetrical case . it is assumed that the expansion of the gas is adiabatic and the flow stationary . the several assumptions necessary on the nozzle shape effect, in general, no essential limitation on the conventional nozzles . the method is applicable throughout the entire speed range,. the velocity of sound itself plays no singular part . the principal weight is placed on the treatment of the flow near the throat of a converging-diverging nozzle . for slender nozzles formulas are derived for the calculation of the velocity components as function of the location .
intensity, scale and spectra of turbulence in mixing region of free subsonic jet . the intensity of turbulence, the longitudinal and lateral correlation coefficients, and the spectra of turbulence in a 3.5 inch-diameter free jet were measured with hot-wire anemometers at exit mach numbers from 0.2 to 0.7 and reynolds numbers from the results of these measurements show the following .. (1) near the nozzle (distances less than 4 or 5 jet diam downstream of the nozzle) the intensity of turbulence, expressed as percent of core velocity, is a maximum at a distance of approximately increasing mach and or reynolds number . at distances greater than 8 jet diameters downstream of the nozzle, however, the maximum intensity moves out and decreases in magnitude until the turbulence-intensity profiles are quite flat and approaching similarity . (2) the lateral and longitudinal scales of turbulence are nearly independent of mach and or reynolds number and in the mixing zone near the jet vary proportionally with distance from the jet nozzle . (3) farther downstream of the jet the longitudinal scale reaches a maximum and then decreases approximately linearly with distance . (4) near the nozzle the lateral scale is much smaller than the longitudinal and does not vary with distance from the centerline, while the longitudinal scale is a maximum at a distance from the centerline of about mum moves out from the centerline . (6) a statistical analysis of the correlograms and spectra yields a /scale/ which, although different in magnitude from the conventional, varies similarly to the ordinary scale and is easier to evaluate .
on the strength distribution of noise sources along a jet . the spatial distribution of noise sources along a jet is investigated by application of lighthill’s theory to regions of ‘similar’ profiles . the analysis refers to the noise power emitted by a ‘slice’ of jet (section between two adjacent planes normal to the axis) as a function of distance x of the slice from the nozzle . it is found that this power is essentially constant with x in the initial mixing region (x law), then further downstream (say 8 or 10 diameters from the nozzle) falls off extremely fast (x law or faster) in the fully developed jet . because of this striking attenuation of strength with distance, it is concluded that the mixing region produces the bulk of the noise and must dominate in muffler behavior,. conversely, the ‘fat’ part of the jet must contribute much less to the total noise power than is commonly supposed . powell’s experiments on the effects of nozzle velocity profile on total noise power are interpreted qualitatively . the behavior of multiple-nozzle or corrugated mufflers, both as to overall quieting and frequency-shifting, is also interpreted in the light of the results . the possibility emerges that such mufflers may be improved without serious thrust loss by the addition of a sound-attenuating shroud .
a general purpose analogue correlator for the analysis of random noise signals . a large proportion of the current research programme of the department of aeronautics and astronautics is concerned with the study of jet noise and boundary layer pressure fluctuations and their effect on aircraft structures . early in the work it was decided that for a complete description of the random processes involved it would be necessary in the experimental programme to make correlation measurements in addition to the more standard spectrum and amplitude distribution measurements . it was also felt that it would be desirable from the university point of view to construct a general purpose correlator which could later be used on other types of work . to this end it was decided to give the correlator a wider bandwidth than might strictly have been necessary for the problems on hand . subsequent development work has amply justified this decision .
a theoretical study of annular supersonic nozzles . this paper is concerned with the design of annular supersonic nozzles to produce uniform flow in supersonic wind tunnels which are axi-symmetrical and which have an internal coaxial circular cylinder throughout . symmetrical two-dimensional and conventional axi-symmetrical nozzles are special cases of annular nozzles . proposals are made for design criteria sufficient to ensure that the flow inside a nozzle is free from limit lines and shock waves,. the criteria for (symmetrical) two-dimensional and (conventional) axi-symmetrical nozzles are new . the two outstanding procedures for designing two-dimensional and axi-symmetrical nozzles are generalised to apply to annular nozzles . one of the design procedures is mainly analytical and the other is mainly numerical,. the analytical expressions in both procedures are made much more complicated by the presence of the internal cylinder but the numerical process criteria and the mainly numerical design procedure are successfully applied to the design of a particular annular nozzle .
the flow over delta wings at low speeds with leading edge separation . a low speed investigation of the flow over a 40 apex angle delta wing with sharp leading edges has been made in order to ascertain details of the flow in the viscous region near the leading edge of the suction surface of the wing . a physical picture of the flow was obtained from the surface flow and a smoke technique of flow visualization, combined with detailed measurements of total head, dynamic pressure, flow directions and vortex core positions in the flow above the wing . surface pressure distributions were also measured and integrated to give normal force coefficients . the results of this investigation were compared with those of other experimental investigations and also with various theoretical results . in particular, the normal force coefficients, vortex core positions and attachment line positions were compared with the theoretical results of mangler and smith, reference 19 . it was found that .. exist on the upper surface of the wing outboard of and below the main vortices . these secondary vortices are formed as a result of separation of the boundary layers developing outboard of the top surface attachment lines .
a note on the theory of the stanton tube . existing theories for the stanton tube are critically reviewed, and the paper then outlines a simple method which predicts the calibration function at high reynolds numbers to the right order of magnitude .
quasi-cylindrical surfaces with prescribed loadings in the linearised theory of supersonic flow . a formula for the velocity field in terms of a given surface distribution of vorticity is applied to points lying on the surface . an equation giving the shape of a quasi circular-cylindrical surface in terms of a prescribed loading is derived . as an example a half ring wing with prescribed loading is discussed .
elliptic cones alone and with wings at supersonic speeds . to help fill the gap in the knowledge of aerodynamics of shapes intermediate between bodies of revolution and flat triangular wings, force and moment characteristics for elliptic cones have been experimentally determined for mach numbers of 1.97 and sectional axis ratios from 1 through 6 and with lengths and base areas equal to circular cones of fineness ratios 3.67 and 5 have been studied for angles of bank of 0 and 90 . elliptic and circular cones in combination with triangular wings of aspect ratios 1 and 1.5 also have been considered . the angle-of-attack range was from 0 to about 16, and the reynolds number was 8×10, based on model length . in addition to the forces and moments at angle of attack, pressure distributions for elliptic cones at zero angle of attack have been determined . the results of this investigation indicate that there are distinct aerodynamic advantages to the use of elliptic cones . with their major cross-sectional axes horizontal, they develop greater lift and have higher lift-drag ratios than circular cones of the same fineness ratio and volume . in combination with triangular wings of low aspect ratio, they also develop higher lift-drag ratios than circular cones with the same wings . for winged elliptic cones, this increase in lift-drag ratio results both from lower zero-lift drag and drag due to lift . visual-flow studies indicate that, because of better streamlining in the crossflow plane, vortex flow is inhibited more for an elliptic cone with major axis in the plane of the wing than for a circular cone with the same wing . as a result, vortex drag resulting from lift is reduced . shifts in center of pressure with changes in angle of attack and mach number are small and about the same as for circular cones . comparisons of theoretical and experimental force and moment characteristics for elliptic cones indicate that simple linearized (flat plate) wing theory is generally adequate even for relatively thick cones . zero-lift pressure distributions and drag can be computed using van dyke’s second-order slender-body theory . for winged circular cones, a modification of the slender-body theory of naca rep. 962 results in good agreement of theory with experiment .
aerofoil theory of a flat delta wing at supersonic speeds . lift, drag, and pressure distribution of a triangular flat plate moving at a small incidence at supersonic speeds are given for arbitrary mach number and aspect ratio . the values obtained for lift and drag are compared with the corresponding values obtained by strip theory . the possibility of further applications of the analysis leading up to the above results is indicated .
a technique for improving the predictions of linearised theory on the drag of straight edge wings . the curve of drag against mach number for straight-edged wings, calculated by using the linearised theory of supersonic flow, displays discontinuities in slope at the various mach numbers for which the edges are sonic . these features, which are not observed in practice, are due to the fact that linearised theory predicts an infinite pressure along a subsonic or sonic edge . it is shown that if the linearised equation of supersonic flow is used to determine the flow over straight-edged wings, but the linearised boundary condition is replaced by the full placed by plausible values . on this basis a simple method is derived for improving the linearised predictions of the drag of straight-edged wings which exhibits satisfactory agreement with experimental results . while the technique is not directly applicable to ridge lines, an artifice renders them amenable to similar treatment .
navier-stokes solutions at large distances from a finite body . this paper is concerned with a theoretical investigation of the flow field at large distances from an object moving through a viscous fluid . the discussion will be restricted to the case of two-dimensional stationary incompressible flow . the object will be assumed to be of finite size . the domain of the fluid is infinite and it is assumed that there are no other boundaries for the fluid except that of the given object . the reynolds number will be assumed to have a fixed value., thus we shall not consider the limiting cases of the reynolds number tending to zero or to infinity .
interference between the wings and tail surfaces of a combination of slender body, cruciform wings and cruciform tail set at both incidence and yaw . the interference between the wings and the tail surfaces of a combination of circular body, low aspect ratio cruciform wings and cruciform tail in an inviscid flow is analysed using the slender body theory . the system may be subjected to both incidence and yaw and, in general, the tail fins may be staggered angularly with respect to the main wings . the method is a development of that used by owen and maskell in r.a.e. report no. aero.2441 to analyse similar effects on a system set at zero yaw . simple expressions to determine the strengths and positions of the trailing vortices (supposed to be rolled-up) downstream of the main wings are given, and from them the forces on the tail are deduced . when the tail surfaces are triangular and of low aspect ratio an exact solution is obtained from slender body theory .. but for rectangular tail surfaces of moderate or high aspect ratio, it is suggested that the changes in lift and sideforce on the tail caused by the wing vortex field can be estimated approximately from the mean upwash and sidewash angles evaluated over the respective tail spans . formulae for these means angles are presented .
interference between the wings and tail plane of a slender wing-body tailplane combination . an approximate method of predicting the interference between the wings and the tailplane of a slender wing-body-tailplane combination in an inviscid flow is developed, in order to explain the change in centre of pressure position with incidence which has been found to occur in wind tunnel and flight tests on guided weapons . incidence changes in one plane only, normal to the plane containing the wings and the tail surfaces, have been considered . the method is based on slender body theory and the assumption that the wing trailing vortices roll-up completely before they reach the tailplane,. it is, therefore, applicable to weapons equipped with low aspect ratio wings far separated from the tail surfaces . when the tail surfaces are triangular and of low aspect ratio, an analytical solution is given for the effect of the wing downwash field on the tail lift . for high aspect ratio, rectangular tail surfaces it is suggested by comparison with experimental data, that the tail lift may be estimated approximately from the value of the mean downwash angle across the tail span . a summary of the method is given in para.5 which, in conjunction with the introduction, may be read independently of the rest of the report .
practical calculation of second-order supersonic flow past non-lifting bodies of revolution . calculation of second-order supersonic flow past bodies of revolution at zero angle of attack is described in detail, and reduced to routine computation . use of an approximate tangency condition is shown to increase the accuracy for bodies with corners . tables of basic functions and standard computing forms are presented . the procedure is summarized so that one can apply it without necessarily understanding the details of the theory . a sample calculation is given, and several examples are compared with solutions calculated by the method of characteristics .
accuracy of approximate methods for predicting pressure on pointed non-lifting bodies of revolution in supersonic flow . the accuracy and range of applicability of the linearized theory, second-order theory, tangent-cone method, conical-shock-expansion theory and newtonian theory for predicting pressure distributions on pointed bodies of revolution at zero angle of attack are investigated . pressure distributions and integrated pressure drag obtained by these methods are compared with standard values obtained by the method of characteristics and the theory of taylor and maccoll . three shapes, cone, ogive, and a modified optimum body, are investigated over a wide range of fineness ratios and mach numbers . it is found that the linearized theory is accurate only at low values of the hypersonic similarity parameter number to body fineness ratio) and that second-order theory appreciably extends the range of accurate application . the second-order theory gives good results on ogives when the ratio of the tangent of maximum surface angle to the tangent of the mach angle is less than 0.9 . tangent-cone method cannot be widely applied with good accuracy . in general, the conical-shock-expansion theory predicts pressure and drag within engineering accuracy when the hypersonic similarity parameter is greater than 1.2 . although newtonian theory gives good accuracy, except for cones, at the highest values of the hypersonic similarity parameter investigated, it is less accurate than the conical-shock-expansion theory .
the theoretical wave drag of some bodies of revolution . this report investigates the wave drag of bodies of revolution with pointed or open-nose forebodies and pointed or truncated afterbodies . the ‘quasi-cylinder’ and ‘slender-body’ theories are reviewed, a reversibility theorem is established, and the concept of the interference effect of a forebody on an afterbody is introduced . the theories are applied to bodies whose profiles are either straight or parabolic arcs, formulae and curves being given for forebody and afterbody drag, and for the interference drag . the results of the two theories are compared and are seen to agree well in the region of geometries where both theories are applicable .
a second order shock-expansion method applicable to bodies of revolution near zero lift . a second-order shock-expansion method applicable to bodies of revolution near zero lift is developed . expressions defining the pressures on noninclined bodies are derived by the use of characteristics theory in combination with properties of the flow predicted by the generalized shock-expansion method . this result is extended to inclined bodies to obtain expressions for the normal-force and pitching-moment derivatives at zero angle of attack . the method is intended for application under conditions between the ranges of applicability of the second-order potential theory and the generalized shock-expansion mehtod – namely, when the ratio of free-stream mach number to nose fineness ratio is in the neighborhood of 1 . for noninclined bodies, the pressure distributions predicted by the second-order shock-expansion method are compared with existing experimental results and with predictions of other theories . for inclined bodies, the normal-force derivatives and locations of the center of pressure at zero angle of attack predicted by the method are compared with experimental results for mach numbers from 3.00 to 6.28 . fineness ratio 7, 5, and 3 cones and tangent ogives were tested alone and with cylindrical afterbodies up to 10 diameters long . in general, the predictions of the present method are found to be in good agreement with the experimental results . for non-inclined bodies, pressure distributions predicted with the method are in good agreement with existing experimental results and with distributions obtained with the method of characteristics . for inclined bodies, the normal-force derivatives per radian (for normal-force coefficients referenced to body base area) are predicted within 0.2 and the locations of the center of pressure are predicted within 0.2 body diameters . on the basis of these results, the second-order shock-expansion method appears applicable for values of the ratio of free-stream mach number to nose fineness ratio from 0.4 to 2 .
on the minimisation and numerical evaluation of wave drag . a fourier analysis of the linearised theory expression for the zero-lift wave drag of a smooth, slender body in terms of its cross-sectional area distribution is used to derive the area distribution which minimises the expression for given length, volume, nose area, base area and n intermediate areas . another minimal deduced from this by relaxing the restriction on volume is used to evolve a method for the numerical evaluation of the original expression . two practical applications of these results are discussed . the first is in the design of wing-body combinations to have small drag rise at transonic speeds . the second is in the calculation of the wave drag of wing-body combinations at zero lift,. an example is constructed to illustrate the method and to give an indication of its accuracy .
criteria for thermodynamic equilibrium in gas flow . when gases flow at high velocity, the rates of internal processes may not be fast enough to maintain thermodynamic equilibrium . by defining quasi-equilibrium in flow as the condition in which the temperature, pressure, density, and velocity deviate by less than a fixed, small percentage from what they would be if the flowing gas could actually be in thermodynamic equilibrium, criteria are derived for determining whether quasi-equilibrium is a stable condition in the flow . by use of excitation of molecular vibration as an example, the general properties of criteria curves are discussed and interpreted . a discussion is given of how to use these results to determine definitely whether a flow is or is not in thermodynamic equilibrium . applications to dissociating gases, to mixtures, and to the phenomenon of /choking/ in a laval nozzle are given special consideration . for cases when application of the criteria predict nonequilibrium, equations are provided in a form useful for numerical forward integration along streamlines .
a compressor routine test code . the routine testing of aircraft-type compressors.dash in the main, axial-flow, multi-stage compressors.dash requires a compromise between research accuracy and the practical considerations . this test code is the outcome of a survey of compressor testing techniques and instrumentation, initiated and subsequently discussed and endorsed by the aerodynamics sub-committee of the gas turbine collaboration committee . the code aims at defining methods of measurement and weighting whereby compressor performance can be obtained sufficiently accurately for a realistic and direct comparison to be made between one compressor and another . the measurement of a quantity at a point in the fluid flow, and the averaging and weighting of such measurements have been treated separately as far as is possible . the recommendations are given in the main text, whilst additional discussion on these is put into the appendices .
on a determination of the pitot-static tube factor at low reynolds numbers, with special reference to the measurement of low air speeds . reasons for enquiry–to provide a standard instrument for the calibration of low speed anemometers .
design and calibration at low speeds of a static tube and a pitot-static tube with semi-ellipsoidal nose shapes . a new static tube and a new pitot-static tube have been designed and calibrated in the no.1 and the no.2 11 ft x 8 ft wind tunnels of the r.a.e., using a long static tube, the error of which is believed to be very small, as a standard for comparison . the results show that the static pressure measured by these tubes is in error due to the supporting strut and to the nose shape of the tube by an amount which may be calculated for positions of the static slot, or holes, greater than 10 tube diameters ahead of the strut . the readings show no measurable scale effect in the speed range 100-230 ft sec . the static tube is insensitive to yaw in the range 1 with a square-edged slot and is even less sensitive to yaw when the slot edges are rounded . the turbulence of the tunnel has an effect on the static pressure reading .
a theoretical analysis of heat transfer in regions of separated flow . the flow field analyzed consists of a thin, constant pressure viscous mixing layer separated from a solid surface by an enclosed region of low-velocity air (/dead air/) . the law of conservation of energy is employed to relate calculated conditions within the separated mixing layer to the rate of heat transfer at the solid surface . this physical speed is app ied to alminar separations in compressible flow for various prandtl numbers, including consideration of the case where air is injected into the separated region .
laminar mixing of a non-uniform stream with a fluid at rest . a theoretical analysis is made of the constant pressure laminar mixing process between a stream having an initial boundary layer velocity profile, and a fluid at rest . the present theory follows the methods of w. tollmien and s. i. pai with certain modifications . the results apply to incompressible flow, but can be extended to the compressible case without difficulty .
an approximate theory of base pressure in two dimensional flow at supersonic speeds . an approximate theory of the base pressure in two-dimensional flow at supersonic speeds is presented using asimplified representation of the flow and some of the findings of tollmien’s work on turbulent mixing in incompressible flow . good qualitative predictions of the effects of a boundary layer, of bleed air and of boat-tailing are obtained .
investigation with an interferometer of the turbulent mixing of a free supersonic jet . the free turbulent mixing of a supersonic jet of mach number of which a description is given, was used for the investigation . density and velocity distributions through the mixing zone have been obtained . it was found that there was similarity in distribution at the cross sections investigated and that, in the subsonic portion of the mixing zone, the velocity distribution fitted the theoretical distribution for incompressible flow . it was found that the rates of spread of the mixing zone both into the jet and into the ambient air were less than those of subsonic jets .
an improved smoke generator for use in the visualisation of airflow, particularly boundary layer flow at high reynolds numbers . and rapid method by which boundary layer flow was rendered visible has been previously described in the journal of the royal aeronautical society . it gave promise of being useful at the highest tunnel speeds provided a denser smoke could be obtained, which at the same time was free from the troublesome deposits associated with the wood smoke . of the aerodynamics division attempts were made by the fuel research station to improve the density of the wood smoke and to reduce the deposits . these they showed were conflicting requirements, and whilst some improvement was effected, it was not sufficient for observation in the new tunnels at high speeds . the staff of the director-general of scientific research and development, ministry of supply, was then approached and it was decided to develop an oil smoke generator from a simple generator of this type which was demonstrated to us . this has been done successfully . the final apparatus in contrast to the wood smoke generator is light and compact . it takes only a few minutes to start and can be run as long as desired . improvement on the wood smoke both as regards density and freedom from deposits, which cause premature transition . the density and quality of the smoke are now under control . smokes ranging from a light smoke of bluish white colour to a heavy smoke dense white in appearance can be obtained . the oil smoke retains the advantages of the wood smoke in that it is non-corrosive and non-irritant, and the smell can be tolerated even when it is present in a considerable concentration . a certain amount of condensation is inevitable with oil smokes, but with suitable precautions troubles arising from this can be avoided . a dry solid smoke made by melting a hard wax was successfully generated with the same apparatus . unfortunately because of its flocculent nature this smoke gave rise to solid deposits when passed through bore tubing, leading eventually to complete blockage . this seems to be a feature of solid smokes . the apparatus has been used to determine transition and laminar separation points on model wings in a number of the national physical laboratory tunnels . smoke filaments have been maintained in the laminar state up to wind speeds of 180 ft. sec. in the new tunnels . there is much to be said for making a standard practice of visualising boundary layer flow on models, particularly as the technique is simple and rapid . it would greatly assist the interpretation of force measurements and the more detailed explorations of the boundary layer by total head tubes and hot wires . the use of oil smoke is not limited to boundary layer flow visualisation . the apparatus described in this report would seem to be particularly suited for educational work in small demonstration tunnels .
the ground effect on the jet flap in two dimensions . this paper presents the results of the first part of an experimental investigation of the ground effect on simple jet flap aerofoils . in this part of the work an aerofoil having a 58.1 deg jet flap was tested under two-dimensional conditions . the pressure lift on the aerofoil was measured, with the ground at fixed positions, for varying jet momentum coefficients . it was found that the effect of the ground on the pressure lift was very small up to a certain critical jet coefficient . on increasing the jet coefficient beyond the critical value a marked loss of pressure lift was observed . this critical value referred to is approximately the same as the jet coefficient at which the jet first hits the ground . some significant, though highly tentative comments, are made regarding the practical application of this work to the take-off characteristics of a jet flapped aircraft .
the design of minimum drag tip fins . with an appendix – on the conformal transformation of a wing with a fin . the report describes an investigation into the design of minimum drag tip fins by lifting line theory . the work is based on an exact solution of the conformal transformation which is applicable to this problem following the method of trefitz . three types of solution are treated, corresponding to symmetrical upper and lower fins, single upper or lower fins, and unequal upper and lower fins . a representative range of solutions for circulation distribution along wing and fins has been calculated for each of the three cases by the use of elliptic and theta functions . a detailed account is given, with examples, of the procedure for calculating the plan of wing and fins, the lift and induced drag, and the setting of the fins .
the calculation of the pressure distribution on thick wings of small aspect ratio at zero lift in subsonic flow . the method of expressing the velocity increment over aerofoils directly in terms of the section ordinates wings of finite aspect ratio . the wings considered are untapered in plan-form but may be tapered in thickness . the section can be of any given shape so that in this sense the analysis is more general than that of refs. 3 to 6 which deal with wings of biconvex section . the coefficients required in the calculation are tabulated for the centre-section of straight and swept-back wings of aspect ratios 0.5,. 1,. 2,. and 4, the wing of infinite aspect ratio having been treated in ref. 1 . the remaining calculations can be made very quickly . since wings of very small aspect ratio can be treated also by the method of slender-body theory, the relations between linear theory, slender-body theory, and linearised slender-body theory are discussed . for the special case of ellipsoids, the results obtained from the various methods are compared with the exact solution .
the application of lighthill formula for numerical calculation of pressure distributions on bodies of revolution at supersonic speed and zero angle of attack . an integral expression, given by lighthill and based on linearized theory, for the external supersonic flow over the surface of slender pointed or ducted bodies of revolution at zero angle of attack is shown to give a good approximation of the exact flow for a much wider mach number and thickness range than could be expected from linearized theory . a numerical method, based on this expression, is developed and applied for digital computing . some results from applying the digital computing procedure for determining the pressure distribution and wave drag for various bodies of revolution are given .
formulae for the computation of the functions employed for calculating the velocity distribution about a given aerofoil . in order to determine the velocity distribution about an arbitrary aerofoil, it is necessary to evaluate the functions and (in the notation of aerofoil theory) when is given numerically . if the values of are specified at 2n points equally spaced about the circle into which the aerofoil is transformed, the formulae obtained here may be used to calculate these functions at the same points . formulae are also given for calculating the integrals of or, since these have application to the design of aerofoils by thwaites’s numerical method . the simplicity of the formulae for and enables the effect on the velocity distribution of a local change of shape readily to be determined by making n large . this is discussed in 3 . the formulae are collected in the appendix, and a table of the coefficients for the case n = 20 is given .
pressure distributions at zero lift for delta wings with rhombic cross sections . the linearised theory of thin wings is used to calculate pressure distributions over delta wings with rhombic cross sections . a deuce programme has been written for the calculation and some of the results are compared with those of slender thin wing theory .
a collection of longitudinal stability derivatives of wings at supersonic speeds . a collection has been made of theoretical data, for wings alone, on those stability derivatives that govern the short-period oscillation of aircraft travelling at supersonic speeds . all the derivatives available have been obtained by means of the linear theory, and so the information given is subject to the usual limitations . the information has been presented in what is hoped is the most convenient form to show its extent, and to expose the parts of the field where experimental investigation is most needed .
an investigation of interference effects on similar models of different size in various transonic tunnels in the u.k. . details are given of a programme of tests being made on similar swept-wing models in transonic tunnels of different types . force measurement results at subsonic speeds in the r.a.e. 3 ft. by 3 ft. slotted tunnel show only small interference effects for models of moderate blockage at low incidence., at higher incidences, the interference effect on lift becomes appreciably greater than estimated by theory, and significant pitching moment differences occur, apparently due to wall interference on the wing flow field . comparable but smaller effects are evident in the results from the a.r.a. 9 ft. by 8 ft. perforated tunnel . at speeds just above m = 1, the force fluctuates as speed is increased, because of wave reflection interference . the magnitude of the fluctuations diminishes as speed is further increased and this reduction is more marked in the perforated tunnel . pressure measurements along the top of the body at zero incidence show delay in shock movements at high subsonic speeds indicating a blockage effect on speed., the effect is larger in the perforated tunnel though smaller than predicted by theory . above m = 1, both expansion and shock waves are strongly reflected in the slotted tunnel but considerable alleviation, particularly of shock waves, is achieved in the perforated tunnel, for which an analysis of the effects is given, showing for example, the effect of the open-area distribution of the walls .
on the ground level disturbance from large aircraft flying at supersonic speeds . the whitham-walkden theory for the estimation of the strength of shock waves at ground level from aircraft flying at supersonic speeds is applied to the case of a typical projected supersonic civil transport aeroplane . if a figure of 2 lb sq.ft. (including a factor of 2 for ground reflection) is taken as an upper limit for the acceptable strength of the bow wave from such an aircraft it is shown that restrictions on the climb and flight plan will be involved . the advantage of the employment of larger engines with or without afterburning is discussed, with reference also to the penalties involved owing to the increase in weight of the aircraft and its direct operating costs . finally it is suggested that an aircraft of given volume could be designed, by suitable choice of thickness and lift distribution, to minimise the strength of the shock waves in the far field .
boundary layers with suction and injection . a review of published work on skin friction . available data on the effects of suction and injection on skin friction are summarised and compared . it is shown that injection into a turbulent boundary layer can produce a skin friction coefficient lower than the laminar value at the same reynolds number on an impermeable plate .
an approximate solution of the turbulent boundary layer equations in incompressible and compressible . if over the ‘outer region’ of the boundary layer, where the mean velocity varies but little from its value outside the shear layer, a virtual eddy viscosity is defined, which is constant over the outer region but varies in the direction of the mainstream, a solution of the turbulent boundary layer equations can be found which satisfies the appropriate boundary conditions . the solution leads to a compatibility condition for the virtual eddy viscosity in terms of the wall shear stress, the boundary layer momentum thickness and the mainstream velocity, at least for the case of a constant external velocity . this compatibility condition, which can be expressed as for moderate to high reynolds numbers, where is the shear velocity, is the boundary layer thickness and is the virtual eddy (kinematic) viscosity, is just the condition townsend (1956) found for the equilibrium of the large eddies . the numerical value of the constant derived by townsend agrees with ours for reynolds numbers (based on x) of about . with this relation for an equation, analoguous to the momentum integral equation solution, can be found for as a function of local freestream velocity, with one disposable parameter .
an experimental study of the glancing interaction between a shock wave and a turbulent boundary layer . an experimental study has been made at mach numbers from 1.6 to 2.0 of the interaction between the turbulent boundary layer on a side wall of a wind tunnel and the shock wave produced by a plate mounted on the wall . under these conditions the shock wave boundary layer interaction was three dimensional at least over the region investigated (up to 10 boundary layer thicknesses from the plate) . it was found that the boundary layer was separated by a shock wave of strength type occur on the sides of fuselages at the wing fuselage junction and may therefore be important with regard to the design of waisted shapes .
on turbulen flow between parallel plates . the reynolds equations of motion of turbulent flow of incompressible fluid have been studied for turbulent flow between parallel plates . the number of these equations is finally reduced to two . one of these consists of mean velocity and correlation between transverse and longitudinal turbulent-velocity fluctuations only . the other consists of the mean pressure and transverse turbulent-velocity intensity . some conclusions about the mean pressure distribution and turbulent fluctuations are drawn . these equations are applied to two special cases .. one is poiseuille flow in which both plates are at rest and the other is couette flow in which one plate is at rest and the other is moving with constant velocity . the mean velocity distribution and the correlation can be expressed in a form of polynomial of the co-ordinate in the direction perpendicular to the plates, with the ratio of shearing stress on the plate to that of the corresponding laminar flow of the same maximum velocity as a parameter . these expressions hold true all the way across the plates, i.e., both the turbulent region and viscous layer including the laminar sublayer . these expressions for poiseuille flow have been checked with experimental data of laufer fairly well . it also shows that the logarithmic mean velocity distribution is not a rigorous solution of reynolds equations .
the effect of turbulence on slider bearing lubrication . based on prandtl’s mixing-length mechanism, the pressure equation for turbulent flow in slider-bearing lubrication is derived . an analytical solution is given and compared with the one for laminar flow . it is found that the turbulent effect increases the pressure and consequently, the load-carrying capacity . however, the power loss also increases .
second order theory for unsteady supersonic flow past slender pointed bodies of revolution . the thermodynamic properties (z = pv rt, e rt, h rt, s r, and pressure) are given for equilibrium mixtures of dissociated and ionized molecules and atoms of the elements nitrogen and oxygen having the low temperature composition of .78847 n and .21153 o . the tabulated properties of this mixture (a close approximation to the properties of air) are given at close intervals from 2000 to and 10 times the normal density . the results are based on chemical equilibria between the species o, o, n, n, no, no, no, no, o, o, o, o, n, n, n and electrons . the method of presentation permits later corrections for the effect of argon and co and the contribution of intermolecular forces . the calculations are based on 9.758 e.v. as the dissociation energy of molecular nitrogen and 1.45 e.v. as the electron affinity of atomic oxygen .
a critical review of skin friction and heat transfer solutions of the laminar boundary layer of a flat plate . a review is made of existing literature concerned with the analytical investigation of the velocity and temperature distributions in the boundary layers of a heated (or cooled) flat plate . the plate is postulated infinitely thin and is parallel to a uniform fluid stream . the more recent solutions include the combined effects of frictional dissipation and variable fluid properties . only the results pertaining to the transfer phenomena occurring at the plate surface are included, i.e., skin drag and over-all heat transfer,. the individual temperature and velocity distributions leading to these results are omitted .
experiments on axi-symmetric boundary layers along a long cylinder in incompressible flow . experiments on axi-symmetric boundary layers along a long cylinder were made especially to investigate the effect of transverse curvature on the velocity profile . laminar velocity profiles were measured and compared with theoretical ones with good accuracy . a representative profile was plotted to see the effect of transverse curvature, which showed small, but obvious effect accompanied by increasing skin friction . the transition of the flow from laminar to turbulent was observed, and its reynolds number was estimated to occur at 1.2 1.8×10 in the present experiment . the turbulent profile was also measured and plotted by using the coordinates to express the wall law deduced by richmond, from which it was estimated that, as the ratio of the momentum thickness to body radius increases, the profile near the outer layer tends to bend down relative to the line of logarithmic wall law .
the formation of a blast wave by a very intense explosion . this paper was written early in 1941 and circulated to the civil defence research committee of the ministry of home security in june of that year . the present writer had been told that it might be possible to produce a bomb in which a very large amount of energy would be released by nuclear fission–the name atomic bomb had not then been used–and the work here described represents his first attempt to form an idea of what mechanical effects might be expected if such an explosion could occur . in the then common explosive bomb mechanical effects were produced by the sudden generation of a large amount of gas at a high temperature in a confined space . the practical question which required an answer was .. would similar effects be produced if energy could be released in a highly concentrated form unaccompanied by the generation of gas .qm this paper has now been declassified, and though it has been superseded by more complete calculations, it seems appropriate to publish it as it was first written, without alteration, except for the omission of a few lines, the addition of this summary, and a comparison with some more recent experimental work, so that the writings of later workers in this field may be appreciated . an ideal problem is here discussed . a finite amount of energy is suddenly released in an infinitely concentrated form . the motion and pressure of the surrounding air is calculated . it is found that a spherical shock wave is propagated outwards whose radius r is related to the time t since the explosion started by the equation where is the atmospheric density, e is the energy released and s a calculated function of, the ratio of the specific heats of air . the effect of the explosion is to force most of the air within the shock front into a thin shell just inside that front . as the front expands, the maximum pressure decreases till, at about 10 atm., the analysis ceases to be accurate . at 20 atm. 45 of the energy has been degraded into heat which is not available for doing work and used up in expanding against atmospheric pressure . this leads to the prediction that an atomic bomb would be only half as efficient, as a blast-producer, as a high explosive releasing the same amount of energy . in the ideal problem the maximum pressure is proportional to r, and comparison with the measured pressures near high explosives, in the range of radii where the two might be expected to be comparable, shows that these conclusions are borne out by experiment .
cylindrical shock waves produced by instantaneous energy release . taylor’s analysis of the intense spherical explosion has been extended to the cylindrical case . it is found that the radius r of a strong cylindrical shock wave produced by a sudden release of energy e per unit length grows with time t according to the equation where is the atmospheric density and is a calculated function of the specific heat ratio . for is found to be approximately unity . for this case, the pressure behind the shock wave decays with radius r according to the relation . applying the results of this analysis to the case of hypersonic flight, it can be shown that the shock envelope behind a meteor or a high-speed missile is approximately a paraboloid given by where d and v denote the total drag and the velocity of the missile, respectively, and x is the distance behind the missile .
asymptotic solution of the two dimensional oscillating aerofoil problem for high subsonic mach numbers . a new method has been given, for obtaining asymptotic solutions of a boundary value problem for the wave equation . the method is simpler than the method previously given by burger, and leads to a result identical with burger’s result .
some instabilities arising from the interaction between shock waves and boundary layer . a brief review is made of the available information concerning the flow fluctuations and instabilities arising from shock-induced separation in the flow over aerofoils and wings . the influence this phenomenon has on the oscillatory behaviour of aerofoils and control surfaces is also briefly discussed . a more detailed consideration is devoted to a recent investigation at the n.p.l. into the part played by shock-induced separation in the instability of a control surface .
exact solution of the neumann problem . calculation for non- circulatory plane and axially symmetric flows about or within arbitrary boundaries . an exact general method of solving the neumann or second boundary-value problem has been developed and has been applied to the calculation of low-speed flows about or within bodies of almost any shape, provided the flow is either plane or has axial symmetry . solid-body, inlet, and purely internal flow problems can be solved . the method is capable of dealing with several bodies at once in the presence of one another, and consequently interference problems can be treated with ease . boundaries need not be solid, that is, flows involving area suction can be calculated . velocities can be computed not only for points on the surface of the body but for the entire flow field . a surface source distribution is used as a basis for solution . this leads to a fredholm integral equation of the second kind, which is solved as a set of linear algebraic equations, usually by a modified seidel method . at the present time the solution is programed on the ibm 704 edpm to solve the flow about any body that has the previously mentioned characteristics and whose profile can be defined satisfactorily by no more than 300 coordinate points . a number of solutions are presented, to show both the scope of the method and its accuracy . computations require from three minutes to two hours, depending upon the shape of the body and the number of points used to define it .
steady and transient free convection of an electrically conducting fluid from a vertical plate in the presence of a magnetic field . an analysis is made for the laminar free convection and heat transfer of a viscous electrically conducting fluid from a hot vertical plate in the case when the induced field is negligible compared to the imposed magnetic field . it is found that similar solutions for velocity and temperature exist when the imposed magnetic field (acting perpendicular to the plate) varies inversely as the fourth root of the distance from the lowest end of the plate . explicit expressions for velocity, temperature, boundary layer thickness and nusselt number are obtained and the effect of a magnetic field on them is studied . it is found that the effect of the magnetic field is to decrease the rate of heat transfer from the wall . in the second part, the method of characteristics is employed to obtain solutions of the time-dependent hydromagnetic free convection equations (hyperbolic) of momentum and energy put into integral form . the results yield the time required for the steady flow to be established, and the effect of the magnetic field on this time is studied .
several magnetohydrodynamic free-convection solutions . the influence of transverse magnetic fields on the laminar free-convection flow of liquid metals over a vertical flat plate and between vertical parallel plates is examined for specific wall temperature variations and prandtl numbers . the extent of influence on the flow and temperature fields is determined by the magnitude of a nondimensional influence parameter which is the ratio of the magnetic force to the buoyant force . in general, increasing the magnetic field strength decreases the magnitude of the velocity, wall shear, and surfaces heat transfer and increases the temperature throughout the fluid . analytical results demonstrate that magnetic fields of practical strengths exert considerable influence on liquid metal free-convection flow fields .
on a laminar free-convection flow and heat transfer of electrically conducting fluid on a vertical flat plate in the presence of a transverse magnetic field . the free-convection flow and heat transfer of an electrically conducting fluid on a vertical plate in the presence of a transverse magnetic field is analysed for a magnetic field fixed to the electrically non-conducting wall . the boundary layer equations for self-preserving flows are integrated numerically for the prandtl number of unity, and the effect of the transverse magnetic field on the velocity profile, temperature profile and rate of heat transfer is discussed . it is concluded that the heat transfer rate is reduced as the magnetic field intensity is increased .
on combined free and forced convection laminar magnetohydrodynamic flow and heat transfer in channels with transverse magnetic field . combined free and forced convective heat transfer in vertical channels has been studied by many researchers . due to the need for engineering design information there have been many papers concerning cases of fully developed flow with varying wall temperature . forced flows in a channel of electrically conducting fluid with a transverse magnetic field have been studied and the large effects of a magnetic field on the flow pattern have been established . flows of combined free and forced convection in electrically conducting fluids in vertical channels with a transverse magnetic field are expected to attract attention in future engineering applications, for example, in a magneto-hydrodynamic generator or in plasma studies . however, except for a report by gershuni and zhukhovitskii (1) concerning a particular case, no general study has been published . this paper is a general treatment of fully developed, free and forced convective, laminar, magneto-hydrodynamic flow in a vertical channel with a transverse magnetic field . it includes combined free and forced convective flows in channels without a magnetic field reported by ostrach (2), tao (3), etc. as special cases . hartmann flow (4) is included in the other limit .
an experimental test of compressibility transformation for turbulent boundary layer . discussion of various turbulent-boundary-layer theories, in the light of experimental measurements by matting and co-workers . the application of (1) the mager insulated-wall transformation, and and illustrated graphically .
oscillatory aerodynamic coefficients for a unified supersonic hypersonic strip theory . the shock tube is shown to be a feasible research tool for conducting boundary-layer transition experiments . the use of the shock tube permits the study of transition with highly cooled boundary layers, as may be encountered on hypersonic vehicles . boundary-layer transition investigations have been made on optically polished pyrex hemisphere-cylinder and ellipse-cylinder models with stagnation-to-wall enthalpy ratios between 4.5 and roughness estimated to be less than 1 microinch (rms) . transition was detected by measurements of the heat-transfer rates on the model surface . the shock tube experiments indicated that a characteristic feature of transition of a highly cooled boundary layer on a hemisphere was the simultaneous occurrence of transition over the entire supersonic portion of the hemisphere . this implies that transition first occurred in the sonic region . the transition reynolds number (based on local fluid properties at the outer edge of the boundary layer and the momentum thickness) in the sonic region increased from about 225 to 325 as the stagnation- to-wall enthalpy ratio increased from about 9.5 to 29.5 . transition occurred along the cylindrical portion of the hemisphere-cylinder model at a nearly constant momentum thickness reynolds number, increasing from about 400 to 625 as the stagnation- to-wall enthalpy ratio increased from about 9.5 to 29.5 . the highly cooled boundary layers obtained on the cylindrical portion of the shock tube hemisphere-cylinder model provided an extension of nasa transition results obtained on a cooled hemisphere-cone-cylinder model in a wind tunnel . the transition reynolds numbers obtained from these shock tube data were of the same order of magnitude as the minimum transition reynolds numbers obtained in the wind-tunnel experiments . the results indicate that, for practical purposes, boundary-layer cooling is not a critical transition parameter for blunt bodies with a highly cooled boundary layer resulting from a stagnation- to-wall enthalpy ratio of about 3 to 30 . that is, the transition reynolds number did not vary significantly with boundary- layer cooling in this cooling range, but transition always occurred at a low reynolds number (between about 350,000 and 750,000 based on local external properties and a distance along the body surface from the stagnation point) . the boundary-layer history (body shape history) appeared to be an important parameter affecting the magnitude of the reynolds number for transition and the amount of increase in the transition reynolds number with increased boundary-layer cooling . that is, transition occurred at a lower reynolds number on the ellipse-cylinder configuration than on the hemisphere-cylinder . also, the increase in transition reynolds number with an increase in boundary-layer cooling was even less significant for the ellipse-cylinder than the hemisphere-cylinder .
flow past slender blunt bodies – a review and extension . a numerical solution of the inviscid flow field about slender blunt bodies of revolution has been developed through a combination of two methods .. the van dyke solution in the subsonic flow region at the nose, and the method of characteristics in the supersonic region . the results are compared with second-order blast wave theory and with experimental data,. and the respective merits and deficiencies of the two theoretical methods are pointed out . the results of the numerical solution are further used in a discussion of the entropy layer, to propose a possible criterion of entropy layer thickness .
analysis of quartz and teflon shields for a particular re-entry mission . the transient performance of ablation type heat protection shields is treated herein for the surface of a vehicle returning from outer space to the earth . the vehicle weighs 8640 kg, has a ballistic factor of 500 lb ft, re-enters with a speed of 11 km sec at ratio of 0.5, and is subjected to a maximum deceleration of 7.7 times the gravity constant . by use of well known equations for the heat transfer and the mass transfer at a heated surface, a numerical calculation method is derived which, for the investigated ablation processes, yields exact transient solutions of the fundamental system of partial differential equations . the method is applied to various quartz shields and to one teflon shield, which all evaporate so readily under the conditions of the problem at hand that practically no flow of molten shield material exists . the solutions also show comparatively small temperature changes parallel to the surface . the results show that the nose of the vehicle is cooled predominantly by the evaporation of the quartz or the teflon,. the rest of the vehicle’s surface is cooled by radiation of the quartz or evaporation of the teflon . the large mass transfer effects on the nose of the vehicle are detrimental since the resulting low surface temperatures prevent the radiative heat transfer out of the shield, which does not involve any mass loss, from being the desirable governing cooling factor .
the effect of lift on entry corridor depth and guidance requirements for the return lunar flight . corridors for manned vehicles are defined consistent with requirements for avoiding radiation exposure and for limiting values of peak deceleration . use of lift increases the depth of the entry corridor . mid-course guidance requirements appear to be critical only for the flight-path angle . increasing the energy of the transfer orbit increases the required guidance accuracy for the flight-path angle . corrective thrust applied essentially parallel to the local horizontal produces the maximum change in perigee altitude for a given increment of velocity . energy required to effect a given change in perigee altitude varies inversely with range measured from the center of the earth .
reaction tests of turbine nozzles for supersonic velocities . a machine for testing turbine nozzles by the reaction method, which was described in a previous paper, was used to test a series of convergent-divergent turbine nozzles . the results of these tests, along with the test of a convergent turbine nozzle, are compared with each other and with analytical values . two kinds of analytical values are employed, namely, the usual values obtained from an assumed isentropic expansion from inlet state to exhaust pressure, and the values obtained from the assumption that the processes in the nozzle are isentropic except for a normal shock which takes up a position in the nozzle such as to cause the stream to fill the exit area at the exhaust pressure whenever possible . this latter kind of analytical value involves no shock when the exit area can be filled at the exhaust pressure by means of isentropic processes only, or when the exhaust pressure is lowered so far that the shock has passed out of the passage . the agreement of the test results with the calculated results of this latter kind is good, and the disagreement which exists can be attributed largely to separation at the shock and to transmission of exhaust-pressure effects upstream through the boundary layer .
study of flow conditions and deflection angle at exit of two-dimensional cascade of turbine rotor blades at critical and supercritical pressure ratios . an analysis was made of the flow conditions downstream of a cascade of turbine rotor blades at critical and supercritical pressure ratios . the results of five theoretical methods for determining the deflection angle are compared with those of an experimental method using the conservation-of-momentum principle and static-pressure surveys, and also are compared with an analysis of schlieren photographs of the flow downstream of the blades . a two- dimensional cascade of six blades with an axial width of 1.80 inches was used for the static-pressure surveys and for some of the schlieren photographs . in order to determine the flow conditions several blade chords downstream of the cascade, schlieren photographs were taken of the flow through a cascade of 18 blades having an axial width of 0.60 inch . for the blade design studied, even at static-to-total pressure ratios considerably lower than that required to give critical velocity at the throat section, the flow was deflected in the tangential direction as predicted for the incompressible case . as the pressure ratio was lowered further, the aerodynamic loading of the rear portion of the blade reached a maximum value and remained constant . after this condition was attained, the expansion downstream of the cascade took place with a constant tangential velocity so that no further increase in the amount of turning across the blade row and no further increase in the loading of the blade was available .
on source and vortex distributions in the linearised theory of steady supersonic flow . the hyperbolic character of the differential equation satisfied by the velocity potential in linearized supersonic flow entails the presence of fractional infinities in the fundamental solutions of the equation . difficulties arising from this fact can be overcome by the introduction of hadamard’s finite part of an infinite integral . together with the definition of certain counterparts of the familiar vector operators this leads to a natural development of the analogy between incompressible flow and linearized supersonic flow . in particular, formulae are derived for the field of flow due to an arbitrary distribution of supersonic sources and vortices . applications to aerofoil theory, including the calculation of the downwash in the wake of an aerofoil, are given in a separate report (ref. 9) .
supersonic drag calculations for a cylindrical shell wing of semicircular cross section combined with a central body of revolution . a semi-circular ring wing with a body of revolution on the axis is studied to find the wave and the vortex drag for various chordwise lift distributions and for three values of a parameter describing the wing geometry . using the wave drag obtained from the chordwise loading that gives the least drag, together with the vortex and skin friction drags, the maximum lift to drag ratio for each wing geometry is computed . compared to the estimates made by lomax and heaslet, somewhat lower drags are found .
the surface oil flow technique as used in high speed wind tunnels in the united kingdom . an examination has been made of the various versions of the surface oil flow technique used in different high speed wind tunnels . to provide background information for this investigation some systematic tests were made on a simple model in a small supersonic tunnel . the experience gained made it possible to explain many of the variations in terms of the different operating conditions of the tunnels . the time taken to form a pattern on a typical model is, to a first approximation, directly proportional to the value of the parameter, the factor being 36,000 12,000 . the time taken appears to be independent of the initial thickness of the oil sheet . a general procedure for the development of oil mixtures for any purpose is suggested .
higher order approximations for relaxation oscillations . the problem of solving asymptotic developments for all quantities involved in relaxation oscillations has been solved by haag . this paper indicates how one can carry out such developments in a case which is simple enough to be treated explicitly .
jet effects on base pressure of conical afterbodies at mach 1. 91 and 3. 12 . data are presented which show the effect of a jet on base pressure for a series of conical afterbody-jet-nozzle combinations having boat-tail angles that varied from 0 to 11 and base-to-jet diameter ratios that varied from 1.11 to 2.67 . the jet nozzles had exit angles from 0 to 20 and were designed for exit mach numbers from 1.0 to 3.2 . pressure ratios up to 30 were tested for both a cold (air) and a hot numbers of 1.91 and 3.12 . in general, base pressure increased for increasing values of boat-tail angle, nozzle angle, jet temperature, and jet total pressure and for decreasing values of base-to-jet diameter ratio, jet mach number, and free-stream mach number . the addition of tail surfaces produced only small changes in base pressure . for all variables, base pressure is governed by the maximum pressure rise that can be supported by the wake fluid in the region of the trailing shock . the wake pressure ratio is in turn governed by the jet and free-stream mach numbers adjacent to the wake region and by the state of the boundary layer on the boattail and on the nozzle . values of wake pressure ratio computed using the theory of korst, page, and childs were in good agreement with experimental values for convergent nozzles .
laminar heat transfer around blunt bodies in dissociated air . a method of predicting laminar heat-transfer rates to blunt, highly cooled bodies with constant wall temperature in dissociated air flow is developed . attention is restricted to the case of axisymmetric bodies at zero incidence, although two-dimensional bodies could be treated the same way . the method is based on the use of the /local similarity/ concept and an extension of the ideas used by fay and riddell . a simple formula is given for predicting the ratio of local heat-transfer rate to stagnation-point rate . it depends on wall conditions and pressure distribution, but not on the thermodynamic or transport properties of the hot external flow, except at the stagnation point . experimental heat-transfer rates obtained with correct stagnation-point simulation and high wall cooling in shock tubes are also presented and compared with the theoretical predictions . on the whole, the agreement is good, although in regions of rapidly varying pressure there is evidence that the local similarity assumption breaks down, and the theory underestimates the actual heat-transfer rate by up to 25 per cent .
the divergence of supersonic wings including chordwise bending . the static aeroelastic stability or divergence problem is investigated for thin supersonic wings when not only the spanwise bending and twist are taken into account but also the chordwise bending . the problem is treated in successive phases of increasing complexity from the two-dimensional curling-up of the leading edge to the three-dimensional stability of the cantilever wing . several methods of approach are developed including the nonlinear aspects of the structure and the aerodynamics . results indicate a strong dependence of stability on poisson’s ratio and the magnitude of the deformation .
on the flutter of panels at high mach numbers . there have recently arisen some questions as to the possibility of panel flutter at high dynamic pressures and mach numbers . in addition, some doubts have been raised about the convergence of the galerkin method when applied to such problems . this note is intended to shed light on these matters .
effect of roll on dynamic instability of symmetric missiles . this note attempts to extend the discussion by stating a slightly neater form of generalized stability conditions and describing certain experimental results on dynamic instability .
some theoretical low-speed loading characteristics of swept wings in roll and sideslip . the weissinger method for determining additional span loading for incompressible flow is used to find the damping in roll, the lateral center of pressure of the rolling load, and the span loading coefficients caused by rolling for wing plan forms of various aspect ratios, taper ratios, and sweep angles . in addition, the applicability of the method to the determination of certain other aerodynamic derivatives is investigated, and corrections for the first-order effects of compressibility are indicated . the agreement obtained between experimentally and theoretically determined values for the aerodynamic coefficients indicates that the method of weissinger is well suited to the calculation of the additional span loading caused by rolling and for the calculation of such resulting aerodynamic derivatives of wings as do not involve considerations of tip suction .
the rolling up of the trailing vortex sheet and its effect on the downwash behind wings . the motion of the trailing vortices associated with a lifting wing is investigated by theoretical and visual-flow methods for the purpose of determining the proper vortex distribution to be used for downwash calculations . both subsonic and supersonic speeds are considered in the analysis . it is found that the degree to which the vortices are rolled up depends upon the distance behind the wing and upon the lift coefficient, span loading, and aspect ratio of the wing . while the rolling up of the trailing vortices associated with high aspect-ratio wings is of little practical importance, it is shown that, with low-aspect-ratio wings, the trailing vortex sheet may become essentially rolled up into two trailing vortex cores within a chord length of the trailing edge . the downwash fields associated with the two limiting cases of the flat vortex sheet and the fully rolled-up vortices are investigated in detail for both subsonic and supersonic speeds . the intermediate case in which the rolling-up process is only partially completed at the tail position is also discussed .
a theoretical study of the aerodynamics of slender cruciform-wing arrangements and their wakes . a theoretical study is made of some cruciform-wing arrangements and their wakes by means of slender-body theory . the basic ideas of this theory are reviewed and equations are developed for the pressures, loadings, and forces on slender cruciform wings and wing-body combinations . the rolling-up of the vortex sheet behind a slender cruciform wing is considered at length and a numerical analysis is carried out using 40 vortices to calculate the wake shape at various distances behind an equal-span cruciform wing at 45 bank . analytical expressions are developed for the corresponding positions of the rolled-up vortex sheets using a 4-vortex approximation to the wake, and these positions are compared with the positions of the centroids of vorticity resulting from the numerical analysis . the agreement is found to be remarkably good at all distances behind the wing . photographs of the wake as observed in a water tank are presented for various distances behind a cruciform wing at 0 and 45 bank . for 45 bank, the distance behind the wing at which the upper two vortices pass between the lower two is measured experimentally and is found to agree well with the the calculation of loads on cruciform tails is considered in some detail by the method of reverse flow, and equations are developed for the tail loads in terms of the vortex positions calculated in the earlier analyses .
dynamic stability of a missile in rolling flight . the paper sets down the equations of motion for a symmetric rolling missile with respect to axes attached to the missile . the missile may be jet (or rocket) propelled or coasting under accelerating or decelerating conditions, respectively, wherein the variable rolling velocity is derived from intentionally or unintentionally /canted/ fins and or wings . the equations contain a force and moment system that includes, in addition to the usual forces and moments, those due to magnus effects, misaligned surfaces, canted surfaces, jet misalignment, and the linear accelerations in the plane normal to the missile axis . the results present general stability criteria for a rolling missile which are summarized in the /discussion of stability ./
sweepback effects in the turbulent boundary-layer shock-wave interaction . experiments are reported on the interaction of turbulent boundary layers and shock waves with sweptback configurations . they show that the peak pressure rise at separation, the upstream influence ahead of separation, and the pressure rise at reattachment for moderate sweep angles can all be understood by simple extensions of available two-dimensional theories .
rapid laminar boundary layer calculations by piece-wise application of similar solutions . a method is presented for the rapid calculation of the incompressible laminar boundary layer in an arbitrary flow around either a two-dimensional or a rotationally-symmetrical body . the solution is obtained without recourse to von karman’s momentum equation by means of a coarse step-by-step procedure in which each segment of the velocity distribution is approximated by one of the falkner-skan family of similar flows . solutions have at least as much accuracy as those of any other one-parameter approximate method, and in certain cases the solutions become exact . in regions of accelerating velocity, the accuracy appears to be very high . in decelerating flows, separation is predicted somewhat early compared with exact solutions that is, the method is conservative in contrast to the von karman-pohlhausen procedure which sometimes fails to predict separation that actually exists . the method is the most rapid hand procedure known to the author, provided the full history of the boundary layer is required . if only a thickness such as is needed at one point on a surface, then it is about equal in speed to the quadrature method . but, if several values of or other properties along a surface are required, it is appreciably faster than the quadrature method . characteristically, only four steps are needed between the forward stagnation point and the pressure peak . once the velocity-distribution data are available, each step in a two-dimensional calculation requires about 5 minutes, using a slide rule .
recent studies on the effect of cooling on boundary layer transition at mach 4. the advent of high-speed flight has necessitated the study of boundary-layer transition on highly cooled bodies . investigations such as those of references 1-4 have concentrated on this problem and have indicated, contrary to the trends predicted by small-disturbance theory, that premature transition can be found with cooling . this phenomenon, commonly called detail in references 2-5 . the purpose of this note is to report some recent transition data obtained on a cooled cone in a mach 4 wind tunnel . the model, a sharp-tip cone (included angle 13.5), was cooled by liquid nitrogen to a temperature of -340 f . the cooling method and the data analysis are similar to that described in reference 3 .
an investigation of laminar transitional and turbulent heat transfer on blunt-nosed bodies in hypersonic flow . laminar, transitional, and turbulent heating rates have been measured by means of the shrouded model technique . the reynolds number was varied over a ninefold range,. the enthalpy ratio (stagnation to wall) varied from 2.3 to approximately 1.5 . two different pressure distributions were imposed on the model which consisted of a spherically capped cone . the experimental data are compared to the laminar hypersonic boundary-layer theory and shown to be in good agreement on the conical portion of the model . on the spherical portion the data are approximately 20 per cent higher than the theoretical prediction . some of this discrepancy can be attributed to radiation to the nose of the model . the fully developed turbulent heat-transfer data are compared to two theories .. (1) a relatively simple turbulent theory which is based on recent theoretical work and which takes into account the upstream history of the boundary layer, and (2) the flat-plate reference-enthalpy theory, which depends on only /local/ conditions . although both theories are in reasonable agreement with the data, the latter method is simpler and somewhat more accurate . for transitional flow the theory mentioned first can be readily modified in order to permit reasonable estimates of transitional heat transfer to be obtained . on this basis it is possible to estimate laminar, transitional, and fully developed turbulent heat transfer under hypersonic blunt-body conditions . the behavior of transition reynolds number based on momentum thickness is also discussed and shown to be in quantitative agreement with recent shock-tube measurements .
a note on transitional heat transfer under hypersonic conditions . in references 1 and 2 there were presented experimental data on transitional heat transfer on a blunt body under hypersonic-flow conditions obtained by the shroud technique . the data were compared with a theoretical prediction of transitional heat transfer based on a suggestion of persh . the agreement between theory and experiment in the transitional region was found to be ‘qualitatively good and quantitatively fair’ . it is the purpose of this note to present some additional transitional data obtained in conventional wind-tunnel tests and to indicate a means for improving somewhat the agreement between transitional theory and experiment .
notes on waves through gases at pressures small compared with the magnetic pressure, with applications to upper atmosphere aerodynamics . most treatments of magnetohydrodynamic waves have confined physical interpretation to cases when the alfven velocity a is small compared with the sound velocity a . here we consider the ‘low-beta situation’, in which a is much larger than a . then, except for two modes with wave velocity a the only possible waves are longitudinal ones, propagated unidirectionally along lines of magnetic force with velocity a . these can be interpreted as sound waves, confined to effectively rigid magnetic tubes of force . hall-current effects do not alter these conclusions (in contrast to the high-beta situation), and finite conductivity introduces only small dissipation . an application is made to the flow pattern around a body moving through the f layer of the ionosphere, where, although neutral particles have a very large mean free path, charged particles interact electrostatically and, it is argued, may be regarded as forming a continuous fluid whose movement is independent of that of the neutral particles . a body moving at satellite speed or below would then excite the above-mentioned unidirectional sound waves, but no waves at much faster alfven velocity . these considerations suggest that its movement would be accompanied by a v-shaped pattern of electron density (figure 2), which might be in part responsible for some anomalous radar echoes that have been reported .
compressibility effects in magneto-aerodynamic flows past thin bodies . the effects of compressibility on the steady motion of a highly conducting fluid past thin cylindrical bodies in the presence of a magnetic field are studied . procedures are developed for the solution of this class of magnetoaerodynamic problems over the entire mach number range and for all ratios of magnetic to fluid-dynamic pressure . the results obtained are analogous either to the ackeret theory or the prandtl-glauert rule of conventional aerodynamics, depending on the relative values of the flow speed and the appropriate speed of propagation of magnetoacoustic disturbances . the methods used and the physical interpretation of the solutions obtained vary according to the orientation of the magnetic field with respect to the flow direction . the results of the theory are explained in terms of the anisotropic propagation of magnetoacoustic pulses studied previously by several authors .
incompressible wedge flows of an electrically conducting viscous fluid in the presence of a magnetic field . the purpose of this note is to discuss the two-dimensional flow of an electrically conducting viscous fluid past a wedge in the presence of a magnetic field . the governing differential equations and boundary conditions are given and analyzed .
magnetohydrodynamic flow past a semi-infinite plate . the flow of viscous electrically conducting fluid past a semi-infinite plate is considered . the applied constant magnetic field and the constant on-coming velocity of the fluid are in the direction parallel to the plate . in addition to reynolds number the flow in the boundary layer depends on two parameters and . the two simultaneous ordinary nonlinear differential equations are solved by the asymptotic method for the cases when and respectively . the main results obtained are as follows . the equations can be solved exactly for and . the perturbation effect from infinity when k is large depends on, whereas the perturbation effect from zero when k is small depends on . for large k, including there is no solution for . it is assumed that the fluid is incompressible with constant physical properties .
on a particular class of similar solutions of the equations of motion and energy of a viscous fluid . by introducing the similarity concept to the two-dimensional, incompressible navier-stokes equations and energy equation, a particular class of solutions is found . two general types of flows are considered .. (1) laminar free convection–i.e., flows which take place due to a body force–and (2) laminar forced convection . for free convection on vertical plates, similar solutions are obtained for two different power-law surface temperature variations, and it is shown that one of these solutions constitutes a new type of boundary problem . results of numerical integrations of the equations are compared with solutions of the similar boundary-layer equations for free convection, and it is demonstrated that a range of surface temperature variations exists for which the boundary layer equations are no longer valid . for forced convection, it is shown that the use of similarity transformations provides an alternate method of deriving the ordinary differential equations for some well-known solutions, such as couette and stagnation point flows . solutions are obtained for radial converging or diverging flows between plane surfaces when the temperatures of the surfaces vary as arbitrary powers of the distance from the orgin . results of numerical integrations of the ordinary differential equations are presented for prandtl numbers of 0.01 and 1.0 and for linear surface temperature variations . some rather surprising results are obtained for diverging flows when separation occurs and some revealing comparisons with results from boundary-layer theory are made .
approximate design of sharp-cornered supersonic nozzles . a modified parabolic curve appears to be in close proximity to that obtained by either the method of characteristics or the wave method . thus an attempt has been made to use analytic geometry to determine approximately the contour of a two-dimensional, sharp-cornered supersonic nozzle in a very short time .
approximations for the thermodynamic and transport properties of high temperature air . the thermodynamic and transport properties of high-temperature air are found in closed form starting from approximate partition functions for the major components in air and neglecting all minor components . the compressibility, enthalpy, entropy, the specific heats, the speed of sound, the coefficients of viscosity and of thermal conductivity, and the prandtl numbers for air are tabulated from 500degree to 15,000degree k over a range of pressure from 0.0001 to 100 atmospheres . the energy of air and the mol fractions of the major components of air can be found from the tabulated values for compressibility and enthalpy . it is predicted that the prandtl number for fully ionized air, which is in complete equilibrium, will become small compared to unity, the order of transparent to heat flux .
effect of variable heat recombination on stagnation point heat transfer . earlier studies assume an average heat of formation of atoms based upon external flow conditions . it is shown that equilibrium heat transfer decreases by 35 for a typical mach number 24 case when allowance is made for the proportions of air components . the variable recombination energy also results in atom mass fractions which are realistically less for equilibrium than frozen situations throughout the cold-wall boundary layer .
first-order approach to a strong interaction problem in hypersonic flow over an insulated flat plate . the present paper concerns with the strong interaction phenomenon over an insulated semi-infinite flat plate with a sharp leading edge . in particular the main interest is in the consistent treatment in which the boundary-layer solution may be joined continuously with the inviscid solution regarding flow variables including pressure, normal velocity, temperature (or streamwise velocity) and density . it is shown that the behavior of the inviscid solution may be consistent with that of the boundary-layer solution to at least first-order approximation that is correct to the order of, where m is the mach number of undisturbed flow, r the reynolds number based on the distance from leading edge and the ratio of specific heats . then the first-order boundary-layer problem is formulated under such an external circumstance and an attempt is made for arriving at the solution . actual calculations are carried out for both cases of air and helium . from the solution it is found that the region in which the viscous effect plays a significant role is ranged over from 0 to a certain finite value of n, say n, in terms of the similarity coordinate n in the corresponding incompressible boundary layer . the numerical results moreover indicate that the induced pressure is considerably smaller than the estimate of lees (7) obtained by his approximate method in which the effect of the first-order induced pressure on the boundary layer is ignored and no survey of the first-order boundary-layer equation is made . the present results are also found to be in excellent agreement with experimental data recently obtained in helium flow by erickson (15) .
hypersonic strong viscous interaction on a flat plate with surface mass transfer . the present report gives an account of the development of an approximate theory to the problem of hypersonic strong viscous interaction on a flat plate with mass-transfer at the plate surface . the disturbance flow region is divided into inviscid and viscous flow regions . the hypersonic small perturbation theory is applied to the solution of the inviscid flow region . the method of similar solutions of compressible laminar boundary layer equations is applied to the treatment of the viscous flow region . the law of surface mass-transfer for similar solutions is derived . the pressure and the normal velocity are matched between the inviscid and viscous flow solutions . formulas for induced surface pressure, boundary layer thickness, skin friction coefficient, and heat transfer coefficient are obtained . numerical results and their significance are discussed . future improvements are indicated .
second approximation to laminar compressible boundary layer on flat plate in slip flow . the first-order solution for the laminar compressible boundary-layer flow over a flat plate at constant wall temperature is given . the effect of slip at the wall as well as the interaction between the boundary-layer flow and the outer stream flow are taken into consideration . the solution is obtained explicitly in terms of the known zero order, or continuum, solution . no assumptions regarding the prandtl number or viscosity-temperature law need be made . it is found that the first-order solution gives a decrease in heat transfer and, for supersonic flow, an increase in skin friction . for subsonic flow there is no first-order shear effect . the change in heat transfer is due to slip and the change in friction is due to the interaction of the zero- and first-order velocities at the outer edge of the boundary layer .
an approximate solution of hypersonic laminar boundary layer equations and its application . approximate formulae of the displacement thickness and the skin friction of the hypersonic laminar boundary layer are derived by use of von karman’s integral method, assuming the heat-insulated wall, the prandtl number of unity and chapman and rubesin’s formula for the variation of viscosity with temperature . the results obtained are compared with some exact solutions . because of the good agreement, it seems that these formulae are very useful . these formulae, together with the tangent-wedge-approximation, are applied to the viscous flow over slender bodies with a sufficiently sharp leading edge . as an example, the pressure distribution over a flat plate is calculated numerically over the entire region of the surface . comparison with other author’s theoretical results as well as experimental values is made .
on the hypersonic viscous flow past a flat plate with suction or injection . the hypersonic viscous flow past a flat plate with suction or injection is dealt with by karman-pohlhausen’s method in special cases when suction or injection velocity proportional to, especially for the region of strong interaction between the shock wave and the boundary layer, were p is the pressure on the plate and x is the distance measured along the plate from its leading edge . several numerical examples are given, which shows similar effects of injection to those in the case of incompressible flow that the injection makes all the height of the shock wave, the thickness of the boundary layer and the pressure on the plate larger than those in the case of no injection . on the contrary, in the case of suction no remarkable change both in the height of the shock wave and the pressure on the plate can be seen and only the velocity profile in the boundary layer is affected by the suction .
on the motion of a flat plate at high speed in a viscous compressible fluid, ii, steady motion . the theory of the steady flow of a viscous compressible fluid past a flat plate at high mach number due to lees and probstein is extended by a more complete discussion of the flow in the inviscid layer between the shock wave and the boundary layer . it is shown that similar solutions exist in this layer, analogously to those found by li and nagamatsu in the boundary layer, and that the two may be joined to give, allowing one minor assumption, a full account of the flow . it is shown that the boundary-layer equations may be reduced to those for an incompressible fluid and that the von karman-pohlhausen method describes the flow in it with good accuracy . the tangent wedge approximation for the pressure on the plate, used by lees and his collaborators, is found to be in deficit by 10 per cent for air . finally, it is shown that the theory for weak interaction cannot be extended further without a complete knowledge of the flow .
hypersonic viscous flow over a flat plate . in dealing with the steady laminar viscous flow over a semi-infinite flat plate some of the following topics are discussed . the streamline in the boundary layer over a leading edge of given thickness . the rate of growth of the boundary layer in the main stream, and causes of pressure variations . asymptotic solutions for thn downstream flow region, including the joining interaction of shock waves at the leading edge . pressure variations in the interanl viscous flow layer and in external inviscid flow considered as prandtl meyer flow . in cases of streamline deflection, the free stream mach number, zero pressure gradient, and surface pressure distribution . asymptotic solutions for cases of fluid injection of a cool gas . prandtl heat transfer . the joining interaction between the external inviscid flow and the internal viscous flow layer . steady laminar hpyersonic viscous flow over a flat wedge and a cone .
a method for predicting the onset of buffeting and other separation effects from wind tunnel tests on rigid models . the method is based on the observation of the divergence that occurs in the variation of mean static pressure at the trailing edge of an aircraft wing at the critical stage in the development of boundary-layer separation when its influence first spreads to the trailing edge and thereby to the overall flow . the significance of the trailing-edge pressure variations and their connection with the effects that separation has on the mean and unsteady loads is discussed for various types of separation . good prediction can be obtained from wind-tunnel tests, or warning provided in flight, for low-speed separations and for shock-induced ones up to the stage at which the shock wave reaches the trailing edge . related divergences in wake width, lift coefficient, or shock position can also be used . pressure measurements at other isolated points often indicate the type of separation . certain special considerations apply for swept wings . the various flow changes that are considered are illustrated by schlieren photographs and described in an appendix .
chordwise pressure distributions over several naca 16 series airfoils at transonic mach numbers up to 1.25 . a two-dimensional wind-tunnel investigation of the pressure distributions over several naca 16-series airfoils with thicknesses of and design lift coefficients of the langley airfoil test apparatus at transonic mach numbers from 0.7 to number from 2.4 x 10 to 2.8 x 10 and in angle of attack from -10 to and schlieren flow photographs are presented without analysis .
on alternative forms for the basic equations of transonic flow theory . attention has been called by numerous authors to the possibility of certain alternative forms for the equations for transonic flow about thin wings . it is the purpose of this note to contribute to this discussion and to indicate some reasons for the selection of one form of these in preference to another more widely used form .
simplified method for determination of the critical height of distributed roughness particles for boundary layer transition at mach numbers from 0 to 5. a simplified method has been devised for determination of the critical height of three-dimensional roughness particles required to promote premature transition of a laminar boundary layer on models of airplanes or airplane components in a wind tunnel with zero heat transfer . a single equation is derived which relates the roughness height to a reynolds number based on the roughness height and on local flow conditions at the height of the roughness, and charts are presented from which the critical roughness height can be easily obtained for mach numbers from 0 to 5 . a discussion of the use of these charts is presented with consideration of various model configurations . the method has been applied to various types of configurations in several wind-tunnel investigations conducted by the national advisory committee for aeronautics at mach numbers up to 4, and in all cases the calculated roughness height caused premature boundary-layer transition for the range of test conditions .
scale effects at high subsonic and transonic speeds and methods for fixing transition in model experiments . the major scale effects at high subsonic and transonic speeds arise from differences between the conditions under which laminar and turbulent boundary layers separate, and in how they behave after separation . for turbulent boundary layers, these conditions and behaviour do not vary greatly as the reynolds number is changed and in many examples, it has been shown that they are similar for the turbulent layers that occur naturally at high reynolds number and for boundary layers in which transition to turbulent flow is fixed artificially . the scale effects arising in wind-tunnel tests made at low reynolds number may, therefore, often be minimised by fixing transition to turbulent flow by introducing an artificial disturbance such as that produced by excrescences attached to the surface . the fact that the effects of separation are often less severe for laminar layers than for the turbulent layers that are likely to be encountered at full scale, makes it all the more important to do this whenever possible . several methods which can be used to fix transition are described, and the results obtained by using them are compared . in general, in experiments in two-dimensional flow, good agreement is found, and explanations can be advanced for cases in which discrepancies occur . several uncertainties and difficulties that arise in fixing transition are discussed and illustrated by examples . in particular, special care is needed in interpreting the results obtained with transition fixed at very low reynolds numbers (say, less than about r = 1 x 10 based on local chord for wings of about 0.1 thickness chord ratio and possibly higher reynolds numbers for thinner wings) . the difficulties of fixing transition satisfactorily are increased for three-dimensional wings, particularly if they are swept-back or highly tapered (i.e., small chord and reynolds number near the tip) and if the tests cover a large range of incidence including high incidences for which the flow may separate from very close to the leading edge . under these circumstances, it is frequently necessary to place the excrescences at different chordwise positions for low and high angles of incidence, and this is inconvenient in practice . more research is needed before sound recommendations can be made as to how and where transition should be fixed on such models, particularly since in routine testing, it is often not possible to check the effects of transition-fixing fully . in the sections dealing with three-dimensional tests, examples are given of the spurious results that have been avoided successfully by fixing transition, of the conditions where even at low reynolds numbers artificial fixing of transition may not be necessary to give a turbulent boundary layer ahead of the shock, and of the conditions under which there are some doubts whether the methods used for fixing transition have been satisfactory .
the occurrence and development of boundary layer separations at high incidences and high speeds . this note describes the manner in which the onset of the effects of boundary-layer separation varies with mach number for two-dimensional aerofoils, and discusses the influence of section shape as far as it is known . a brief qualitative description is given of the mechanism underlying the development of the separated flow and its effects, followed by a discussion of some of the ways in which this is likely to differ for swept-back wings at high speeds . finally, the need is emphasized for continued work in a broadening field .
non-equilibrium flow of an ideal dissociating gas . the theory of an’ideal dissociating’gas developed by lighthill/1957/for conditions of thermodynamic equilibrium is extended to non-equilibrium conditions by postulating a simple rate equation for the dissociation process/including the effects of recombination/ . this equation contains the’equilibrium’parameters of the lighthill theory plus a further dissociation phenomena . the behaviour of this gas is investigated in flow through a strong normal shock wave and past a bluff body . the assumption is made that the gas receives complete excitation of its rotational and vibrational degrees of freedom in an infinitesimally thin region according to the familiar rankine-hugoniot shock wave relations before dissociation begins . the variation of the relevant thermodynamic variables down-stream of this region is then computed in a few particular cases . the method used in the latter case is an extension of the’newtonian’theory of hypersonic inviscid flow . in particular, the case of a sphere is treated in some detail . the variation of the shock shape and the sphere diameter to the length scale of the dissociation process, is exhibited for conditions extending from completely undissociated flow to dissociated flow in thermal equilibrium . results would indicate that significant and observable changes from the undissociated values occur, although values for the non-equilibrium parameter are not, at present, available .
inviscid hypersonic flow past blunt bodies . two methods are shown for the calculation of the flow field between a blunt body and the shock associated with it for the case of hypersonic flow . real gas effects are included . the solutions consider only symmetric flows–that is, symmetric bodies at zero incidence . one method consists in tracing successive stream tubes around the body and leads to iterations on the initially assumed position of the shock . the second is an integral method closely analogous to the karman-pohlhausen procedure for boundary layers . a distinction is made between round-nosed and flat-nosed bodies, and both cases are discussed . a specific example corresponding to a re-entry missile situation is calculated,. the two methods agree within a few per cent . comparison is also made with other known solutions in the stagnation region .
propagation of weak disturbances in a gas subject to relaxation effects . a generalized wave equation is derived for sound disturbances in a gas when relaxation effects connected with, for example, molecular vibration or dissociation are important . solutions involving discontinuous wave fronts are presented, and it is shown that, under certain assumptions, the complete wave equation reduces to a variant of the telegraph equation . detailed solutions are presented for disturbance fields produced by a wavy wall in subsonic and supersonic flow and a simple wedge in supersonic flow . this study is viewed as a step in the development of a theory of small disturbances of a high-temperature gas, as is found behind the shock in hypersonic flight .
comment on improved numerical solution of the blasius problem with three-point boundary conditions . attention is drawn to a previous accurate solution to the problem .
improved numerical solution of the blasius problem with three-point boundary conditions . the blasius equation describes the velocity distribution resulting from laminar, constant-pressure mixing of a stationary fluid layer and a moving stream . in connection with a numerical procedure for the univac based on analytic continuation of the function f’ . high-speed computers now make it feasible to use analytic continuation for numerical integration of single-point boundary-value problems such that, within the limits of taylor’s expansion, truncation error may be made arbitrarily small . a brief description of the application of the routine is given .
on the numerical solution of the blasius problem with three-point boundary conditions . relates to a technique for approximate determination of the initial parameters . the technique is an application of the asymptotic integration method introduced by meksyn and has been applied to the computation of the skin friction for shock-generated boundary-layer flow .
vorticity interaction at an axisymmetric stagnation point in a viscous incompressible fluid . the purpose of the present note is to give an exact solution of the incompressible navier-stokes equations at an axisymmetric stagnation point with vorticity in the oncoming flow which varies linearly with distance from the axis . this solution has application to the hypersonic axisymmetric blunt body problem, for which lighthill has shown the vorticity in the inviscid shock layer is very nearly of this form .
vorticity effect on the stagnation point flow of a viscous incompressible fluid . the effect of vorticity on axisymmetric stagnation point boundary layer calculations is investigated by calculating a perturbation to the stagnation point flow . the shear caused by the vorticity effect is found to be surprisingly large,.the slope of the shear curve /at zero vorticity/ as calculated by kemp agrees perfectly with the value deduced in this note .
heat transfer to constant property laminar boundary layer flows with power function free stream velocity and wall temperature variation . numerical computations have been performed for the boundary-layer form of the energy equation for incompressible flows with power-function variation of free-stream velocity (u = cx) and of wall temperature (t = ax), the pertinent solutions of the momentum equation in this case being those of hartree . the numerical computations given herein are to some extent a repetition of those given by schuh and by chapman and rubesin, the object of the present computations being the resolution of discrepancies appearing in the previous solutions and an extension of their range . ibm machine calculations were employed in the finite difference calculation presently utilized, the results thereof covering a range of wall-temperature function exponents from values of m(4, 1, 0, -0.0904) . the accuracy of the numerical computations is examined in detail, and the accuracy of the computed functions at the wall, which determine the heat-transfer rate, is estimated to be within 2 per cent . examination of the results reveals that the results of schuh for the flat plate are in error . for the range of the calculations, it was found that the local heat-transfer coefficient can, with the exception of large negative values, be expressed within 5 per cent as where the exponent of the prandtl number varies from 0.254 to 0.367 for -0.0904 and where the function can be approximated by the equation
forst-order slip effects on the compressible laminar boundary layer over a slender body of revolution in axial flow . analysis of the compressible boundary layer with transverse curvature in first order slip flow . no boundary-layer interaction effects are considered and only the zero pressure-gradient case is examined .
on local flat plate similarity in the hypersonic boundary layer . a study is made of lees’ /local flat-plate similarity/ rule for the hypersonic laminar boundary layer . it is shown that this rule is exact under assumptions commonly invoked in the inviscid theory of hypersonic flow . beginning from this theoretical basis, a modified local flat-plate similarity scheme is derived, involving separate rules for velocity and enthalpy profiles, and is compared with exact similarity solutions and with the existing theory of hypersonic leading-edge interaction .
the boundary layer near the stagnation point in hypersonic flow past a sphere . flow properties behind shock waves caused by bluff bodies traveling at supersonic speeds are of major importance in missile and high-speed aircraft design . paper presents a mathematical solution for the laminar boundary layer near the stagnation point of a sphere . surface temperature is free-stream static and shock is strong . air is assumed calorically and thermally perfect with a prandtl number of 0.72 and a dynamic viscosity directly proportional to temperature . based on work of homann (zamm 16, p. 153, 1936) and lighthill simultaneous differential equations for the velocity and temperature profiles . these are solved by numerical integration along a normal to the surface using a digital computer . results are presented as functions of free-stream mach number, reynolds number, and specific heat ratio . as increases, boundary-layer thickness is shown to decrease while shock stand-off distance increases . stand-off distance also decreases with increasing and decreasing specific heat . for constant and specific heat ratio, the product of skin-friction coefficient and the square root of decreases with increasing only approaching a constant value at greater than 10,000 . reviewer’s comment is concerned with the perfect gas assumption for air . author suggests that the effects of dissociation on flow properties are accounted for by a proper choice of specific heat ratio . a consideration of the kinetics of chemical reaction in the cooled boundary layer emphasizes the oversimplification of this approach . the effect on transport properties could have been approximated in present analysis by changing the prandtl number to one more representative of the existing pressures and temperatures .
various aerodynamic characteristics in hypersonic rarefied gas flow . this paper considers the problem of calculating viscous aerodynamic characteristics of blunt bodies at hypersonic speeds and at sufficiently high altitudes where the appropriate mean free path becomes too large for the use of familiar boundary-layer theory but not so large that free molecule concepts apply . results of an order-of-magnitude analysis are presented to define the regimes of rarefied gas flow and the limits of continuum theory . based on theoretical and experimental evidence, the complete navier-stokes equations are used as a model, except /very close/ to the free molecule condition . this model may not necessarily give the shock wave structure in detail but satisfies overall conservation laws and should give a reasonably accurate picture of all mean aerodynamic quantities . in this /intermediate/ regime there are two fundamental classes of problems .. a /viscous layer/ class and a /merged layer/ class, the latter corresponding to a larger degree of rarefaction . for the viscous layer class there is a thin shock wave, but the shock layer region between the shock and the body is fully viscous, although the viscous stresses and conductive heat transfer are small at the shock wave boundary . here, the use of the navier-stokes equations with outer boundary conditions given by the hugoniot relations is justified . for the merged layer class, the shock wave is no longer thin, and the navier-stokes equations can be used to give a solution which includes the shock structure and has free-stream conditions as outer boundary conditions . a simpler procedure is presented for /incipient merged/ conditions where the shock may no longer be considered an infinitesimally thin discontinuity but where it has not thickened sufficiently to entail the /fully merged layer/ analysis . in this case we approximate the shock by a discontinuity obeying conservation laws which include curvature effects, viscous stresses, and heat conduction . for a sphere and cylinder it is shown that the navier-stokes equations can be reduced to ordinary differential equations for both the viscous and merged layer class of problems . solutions of these equations, when used in connection with hypersonic flow problems, are in general only valid in the stagnation region . to illustrate the viscous layer solutions, numerical calculations have been performed for a sphere and cylinder with the assumption of constant density in the shock layer, which is a useful approximation at hypersonic speeds . to illustrate the merged layer solution, calculations have been carried out for a sphere using the incipient merged layer approximation . results are presented for detachment distance, surface shear, and heat-transfer rate in the stagnation region of a highly cooled sphere flying at hypersonic speed . with decreasing reynolds number, the shear and heat transfer are shown to increase above the extrapolated boundary-layer values in the viscous layer regime and then to begin falling in the incipient merged regime . as the reynolds number decreases in the incipient merged regime, the density in the shock layer increases, and the static and stagnation enthalpy behind the shock decrease . calculations performed for an insulated sphere show that, with decreasing reynolds number in the incipient merged regime, the density in the shock layer decreases,. the total enthalpy behind the shock and at the stagnation point increase so that they are higher than the free-stream total enthalpy,. and the stagnation-point pressure behaves like the total enthalpy . for the highly cooled cylinder in the viscous layer regime, the same quantities are presented as for the sphere . the increase found in shear and heat transfer above extrapolated boundary-layer theory is small, in agreement with vorticity interaction theory . a discussion is given of the behavior of available experimental data for viscous flow quantities in the intermediate regime and the behavior predicted by the results of the present calculations . qualitative agreement is indicated .
taylor instability of finite surface waves . the instability of the accelerated interface between a liquid (methanol or carbon tetrachloride) and air has been investigated experimentally for approximate sinusoidal disturbances of wave-number range from well below to well above the cut-off . the growth rates are measured and compared with theoretical results . a third-order theory shows the phenomena of overstability which is found in the experimental results . some measurements of later stages of growth agree moderately well with the available theory and disclose some additional phenomena of bubble competition, helmholtz instability with transition to turbulence, and jet instability with production of drops .
effects of surface tension and viscosity on taylor instability . the model used is that of two fluids of infinite depth, with the interface initially in the form of a sine wave with amplitude small compared to wave length . the fluids are considered incompressible, and only the linear terms in the equations of hydrodynamics are used . the first four sections discuss the effects of surface tension and viscosity . the fifth gives a few numerical results to illustrate the main points of the preceding sections .
similitude of hypersonic real-gas flows over slender bodies with blunted noses . on the basis of the hypersonic small-perturbation theory, the laws of similitude for hypersonic inviscid flow fields over thin or slender bodies are examined, and the restrictions to ideal gases with constant specific heats and to bodies with pointed noses are removed . only steady plane or axisymmetric flows are considered . inspection of the governing system of equations shows that a similitude law exists for flow fields, under local thermal equilibrium, having the same free-stream atmosphere . for flows of ideal gas with constant specific heats, the requirement of the same free-stream atmosphere–i.e., the same composition, pressure, and density–can be replaced by the requirement of the same ratio of specific heats . for flows over blunted wedges or cones, special laws of similitude can be obtained . application of the similarity rules is examined for the case of hypersonic flows of an ideal gas with over flat plates with blunt leading edges, and for the case of equilibrium air flows over wedges . the possibility of simulating nonequilibrium flows over slender or thin bodies is also pointed out .
boundary-layer interaction on a yawed infinite wing in hypersonic flow . the equations are given for the laminar boundary-layer equations on a yawed infinite wing for constant wall temperature, under the combined howorth and mangler transformation . diagrams show the relatively small influence of yaw, the increase of boundary-layer secondary flow, and the variation of the local heat transfer rate with yaw .
influence of the leading-edge shock wave on the laminar boundary layer at hypersonic speeds . in order to bring out the importance of the leading-edge region at hypersonic speeds, the influence of the leading-edge shock wave on the laminar boundary layer is investigated in two simple cases of steady flow over a semi-infinite, insulated flat plate.. (1) sharp leading edge., (2) blunt leading edge, as approximated by a normal shock wave . the streamlines that enter the boundary layer over a large region of the plate surface has previously crossed the shock wave very near the leading-edge, where the shock is strong and highly curved . consequently, the temperature at the outer edge of the boundary layer is appreciably higher than free-stream temperature, and the vorticity there is not zero . the effects of this shock-wave larger than the usual /errors/ made in the boundary-layer theory, and an estimate of these effects can therefore be obtained within the framework of that theory . the numerical magnitude of the shock-wave influence is found to be appreciable . for the case of the blunt leading edge the slope of the curve of induced pressures plotted against the hypersonic interaction parameter closely approaches the experimental data of hammitt and bogdonoff obtained in helium at large values of this parameter . these approximate results show that the influence of the leading-edge region at hypersonic speeds requires careful theoretical and experimental study .
the interaction between boundary layer and shock waves in transonic flow . experiments of transonic flow past a circular arc profile show that the shock-wave pattern and the pressure distribution are strongly dependent upon the state of the boundary layer . a change from laminar to turbulent boundary layer at a given mach number changes the flow pattern considerably . shock waves can interact with the boundary layer in a manner similar to a reflection from a free jet boundary . these shock waves are not distinctly discernible from pressure distribution measurements .
simplified laminar boundary layer calculations for bodies of revolution and for yawed wings . since the introduction of momentum methods in boundary-layer calculations by von karman and pohlhausen, many improvements have been proposed . an especially simple solution reduces the problem to a quadrature . here, it is proposed to extend these methods to elementary three-dimensional cases and to compressible laminar boundary-layer calculations . for comparison, the corresponding problems for the turbulent boundary layer are also discussed briefly .
boundary layer transition with gas injection . the mass-injection process has been proposed as a method of cooling aerodynamic surfaces, and, since the amount of coolant required to maintain practical wall temperatures is considerably larger for turbulent than for laminar boundary layers, knowledge of the effect of the cooling method on the transition process is certainly important . exploratory studies reported here were conducted at mach number 3.7 to ascertain the effects of gas injection on the stability of the laminar boundary layer on a conical surface .
mass transfer cooling at mach number 4. 8. mass-transfer experiments on a 5 mil wire porous cone of 20 total angle have been conducted at using air and helium injection . details of the experimental technique are described in references 1 and 2 . in the laminar boundary layer the recovery factors and heat-transfer coefficients measured with zero injection agreed within per cent with theory . transition reynolds numbers observed on the porous cone with zero injection were half as large as observed on a smooth, impermeable model of identical geometry in the same channel, but injection of large amounts of air or helium did not cause transition to move forward from its zero-injection position on the porous cone . distributed roughness of this type apparently does not disturb impermeable wall theory, but it masks whatever effective roughness may be caused by discrete pore injection .
experimental evaluation of heat transfer with transpiration cooling in a turbulent boundary layer at m=3 .2. it is found that for prescribed velocity field, electrical field and conductivity, the current can be calculated by integration . work is related to analytic investigation of the boundary layer in a physically reasonable accelerator .
analysis of effects of diffusion of a foreign gas into the laminar boundary layer of a supersonic flow of air in a tube . adiabatic wall temperatures and recovery factors are calculated for pipe flows with an entrance mach number of 5 and with uniform injection of helium . predicted values of the recovery factor increase slowly with increasing injection rate and with increasing distance from the tube entrance .
the analytical design of an axially symmetric laval nozzle for a parallel and uniform jet . the equations for the nozzle’s contours are derived by integration of the characteristic equations of the axially symmetric flow . since it is not possible to integrate these equations mathematically in an exact form, it was necessary to find a way to approximate the calculations . the approximation offers itself by considering and comparing the conditions of the flow in a cone with those in a nozzle, as a linearization of the characteristic equations . the first part of the report deals with equations for the transition curve by which the conical source flow is converted into a parallel stream of uniform velocity . the equations are derived by integration along a mach line of the flow in the region where the conversion takes place . a factor f is introduced expressing a relation between the direction and the velocity of the flow along a certain mach line . f remains undetermined and is not involved in the final equations . in the second part of the report, the spherical sonic flow section is converted into a plane circular section of the throat . the nozzle’s contour adjacent to the throat is formed by the arc of a circle connected with the transition curve by a straight line . the gas dynamic properties of the boundary mach line are calculated in table 1, the use of which shortens the calculations considerably .
effect of diffusion fields on the laminar boundary layer . a theory is developed which describes the effect of a general diffusion field on the dynamic and thermal characteristics of a laminar boundary layer on a flat plate in steady compressible flow . fluid properties are considered as functions of temperature and local concentration of the foreign gas . the diffusion field is described by a differential equation that relates convective and diffusion transfer and which considers diffusion currents arising from gradients of concentration and temperature . by means of the usual transformations the system is reduced to a set of ordinary differential equations, which in turn are transformed into a set of integral equations . the latter is amenable to solution by the method of successive approximations . the theory and results have bearing on the problem of control and reduction of aerodynamic heating at hypersonic speeds . the special feature of this approach lies in the utilization of diffusion fields for the purpose of reducing the detrimental effects of viscous dissipation . although the theory is adapted to a fuller investigation of this problem, the numerical examples considered involve mainly diffusion fields of helium, with which good results have been achieved at mach numbers 8 and 12 . whereas at the higher mach number the influx of heat was practically eliminated, a reversal in the direction of heat flow has been effected at the lower mach number .
transpiration cooling experiments in a turbulent boundary layer at m=3 . turbulent recovery factor and heat-transfer measurements have been made on a porous flat wall section at a nominal mach number of 3.0 and a reynolds number of approximately 4 x 10 using both air and helium as the transpired gas . measured heat-transfer coefficients correlate well with the compressible theory of rubesin for air and qualitatively with simple film theory for either coolant, indicating that the heat transfer from a turbulent boundary layer can be reduced by transpiration cooling to well below that of the uncooled boundary layer at the same reynolds number .
some experimental techniques in mass transfer cooling . author introduces his survey by a brief review of the history of investigations dealing with boundary layers on impermeable solid surfaces, and notes that no true theory exists for turbulent boundary layers, the success of studies in this area having been due to the introduction of artificial, if ingenious, assumptions which permitted empirical correlations fd data . the terminology introduced by the author for distinguishing the different situations involving mass transfer from the wall to the stream may give rise to some objections . for instance, /film cooling/ need not refer only to the injection of a liquid, since applications involving gas film cooling exist . also, his restriction of the term /transpiration cooling/ to refer to the injection through a porous surface of a gas only of the same composition as the exterior stream does not enjoy universal usage . the influence of mass transfer on heat transfer through laminar boundary layers and on the transition from laminar to turbulent flow is described, with consideration given to the question of the net effect of the stabilizing influence of surface cooling and the destabilizing influence of injection . reviewer suggests that author’s inaccurate statement to the effect that /thus far the higher energy conditions do not threaten to involve turbulent injection, so turbulent boundary-layer research enjoys a fairly academic serenity broken only by its own frustrations/ be excused on grounds of poetic license, although it ignores the efforts being devoted to the pressing practical problems of erosive burning of solid propellants (possibly the most common example of a complete /aerothermochemical/ problem involving distributed surface heat and mass transfer with chemical reaction in a flow system) and of effusion cooling of rocket nozzles, both of which involve turbulent boundary-layer conditions . author emphasizes the tedious experimental problems involved in research on boundary layers with blowing, and notes the desirability of velocity distribution measurements, especially in turbulent injection layers . the observation that no good data on concentration profiles in the case of the diffusion boundary layer have been published may be an overstatement, since author’s bibliography overlooks the work of j. berger (/contribution a l’etude de l’injection parietale,/ doctor’s thesis, university of paris, memorial des poudres 38 (annex), p. 1,. paris, imprimerie nationale, 1956) .
the interaction of shock waves with boundary layer on a flat surface . the development of supersonic compressors, supersonic diffusers, and high-speed aircraft points to the increasing importance of the interaction between shock waves and boundary layers . the experimental work reported here is intended to (1) provide a better understanding of the nature of the shock boundary-layer interaction, (2) serve as a guide and stimulus to theoretical work, and (3) develop an empirical method for predicting the effects of the interaction . experiments were performed on the reflection of an oblique shock from a boundary layer on a flat surface at a mach number of 2.05 . the effects of shock strength and boundary-layer regime were explored . the results are in the form of schlieren photographs, constant-density contours found from interferometer photographs, and static pressure distributions at the plate surface .
measurements of turbulent friction on a smooth flat plate in supersonic . direct measurements of supersonic local skin friction, using the floating-element technique, are presented for mach numbers from bulent flow and transition are emphasized, although some measurements in the laminar regime are included . the observed effect of compressibility is to reduce the magnitude of turbulent skin friction by a factor of two at a mach number of 4.5 and a reynolds number of about 10 . the boundary-layer momentum-integral equation for constant pressure is verified within a few per cent by two experimental methods . typical static pressure measurements are presented to show that transition can be detected by observing disturbances in pressure associated with changes in displacement thickness of the boundary layer . it is found that the turbulent boundary layer cannot be defined experimentally for values of less than about 2,000, where is the momentum thickness . for larger values of there is a unique relationship between local friction coefficient and momentum-thickness reynolds number at a fixed mach number . the appendix compares the present measurements at m = 2.5 with experimental data from other sources .
boundary layer measurements in hypersonic flow . experimental data are presented on boundary-layer formation, heat transfer, and skin-friction coefficient at mach numbers of the wall of a conical nozzle in the presence of a favorable pressure gradient and several rates of heat transfer . the reynolds number based on momentum thickness varied from 1,500 to 3,500 . comparison is made with data at lower mach numbers and with the semiempirical theory of von karman . the existing data up to mach numbers of nine indicate agreement to within 5 per cent when compared with a form of the wilson theory, but it is clear that the effects of heat transfer and pressure gradients present problems which require extensive study and experiment in the future .
turbulent boundary layer in compressible fluids . the continuity, momentum, and energy differential equations for turbulent flow of a compressible fluid are derived, and the apparent turbulent stresses and dissipation function are identified . a general formula for skin friction, including heat transfer to a flat plate, is developed for a thin turbulent boundary layer in compressible fluids with zero pressure gradient . curves are presented giving skin-friction coefficients and heat-transfer coefficients for air for various wall-to-free-stream temperature ratios and free-stream mach numbers . in the special case when the boundary layer is insulated, this general formula yields skin-friction coefficients higher than those given by the von karman wall-property compressible-fluid formula but lower than those given by the von karman incompressible-fluid formula . heat transfer from the boundary layer to the plate generally increases the friction and heat-transfer coefficients .
numerical solution of the boundary layer equations without similarity assumptions . the crocco transformation combined with a mangler transformation is used to carry the boundary-layer problem for axially symmetric blunt bodies into a form suitable for direct numerical computation without introduction of similarity assumptions . conditions which in the original problem appear at infinity now are brought to a finite straight line, and the body is transformed to a parallel line . data can be generated on the stagnation line the equations are a parabolic system of two second-order equations, the boundary-value problem is analogous to the slab problem for the heat equation . an implicit difference equation is used to reduce stability difficulties . special techniques in forming the difference equation result in a linear system of algebraic equations to be solved on any given line of integration, and these solutions are computed from recursion relations generated by back substitution . for bluntnosed bodies with approach flow mach numbers greater than 8 (approximately), large temperature gradients occur across a thin boundary layer of dissociated gas, and it is necessary to use real-gas effects, approximated here by certain fits to the gas tables . a case is computed, however, for a lower mach number approach flow using perfect-gas theory to provide a standard against which similarity solutions may be tested .
laminar jet mixing of two compressible fluids with heat release . the laminar jet mixing problems with heat release have been formulated . a general discussion of the solution of these problems is also given . the important parameters of these problems are brought out . some specific cases of the jet mixing problem, such as jet mixing of one compressible fluid, isothermal jet mixing of two compressible fluids, and isovel jet mixing of two compressible fluids with heat release, are discussed in detail.
thermal distributions in jeffrey-hamel flows between nonparallel plane walls . the authors give the exact solution for the thermal distributions for the steady laminar flow of a viscous incompressible fluid between non-parallel plane walls held at a constant temperature . the velocity profiles are determined with the aid of jacobian elliptic functions by using the jeffery-hamel solution of the hydrodynamic problem . it is shown that in this special case the energy equation giving the temperature profiles can be reduced to an ordinary linear differential equation with variable coefficients . after the introduction of dimensionless parameters, numerical solutions are given for diverging and converging channels with total openings of 10degree for the possible combinations of three reynolds numbers and five prandtl numbers .
on heat transfer over a sweat-cooled surface in laminar compressible flow with a pressure gradient . a simple expression is derived for the normal injection velocity distribution theoretically required to maintain a given uniform temperature along a porous surface in the laminar boundary-layer region of a compressible flow with a given velocity distribution outside of the boundary layer . this expression is valid for any given free-stream mach number but is based on a prandtl number of unity and on the assumption that the viscosity coefficient varies linearly with the temperature . by using the dorodnitsyn type of transformation, the variation of fluid properties even in the case of zero mach number is taken into account . this study is of particular practical interest in connection with the sweat-cooling of turbine blades and of airfoil surfaces in high speed flow . the method of analysis consists of applying the karman-pohlhausen method to both the momentum and energy boundary-layer equations and of using an additional heat balance equation, involving the coolant temperature . a closed-form approximate solution of the equations is then derived . numerical examples for flow in the immediate vicinity of a stagnation point and for a typical type of flow over a turbine blade are given .
the effect of helium injection at an axially symmetric stagnation point . an effective means of protecting the surface of a hypersonic re-entry vehicle is to inject small quantities of a lightweight gas into the boundary layer through a porous wall . this process, which is known as mass-transfer cooling, protects the surface in two ways . first of all, as the injected gas or coolant passes from the reservoir through the wall to the surface, a considerable quantity of heat is absorbed as its temperature is raised from the reservoir temperature to the wall surface temperature . characteristically, lightweight gases have relatively high specific heats . secondly, the transfer of mass and enthalpy by convection and diffusion normal to the surface alters the characteristics of the boundary layer in such a manner as to reduce the temperature gradient at the wall, and, hence, the conductive heat transfer at the wall . this is sometimes referred to as the blowing effect .
laminar heat-transfer and pressure measurements over blunt-nosed cones at large angle of attack . tests have been conducted at a mach number of 6, in the pibal hypersonic facility, in order to determine the heat-transfer and pressure distributions over a slender blunted cone at angles of attack of erature ratio, stagnation to wall, was approximately 2.3 . the model tested has a sperical nose diameter of 1.0 in., a base diameter of 3.75 in., and a cone half-angle of 20 degrees . the measurements were made at 5 peripheral stations on the model . in this note the experimental results at a 15 degree angle of attack are presented . a more detailed analysis of the results for all angles of attack is presented in reference 1 .
the injection of air into the dissociated hypersonic laminar boundary layer . in first approximation, dissociated air may be treated as a binary mixture of air atoms and air molecules . in order to include the effects of mass transfer into the boundary layer, it becomes necessary to introduce a third chemical species and hence a second diffusion equation . we have avoided this complexity by considering the injection of air molecules into the boundary layer, and hence the theoretical treatment is accomplished within the framework of a binary mixture gas .
on optimum nose curves for missiles in the super-aerodynamic regime . author shows that the differential equations defining the minimum drag body shapes for free molecule flow that were developed and numerically integrated by w. j. carter (amr 11 (1958), rev. realized, however, that numerical or analytical integration of the second-order differential equation is unnecessary since, for the flow conditions considered, the first integral to the euler equation can be written prior to the substitution of the expression defining the pressure coefficient .
optimum nose shapes for missiles in the super-aerodynamic region . the mechanics of the kinetic theory of gases is employed to describe the drag force on the nose of a missile moving in the super-aerodynamic region of the atmosphere . three separate cases are considered–ideal specular reflection, specular-type reflection from a slightly rough surface, and surface absorption followed by random emission of the striking molecules . the calculus of variations is employed to obtain the differential equation of the nose shape which minimizes the drag force for each of the three cases . the resulting differential equations are then solved by a numerical procedure . the drag coefficients for the optimum nose shapes are likewise determined and these are compared with the drag coefficients given by other nose shapes . it is further shown that the drag coefficients arising when specular-type reflections occur are significantly dependent on the nose shape . when surface absorption followed by random emission occurs, the drag coefficient is not strongly dependent on either the missile nose shape or the fineness ratio of the nose .
on the model of the free shock separation, turbulent boundary layer . by free shock-separated boundary layers, one means that type of separation where the flow downstream of the separation region is free to adjust to any direction that may result from the shock-boundary-layer interaction process . a detailed model of the free shock-separated turbulent boundary layer is postulated, and the pressure rise following from this model is estimated and compared with experiments . the results are applied to the prediction of separation in an overexpanded nozzle .
note on the hypersonic similarity law for an unyawed cone . it is now known that the hypersonic similarity law derived for slender cones and ogival bodies under the assumption, is applicable for mach numbers as low as 3 . this note makes use of a series development to infer the hypersonic similarity law for unyawed cones from the taylor-maccoll differential equations and associated boundary conditions . a simple approximate formula for the function of the similarity law is obtained, and the drag function computed with this formula is compared with kopal’s numerical results and, for very slender cones, with von karman’s linearized formula .
lift on inclined bodies of revolution in hypersonic flow . the importance of body lift lies in the fact that at moderate angles of attack and high mach number it can constitute an appreciable part of the total lift of a winged missile . in this paper an attempt has been made to analyze body lift in hypersonic flow by an approximate method and, together with a correlation of existing experimental data, to indicate the probable variation of body lift over a wide range of mach numbers extending from low supersonic to hypersonic . the method of analysis of hypersonic flow over inclined bodies of revolution employed herein has been denoted as the hypersonic approximation . it is an improvement on the newtonian corpuscular theory of aerodynamics, since it considers the centrifugal forces resulting from the curved paths of the air particles in addition to the impact /newtonian/ forces .
the flow of a viscous liquid past a flat plate at small reynolds number . the authors repeat the earlier calculations of piercy and winny (proc. roy. soc. london. ser. a. 140 (1933), earlier works were known to be different from each other . the careful analysis of the present authors shows that the skin-friction coefficient up to the second approximation agrees perfectly with that of piercy and winny .
three-dimensional effect of flutter in a real fluid . in ref. 1, an alternative semi-empirical formulation for flutter in a real fluid is given . for more accurate determination of the empirical coefficients, the three-dimensional effect of finite span should be taken into account . following reissner’s approximation for large-aspect-ratio rectangular wings, the boundary-value problem governing the downwash w and the vorticity distribution .
an alternative formulation of the problem of flutter in real fluids . it is well known, in steady flow, that the actual lift curve slope is somewhat less than that predicted by inviscid flow theory, even at small angles of attack . as the stall angle is approached, the lift curve slope continually decreases and thus deviates even more from the theoretical value . pinkerton employed the measured circulation to determine the pressure distribution and found that the resulting prediction of the moment is considerably improved over that given by the classical theory . this amounts to replacing the conventional kutta-joukowski condition with the condition that the total lift should agree with the measured value, and this, in turn, completely determines the flow pattern . practically, this is accomplished by giving a fictitious camber to the profile . since potential flow theory is valid outside of the boundary layer, once the boundary-layer thickness is known, the potential flow may be corrected for the displacement thickness and the viscous wake by appropriate source distributions . the boundary layer cannot be evaluated, of course, until the potential flow is known and the circulation is applied . a criterion to determine the circulation, by generalizing the kutta-joukowski condition, was proposed by preston and spence by assuming that the pressure at the trailing edge shall have the same value when determined from the potential-flow values above and below the airfoil . this procedure gives qualitative information concerning viscous effects in steady flow .
a method for analysing the insulating properties of the laminar compressible boundary layer . in some cooling problems associated with high energy flows it may be convenient to localize strongly the cooling, as for example by injecting a coolant through an upstream porous strip, and to depend on the insulating properties of the boundary layer to reduce, or to eliminate completely the need for further cooling on the surface downstream of the highly cooled section . this upstream cooling technique may be of interest in connection with optical windows in hypersonic wind tunnels, and on radomes, wings, and bodies of high-speed aircraft and missiles . in this paper a method for investigating the insulating properties of a laminar compressible boundary layer on a two- dimensional surface with zero heat transfer is presented . the physical situation considered thus corresponds to the case in which the heat transfer downstream of the strongly cooled section is completely eliminated . of practical concern is how the temperature of the uncooled surface varies in the downstream direction from its low initial value and thus how the low energy layer established by the upstream cooling insulates the downstream surface . the karman integral method extended to both the momentum and energy partial differential equations of the boundary layer has been used . the station, at which cooling and or injection ceases, corresponds to a discontinuity in boundary conditions and thus in solutions . at this point the flux of mass, momentum, and energy within the boundary layer has been made continuous by the introduction of three additional parameters in the velocity and stagnation enthalpy profiles . thus the velocity and stagnation enthalpy profiles have both been taken as sixth degree polynomials . the resulting two integral-differential equations are then solved for two unknown functions of the distance along the wall . these two functions are related to the boundary-layer thickness and to the wall temperature . initial conditions corresponding to a given initial wall temperature and an initial boundary-layer thickness are prescribed . exact closed-form solutions for the case of zero axial pressure gradient are obtained . for flows with significant pressure gradients, numerical solutions are required in general . several numerical examples of practical interest are presented .
the homogeneous boundary layer at an axisymmetric stagnation point with large rates of injection . this report presents a theoretical analysis of the boundary layer at an axisymmetric stagnation point with large rates of air injection . the results of a previous investigation indicated that for localized mass transfer in the stagnation region, the rates of injection are considerably greater than those usually treated . the exact stagnation-point boundary-layer equations are integrated numerically for an approximate representation of the gas properties . the two-point boundary conditions are treated in a new manner which is useful for various boundary-layer and mixing problems . the exact solutions indicate that for large rates of injection the boundary layer is closely represented by an inner isothermal shear flow and by and exterior, relatively thin region, in which the flow variables change to their free-stream values . an integral method based on profiles suggested by the exact solutions is developed and shown to lead to accurate predictions of the integral thicknesses which are of interest for a study of the downstream influence of the stagnation-point mass transfer .
helium injection into the boundary layer at an axisymmetric stagnation point . this report presents a theoretical analysis of the boundary layer at an axisymmetric stagnation point with large rates of helium injection . the exact stagnation-point boundary-layer equations are integrated numerically with approximate representations of the gas properties . the treatment of the two-point boundary-value problem employed herein is shown to be useful for various boundary-layer and mixing problems . the exact solutions indicate that for large rates of injection the boundary layer can be represented by a thick, inner layer of constant shear, temperature, and composition and by a relatively thin outer region in which the flow variables adjust to their free-stream values . an inviscid-flow model is shown to lead to accurate predictions of this shear layer and will thus provide sufficiently accurate profiles for use in the study of the downstream influence of stagnation-point mass transfer . the heat transfer to the stagnation point is also considered . tabulations of the eigenvalues for a variety of wall conditions and injection rates are given .
control system and analysis and design via the second method of lyapunov . the/second method/of lyapunov is the most general approach currently in the theory of stability of dynamic systems . after a rigorous exposition of the fundamental concepts of this theory, applications are made to/a/ stability of linear stationary, linear nonslationary, and nonlinear systems,./b/estimation of transient behavior,./c/control-system optimization,./d/design of relay servos . the discussion is essentially self-contained, with emphasis on the thorough development of the principal ideas and mathematical tools . only systems governed by differential equations are treated here . systems governed by difference equations are the subject of a companion paper .
some problems of polar missile control . a polar-controlled missile is one in which manoeuvre is carried out by rotations about roll and pitch axes, that is, in the manner of a conventional aeroplane . this paper discusses some problems in the application of this form of control to homing missiles . in comparison with the alternative cartesian configuration, this method presents some special design problems . in the former case, it is often possible to resolve the motion into two planes and consider the pitch and yaw control systems as independent two-dimensional problems . this simplification is not possible in the case of polar control and it is usually necessary to consider the whole three-dimensional system . the equations of motion which result are, in general, not susceptible to analysis . because of this, the design of control systems requires extensive use of simulators .
an approximate solution of the supersonic blunt body problem for prescribed arbitrary axisymmetric shapes . the integral method of belotserkovskii has been carried out to the first approximation for arbitrary blunt axisymmetric bodies in supersonic or hypersonic flight . this method is direct, in that it gives the surface-pressure distribution and shock shape for a prescribed body . results obtained by numerical integration for several body shapes at several mach numbers are compared to experimental results with good agreement . it is also shown that the method can be successfully applied to pointed bodies with attached shock . in the stagnation region, simple relationships are found from the equations of the first approximation which connect the surface-velocity gradient, shock curvature, shock-detachment distance, and body curvature . these relations are also correlated with experiment for a variety of shapes as a function of mach number . the correlations permit a rapid estimate of the stagnation-point velocity gradient, important for heat-transfer calculations, for any blunt body from the shock stand-off distance . a method for a higher approximation is described, for which, in contrast to the higher approximations of belotserkovskii, a large number of simultaneous total differential equations with unknown parameters does not occur . one form of this method has been studied numerically . results are given which, though only partially successful, indicate the amount of improvement to be expected from a higher approximation .
theoretical pressure distribution on a hemisphere-cylinder combination . in recent years great use has been made of approximate methods for the determination of the pressure distribution on blunt-nosed bodies and afterbodies at high mach numbers . for quasi-spherical bodies it has been suggested that modified newtonian theory in combination with a prandtl-meyer expansion be used on the nose portion, the two laws being matched at the point where the pressure gradients are equal . no simple approximation, however, has been found for flat-nosed bodies . as for the pressure distribution on the afterbody, the blast-wave analogy has been suggested for general nose shapes but particular afterbody profiles . the purpose of the present note is to compare these approximate estimates with a more accurate determination of the flow field about a hemisphere-cylinder in an ideal gas flow . it was felt that since experimental investigations in air at this mach number are scarce and very difficult to obtain, the comparison would be of interest . the basis of comparison is the flow field as it results from a numerical integration of the exact equations governing the motion of the ideal fluid .
note on tip-bluntness effects in the supersonic and hypersonic regimes . in a recent letter, m. h. bertram presents some data on flows at m = 6.85 around 10 half-angle cones with blunted tips . since the demarcation between the supersonic and hypersonic regimes is not sharp and since one expects hypersonic flows to be generally similar to those at lower mach numbers– especially where viscous effects do not predominate throughout the entire field of interest–it is of some value to compare bertram’s results with those obtained by giese and bergdolt for 15 half-angle cones at m = 2.45 . following the observation by charters and stein that drag coefficient measurements on blunted cones imply a reynolds number effect, giese and bergdolt study the convergence to conical flow of the perturbed flow about a cone with truncated tip . they employ the mach-zehnder interferometer and the conical flow criterion as analytical tools .
an experimental investigation of flow about simple blunt bodies at a nominal mach number of 5. 8. an experimental investigation was conducted in the galcit hypersonic wind tunnel to determine flow characteristics for a series of blunt bodies at a nominal mach number of 5.8 and free-stream reynolds numbers per in. of measured values for the pressure coefficient distributions are compared with a modified newtonian expression . the agreement is very good for the three-dimensional bodies and is fair for the circular cylinder transverse to the free-stream flow direction . a complete report of the investigation is given in a galcit hypersonic wind tunnel memorandum .
the generalized expansion method and its application to bodies travelling at high supersonic airspeeds . it is demonstrated that the shock-expansion method can be generalized to treat a large class of hypersonic flows, only one of which is flow about airfoils . this generalized method predicts the whole flow field, including shock-wave curvatures and resulting vorticity, providing that (1) disturbances originating on the surface of an object are largely absorbed in shock waves with which they interact and (2) disturbances associated with the divergence of stream lines in tangent planes to the surface are of secondary importance compared to those associated with the curvature of stream lines in planes normal to the surface . it is shown that these conditions may be met in three-dimensional as well as two-dimensional hypersonic flows . when they are met, surface streamlines may be taken as geodesics, which, in turn, may be related to the geometry of the surface . the validity of the generalized shock-expansion method for three-dimensional hypersonic flows is checked by comparing predictions of theory with experiment for the surface pressures and bow shock waves of bodies of revolution . the bodies treated are two ogives having fineness ratios of 3 and 5 . tests were conducted at mach numbers from 2.7 to 6.3 and angles of attack up to 15 degrees in the 10- by 14-in. supersonic wind tunnel of the ames aeronautical laboratory . at the lower angles of attack, theory and experiment approach agreement when the ratio of mach number to fineness ratio–that is, the hypersonic similarity parameter–exceeds 1 . at the larger angles of attack, theory tends to break down, as would be expected, on the leeward sides of the bodies . as a final point, it is inquired if the two-dimensionality of inviscid hypersonic flows has any counterpart in hypersonic boundary-layer flows . the question is answered in the affirmative, and results of experiment are employed to provide a partial check of this conclusion .
an investigation of optimum zoom climb techniques . the problem of optimal zoom climb maneuvering of a turbojet aircraft has been investigated using the mayer formulation of the calculus of variations . the euler-lagrange equations governing optimum symmetric flight have been integrated numerically by digital computation . discontinuities in thrust arising from turbojet afterburner blowout have been treated, and conditions which must be satisfied across the interface generated by the discontinuity have been derived . arbitrary control techniques have been compared with the optimum, and it has been found that performance is relatively insensitive to piloting technique unless a time limitation is imposed which requires high maneuvering load factors .
steady flow in the laminar boundary layer of a gas . if the boundary-layer equations for a gas are transformed by mises’s transformation, as was done by karman tsion for the flow along a flat plate of a gas with unit prandtl number, the computation of solutions is simplified, and use may be made of previously computed solutions for an incompressible fluid . for any value of the prandtl number, and any variation of the viscosity with the temperature t, after the method has been applied to flow along a flat plate (a problem otherwise treated by crocco), the flow near the forward stagnation point of a cylinder is calculated with dissipation neglected, both with the effect of gravity on the flow neglected and with this effect retained for vertical flow past a horizontal cylinder . the approximations involved by the neglect of gravity are considered generally, and the cross-drift is calculated when a horizontal stream flows past a vertical surface . when, and the boundary is heat-insulated, it is shown that the boundary-layer equations for a gas may be made identical, whatever be the main stream, with the boundary-layer equations for an incompressible fluid with a certain, determinable, main stream . the method is also applied to free convection at a flat plate variation with altitude of the state of the surrounding fluid neglected) and to laminar flow in plane wakes, but for plane jets the conditions, previously imposed by howarth, are also imposed here in order to obtain simple solutions .
transformation between compressible and incompressible boundary layer equations . it is proposed to show that the boundary-layer equation of compressible flow can be reduced to that of incompressible flow . such work was initiated by stewartson and by rott and crabtree . in the following some of the restrictions imposed by references 1 and 2 will be removed, and it will be shown that the transformation from compressible boundary layer to incompressible boundary layer can be applied to the laminar, as well as turbulent, case . a direct method will be used for this purpose .
a turbulent analog of the stewartson-illingworth transformation . the stewartson-illingworth transformation is applied to the integral momentum equation for compressible boundary-layer flow, leaving the x-coordinate transformation unspecified, however . it is shown that the transformed equation is the integral momentum equation for incompressible flow if (a) the effect of compressibility on the boundary-layer shape parameter h can be represented by and (b) the x-coordinate transformation is chosen to be suitably related to the ratio of skin-friction coefficients in compressible and incompressible flows . experimental evidence is presented which shows that condition (a) is satisfied for turbulent boundary layers up to m = 5 . an x-transformation is chosen according to (b) and an equation is presented which gives the turbulent boundary-layer growth in compressible flow in terms of a simple quadrature . the predictions of this equation are then compared with some measurements on wind-tunnel nozzles .
engineering relations for friction and heat transfer to surfaces in high velocity flow . in calculations of thermodynamic heating for high speed missiles parameters have been used based on relationships which hold for constant-property fluids . the validity of this procedure has been verified recently in a survey of heat transfer in which a relationship for the reference temperature was developed . a calculation procedure for laminar and turbulent boundary layers, based on this relationship, is given .
reverse flow and variational theorems for lifting surfaces in nonstationary compressible flow . a reverse-flow theorem for compressible nonsteady flow, valid within the limits of linearized theory, is derived . this theorem gives a general class of relations between linearized solutions for lifting surfaces in direct and reverse flow . based on the same considerations used to establish the theorem, an adjoint variational principle, which may be useful in approximate solutions of non-steady lifting surface problems, is obtained . to illustrate the uses of the reverse-flow theorem, it is applied to the determination of relations between aerodynamic coefficients in direct and reverse flow and to the obtaining of influence functions for total lift, pitching moment, and rolling moment for a wing oscillating with arbitrary motion and surface deformation, in terms of the pressure distributions for simpler cases in reverse flow .
effect of quasi-steady air forces on incompressible bending-torsion flutter . explicit solutions are obtained for the bending-torsion flutter of a two-dimensional airfoil in incompressible flow under the assumptions that the theodorsen function, c(k) is set equal to a real constant, and the diagonal virtual mass terms are negligible . for the case of small bending to torsion frequency ratio, a comparison is made of these quasi-steady solutions with an earlier empirical expression suggested by theodorsen and garrick for the nonsteady case, and the effect of the c(k) function is indicated . the importance of the c.g. location for these small cases is re-emphasized, and the possibility of flutter at zero air speed is indicated .
the axisymmetric boundary layer on a long thin cylinder . the laminar boundary layer in axial flow about a long thin cylinder is investigated by two methods . one (2) is a pohlhausen method, based on a velocity profile chosen to represent conditions near the surface as accurately as possible . the other (3) is an asymptotic series solution, valid far enough downstream from the nose for the boundary-layer thickness to have become large compared with the cylinder radius . another series solution (due to seban, bond and kelly) is known, valid near enough to the nose for the boundary layer to be thin compared with the cylinder radius . the pohlhausen solution shows good agreement with both series, near and far from the nose, and enables an interpolation to be made (4) between them in the extensive range of distances from the nose for which neither is applicable . the final recommended curves, for the variation along the cylinder of skin friction, boundary-layer displacement area and momentum defect area, are displayed in graphical and tabular form (figure 1 and table 1) and are expected to be correct to within about 2 . the velocity near the wall is closely proportional to the logarithm of the distance from the axis,. this is the profile used in the pohlhausen method . the analogy with the distribution of mean velocity in turbulent flow over a flat plate is discussed at the end of 2 .
a note on the laminar boundary layer on a circular cylinder in axial incompressible flow . a correction is made for the equation to compute the ratio of the displacement thickness on a cylinder to the corresponding thickness on a flat plate .
integration of the boundary layer equations for a plane in compressible flow with heat transfer . the equations of motion of compressible viscous flow with vanishing pressure gradient past a plane are integrated in semi-convergent expressions, for the case when the physical constants depend on temperature and the prandtl number is close to unity . simple expressions are obtained for the temperature and velocity distributions in the boundary layer, the drag coefficient, and their dependence on the physical constants,.they contain the well-known results and several new ones . for the case when the temperature of the boundary is either above, or not much below, the temperature of the main flow, the results obtained closely agree with crocco’s numerical computations .
application of second-order shock-expansion theory to several types of bodies of revolution . second-order shock-expansion theory is utilized to obtain equations for the initial normal force curve slope, initial pitching moment curve slope, and zero-lift wave drag for several type bodies of revolution . bodies considered are the cone-cylinder, cone-cylinder-frustum, cone-cylinder-frustum-booster, cone-frustum, and cone-frustum booster .
on a generalised porous-wall ?couette type? flow . in a recent paper, the problem of a /couette-type/ flow in which the fixed wall is porous has been considered . the results quoted in the above reference can be obtained rigorously by the method stated below in which a different interpretation to one of the parameters is made .
a generalised porous-wall ?couette type? flow . recently, it was observed that the two existing boundary-layer texts (references 1 and 2) did not contain a solution for the case of couette flow with a constant, uniformly distributed suction or blowing . thus, the following analysis considers a /couette-type/ flow between a stationary flat surface and a slightly inclined flat plate moving at a constant velocity . in addition, the flow is subjected to a constant, uniformly distributed suction or blowing at the fixed surface .
heat transfer for laminar flow in an annulus with porous wall . temperature profiles and heat-transfer rates of established incompressible flow through an annulus channel with porous walls of constant temperatures are determined at different injection rates . axial conduction and viscous dissipation are, as usual, neglected . injecting fluid is tacitly assumed to have the same temperature as the porous wall .
the pressure gradient induced by shear flow past a flat plate . article is a continuation of an earlier note on papers by li on a semiinfinite plate in a uniform shear flow . li had deduced from the form of his equations that stream vorticity caused an induced pressure gradient in the flow . later papers by li and murray (amr 15(1962), rev. 7157) support the induced pressure gradient theory . the author notes, however, that the mathematics used by li and murray are not acceptable and the problem thus not resolved . the present note sets up simple models of complete flows examinable by elementary means . author holds that analyses demonstrate conclusively that no pressure gradient is induced in the boundary layer on a flat plate in a limited region of shear flow . he notes that the original question in the case of unbounded shear remains obscure–and anyway an unlimited shear layer is not of great practical importance .
simple shear flow past a flat plate in a compressible viscous fluid . by transformation of variables, the problem of a simple shear flow of a compressible fluid over a flat plate is reduced to the corresponding problem for an incompressible fluid . the prandtl number of the compressible fluid is assumed to be unity and its viscosity to be a linear function of temperature .
some panel-flutter studies using piston theory . the use of piston theory was recently advocated for supersonic aeroelastic analyses, including the problem of panel flutter, and this has stimulated the investigation reported here . linear piston theory is mainly considered, but some effects of introducing higher order terms are discussed . flutter of rectangular simply supported panels and of elliptically shaped clamped-edge panels is considered, and some justification is provided for the use of /static/ aerodynamic forces and the neglect of aerodynamic damping . hence, it is concluded that ackeret loading gives more exact results than piston theory . solution of the flutter equations is made by applying galerkin’s method to a rayleigh-type analysis using assumed modes of deformation .
flutter of rectangular simply supported panels at high supersonic speeds . the problem of panel flutter of rectangular simply supported plates subjected to supersonic flow over one surface is treated theoretically . the assumption is made, and subsequently verified, that the /static/ approximation to the aerodynamic flutter forces yields flutter boundaries with satisfactory accuracy for mach numbers greater than about 2 . two panel flutter analyses are performed using this static approximation in conjunction with thin-plate theory–one employs aerodynamic strip theory, the other aerodynamic surface theory . the influence of mach number, dynamic pressure, panel aspect ratio, and midplane stress on the panel thickness required to prevent flutter is determined for extensive ranges of these parameters .
natural frequencies of rectangular plates with edges elastically restrained against rotation . plates with attachments to heavier members along the edges can be described as having edges elastically restrained against rotation, in many cases uniformly along each edge . at the edges, setting slope, when is the edge bending moment with always positive, the elastic restraint can be analytically defined with describing respectively, clamped and simply supported edges . in this note natural frequencies of such plates are calculated mainly following the nomenclature of dana young .
the shear flow along a flat plate with uniform suction . recently, several authors have investigated the boundary layer in a shear flow . in this note, an exact solution of the navier-stokes equations will be presented, which represents the boundary layer along an infinite flat plate with uniform suction situated in a shear flow .
the viscous flow near a stagnation point when the external flow has uniform vorticity . in view of the recent controversy between li and glauert on the nature of the solution of the boundary-layer equations when the external flow is rotational, it seems worthwhile to draw attention to a certain exact solution of the navier-stokes equations which lends support to glauert’s point of view .
new methods in heat flow analysis with application to flight structures . new methods are presented for the analysis of transient heat flow in complex structures, leading to drastic simplifications in the calculation and the possibility of including nonlinear and surface effects . these methods are in part a direct application of some general variational principles developed earlier for linear thermodynamics . they are further developed in the particular case of purely thermal problems to include surface and boundary-layer heat transfer, nonlinear systems with temperature-dependent parameters, and radiation . the concepts of thermal potential, dissipation function, and generalized thermal force are introduced, leading to ordinary differential equations of the lagrangian type for the thermal flow field . because of the particular nature of heat flow phenomena, compared with dynamics, suitable procedures must be developed in order to formulate each problem in the simplest way . this is done by treating a number of examples . the concepts of penetration depth and transit time are introduced and discussed in connection with one-dimensional flow . application of the general method to the heating of a slab, with temperature-dependent heat capacity, shows a substantial difference between the heating and cooling processes . an example of heat flow analysis of a supersonic wing structure by the present method is also given and requires only extremely simple calculations . the results are found to be in good agreement with those obtained by the classical and much more elaborate procedures .
variational and lagrangian thermodynamics of thermal convection-fundamental shortcomings of the heat transfer coefficient . extension of previous analyses, indicating the possibility of extending the thermodynamics of irreversible processes to systems which are not in the vicinity of an equilibrium state and for which onsager’s relations are not verified . this involves generalizations beyond the narrow field of heat transfer and to principles of wider range than those of current nonequilibrium thermodynamics .
a sublayer for fluid injection into the incompressible turbulent boundary layer . a sublayer region is introduced in which the intensity of turbulence grows at a prescribed rate . the decrease in wall shear stress due to fluid injection into the boundary layer is found under the hypothesis that the effect of injection is restricted to the sublayer region . experimental measurements of the velocity profiles with fluid injection substantiate this hypothesis . the theoretical decrease in wall shear stress is in good agreement with experiment,. the solution is particularly simple and for small values of the injection parameter it contains no arbitrary parameters . the theory provides a similarity parameter which differs from the one in general use .
heat transfer in turbulent shear flow . the problems of heat transfer in turbulent shear flow along a smooth wall are discussed from the point of view of von karman’s well-known 1939 paper on the analogy between fluid friction and heat transfer . methods for extending the analysis to higher prandtl numbers are suggested .
conduction of heat in composite slabs . a method of calculating the total quantity of heat that passes through a unit area from zero time to time t is developed . allowance is made for surface resistance by regarding each contact resistance as an additional layer of the appropriate thermal resistance and zero heat capacity
buckling stress of clamped rectangular plates in shear . by consideration of antisymmetrical, as well as symmetrical, buckling configurations, the theoretical shear buckling stresses of clamped rectangular flat plates are evaluated more correctly than in previous work . the results given, which represent the average of upper and lower-limit solutions obtained by the lagrangian multiplier method, are within percent of the true buckling stresses .
inviscid hypersonic airflows with coupled non-equilibrium processes . analyses have been made of the effects of coupled chemical rate processes in external inviscid hypersonic airflows at high enthalpy levels . exact (numerical) solutions have been obtained by the inverse method for inviscid airflow over a near-spherical nose under flight conditions where substantial nonequilibrium prevails through the nose region . typical conditions considered include nose radii of the order of 1 ft at an altitude of 250,000 ft and velocities of 15,000 and 23,000 ft per sec . the results illustrate the general importance of the coupling among the reactions considered . these included dissociation-recombination, bimolecular-exchange, and ionization reactions . the exact solutions show the bimolecular, no exchange reactions to be important in blunt-nose flow for the kinetics of no and n, as they are in the case of a plane shock wave . an important difference between blunt-nose flow and plane shock flow, however, is the gasdynamic expansion in the curved shock layer of the former . this expansion reduces post-shock reaction rates . as a consequence, in the regime studied the oxygen and nitrogen-atom concentrations tend to freeze in the nose region at levels below those for infinite-rate equilibrium . the reduction below the equilibrium dissociation level can be large, particularly for nitrogen dissociation at higher velocities . in the regime considered, the chemical kinetics are dominated by two-body collision processes . the inviscid nose flow, including coupled nonequilibrium phenomena, is thus amenable to binary scaling for a given velocity . the binary scaling is demonstrated for a range of altitude and scale by correlation of the exact solutions for given velocity and a constant product of ambient density and nose radius . this similitude, which can also scale viscous nonequilibrium and radiation phenomena in the shock layer, provides a useful flexibility for hypersonic testing where it is applicable . the afterbody inviscid-flow problem is briefly discussed in the light of the results for the nose flow .
magnetohydrodynamics shocks . a mathematical treatment of the coupled motion of hydrodynamic flow and electromagnetic fields is given . two simplifying assumptions are introduced .. first, the conductivity of the medium is infinite, and second, the motion is described by a plane shock wave . various orientations of the plane of the shock and the magnetic field are discussed separately, and the extreme relativistic and unrelativistic behavior is examined . special consideration is given to the behavior of weak shocks, that is, of sound waves . it is interesting to note that the waves degenerate into common sound waves and into common electromagnetic waves in the extreme cases of very weak and very strong magnetic fields .
magnetohydrodynamic shock waves . an interpretation of the de hoffman-teller shock-wave equations for an infinitely conducting medium is given analogous to the classical interpretation of the ordinary hydrodynamic shock-wave equations of rankine and hugoniot . two cases of interest are considered as a consequence of this theory . it is shown that weak magnetic fields in interstellar clouds will be amplified, and, if external mechanisms are available to reduce the compressional effects of shock waves, the field will reach a value, where p is the pressure . also, some aspects of the internal motions of prominences are considered ,. it is shown that gauss will yield results in accord with the observational material .
two dimensional transonic flow past airfoils . this report concerns the problem of constructing solutions for transonic flows over symmetric airfoils . the aspect of the problem emphasized is, of necessity, not how to form a solution for compressible flow but how to simplify the initial phase of the problem, namely, the mapping of the incompressible flow . in the case of the symmetric joukowski airfoil without circulation, the mapping is relatively simple, but the coefficients in the power series are difficult to evaluate . as a result, the problem requires simplification . instead of the exact incompressible flow past the airfoil, an approximate flow is used, which is derived from a combination of source and sink . this flow differs only slightly from the exact one when the thickness is small . by the same method, the flow with circulation is also considered . after the incompressible-flow functions are approximated in this fashion, the numerical calculation of the corresponding compressible flow, by the hodograph theory, does not present any essential difficulty .
tables of thermal properties of gases . tables of thermodynamic and transport properties of air, argon, carbon dioxide, carbon monoxide, hydrogen, nitrogen, oxygen, and steam .
on the behaviour of boundary layers at supersonic speeds . this paper considers the implications of recent advances in knowledge of the behaviour of boundary layers in supersonic flow . only the simplest case is considered-dashthat of the two-dimensional boundary layer on a flat plate, with nominal zero longitudinal pressure and temperature gradients . it is shown that the empirical/intermediate enthalpy/used with success in approximations for skin friction, etc., of laminar boundary layers is closely the same as the mean enthalpy with respect to velocity . furthermore, the mean enthalpics of laminar and turbulent boundary layers may be the same . a nonrigorous approach is made to the problems of self-induced pressure gradients, and the indications are that their effects on laminar skin friction, etc., may become noticeable at mach numbers greater than 5 and they increase as the surface temperature builds up towards zero heat-transfer conditions . the effects with turbulent boundary layers may not be so severe . finally, the results are applied to give an idea of the magnitude of the drag and aerodynamic heating problems up to m 10, and one result is that, if there is any conflict at the higher mach numbers between surface conditions required for high radiative emissivity and those which may be thought necessary for preserving a laminar boundary layer, then it may be better to choose the former .
stationary convection flow of an electrically conducting liquid between parallel plates in a magnetic field . a study is made of the stationary convection of an electrically conducting liquid in the space between two parallel plates, heated to different temperatures, in the presence of a magnetic field . the distribution of velocity, temperature, and induced fields are found, and the convective heat flow is calculated .
on convective motion of a conducting fluid between parallel vertical plates in a magnetic field . stationary convective motion of a conducting fluid between vertical parallel plates in a magnetic field is considered . an exact solution of the magnetohydrodynamic equations is obtained for the case of a constant vertical temperature gradient . the critical value of grasshof’s number is determined for the case when the temperature of both plates is the same .
on the base pressure resulting from the interaction of a supersonic external stream with a sonic or subsonic jet . it is shown that the two-dimensional base pressure problems relating to base bleed into the wake of blunt-trailing-edge airfoils, or the interaction between an external supersonic or sonic slipstream with a sonic or subsonic jet stream of a jet engine, can be calculated by theoretical considerations . constant-pressure, isoenergetic, turbulent mixing between the streams and the stagnant fluid in the wake is assumed . the theoretical calculations are in good agreement with the experimental results .
the supersonic flow about a blunt body of revolution for gases at chemical equilibrium . the supersonic flow about a blunt body of revolution for gases at chemical equilibrium . a method to determine the shock wave, and its location, about a body of revolution moving at supersonic speeds is given . the method provides also the means to compute the flow characteristics in the shock layer . the fluid in which the motion takes place is assumed to be in chemical equilibrium within the shock layer ,. its thermochemical properties must be known . the essential new features of the method are .. a) it solves the direct problem, i. e., the initial data are the conditions upstream and the body shape ,. b) the integration of the fundamental equations is done in the physical plane and the difficulties inherent to other, less direct, mathematical formulations of the problem are avoided . a physical interpretation of the method is made which is in accord with the analytical definition of the problem .
data on shape and location of detached shock waves in cones and sphere . accurate experimental data are given on the shape and the location of detached shock waves on cones and spheres at mach numbers from 1.17 to 1.81 . the data are correlated to obtain equations that describe the shock waves . this knowledge of the shock waves should be useful in calculations of the pressure distribution and the pressure drag of the fore part of cones and spheres . the experimental data on shock waves are compared with theory .
critical combinations of shear and transverse direct stress for an infinitely long flat plate with edges elastically restrained against rotation . an exact solution and a closely concurring approximate energy solution are given for the buckling of an infinitely long flat plate under combined shear and transverse direct stress with edges elastically restrained against rotation . it was found that an appreciable fraction of the critical stress in pure shear may be applied to the plate without any reduction in the transverse compressive stress necessary to produce buckling . an interaction formula in general use was shown to be decidedly conservative for the range in which it is supposed to apply .
turbulent skin friction at high mach numbers and reynolds numbers in air and helium . nasa r82, 1960 . results are given of local skin-friction measurements in turbulent boundary layers over an equivalent air mach number range from 0.2 to 9.9 and an over-all reynolds number variation of 2×10 to 100×10 . direct force measurements were made by means of a floating element . flows were two-dimensional over a smooth flat surface with essentially zero pressure gradient and with adiabatic conditions at the wall . air and helium were used as working fluids . an equivalence parameter for comparing boundary layers in different working fluids is derived and the experimental verification of the parameter is demonstrated . experimental results are compared with the results obtained by several methods of calculating skin friction in the turbulent boundary layer .
the problem of resistance in compressible fluids . this report is restricted to the resistance of bodies of revolution and of cylindrical bodies of infinite length moving with uniform velocity in a compressible fluid . in the case of bodies of revolution it will be assumed that the direction of the movement is parallel to the axis of symmetry . it will be assumed that the fluid satisfies the equation of state of perfect gases, i. e. const., where p denotes the pressure, the density and t the absolute temperature . in addition to obeying this equation the fluid is characterized by the statement that the intrinsic energy of the unit mass amounts to where for simplicity’s sake the specific heat will be expressed in work rather than heat units . the ratio between the specific heat at constant pressure and the specific heat at constant volume will be denoted by . it is known that the value of x depends upon the number of degrees of freedom of the molecules,. if this number is denoted by n . for air the value x = 1.4 will be used . the limiting case x = 1 will be referred to as that of a assumed that in the range considered and are independent of the temperature .
the aerodynamic design of section shapes for swept wings . an extension of work of lock and rogers and the result of cooperation by n.p.l., r.a.e. and members of the british aircraft industry to achieve a satisfactory design for an aircraft cruising at low supersonic speeds . knowledge of shock-wave prediction, onset of wave drag and shock-induced separation allows the basic design to be generalized for a wide range of parameters . unpublished work by bagley on the relation of aerodynamic coefficients and geometry is used . the role of upper surface velocity distribution is noted and methods for predicting pressure distributions with shock waves are reviewed for both subsonic and transonic flows .
methods of boundary-layer control for postponing and alleviating buffeting and other effects of shock-induced separation . the use of boundary-layer control to increase the separation-free margins of mach number and lift coefficient beyond the cruise point of high-speed aircraft may often be preferred to design changes that impair the cruising performance or the landing and take-off characteristics . the factors that influence the choice of method and details of its application are discussed, emphasising particularly the need to maintain effectiveness over most of the chord to cover the wide range of separation positions encountered as the shock moves over the wing with changing flight conditions . research at the national physical laboratory that has embraced high-velocity blowing, vane and air-jet vortex generators, and, in a preliminary way, distributed suction, is briefly described . the relative merits of the various methods are discussed, and some results achieved in their application are given . for vortex generators, the importance is stressed of the vortex paths determined by the interactions of neighbouring vortices and their images . thus, systems of counter-rotating vortices always leave the surface in pairs and lose their effectiveness . co-rotating systems are therefore preferred for many applications . blowing, which in wind-tunnel tests gives results as good as or better than vortex generators and does not have the disadvantage of a drag penalty at cruise, has not yet been assessed in flight . air-jet vortex generators, which would also avoid the drag penalty, show promise of producing significant effects with relatively small blowing pressures and quantities .
on the stability of two dimensional parallel flows . this is the last part of the author’s theory of the stability of plane laminar motion . (for parts 1 and 2, cf. the same quart. 3, 117-142, 218-234 (1945),. these rev. 7,225,226 .) the stability character of a viscous fluid is considered in detail . the author proceeds first to give a proof of a criterion of stability due to heisenberg .. if a velocity profile has an number and phase velocity, the disturbance with the same wave number is unstable in the real fluid when the reynolds number is sufficiently large . this destabilizing effect of viscosity is one of the most interesting phenomena in the general stability theory,. its physical and mathematical significance is carefully discussed . the author then discusses the behavior of the so-called neutral curve for the two characteristic types of velocity distribution, the boundary layer type profile and the symmetrical profile . the asymptotic behavior of the neutral curve is discussed first . the main difference between profiles with and without a point of inflection is that the two branches of the neutral curve approach and for profiles with a flex, but both converge to for the profile without a flex . the most important results are as follows . for sufficiently large reynolds number r . (2) there always exists a minimum r below which the motion is stable . a similar result was obtained by synge from energy considerations . synge found a limiting curve below which the motion is necessarily stable . the author’s discussion of the asymptotic behavior of the curves shows further that there always exists a maximum value of a beyond which the motion is stable for all reynolds numbers . hence the qualitative shape of the curve is determined . the author proceeds to show that simple approximate expressions for the stability limit can be obtained from his general analysis for a given velocity profile . these approximate stability limits for plane poiseuille flow and blasius flow are found to be r=5906 and r=502 . the reynolds numbers are based on the width of the channel and the displacement thickness, respectively . finally, the method for computing the complete instability curve is presented and the plane poiseuille case and the blasius problem worked out in detail . the stability limit for blasius flow had been given before by tollmien and schlichting . the present more exact computations agree well with tollmien’s result as far as the minimum critical reynolds number is concerned . the value found here is r=420 . the neutral curve for poiseuille motion had not been obtained before . the minimum critical number here is found to be r=5314 . the agreement with the estimate from the simple criterion mentioned above is thus very good . a discussion of the physical significance of the viscous effects and of future developments concludes the paper .
transition form laminar to turbulent shear flow . recent experimental studies of transition from laminar to turbulent shear flows are reviewed . certain common features are emphasized and related to the stability theories of viscous shear layers . the three-dimensional character, the unsteadiness, and the nonlinear and random behavior of the latter stages of the transition process are also examined .
the design of intermediate vertical stiffeners on web plates subjected to shear . the correct design of intermediate vertical stiffeners on web plates subjected to shear becomes very important when the web plates are designed to operate at loads close to their buckling loads . this paper presents details of an extensive series of tests conducted on stiffened web plates subjected to shear . from the analysis of the results obtained from these tests, new empirical relationships between the flexural rigidity and spacing of the intermediate stiffeners and the buckling stress of the stiffened web plate have been obtained . one interesting and important feature of these new relationships is that they define more clearly than hitherto the difference in the behaviour of single-and double-sided stiffeners .
an experimental study of the flow field about swept and delta wings with sharp leading edges . a series of experiments was performed to define the flow field on the upper surface of high aspect ratio swept wings and narrow delta wings at high angles of attack . it was found that near the root section of either type of wing the flow is conical . the edge of the vortex sheet which originates at the leading edge is a straight line whose position relative to the leading edge depends only on incidence . on swept wings, the vortex edge turns down-stream as soon as the vortex sheet covers the front half of the wing chord, and the flow under the vortex sheet outboard of that turning point is uniform and parallel to the leading edge of the wing . on narrow delta wings, the conical symmetry persists almost to the trailing edge .
analytic study of induced pressure on long bodies of revolution with varying nose bluntness at hypersonic speeds . a systematic study of induced pressures on a series of bodies of revolution with varying nose bluntness has been made by using the method of characteristics for a perfect gas . the fluid mediums investigated were air and helium and the mach number range was from 5 to 40 . a study of representative shock shapes was also made . flow parameters obtained from the blast-wave analogy gave good correlations of induced pressures and shock shapes . the induced-pressure correlations yielded empirical equations for air and helium which cover the complete range of nose bluntness considered . (nose fineness ratios varied from 0.4 to 4.) available experimental results were in good agreement with the characteristics solutions . properties connected with the concept of hypersonic similitude enabled correlations of the calculations to be made with respect to nose shape, mach number, and ratio of specific heats .
bending of a square plate with two adjacent edges free and the others clamped or simply supported . the title problems were solved for the two cases .. (1) uniform transverse loading, (2) a concentrated force at the free corner . a function is chosen to exactly satisfy the biharmonic equation while the boundary conditions are enforced at a number of points plied at discrete points around the boundary for each of the four problems and the resulting 35 simultaneous equations were solved on an ibm 704 . tables listing the values of deflection and bending moments are presented . this paper provides useful information on the solution of these problems which are intractable by analytical methods .
an experimental investigation of the flow over blunt-nosed cones at a mach number of 5. 8. shock shapes were observed and static pressures were measured on spherically-blunted cones at a nominal mach number of 5.8 over a range of reynolds numbers per inch from 97,000 to 238,000, for angles of yaw from 0 to 8 . six combinations of the bluntness ratios 0.4, 0.8, and 1.064 with the cone half angles 10, 20, and 40 were used in determining the significant parameters governing pressure distribution . the pressure distribution on the spherical nose for both yawed and unyawed bodies is predicted quite accurately by the modified newtonian theory given by, where is the angle between the normal to a surface element and the flow direction ahead of the bow shock . cone half angle was found to be the significant parameter in determining the pressure distribution near the nose-cone junction and over the conical afterbody . on the 40 spherical nosed cone models the flow overexpanded with respect to the taylor-maccoll pressure in the region of the spherical-conical juncture, after which the pressure returned rapidly to the taylor-maccoll value . for models with smaller cone angles the region of minimum pressure occurred farther back on the conical portion of the model, and the taylor-maccoll pressure was approached more gradually . the shape of the pressure distributions as described in nondimensional coordinates was independent of the radius of the spherical nose and of the reynolds number over the range of reynolds number per inch between .97 x 10 and 2.38 x 10 . integrated results for the pressure foredrag of the models at zero yaw compared very closely with the predictions of the modified newtonian approximation, except for models with large cone angles and small nose radii, where the drag approaches the value given by the taylor-maccoll theory for sharp cones .
cantilever plate with concentrated edge load . the author gives, by the method of finite differences, an approximate solution of the problem of a finite length of a cantilever plate which bears a concentrated load at the longitudinal free edge . all the boundary conditions are taken into account, and the plate action is determined approximately at all points of the plate . the author points out that a secondary maximum transverse stress occurs at the clamped edge nearest the loading point, and that the longitudinal stress is greatest directly under the loading point .
the solution of elastic plate problems by electrical analogies . a dynamic-analogy method for the solution of elastic plate problems is described in this paper . the electrical circuits developed here can be set-up and studied on an electric-analog computer . problems involving deflections under constant load, transient vibrations, or normal modes can be solved in this way . the method of applying boundary conditions to plates with irregular edges is given, together with a detailed description of the representation of the boundary conditions for a rectangular variable-thickness plate . solutions that have been obtained on the cal tech electric-analog computer are presented for the static deflections and normal modes of a rectangular cantilever plate .
preliminary analysis of axial flow compressors having supersonic velocity at the entrance of the stator . a supersonic compressor design having supersonic velocity at the entrance of the stator is analyzed on the assumption of two-dimensional flow . the rotor and stator losses assumed in the analysis are based on the results of preliminary supersonic cascade tests . the results of the analysis show that compression ratios per stage of 6 to 10 can be obtained with adiabatic efficiency between 70 and 80 percent . consideration is also given in the analysis to the starting, stability, and range of efficient performance of this type of compressor . the desirability of employing variable-geometry stators and adjustable inlet guide vanes is indicated . although either supersonic or subsonic axial component of velocity at the stator entrance can be used, the cascade test results suggest that higher pressure recovery can be obtained if the axial component is supersonic .
flow of gas through turbine lattices . paper is a translation of chap. 7 of the book /technical gas-dynamics/ (see amr 9, rev 1869) . the topics treated are best shown by the list of paragraph headings . they are .. 7-1 . geometrical and gasdynamical parameters of the lattices,. fundamentals of flow through lattices,. 7-2 . theoretical methods of investigation or plane potential flow of incompressible fluid through a lattice,. 7-3 . electro-hydrodynamic analogy,. 7-4 . forces acting on an airfoil in a lattice,. theorem of joukowsky for lattices,. 7-5 . fundamental characteristics of lattices,. 7-6 . friction losses in plane lattice at subsonic velocities,. 7-7 . edge losses in plane lattice at subsonic velocities,. 7-8 . several results of experimental investigations of plane lattices at small subsonic velocities,. 7-9 . flow of gas through lattice at large subsonic velocities,. critical mach number for lattice,. 7-10 . profile losses in lattices at large subsonic velocities,. 7-11 . flow of a gas through reaction lattices at supersonic pressure drops,. 7-12 . impulse lattices in supersonic flow,. 7-13 . losses in lattices at near sonic and supersonic velocities,. 7-14 . computation of angle of deflection of flow in overhang section of a reaction lattice at supersonic pressure drops,. 7-15 . characteristic features of three-dimensional flow in lattices .
the quasi-cylinder of specified thickness and shell loading in supersonic flow . the methods of the operational calculus are used to obtain a linear approximation to the shape of the mean camber surface of a quasi-cylinder in a supersonic flow in terms of its shell thickness and loading distributions . the analysis deals with a generalised quasi-cylinder ,. that is one which, although lying close to a mean cylinder, need not possess axial symmetry . the quasi-cylinder is also permitted to be within the small disturbance field of other separate components, e.g. a centre-body . because the linearised theory is inadmissable for internal duct flows close to and beyond the first reflected characteristic cone, the present solution is likewise invalid close to and beyond the position where this characteristic meets the mean cylinder . the work given here enables the camber shapes of /ring-wings/, which have been used theoretically to reduce or even nullify the wave-drag of a central slender-body, to be found . an example illustrates the general method .
a description of the r. a. e. high speed supersonic tunnel . an account is given of the high supersonic speed tunnel now nearing completion . the design philosophy is reviewed, the principal features are described and some of the more interesting development problems are noted .
calibration of the flow in the mach 4 working section of the 4ft . x 3ft . high supersonic speed wind tunnel at rae bedford . mach number and flow angle distributions in the working section of the mach 4 nozzle of the 4 ft x 3 ft high-supersonic-speed wind tunnel are presented for a range of total pressure and humidity .
free-flight measurements of the zero-lift drag and base pressure on a wind tunnel interference model (m=0 . 8 – 1. 5) . five free-flight models were flown to measure the zero-lift drag and body base pressure on a standard wind tunnel interference model over a mach number range of 0.84 to 1.48 . roughness bands on the wings and body of the model are shown to produce a small but definite increase in the zero-lift drag at all mach numbers . the measured drag is in fair agreement with corresponding measurements made in various transonic tunnels with differences that could plausibly be explained as the effects of tunnel interference . the effect of a simulated wind tunnel support sting is shown to increase the base pressure . the discrepancy between models with and without a sting is greatest at subsonic speeds and progressively decreases with increasing mach number until at m = 1.4 the sting has no effect on base pressure .
theoretical damping in roll and rolling moment due to differential wing incidence for slender cruciform wings and wing-body combinations . a method of analysis based on slender-wing theory is developed to investigate the characteristics in roll of slender cruciform wings and wing-body combinations . the method makes use of the conformal mapping processes of classical hydrodynamics which transform the region outside a circle and the region outside an arbitrary arrangement of line segments intersecting at the origin . the method of analysis may be utilized to solve other slender cruciform wing-body problems involving arbitrarily assigned boundary conditions . in the present report, the application of the method has shown .. differential incidence of both pairs of opposite surfaces of the cruciform wing-body combinations are practically independent of the body-diameter-maximum-span ratio up to a value of this ratio of 0.3 . arrangement is only 62 percent greater than that for a corresponding planar wing-body combination . dence of both pairs of the opposing surfaces of the cruciform wing-body arrangement, is only 52 percent greater than that for a corresponding planar wing-body combination . unit surface deflection) of the cruciform wing-body arrangement having four equally deflected panels is therefore 94 percent of the corresponding planar wing-body combination .
application of two dimensional vortex theory to the prediction of flow fields behind wings of wing-body combinations at subsonic and supersonic speeds . a theoretical investigation has been made of a general method for predicting the flow field behind the wings of plane and cruciform wing and body combinations at transonic or supersonic speeds and slender configurations at subsonic speeds . the wing trailing-vortex wake is represented initially by line vortices distributed to approximate the spanwise distribution of circulation along the trailing edge of the exposed wing panels . the afterbody is represented by corresponding image vortices within the body . two-dimensional line-vortex theory is then used to compute the induced velocities at each vortex and the resulting displacement of each vortex is determined by means of a numerical stepwise integration procedure . the method was applied to the calculation of the position of the vortex wake and the estimation of downwash at chosen tail locations behind triangular-wing and cylindrical-body combinations at supersonic speeds . the effects of such geometric parameters as aspect ratio, angle of attack and incidence, ratio of body radius to wing semi-span, and angle of bank on the vortex wake behind wings of wing-body combinations were studied . the relative importance of wing vortices, the corresponding image vortices within the body, and body crossflow indetermining the the total downwash was assessed at a possible tail location . it was found that the line-vortex method of this report permitted the calculation of vortex paths behind wings of wing-body combinations with reasonable facility and accuracy . a calculated sample wake shape agreed qualitatively with one observed experimentally, and sample results of the line-vortex method compared well with an available exact crossflow-plane solution . an empirical formula was derived to estimate the number of vortices required per wing panel for a satisfactory computation of downwash at tail locations . it was found that the shape of the vortex wake and the ultimate number of rolled-up vortices behind a wing depend on the circulation distribution along the wing trailing edge . for the low-aspect-ratio plane wing and body combinations considered, it appeared that downwash at horizontal tail locations is largely determined except near the tail-body juncture by the wing vortices alone for small ratios of body radius to wing semispan, and by the body upwash alone for large values of that ratio .
contributions of the wing panels to the forces and moments of supersonic wing-body combinations at combined angles . a wind-tunnel investigation was conducted at a mach number of 1.96 and at reynolds numbers (based on the mean aerodynamic chord of the exposed wing) of 0.36 and 1.03 million to determine the normal forces, pitching moments, and rolling moments contributed by each wing panel of a cruciform-wing and body combination over a wide range of combined angles of pitch and roll . the wings were triangular of aspect ratio 2, and the body was an ogive-cylinder combination . the effects of forebody length and roughness and of the presence of the adjacent panels on these panel contributions were determined . the results of the investigation show that large changes in the panel forces and moments can occur as the result of combined angles . a general theoretical method based on slender-body and strip theories was found to yield results in good agreement with the wind-tunnel measurements . these comparisons indicate that the changes in the panel characteristics due to combined angles are caused primarily by a cross coupling between the side-wash velocities due to angle of attack and sideslip and by the presence of forebody vortices due to crossflow separation . it was found that an increase in forebody length increases the effect of the forebody vortices because of the dependence of the strength of these vortices on the forebody length .
application of similar solutions to calculations of laminar heat transfer on bodies with yaw and large pressure gradients in high speed flow . an integral method for the rapid calculation of heat-transfer distributions on yawed cylinders of arbitrary cross-sectional shape and on bodies of revolution in high-speed flows is developed for laminar boundary layers . the method involves the quadrature of a function of the pressure distribution (assumed given) and satisfies the integral energy equation with the assumption of local similarity, wherein the actual boundary-layer profiles at every station are replaced by corresponding profiles from a family of similar solutions . the method is compared with other local similarity methods and with experimental heat-transfer data on a circular cylinder and on a body of revolution designed for large axial pressure gradients . good agreement between theory and data is obtained and it is shown that the present integral method, in both its complete and simplified form, gives generally better agreement with the data than certain other local similarity methods . numerical examples are presented showing that the effect of sweep and gas properties on heat-transfer distribution is small .
heat transfer in planetary atmospheres at super-satellite speeds . the main purpose of this investigation is to examine the dependence of heat transfer in planetary atmospheres on the total enthalpy up to flight velocities of 50,000 ft/sec where a large proportion of the atoms are ionized . the /total thermodynamic and transport property/ concept discussed by hirshfelder /j.chem.phys.,26/2/,feb.,1957/ is used .
hypervelocity stagnation point heat transfer . this analysis includes the specific contributions of atoms, molecules, tions are .. /i/ partially ionized air can be approximated as a four-component gas including n2, n, n and e,. /ii/ the gas is in local thermochemical equilibrium,. /iii/ there is no charge separation,. /iv/ thermal diffusion is neglected,. /v/ no electrical or magnetic fields,. low re effects are neglected .
stagnation point heat transfer measurements at super satellite speeds . brief description of experiments performed by using shock tube techniques for measurement of the stagnation point heating of a blunt body over a stagnation enthalpy range of 650 to 900, corresponding to velocities between 32,000 ft. per sec. and 39,000 ft per sec., respectively . data thus provided are used for comparison with theory .
a factor affecting transonic leading edge flow separation . a change in flow pattern that was observed as the free-stream mach number was increased in the vicinity of 0.8 was described in naca technical note 1211 by lindsey, daley, and humphreys . the flow on the upper surface behind the leading edge of an airfoil at an angle of attack changed abruptly from detached flow with an extensive region of separation to attached supersonic flow terminated by a shock wave . in the present paper, the consequences of shock-wave–boundary-layer interaction are proposed as a factor that may be important in determining the conditions under which the change in flow pattern occurs . when the mach number is high enough, the attached-flow pattern exists because then the shock wave is far enough behind the leading edge to keep the influence of the high pressure behind the shock wave from extending through the boundary layer to the immediate vicinity of the leading edge and affecting the flow there . some experimental evidence in support of the importance of shock-wave–boundary-layer interaction is presented .
compilation of information on the transonic attachment of flows at the leading edge of airfoils . schlieren photographs have been compiled of the two-dimensional flow at transonic speeds past 37 airfoils having variously shaped profiles, some of which are related and vary in thickness and camber . the data for these airfoils were analyzed to provide basic information on the flow changes involved and to determine factors affecting transonic-flow attachment, which is a transition from separated to unseparated flow at the leading edges of two-dimensional airfoils at fixed angles of attack as the subsonic mach number is increased .
evaluation of high angle-of-attack aerodynamic derivative data and stall-flutter prediction techniques . the problem of stall flutter is approached in two ways . first, using the m.i.t.-naca airfoil oscillator, the aerodynamic reactions on wings oscillating harmonically in pitch and translation in the stall range have been measured, evaluated, and correlated where possible with available published data, with the purpose of providing empirical information where no aerodynamic theory exists . the major effects of reynolds number, airfoil shape, and reduced frequency on the aerodynamic reactions have been reaffirmed . no instances of negative damping were observed in pure translatory motion and the ranges of negative damping occurring in pure pitch had the same general trends noted by other experimenters . data on the time-average values in the stall range of both lift and moment are presented for the first time . second, the results of numerous experimental observations of stall flutter have been reviewed and the various known attempts at its prediction have been examined, compared, and extended . the sharp drop in critical speed and change to a predominantly torsional oscillation usually associated with the transition from classical to stall flutter is apparently primarily but not entirely caused by the marked changes in moment due to pitch . fairly good stall-flutter predictions have been reported only when adequate empirical data for this aerodynamic reaction happened to be available for the desired airfoil shape, reynolds number range, and reduced-frequency range . a semiempirical method of predicting the variations of moment in pitch with airfoil shape, reduced frequency, initial angle of attack, and amplitude of oscillation has been presented .
some effects of variations in several parameters including fluid density on the flutter speed of light uniform cantilever wings . an experimental investigation has been made of some effects of variations in several parameters, including fluid density, on the flutter characteristics of light uniform cantilever wings . the assortment of wings tested covered a variety of positions of the elastic axis and center of gravity and values of the aspect ratio of 8, 6, and 4 . the relative-density parameter (where k is representative of the ratio of fluid density to wing mass) was varied over a range of values from 1.2 to nearly 14 . special emphasis has been placed on the lower values . the experimental investigation has been supplemented by an analytical investigation based on the two-dimensional aerodynamic theory for incompressible flow . in a few instances corrections for the effects of finite span have been made . in general, the theoretical results followed the trends indicated by experiment except at very low values of the relative-density parameter . for these low values the analytical considerations employed indicated a freedom from flutter not found experimentally . at higher values of the flutter-speed coefficient is shown to decrease with decreasing values of and to be nearly proportional to the inverse of the square root of the air density .
calculated and measured pressure distributions over the midspan section of the naca 4412 airfoil . pressures were simultaneously measured in the variable-density tunnel at 54 orifices distributed over the midspan section of a 5 by 30 inch rectangular model of the n.a.c.a. 4412 airfoil at 17 angles of attack ranging from -dash 20degree to 30degree at a reynolds number of approximately 3,000,000 . accurate data were thus obtained for studying the deviations of the results of potential-flow theory from measured results technique are presented . it is shown that theoretical calculations made either at the effective angle of attack or at a given actual lift do not accurately describe the observed pressure distribution over an airfoil section . there is therefore developed a modified theoretical calculation that agrees reasonably well with the measured results of the tests of the n.a.c.a. 4412 section and that consists of making the calculations and evaluating the circulation by means of the experimentally obtained lift at the effective angle of attack,. i.e., the angle that the chord of the model makes with the direction of the flow in the region of the section under consideration . in the course of the computations the shape parameter is modified, thus leading to a modified or an effective profile shape that differs slightly from the specified shape .
an approach to the flutter problem in real fluids . an approximate theory of airfoils in unsteady motion in a viscous fluid is proposed, in which viscous effects are accounted for by relaxing the kutta condition and replacing it by a relation derived from experiments in steady flow . applications here, are limited to moderate viscous effects below the stall . the possibility of one-degree- of-freedom flutter is discussed under this assumption . the discussion is partly extrapolated to the domain of stall flutter . some possibilities of further development of this theory for the stalled case are indicated .
on the application of mathieu functions in the theory of subsonic compressible flow past oscillating airfoils . an account is given of explicit solutions in terms of mathieu function functions of the problem of two-dimensional subsonic compressible flow past oscillating airfoils . the results are applied to the calculation of three-dimensional corrections for the two-dimensional theory and the effect of the incorporation of the three-dimensional effects on the mathieu function solution of the two-dimensional problem is shown . the developments are formal and must be supplemented by an appreciable amount of numerical calculations before the theory can be applied to specific problems .
wake of a satellite traversing the ionosphere . the particle treatment is applied to a study of the structure of the wake behind a charged body moving supersonically through a low-density plasma . for the case of a body whose dimensions are considerably smaller than a debye length, a solution is obtained which is very similar in structure to the solution obtained by using the linearized fluid dynamics equation . for the case of a disk whose radial dimensions are much larger than a debye length, two conical regions are found in the wake . at the surface of each of these cones, over thicknesses of the order of a debye length, the ion and electron densities are increased over their ambient values . formulae for the electrohydrodynamic drag on a wire, and on a large disk are obtained .
motion of thin bodies in a highly rarefied plasma . magnetic effects are considered negligible, and the velocity of the body is in a range between the electron and positive ion thermal speeds . the self-consistent field approach is used in which the electron distribution is assumed to be maxwellian, while the positive ion distribution function is given by the /collision-free/ boltzmann equation . it is assumed that the ion reflection at the body surface is specular, and the body is sufficiently thin so that the ion distribution function is a small perturbation of a maxwellian distribution . the solution for the simple case of a dielectric body with a given surface charge, as well as some general properties to be expected for a conducting body are given .
induction drag on a large negatively charged satellite moving in a magnetic-field-free ionosphere . an induction drag, experienced by a charged satellite during its traversal of the ionosphere, has been theoretically postulated by several authors . previous ‘exact’ treatments of the problem are inapplicable to large systems, and the semiempirical approach of jastrow and pearse may yield somewhat questionable results . the present description initially considers the satellite as a completely permeable spherical shell of charge, thus avoiding the difficult boundary conditions introduced by the ‘exact’ linearized treatment . the effects of permeability are then shown to be approximately removable by means of an iterative process . a final result, apparently valid to within an order of magnitude, is obtained for the drag force arising solely from electrical effects . its magnitude is considerably less than that obtained by jastrow and pearse .
interaction of a charged satellite with the ionosphere . the problem of the ion density distribution around a charged satellite has been treated by a numerical method which does not require linearization of the equations or restriction to infinitesimal objects . however, magnetic field effects were not considered, and a number of other simplifying assumptions were required . some sample calculations for spherical satellites are presented, illustrating the general character of the satellite wake . calculations of the so-called /charge drag/ were also made, yielding results qualitatively similar to those previously obtained by jastrow and pearse .
some physical interpretations of magnetohydrodynamic duct flows . this note presents some physical interpretations of magnetohydrodynamic duct flows with various boundary conditions viewed in the light of the effects of conducting walls on the pattern of electric current, taking examples from published results on rectangular ducts . the current patterns are illustrated in fig. 1 for rectangular ducts having various combinations of conducting and nonconducting walls, a uniform magnetic field being applied in the horizontal direction .
liapunov’s methods in automatic control theory . the work of a. m. liapunov and his theory of stability is discussed . the second method of liapunov is shown to have applications for linear equations with real constant coefficients, for a proof of the routh-hurwitz criterion, and linear equations with periodic coefficients . practical examples include non-linear stability problems of control, and the functions have uses in other areas of control systems .
symmetric joukowsky airfoils in shear flow . the velocity components of the fluid far from the airfoil are given by where c is the chord of the airfoil, and k are constants, u and v are velocity components in the directions of the coordinates x and y . the solution is sought in the form of the stream function and satisfies laplace’s equation . a general expression for for vanishing disturbance velocities at points far from the origin is written, and the flows due to a source, a vortex and a solid circular cylinder in shear flow are considered as examples . typical streamline patterns are shown for these cases . from the eulerian equations of motion the author obtains the expression for in terms of the parameter and derivatives of . the general form of is introduced and the appropriate solution for the pressure p is obtained . by integration around a contour enclosing the body, expressions are obtained, analogous to the blasius formulae, for the force and couple on any cylinder in this type of flow . these formulae are applied to the case of a symmetrical joukowsky airfoil . the method of conformal transformation is employed in the determination of . the boundary condition of tangential flow at the airfoil surface must be satisfied by the total flow in the airfoil plane, but this condition leads to a boundary condition for in the transformed plane . the kutta-joukowsky condition of finite velocity at the trailing edge also leads to a condition on in this plane . from these conditions and the general expression for the circulation and the strengths of the doublets and quadruplets required for the force and moment are determined . hence, the formulae for lift and moment coefficient are obtained . these involve, in addition to the usual (potential-flow) terms, terms proportional to . the ten functions that appear in the expressions for the lift and moment coefficients are tabulated for values of the thickness ratio between 0 and 1 . the aerodynamic-center position and the coefficient of the moment about the aerodynamic center are also calculated and are presented graphically as functions of .
the influence of two-dimensional stream shear on airfoil maximum lift . the cornell aeronautical laboratory is conducting a program of theoretical and experimental research on low-speed aerodynamics as applied to stol and vtol aircraft . the objective of this program is to re-examine certain aspects of classical aerodynamic information, in the light of low-speed flight requirements, with the aim of seeking aerodynamic processes which might be exploited to enhance law-speed performance . one aspect of propeller-driven aircraft which has recently received increasing attention is the existence of strong gradients of longitudinal velocity, or shear, in the propeller slipstream . this slipstream shear interacts with a wing surface and can alter the wing characteristics . in theoretical treatments of a wing interacting with a propeller slipstream, the first important simplification is the replacement of the slipstream with an ideal uniform jet, free of all velocity gradients . the application of these theories requires that one equate the actual slipstream to an effective uniform jet . one method employed is to assume the uniform jet has a momentum flux equal to the average in the propeller slipstream . these and similar procedures are well founded on momentum considerations., however, the implicit assumption is that the flow nonuniformity, the shear, does not influence the wing characteristics .
several approximate analyses of the bending of a rectangular cantilever plate by uniform normal pressure . three methods of approximating the deflections and moments occurring in a rectangular cantilever plate subjected to uniform normal pressure over its entire surface are presented in this paper . the first is the application of the well-known finite-difference procedure . the second and third are collocation methods, one based upon polynomial solutions of the lagrange equation, the other employing /mixed/ hyperbolic-trigonometric terms satisfying this equation . in the last two methods the boundary conditions are satisfied exactly along the clamped edge and at a finite number of points along the free edges of the plate . the results obtained for the particular case of a cantilever plate with uniform normal load indicate that the use of a relatively small number of points in the collocation method yields values of deflections and moments that are in substantial agreement with those given by the finite-difference procedure . it cannot be concluded from these results that the collocation method using the assumed functions will give satisfactory results with fewer points than the finite-difference method for cantilever plates with loading different from the one investigated .
modified cross-lees mixing theory for supersonic separated and reattaching flows . re-examination of the crocco-lees method has shown that the previous quantitative disagreement between theory and experiment in the region of flow up to separation was caused primarily by the improper c(k) relation assumed . a new c(k) correlation, based on low-speed theoretical and experimental data and on supersonic experimental results has been developed and found to be satisfactory for accurate calculation of two-dimensional, laminar, supersonic flows up to separation . a physical model which incorporates the concept of the /dividing/ streamline and the results of experiment . according to this physical model, viscous momentum transport is the essential mechanism in the zone between separation and the beginning of reattachment, while the reattachment process is, on the contrary, an essentially inviscid process . this physical model has been translated into crocco-lees languages using a semiempirical approach, and approximate c(k) and f(k) relations have been determined for the separated and reattaching regions . the results of this analysis have been applied to the problem of shockwave, laminar-boundary-layer interaction, and satisfactory a study of separated and reattaching regions of flow has led to quantitative agreement with experiment has been achieved .
a study of flow fields about some typical blunt-nosed slender bodies . complete inviscid flow fields about three model axisymmetric configurations have been determined numerically . configurations decreasing bluntness) and flight conditions have been selected so as to indicate separately effects of nose shape, drag coefficient, flight mach number, and thermodynamic behavior of the gas (either ideal calorically perfect gas or air in equilibrium dissociation) . results are presented for thirteen cases . particular attention is devoted to interpretation and, when possible, correlation of pressure distributions on, and shock shapes about, the cylindrical afterbodies . it is found that .. (a) the correlation of pressure distributions on bodies having nonspherical noses involves interpretive modifications of the law suggested by blast wave analogy . also shocks about these bodies are not described by parabolae,. (b) for all configurations there is substantial influence of gas behavior on shock shape,. this, however, can be correlated in terms of the gas conditions along a generally defined streamline,. (c) the shock layer can generally be divided into two regions (the first bound by the body and the aforementioned streamline, the second delimited by this streamline and the shock) wherein flow properties can either be approximated by simple laws or correlated .. (d) for each configuration knowledge of the complete flow field in one flight condition (even pertaining to ideal gas flow) can be used to estimate features of flows under general flight conditions including those where equilibrium dissociation is encountered .
on laminar boundary-layer flow near a position of separation . singularities are considered in the solution of the laminar boundary-layer equations at a position of separation . a singularity of the type here considered occurred in a careful numerical computation by hartree for a linearly decreasing velocity distribution outside the boundary layer ,. it may occur generally . whenever it does occur, the boundary-layer equations cease to be valid at and near separation on the upstream side, and also downstream of separation . the work suggests that singularities may arise in the solution of non-linear parabolic equations due to their non-linearity . the formulae found may help computers of laminar boundary layers, who desire more than a rough solution, to have an end-point at which to aim .
a new series for calculation of steady laminar boundary layer flows . a new and general method for solving problems of plane and steady laminar boundary layer flows in incompressible fluids with arbitrary outer pressure distribution is developed . this method is based on the introduction of the dimensionless quantities as new independent spatial variables . ordinates, u(x) the given outer velocity distribution, v the kinematic viscosity .) the solution of the boundary layer problem is then given as a power series in e with coefficient functions depending on n . this series is a formally exact solution of the boundary layer problem . the new series solution has the following qualities .. have the significance only of cartesian coordinates, the influence of wall curvature being neglected in boundary layer theory, the new coordinates are adjusted to the data of the special problem in any case of application . the new variables represent a logical development of former efforts in the field of boundary-layer flow calculation . with other series solutions known for some special cases is that the leading term of the new series satisfies exactly the outer boundary condition at all cross-sections along the wall . therefore, the succeeding terms give corrections only in the inner part of the boundary layer . accordingly, taking also no. 1 into account, the zero order term by itself gives a good approximation for the boundary layer flow .
on the solution of the laminar boundary layer equations . the theory of the laminar boundary layer offers a means of determining the skin friction under the assumption of a given velocity distribution outside the boundary layer . owing to the mathematical difficulties, however, exact solutions are possible only when the velocity distribution is expressed as a simple function of the distance along the surface . more complicated velocity distributions necessitate recourse to the method of expansion in series or that of step-by-step calculations, but the labor involved is too great for the methods to be of practical use . approximate method due to pohlhausen (1921), which had long been recommended for general use, gives a reasonably accurate solution in a region of accelerated flow, but recently its adequacy in a region of retarded flow has been questioned . separation of flow may actually occur where the solution of pohlhausen fails to give it . more recently howarth solution, which gives fairly reasonable results in a region of retarded flow . howarth’s solution essentially consists in solving the boundary layer equations for the particular case in which the velocity u outside the boundary layer decreases linearly with the distance x measured along the surface, and utilizing the solution by replacing the actual distribution of u by a circumscribing polygon of infinitesimal sides . therefore, it is assumed that the velocity distribution at any section depends on the velocity gradient du/dx at that section only, being affected by the conditions upstream only in so far as this affects the momentum thickness 0 . in other words, the velocity distribution across the boundary layer is determined by a parameter .
correlated incompressible and compressible boundary layers . the boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the prandtl number is unity . various results in the theory of incompressible boundary layers are then taken over into the compressible theory . in particular, the existence of method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity . a comparison with an exact solution is used to show that this method gives a closer approximation than does pohlhausen’s .
approximate methods fore predicting separation properties of laminar boundary layers . some new solutions for steady incompressible laminar boundary layer flow, obtained by gortler, have been used to test the accuracy of two methods which are commonly used to predict separation . a modification of stratford’s criterion for separation is given in this paper and is probably the most accurate and the simplest of all methods at present in use . modified numerical functions are also given for thwaites’s method of predicting the main characteristics of the boundary layer over the whole surface, which improve the accuracy of the method .
photo-thermoelasticity . this paper summarizes the optical and physical properties of the photoelastic model material paraplex p-43 over the temperature range from room temperature to -40 f . descriptions are presented of techniques and equipment developed to obtain the modulus of elasticity, the material fringe value, and the thermal-expansion coefficient as a function of temperature . experimental investigations were conducted into the plane-stress problems of a disk contracting upon an elastic inclusion and the transient thermal-stress field produced by a temperature differential suddenly applied to the upper edge of a long beam . the data are correlated with theory using the material properties obtained in the calibration phase . also included are photographic results of an exploratory investigation of the thermal-shock phenomenon produced by the sudden application of a temperature differential upon plastic beams of various length-depth ratios .
physical properties of plastics for photo-thermoelastic investigation . the optical and physical properties of paraplex p43, castolite, and epoxy resin hysol 6000-op, which are potentially of interest in photothermoelastic investigations, were investigated over a temperature range from +100 to -60 f . results on the thermal-expansion coefficient, the material fringe value, and the modulus of elasticity as functions of temperature are presented . also evaluated were thermal properties of importance in heat conduction . photothermoelastic figures of merit, which rate the optical sensitivity of materials in photothermoelastic applications, as well as a new method to determine this figure in a relative manner are presented .
flow studies on flat plate delta wings at supersonic speeds . an experimental study has been made to investigate some aspects of the nature of the flow around delta wings . vapor-screen, pressure-distribution, and ink-flow studies were made at a mach number of 1.9 on a series of semispan delta-wing models with slender wedge airfoil sections and very sharp leading edges . the models had semiapex angles ranging from 5 to 31.75 . separated regions of vorticity existed along the chords of all the wings in the series tested . concentrated vortex cores were found only on wings of very small semiapex angles . for wings with medium and large semiapex angles, the separated vorticity was concentrated in a region extending over the outboard part of the span and lying close to the wing upper surface . the results show that theoretical aerodynamic calculations, such as those in naca tn 3430, utilizing a single, separated vortex pair above the wing upper surface to represent the separated vorticity can be applied at supersonic speeds for very slender wings .
slender delta wings with sharp edges at zero lift . several slender wings of delta planform with sharp edges have been investigated theoretically at zero lift at subsonic and at supersonic speeds . most of the wings have diamond-shaped cross sections and are intended to lead to a type of flow with leading-edge separation in the lifting condition . the pressure distributions and overall normal-pressure drags resulting from various theoretical methods are compared with one another and some discussion is included concerning the possibility of achieving the results, calculated for an inviscid stream, in a real flow in the presence of a viscous layer around the body .
development of the vapour screen method of flow visualization in the 3ft tunnel at rae bedford. the vapour screen method of flow visualisation in supersonic wind tunnels is outlined, and the development of a suitable technique for use in the 3 ft tunnel described, together with the associated optical and photographic equipment . the results of tests to determine the humidity required to produce an optimum density of fog in the working section over the mach number range temperature discussed . numerous vapour screen photographs of the flow over and behind delta wings are included and some comparisons made with the corresponding surface oil-flow patterns . the process of condensation, the physical and optical properties of the resulting fog, and the formation of the vapour screen picture are all considered in some detail . the effects of humidity on the mach number and static pressure in the working section were investigated and the results are compared with theoretical estimates at a nominal mach number of 2.0 . it is shown that the adverse effects of condensation on the flow at high mach numbers may be alleviated by the use of liquids with a lower latent heat of evaporation than water, and some results obtained at a mach number of the possibility of extending the vapour screen technique to transonic and subsonic speeds is also considered, and some results obtained at a mach number of 0.85 are included .
thin airfoil theory based on approximate solution of the transonic flow equation . the present paper describes a method for the approximate solution of the nonlinear equations of transonic small disturbance theory . although the solutions are nonlinear, the analysis is sufficiently simple that results are obtained in closed analytic form for a large and significant class of nonlifting airfoils . application to two-dimensional flows with free-stream mach number near 1 leads, for instance, to general expressions for the determination of the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution and gives, furthermore, the correct variation of pressure with mach number at mach number 1 . for flows that are subsonic everywhere, the method yields a pressure-correction formula that is more accurate than the prandtl-glauert rule and compares favorably with existing higher approximations . for flows that are supersonic everywhere, the method yields the equivalent, in transonic approximation, of simple wave theory . results obtained by application of these general expressions are shown to correspond closely to existing solutions and to experimental data for a wide variety of airfoils .
a refinement of the linearised transonic flow theory . a new method is proposed to calculate the velocity and pressure distributions around a thin symmetrical aerofoil or a slender body of revolution flying at transonic speed . it is essentially a refinement of the linearized transonic flow theory due to oswatitsch and maeder, such that a correction term is introduced to take account of the nonlinear character of the transonic flow . as examples of application, a symmetrical circular-arc aerofoil and a circular-arc body of revolution in the sonic flow are dealt with, and the results are found to be in good agreement with experiments, except for the rear portion in the latter case .
linearised transonic flow about slender bodies at zero angle of attack . the simple linearized transonic flow theory as originally proposed by oswatitsch and keune(1) and by the present authors (2) is improved by considering and partially correcting its error . in this manner a theory which is easy to apply and which should be valid for a great number of smooth bodies is obtained . this improved theory predicts shock waves in the lower transonic regions . it is applied to a number of significant body and airfoil shapes and its predictions are compared with experiments and results of other theoretical investigations .
some notes for the small disturbance linear theory of the method of local linearisation of the flow over an airfoil at mach number of unity . in this paper, the pressure distribution at the surface of a symmetrical non-lifting aerofoil with free stream mach number of unity has been investigated by means of the small-disturbance linear theory or the method of local linearization . and by comparing with the calculated results based on an hodograph method, the accuracy of these approximate methods has been evaluated . moreover, when these approximate methods are used for the calculation of the pressure coefficient, some notes necessary to obtain more correct results have been discussed .
waves in supersonic flow . in this chapter we shall mainly consider problems of steady, two- dimensional (plane) supersonic flow . using the fact that in this case there is a steady wave system, we shall find solutions by an indirect approach . that is, we shall first study the conditions under which simple stationary waves may exist in the flow, and then find the flow boundaries to which they correspond or which may be fitted to them . in this procedure the limited upstream influence in a supersonic field is very helpful, for it allows flows to be analyzed or constructed step by step, which is a method that is not possible in the subsonic case .
freeman method . the freeman method (ref. 26) is similar to chester’s method in that the newtonian-plus-centrifugal solution (eq. with the von mises transformation . a method of successive approximations is applied to both plane and axially symmetric blunt-nosed bodies for small and infinite free-stream mach number . formulas for the streamlines, shock shape, and pressure distribution are determined to this approximation . a number of special shapes are treated in ref. 26, and in certain cases the theory has a singular point where the first approximation to the pressure vanishes., that is, for a sphere (see eq. 7-113) . as in chester’s method, the theory is not applicable where the pressure becomes too small .
laminar mixing of a compressible fluid . a theoretical investigation of the velocity profiles for laminar mixing of a high-velocity stream with a region of fluid at rest has been made assuming that the prandtl number is unity . a method which involves only quadratures is presented for calculating the velocity profile in the mixing layer for an arbitrary value of the free-stream mach number . detailed velocity profiles have been calculated for free-stream mach numbers of 0, 1, 2, 3, and 5 . for each mach number, velocity profiles are presented for both a linear and a 0.76-power variation of viscosity with absolute temperature . the calculations for a linear variation are much simpler than those for a 0.76-power variation . it is shown that by selecting the constant of proportionality in the linear approximation such that it gives the correct value for the viscosity in the high-temperature part of the mixing layer, the resulting velocity profiles are in excellent agreement with those calculated by a 0.76-power variation .
the velocity distribution in the laminar boundary layer between parallel streams . a method is given for obtaining the solution of the laminar boundary layer equations for the steady flow of a stream of viscous incompressible fluid over a parallel stream of different density and viscosity . an approximate solution is also obtained by means of the momentum equation . it is shown that the solutions depend only on the ratio of the velocities of the two streams and on the product of the corresponding density and viscosity ratios . numerical results are given, in the case where the lower fluid is at rest, for four values of and also when for one non-zero value of the velocity ratio .
the blasius equation with three-point boundary conditions . the blasius equation subject to three-point boundary conditions, describing the interaction between two parallel streams, is solved by way of a series in terms of ascending powers of the ratio equals /u1 -dash u2//u1, where the u1’s are the outer streams’ velocities . the first three terms of the series are analytically expressed in terms of the repeated integrals of the complementary error function /im erfc / and of the repeated integrals of the square of the successive integrals of the complementary error function /jmin erfc n/ . these functions often appear in problems leading to extended heat-conduction type of equations . a recurrence formula for jmin erfc n is established and formulae relating the functions in erfc /-dashn/ and jmjn erfc to available tabulated values of the functions in erfc /n/ are derived . the first three approximations to the blasius function and to its first two derivatives are also presented in tabulated form with four significant figures . test on the convergence of the series has been made by comparison with some exact solutions obtained by high speed computing machine . the comparison, extended to the physically essential quantities, shows that .. second and first derivatives . yield extremely accurate results . the errors in the first two derivatives of the blasius functions are always contained within less than one per cent .
laminar boundary layers at the interface of co-current parallel streams . the approximate solution of keulegan(1) for the steady flow of a stream of viscous incompressible fluid over another at rest is extended to the case where both fluids are moving co-current but at different velocities . this solution utilizes a sextic polynomial for the velocity distribution in the boundary layers . the solutions depend only on the ratio of the velocities of the two streams and on the product of the corresponding viscosity and density ratios . numerical results are given for seven values of at one value of . lock(2) has published an exact solution with a numerical result for and the sextic polynomial solution is evaluated f40umerical result for and the sextic indicates that in general the sextic polynomial is more accurate than the quartic polynomial but that the advantage is not great .
tabulation of the blasius function with blowing and suction . authors tabulate solutions of f”’ + ff” = 0 for the velocity distribution in a boundary layer . for each solution f'(0) = 0, the third boundary condition is the specification of f(0) . f(n) and its first three derivatives are tabulated to 5d in gaps of 0.1 in n for f(0) = -1.23849, -1.2(0.05) 0.5 (0.1) 1.5, introduction gives method of solution and physical meaning of boundary conditions, etc . lock’s (amr cussed .
on an equation occurring in falkner and skan’s approximate treatment of the equation of the boundary layer . the differential analyser has been used to evaluate solutions of the equation y”’ = -yy” + with boundary conditions y = y’ = 0 at x = 0, as which occurs in falkner and skan’s approximate treatment of the laminar boundary layer (see abstract 1081 (1932)) . a numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about insufficient to specify a unique solution for negative values of,. a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that from below, and as rapidly as possible, as . the condition that from below can be satisfied only for values of greater than a limiting value whose value is approximately -0.199, and which is related to the point at which the laminar boundary layer breaks away from the boundary .
adiabatic wall temperature due to mass transfer cooling with a combustible gas . a recent technical note by sutton (1), with the above title, discusses the influence of the burning of a transpiration coolant on the quantity of coolant necessary to maintain a given wall temperature . the present note discusses the same problem in a way which has been found useful in calculating the burning rates of solid and liquid fuels (2) . consider the transpiration cooling of a porous surface in a gas stream . then a simple modification of the general mass .
mass transfer cooling of a laminary boundary layer by injection of a light weight foreign gas . analytical predictions are given for the development of the velocity, temperature and concentration fields in a laminar air boundary layer on a flat plate in high-speed dissipative flow, the plate being considered porous and cooled by injection of hydrogen from its surface . the admixture of hydrogen, having a low density and high thermal capacity relative to air, is shown to greatly diminish the skin friction and to markedly relieve the adverse thermal effects of intense aerodynamic heating under conditions of hypersonic flow .
a re-examination of the use of the simple concepts for prediction the shape and location of detached shock waves . a reexamination has been made of the use of simple concepts for predicting the shape and location of detached shock waves . the results show that simple concepts and modifications of existing methods can yield good predictions for many nose shapes and for a wide range of mach numbers .
stagnation point shock detachment distance for flow around spheres and cylinder . development of an analytical relation between shock detachment distance and free-stream mach numbers . results are presented graphically for shock detachment distance of cylinders and spheres in air .
the influence of two-dimensional stream shear for airfoil maximum lift . the effects of stream velocity gradients on airfoil maximum lift are defined with experimental data obtained in a simulated two-dimensional slipstream . the experimental results show that when positioned near the slipstream plane of symmetry, the airfoil maximum lift varies markedly with location in the slipstream . in moving the airfoil from above to below the slipstream plane of symmetry through a total distance corresponding to the airfoil thickness, force data and boundary-layer observations show that boundary-layer separation is delayed to higher angles of attack, and the airfoil maximum lift is doubled . it is concluded that the destalling effect observed in the non-uniform slipstream is not associated with slipstream boundary interference, but stems from the influence of the large local slipstream shear on airfoil characteristics . the effects of uniform and nonuniform shear on airfoil lift and pressure distribution are discussed, within the framework of existing first-order, small-shear theory, to show that these effects of shear tend to promote stall . a pohlhausen calculation of the laminar boundary layer in a stream with shear is used to identify and to assess the effects of stream shear on boundary-layer separation criteria . it is demonstrated that these effects are negligibly small, and that the uniform-flow criterion applies . it is concluded on the basis of the experimental data that the observed destalling phenomenon stems from a shear effect of higher order than those treated in the inviscid theories . it is hypothesized that it is a second-order effect, fixed by the product of the stream shear and the derivative of the shear, which was large in the present experiments .
linear heat flow in a composite slab . the temperature is determined as a function of position and time in the case of linear heat conduction in a composite slab of ture throughout, and the two external surface temperatures are considered to be prescribed functions .
similarity laws for aerothermoelastic testing . the similarity laws for aerothermoelastic testing are presented in the range . these are obtained by making nondimensional the appropriate governing equations of the individual external aerodynamic flow, heat conduction to the interior, and stress-deflection problems which make up the combined aerothermoelastic problem . for the general aerothermoelastic model, where the model is placed in a high-stagnation-temperature wind tunnel, similitude is shown to be very difficult to achieve for a scale ratio other than unity . the primary conflict occurs between the free-stream mach number reynolds number aeroelastic parameter heat conduction parameter and thermal expansion parameter . means of dealing with this basic conflict are presented . these include (1) looking at more specialized situations, such as the behavior of wing structures and of thin solid plate lifting surfaces, and panel flutter, where the aerothermoelastic similarity parameters assume less restrictive forms, (2) the use of /incomplete aerothermoelastic/ testing in which the pressure and/or heating rates are estimated in advance and applied artificially to the model, and (3) the use of /restricted purpose/ models investigating separately one or another facet of the complete aerothermoelastic problem . some numerical examples of modeling for the general aerothermoelastic case as well as for the specialized situations mentioned in (1) above are given . finally, extension of the aerothermoelastic similarity laws to higher speeds and temperatures is discussed .
theory for supersonic two-dimensional, laminar, base-type flows using the crocco-lees mixing concepts . a separated flow field, in which the incoming boundary layer is undisturbed up to the separation point, is defined as a / base-type/ flow . examples are the flows over a blunt base and over a backward-facing step . the crocco-lees theory is applied to the supersonic, two-dimensional, laminar, base-type flows defined above . the separated flow is divided into a mixing region and a recompression (or reattachment) region . calculations of base pressure show its dependence on the mach number and on two reynolds-number-dependent variables, and . it is shown that existing base-pressure data can be explained by these results .
a reaction-rate parameter for gasdynamics of a chemically reacting gas mixture . presented note proposes a linearized reaction rate parameter which is applicable to any reacting gas mixture provided all the pertinent reactions and their rate constants are known at the thermodynamic conditions under consideration . linearizing is achieved by expanding equation of rate of chemical reaction in a taylor series and neglecting higher-order terms . author announces that tables of linearized reaction rate parameters for dissociated and slightly ionized air are now in preparation at the space sciences laboratory, general electric co., msvo . comparison of preliminary results with exact calculations published by hall, i. g., et.al., /inviscid hypersonic air-flows with coupled non-equilibrium processes/ (ias paper 62-67, 30th annual meeting, new york, jan. 1962) indicates good agreement .
on calculation of the laminar separation point and results of certain flows . paper studies compressible laminar boundary layer in adverse pressure gradient . after mentioning mathematical instabilities in howarth’s and like solutions, authors quote equation from one of the references, based on the assumptions that zero heat transfer and y = 1.4 . thence authors compute nondimensional distances to separation, comparing with solutions by other workers . results are interesting, though reviewer feels rather unhappy about approximations leading to eq. (4),. more detailed justifications should have been given . thus we have the statement ber, as ./ surely a fuller discussion of effects of letting is warranted . typography in eqs. (2) and (3) is rather confusing and there is a typographical error in heading to table 2 .
normal-shock relations in magnetohydrodynamics . the magnetic-field vector is perpendicular to the flow direction,. thus for normal shocks there is no change of flow direction through the shock front . this class of shocks is included in investigations by several authors (five are referred to here), but the presentation here is thought to be especially convenient . all downstream quantities are given in terms of upstream flow conditions, including the upstream ratio of alfven speed to sound speed, and the shock strength (density ratio) .
on the close relationship between turbulent plane-couette and pressure flows . author studies the velocity profiles measured by others in plane and turbulent couette flow, such as is induced in parallel channels of which one of the walls moves in its own plane . he finds these profiles to be satisfactorily describable in terms of the seventh-power law, which was originally set up for plane and turbulent pressure flow in channels where both walls are stationary . further, he finds the shear law for pressure flow, to be applicable also to the couette flow, in a similar range of reynolds number, r . no attempt is made in this concise contribution to put these findings on a firmer basis through a theoretical explanation .
prediction of ogive-forebody pressures at angles of attack . various approximations are being suggested for obtaining surface pressures on arbitrary bodies at angle of attack . this not presents a method for obtaining an approximate pressure distribution over the lower surface of an ogive forebody at angle of attack by utilizing the calculated pressures for zero angle of attack .
real-gas laminar boundary layer skin friction and heat transfer . the laminar-boundary-layer equations have been integrated for the case of a flat plate over a wide range of free-stream enthalpies and velocities and over a wide range of enthalpies of the gas at the wall . the range of free-stream velocities extended up to 25,000 ft sec at low free-stream enthalpies, corresponding to local conditions on a slender body traveling at high speeds . at low free-stream velocities, the range of free-stream enthalpies extended up to 400,000 btu slug, corresponding to the local conditions on a blunt body traveling at speeds up to 25,000 ft sec . the gas was assumed to be in thermodynamic equilibrium at each point in the boundary layer and diffusion effects were neglected . the solutions to the boundary-layer equations were carried out on a high-speed digital computing machine, both skin-friction and heat-transfer coefficients being obtained from the computations . before presenting the results, the t’ method of rubesin and johnson for computing skin-friction coefficients for the perfect-gas case is reviewed . for the real-gas case, the average temperature, t’, is replaced by the average enthalpy, h’, and the h’ method is then used to compute skin-friction coefficients . these values are in excellent agreement with the computing-machine results . it was found that the recovery factor for the real-gas case can be approximated by, the best results for the cases considered being obtained if a value of pr corresponding to the enthalpy, h’, is used . using this recovery factor and reynolds analogy, heat-transfer rates can be computed which, with a few exceptions, are within 5 percent of values obtained from computing-machine results .
axisymmetric viscous flow plast very slender bodies of revolution . axisymmetric viscous flow past unyawed very slender bodies of revolution is treated within the category of the perfect gas . attention is paid especially to the effect of transverse curvature of the body . from the transformed equations, the similarity conditions are deduced, and the parameter characterizing the effect of transverse curvature is obtained . several numerical solutions of similarity equations for hypersonic flows are presented, and upon the basis of these results, the effect of the transverse-curvature parameter is discussed . a method of applying the local-similarity approximation to obtain the approximate solution for nonsimilar cases is described, as are practical applications to incompressible flow past a long cylinder and to hypersonic flow past a very slender cone . comparison with experimental results shows fair agreement with calculations using the local-similarity approximation in the present range of experimental flow conditions .
on similar solutions for strong blast waves and their application to steady hypersonic flow . the general solution of the strong blast wave is found in the newtonian approximation–i.e., neglecting terms of order the expressions obtained for the pressure, temperature, density, and velocity profiles are simple . the results are applied to power-law bodies in hypersonic flow using the equivalence principle . higher-order approximations for strong blast waves are investigated for the cases in which the shock layer is thin . a simple pressure formula is found, which constitutes an improvement upon the newton-busemann formula, and some of its applications are shown .
a theory of transonic aileron buzz, neglecting viscous effects . usaf-sponsored analysis of the unsteady perturbations of two-dimensional transonic flow around an airfoil, where local supersonic regions terminated by shock waves are present in the vicinity of the airfoil . viscous effects are neglected, and a linearized theory of the perturbations due to harmonic oscillations of an aileron is developed . a series solution for the pressure distribution is obtained, and numerical results for the nonsteady hinge moment, from the first approximation to the solution, are presented . as a result of flutter analysis a stability boundary for transonic aileron buzz is obtained . comparison of the theoretical results with experimental observations shows satisfactory agreement .
theoretical and experimental investigation of thermal stresses in hypersonic aircraft wing structures . a simple and relatively accurate analytic approximation is developed to determine the temperature and thermal-stress distribution in aircraft wing structures . theoretical investigations show that the results of the existing thermal-stress theories which neglect the temperature gradient through the skin thickness may exceed, in the range of higher biot numbers, the true values by more than 30 percent . refined photothermoelastic experiments verify these results and add another significant conclusion . they indicate that thermal stresses in wing structures generated by a variable heat-transfer coefficient coincide with the theoretical predictions which are based on a constant heat-transfer coefficient, as long as the latter represents the arithmetic average over the heating cycle and the variation is in the order of 10 percent . however, even much greater variations in the order of 100 percent produce only relatively small differences .
calculation of potential flow about bodies of revolution having axes perpendicular to the free-stream direction . a general method is described for calculating, with the aid of an electronic computer, the potential flow about arbitrary bodies of revolution whose axes are perpendicular to the free-stream direction . when combined with the solution for the axisymmetric flow about these bodies, this method makes it possible to calculate the pressure distribution on any body of revolution at angle of attack forward of any separated region of the flow, and also to calculate the flow at points off the body surface . after the basic equations of the method have been derived, its accuracy is exhibited by comparison with analytic solutions for ellipsoids of revolution . calculated pressure distributions are then compared with experimental data for a variety of bodies . the agreement is quite satisfactory in all cases . the calculated velocities for other selected bodies are presented to exhibit certain properties of this type of flow .
a closed-form solution for the oscillations of a vehicle entering a planetary atmosphere . author considers the equation of the yawing motion of a missile, derived with a series of customary assumptions and with the distance traveled as the independent variable . his assumptions include the linearity of the aerodynamic forces, the constancy of the aerodynamic coefficients with respect to mach number, the absence of spin, and the absence of gravity . if to these assumptions one could add the common ballistic assumption of a constant air density, the coefficients of this equation would have been con-damped sinusoids . in ballistics any slow variation of these coefstant, and the solution would have been simply the exponentially-ficients is usually treated by adding an approximate correction term to the damping rate (which is spoken of as the wkb perturbation) . however, with a body entering the planetary atmosphere the variation of the air density is apparently of greater essence (this is a point not stated explicitly in this brief communication), and the equation is of the type . the author shows that with a series of further transformations the equation can be reduced to the form the solutions of which are confluent hypergeometric functions . these functions are defined as series involving gamma functions, and with a series of further assumptions can be reduced to laguerre polynomials and bessel functions . it is certainly nice to have an exact solution to a problem which has heretofore been extensively treated by approximations and by the numerical approach . this reviewer is puzzled, however, as to the practical significance of the proposed approach . an idealization is of value in that it facilitates our understanding,. and the numerical approach, in that it allows refinements of the problem, freeing us from the necessity of idealizing . but the proposed solution is certainly more difficult to refine than the original problem,. and it is certainly not simple (the solution of the original equation is not the value of z, but the various /reverse/ transformations of z) . an evaluation of a series in practice must compete with the numerical approach,. and the equation suggested is of the zero) . viewing the problem /afresh/ (in the light of the / computer revolution/ and without the constraints imposed by the prior art), it seems at least equally easy to /standardize/ the solutions of the original equation .
joule heating in magnetohydrodynamic free-convection flows . the steady, fully developed, laminar, free-convection flow of an electrically conducting fluid between two fully submerged open-ended, constant-temperature vertical plates located in a constant, uniformly distributed, transverse magnetic field has been analyzed with the joule heating term retained in the energy equation . analytic results are obtained . such analytic results are useful in estimating the actual magnitude of the influence of joule heating as well as a qualitative description of the manner in which it alters the temperature and flow fields . the present result confirms the usual practice that the influence of joule heating is negligibly small .
stagnation-point shock detachment of blunt bodies in supersonic flow . presentation of stagnation-point shock-detachment distances determined by the exact numerical method of gravalos, edelfelt, and emmons . the results are compared with those from the previously published methods of van dyke and gordon, li and geiger, and serbin, and with experimental data .
on squire’s test of the compressibility transformation . discussion of a previous application, by squire, of the author’s compressibility transformation to the correlation of high-speed boundary-layer data for air and helium . squire’s suggestion that the compressibility transformation is invalid is shown to be incorrect .
theoretical prediction of the transonic characteristics of airfoils . it is shown that the author’s transonic-flow airfoil theory can be used to estimate transonic drag-rise and onset-of- separation-effects mach numbers without reference to experimental results . a simple comparative method is applied to a series of airfoils, and the results are analyzed to determine some of the design features of importance in transonic flow . an improvement to this scheme is shown to give results in good agreement with experiment for both the first appearance of shock waves and the onset of separation effects . application to finite swept wings is briefly considered and illustrated .
stability of compressible boundary layers induced by a moving wave . the problem of determining the stability of compressible viscous flows with nonzero surface velocities is formulated and is shown to be identical to that for conventional boundary layers, with only a redefinition of the mach and reynolds numbers required . specific consideration is given to the wall boundary layer behind a moving shock wave, and the minimum critical reynolds numbers are obtained for various shock velocities . the entire stability map is determined for the limiting case of a weak wave, which is analogous to the rayleigh problem . the minimum critical reynolds number is found to increase monotonically with shock velocity–i.e., with increasing surface cooling and stream mach number combined . for the ratio of wall to stream velocity of 2.92 with (shock mach number of 2.18) the flow is found to be infinitely stable to two-dimensional disturbances . experimental transition data do not follow the trends predicted by the theory . in fact, the transition reynolds numbers are orders of magnitude below the computed minimum critical reynolds numbers . the lack of correlation between theory and experiment is attributed to disturbances which are external to the boundary layer .
transition measurements on cones in free flight ballistics range tests . navy-sponsored experimental investigation of the location of boundary-layer transition on sharp-nosed cones having 10 total angles . the ambient temperature in a portion of the aeroballistics range is varied so as to obtain different adiabatic recovery temperatures at a constant nominal mach number of 3.1 . the location of transition is expressed as a transition reynolds number, and results are presented graphically as a function of the ratio between the wall temperature and the adiabatic recovery temperature .
a note on havelock’s shallow-water wave-resistance curves . in the continuous quest for improved means of transportation, attention is currently focused on the ground-effect machine . as there is no physical contact between the vehicle and the terrain over which it operates, its performance should be similar over land and water . however, over water there is an additional resistance to motion due to the gravity-wave system generated by the supporting or /cushion/ pressure acting on the water surface . estimates of this component can be made using the analysis of t. h. havelock . it is the purpose of this note to present an ibm 650 digital-computer solution of his equations . as shown below, these results differ from havelock’s original results .
energy equation approximations in fluid mechanics . discussion of several forms of the energy equation and of their use for the study of the flow of nearly incompressible fluids .
a correlation of nose-bluntness induced pressures on cylindrical and conical after-bodies at hypersonic speeds . van hise, in his detailed study of the nose-bluntness-induced pressures on cylindrical afterbodies, shows that, starting a few nose diameters aft of the nose-afterbody junction, these pressures are correlated with the parameter as predicted by the blast-wave analogy . chernyi developed a modified form of the blast-wave analogy which takes into account the addition of energy to the flow by a thin afterbody . he showed that for thin afterbodies and hypersonic speeds, the pressure distribution, plotted as should correlate with the parameter . the purpose of this note is to show that the above correlation techniques may be combined into a form such that pressures on cylindrical and conical afterbodies are correlated by one parameter .
a graphical approximation for temperatures and sublimation rates at surfaces subjected to small net and large gross heat transfer rates . considers a material, acted upon by heat of conduction, which changes its state by sublimation at the heated surface . the derived method is most suitable under conditions of severe heating such as space vehicle re-entry .
manoeuvring technique for changing the plane of circular orbits with minimum fuel expenditure . usaf-supported discussion of the use of an intermediate elliptic orbit for changing the plane of a circular orbit . values of the perigee and apogee velocities are calculated for the following cases .. the braking impulse supplied by grazing of the atmosphere,. and (3) re-orbit with 90 of the braking impulse supplied in this manner .
tunnel tests on a double cascade to determine the interaction between the rotor and the nozzles of a supersonic turbine . experimental confirmation has been required that in a supersonic turbine the leading edges of the rotor governs the rotor incidence and, hence, the gas exit angle from the nozzles . evidence has also been required that, once the rotor incidence has been allowed for, there is no adverse effect of the rotors on the nozzle flow, even when the rotors have a large turning angle . the present test cascade represented the stationary configuration of a turbine of 2.5 nozzle mach number and 74 swirl angle, the rotors being designed to operate at 1.9 relative mach number and to provide a turning angle of 140 . in the tests, fully supersonic flow could be established through the system, but the losses were fairly high and an increase in loss of about 25 per cent would have caused choking in the rotor .
quasi-cylindrical surfaces with prescribed thickness distributions . a formula for the supersonic velocity field in terms of a given surface distribution of sources is applied to points lying in the surface . an equation giving the camber shape of a quasi circular-cylindrical surface in terms of a prescribed thickness distribution is derived and the half ring wing with prescribed thickness distribution is discussed as an example .
pressure measurements at supersonic speeds on three uncambered conical wings of unit aspect ratio . pressure measurements were made at mach numbers between 1.3 and 2.8 over a range of incidences on three simple models representing thick conical uncambered wings with sharp leading edges . these tests form part of an investigation into the effects of thickness and camber on slender wings . the aspect ratio of the models was unity in each case, and the spanwise cross sections were bounded by .. the measured pressure distributions are presented, along with overall lift and drag (excluding skin friction and base drag) obtained by integration .
pressure distributions and flow patterns on some conical shapes with sharp edges and symmetrical cross-sections at m=4 .0. results are given of a wind tunnel programme made to study the pressure distributions and flow patterns over a series of simple, conical shapes at a mach number of 4.0 . the results have been compared with various approximate theories and the limitations of these theories are discussed . it is found that at this mach number leading edge separations still have an influence on the suction surface pressure, and that this surface still makes a significant contribution to the overall forces .
self sustained oscillations of a system with non-linear damping of a particular type . the paper deals with self-sustained oscillations of a dynamic system of single degree of freedom, with linear restoring force and non-linear damping force . the latter is supposed to be a function of velocity representable by a simple /polygonal/ graph, such that the damping is negative at small velocities but becomes positive at velocities above a certain value . on these assumptions, a rigorous solution is presented, including the equations of motion, amplitude, maximum velocity and period . a very simple solution is obtained for the limiting case of vanishingly small damping . an approximate solution by series in powers of damping ratio is worked out which gives a satisfactory accuracy for quite large values of .
free-flight measurements of the dynamic longitudinal stability characteristics of a wind tunnel interference model (m=0 .92 to 1. 35) . the dynamic longitudinal-stability characteristics of a standard wind tunnel interference model have been investigated in free flight over a mach number range of 0.92 to 1.35 . measurements of lift-curve slope and manoeuvre margin were obtained, and are compared with results from transonic-tunnel tests under low blockage conditions . the analysis was extended to obtain damping derivatives to allow comparison to be made with possible future dynamic tests in wind tunnels on the standard shape .
reaction-resisted shock fronts . it is shown that shock waves whose structure is determined solely by the effects of chemical reactions (reaction-resisted shock fronts) are possible and completely analogous to relaxation – resisted waves . a single dissociation reaction is considered and numerical results indicate that such waves could be observed experimentally . bulk viscosities equivalent to reaction effects are possibly 10 or more times shear viscosity values . (examples are based on lighthill’s ideal dissociating gas) .
heat conduction through a polyatomic gas . a heat conduction problem is set up which, in essence, simulates the conditions arising when a plane shock wave reflects from a co-planar solid boundary . the gas is assumed to be polyatomic, with one the quantity of primary interest is the temperature of the solid at the interface, since this can be observed experimentally without much difficulty . solutions are obtained for this quantity which cover a range of practically plausible relaxation times and ‘wall effect’ parameters . it is essential to include proper temperature jump boundary conditions for both active and relaxing (or inert) energy modes . thus it is necessary to know accommodation coefficients for these modes of energy storage . the temperature jump effects are found to dominate the (interface) solid’s temperature time history, with relaxation effects playing a very secondary role . the theoretical results are compared with some experimental observations and encouraging agreement is found . as a result of this agreement it proves possible to estimate the accommodation coefficient for the active modes (in this case for the combination platinum air), the pressure being about 15 atmospheres . the pressure sensitivity of accommodation effects is commented on .
base pressure at supersonic speeds in the presence of a supersonic jet . the effects on base pressure of jet mach number, free stream reynolds number and jet to base diameter ratio have been investigated experimentally . it was found that, for jet stagnation pressures greater than that required for the nozzle to reach its design mach number, an increase of jet mach number reduced the base pressure . similarly the base pressure increased with increase of the ratio of jet diameter to base diameter and, at high jet stagnation pressures, base pressures higher than free stream static pressure were found . the base pressure was independent of free stream reynolds numbers greater than 2 x 10 per foot but increased with reduction of reynolds number below 2 x 10 per foot . unsteady wave patterns were found when the jet mach number did not differ markedly from the free stream mach number and the jet had just reached its design conditions .
wing-tail interference as a cause of ‘magnus’ effects on a finned missile . wing-tail interference is shown to cause large /magnus/ effects on a finned missile whose wings are deflected into an aileron setting . a simple experimental method with water as the working medium is used to obtain low-speed magnus data on a rolling missile . the missile is a slender cruciform configuration with all-movable wings and fixed tail fins . magnus data are presented for angles of attack up to 15 and for the one (high) roll rate which accompanies a 30 aileron deflection angle of the wings . tests conducted at zero roll rate but with the wing deflection maintained, revealed large forces in the magnus direction, thereby providing the basis for understanding magnus effects due to wing-tail interference . a semiempirical theory is proposed to explain the experimental data . a simplified model of the wake behind the wings is introduced to predict tail-interference factors . good agreement with the data is obtained . this magnus effect is opposite in direction to the classical magnus lift on a spinning cylinder ,. it is much larger than either that effect or the one on a missile with only one set of fins . wing-tail interference is the predominant source of the effect ,. roll rate only modifies the basic interference mechanism .
a note on application of transonic linearization to an airfoil with a round leading edge . the profile of a symmetric airfoil of unit length with a round leading edge can be expressed, in general, as where p(x) has a finite slope at x = 0 . it is well known that the conventional sub- and supersonic linear theories of compressible flow break down in the neighborhood of such a round leading edge due to the failure of the small-disturbance assumption . the linearized transonic flow theory has the same short-coming, but if the determination of the sonic point on the airfoil plays an important role in any more advanced theory–e.g., spreiter’s local-linearization method or hosokawa’s method of refinement–this theoretical barrier will become more serious because the sonic point is usually located in a flow region near the leading edge that may be greatly affected by the roundness .
laminar, transitional and turbulent heat transfer to a cone-cylinder-flare body at mach 8. 0. an experimental investigation of the laminar, transitional, and turbulent heat transfer rates over a conical cylindrical flared body is presented . regions of favorable, zero, and adverse pressure gradient on the body are investigated . the experimental results are compared with the theories available in the literature . the model chosen for this investigation is a cone-cylinder-flare configuration consisting of a 20 semivertex conical nose portion smoothly blended by a shoulder radius into a long cylindrical body and terminated by a smooth large radius flare . the model was tested at a free stream mach number of 8, over a range of reynolds number from 0.3 x 10 to 1.6 x 10 per inch based on free stream conditions . various stagnation-to-wall temperature ratios were obtained by cooling the model prior to the test with liquid nitrogen . the stagnation-to-wall temperature ratios were 10 and 3.3 . the theoretical predictions gave good results for the heat transfer rates in the laminar region, and fair prediction in the transitional and turbulent regimes extending over the shoulder and forward portion of the cylindrical body . over the aft portion of the cylinder and over the flare the predictions are only qualitatively correct, and underestimate the heating rate by a factor as high as 3 . conversely, the /flat plate reference enthalpy/ over the aft portion of the body, but to increasingly overestimate the heating rates over the forward portion of the cylinder . a modified equation for the heat transfer coefficient in the transitional and fully turbulent region based on the f.p.r.e. method is then presented . this method gives good agreement with the experimental results presented over the entire range of transitional and turbulent flow . from the results the following is concluded .. cooling the wall delayed transition . by expanding the flow rapidly between the cone and the cylinder, the transition reynolds number is reached very rapidly . by making a smooth transition between the cylinder and the flare, no separation occurred at the cylindrical flare junction . the transitional and turbulent heat transfer in the presence of an adverse pressure gradient may be predicted with sufficient accuracy by the f.p.r.e. method .
approximate determination of position of the sonic line for a blunt body in hypersonic flow . the detached shock in front of a blunt body in hypersonic flow tends to acquire the shape of the frontal curvature of the body . thus the curvature of the shock can be assumed to be the same as that of the body, at least up to the sonic point (point a, fig. 1) . if the equation of curvature of the body is known, the equation of curvature of the shock is also known . in this paper, with this assumption, a method is described to determine the approximate position of the sonic line (ao’b, fig. 1) . the shock-detachment distance is assumed known . the method is, of course, general . this can be applied to any detached shock provided its equation of curvature is known corresponding to that of the body . for simplicity the detached shock is assumed to be circular in this paper and the procedure is outlined below with the assumption that the sonic line ao’b is parabolic .
stagnation point heat transfer in partially ionized air . comparison of heat-transfer rates, obtained by using transport properties recently reported by peng and pindroh, with rates based on hansen’s thermodynamic and transport properties . it is shown that the heat-transfer rates based on the peng and pindroh data are 20 to 30 lower for the velocity range of 25,000 to 40,000 ft sec .
on hypersonic viscous flow over an insulated flat plate with surface mass transfer . hypersonic viscous flow over an insulated flat plate with surface mass transfer is studied . the tangent-wedge approximation is used in the inviscid-flow region, and the integral method is applied to the treatment of the laminar boundary layer . the law of surface mass transfer for the present analysis is derived . a continuous transition of the pressure variation is achieved from the strong to the weak pressure-interaction region . first-order formulas for the induced surface pressure and the skin-friction coefficient are obtained for both the strong and weak pressure-interaction regions . results are compared with those calculated from other analyses .
leading edge attachment in transonic flow with laminar or turbulent boundary layers . the transonic flow round a two-dimensional airfoil at incidence is often determined by the type of flow in the leading-edge region . if the flow separates at the leading edge at low speeds it is liable to attach as the speed increases, often quite suddenly . a review of this change with laminar or with turbulent boundary layers re-emphasizes the importance of fixing transition when making model tests at transonic speeds in order to obtain flows closest to full-scale conditions . it is shown that similar airfoils with attached leading-edge flow show transonic similarity upstream of the terminal shock .
note on the three-point boundary layer problem for the blasius equations . in a recent paper a method was presented for obtaining higher accuracy in the numerical solution of the blasius equation with three-point boundary conditions . the well-known blasius equation was previously developed in an investigation of the steady two-dimensional incompressible boundary-layer flow over a flat plate, but it has been extensively used in investigating other fluid flow problems . the three-point boundary-value problem is encountered in the theory of laminar mixing and in approximate analyses of separated and wake flows as noted in ref. 1 .
first-order slip effects on the laminar boundary layer over a slender body of revolution with zero pressure gradient . in reference 1, the analysis given by probstein and elliott for the zero-pressure-gradient, constant-wall-temperature, compressible, laminar boundary layer with transverse curvature was extended to first-order slip flow . this extension was based on a double asymptotic expansion in a transverse-curvature parameter and a slip parameter . the expansion in ref. 1, however, was carried out with the parameter held constant . for and a constant wall temperature, is constant and e varies with x due to the dependence of the local body radius on x . thus, for arbitrary body shapes, e will not be constant . in the present note, the analysis of ref. 1 is re-examined taking into account the variation of e .
some effects of injection of foreign gases in a decelerating laminar boundary layer in supersonic flow . the purpose of this research program was to investigate the effects of a diffusion field on a laminar boundary layer in a supersonic flow . specifically, helium, nitrogen, and argon were uniformly injected into the laminar boundary layer of a high-speed flow in a tube with the objective of determining the effects of such injection on the pressure, temperature, and recovery factor distribution along and downstream of the injection region . a continuously operating axially-symmetric wind tunnel has been designed, constructed, and operated . this tunnel consists of an air supply system, a flowmeter, an upstream stagnation tank, a supersonic nozzle (throat diameter 0.262 and exit diameter 1.400), a test section of variable length (zero to 81 diameters, test section diameter of 1.400), a downstream stagnation tank, an exhaust system, a foreign gas supply system, and all necessary instrumentation . the overall performance of this apparatus in terms of the design specifications was excellent . the tunnel was instrumented with 109 thermocouples . all temperatures except ambient temperatures were automatically measured and recorded by means of a self-balancing recording potentiometer . there was 29 pressure taps distributed along the tunnel, 23 along the test section itself . pressures were measured by means of an interconnected micromanometer and a vacuum referenced manometer system with overlapping ranges . for all of the results reported herein, the overall test section was 41 diameters in length,. composed of a porous test section approximately 7.2 diameters in length (leading edge approximately 1.8 diameters from the nozzle exit plane) and four nylon test sections of 8 diameters each .
an aerodynamic analysis for flutter in oseen-type viscous flow . oseen’s equations for unsteady flow are employed to obtain a linearized solution based on a discontinuous-wake model . the analysis is employed to estimate the viscous correction to unsteady lift and moment at large reynolds number . if the asymptotic solution is not too slowly convergent, the correction is of the order of the ratio of the logarithm of reynolds number to the reynolds number . the theory is preliminary in nature as it is limited by the accuracy of oseen’s equations and is restricted to small angle of attack . however, it also shows that the generalized trailing-edge condition for potential flow is reasonable and might predict the essential correction in a real fluid .
the flow about a moving body in the upper ionosphere . a particle approach is used to study the flow pattern around a body moving in the upper layers of the ionosphere . the effects of distant encounters between charged particles (dynamic friction) and of the earth’s magnetic field are taken into account . it is shown that, when the magnetic lines of force are parallel to the direction of motion of the body, there may be a marked concentration of charged particles in the vicinity of the body and a considerable fraction of the reflected or deflected charged particles may reimpinge on the body surface . a numerical example is given for the size and shape of the charged- particle-density contours in the flow field surrounding a circular disc, and these are compared with the corresponding neutral-particle contours .
pitch-yaw stability of a missile oscillating in roll via the second method of lyapunov . the stability theory of a. m. lyapunov, a popular topic in the u.s.s.r., is receiving increasing attention elsewhere . this note describes lyapunov’s /second method/ very briefly and applies it to an aeronautical stability problem .
stagnation-point shock-detachment distance for flow around spheres and cylinders in air . author discusses the problem of deflection of a cantilevered bar, initially in the shape of a circular arc, subjected to an arbitrarily inclined end load .
consideration of energy separation for laminar slip flow in a circular tube . the energy separation for laminar low-density-nonunity prandtl number flow in circular cross-section tubes is the topic of this note . a conclusion is reached as to the effect of prandtl number on the velocity profiles for these flows . however, in order to reach valid quantitative conclusions the reviewer feels that more detailed analysis is in order, and that the analysis as presented here is of qualitative value only .
shroud design for simulating hypersonic flow over the nose of a hemisphere . following is an analytical method for designing a shroud which will generate the hypersonic pressure distribution on a hemisphere . the method was found to be successful throughout the region of subsonic flow . this shroud was designed as part of a low-turbulence wind tunnel used for investigating the effects of cooling on boundary-layer transition on a hemisphere . the design of the shroud contour was carried out in two steps . first, an approximate solution for the incompressible, irrotational flow field was found in the region, and, second, the resulting contour was corrected for compressibility near the sonic region, assuming one-dimensional flow .
transition in the viscous wakes of blunt bodies at hypersonic speeds . transition from laminar to turbulent flow in the hypersonic wakes of spheres was detected in laboratory measurements of the radiation from the flow field . a hypervelocity gun facility was used to fire models, 0.22-in. in diameter, into a range at velocities from 10,000 to 17,000 ft sec . experiments were performed by changing .. (a) the material of the projectile ,. (b) the ambient gas in the range ,. and (c) the pressure in the range . three optical techniques were used to observe the wake radiation .. which show a turbulent viscous wake as the pressure in the range is decreased from one atmosphere to about 20 cm hg . which show the luminous flow field at pressures between 30 and ence of short luminous streaks, which disappear suddenly as the pressure is decreased below 3 cm hg for air, and below 0.8 cm hg for argon . both air and argon, which show the main features of the flow field . above the transition pressure, the intensity of radiation from the wake is always associated with fluctuations that appear to be the same phenomenon as the drum-camera streaks . the appearance of the streaks in the drum camera and photo-multiplier data is interpreted as transition from laminar to turbulent flow in the viscous wake, because experimental evidence shows that their appearance is not controlled by chemical, radiative, or ablative processes, but depends on aerodynamic effects . this conclusion is supported by other experiments based on optical and schlieren techniques . the transition in the wake at positions very close to the body is given by a local reynolds number of 10 for air, and 3 x 10 for argon . the results indicate a possible local-mach-number effect .
stagnation point viscous hypersonic flow . several methods have been developed for computing the hypersonic, low reynolds number flow in the stagnation region of a blunt body . in general, these methods involve complicated numerical solutions . simultaneous iterations on several parameters are usually required in view of the boundary-value nature of the problem . the purpose of this note is to present an approximate closed-form solution to axisymmetric stagnation point hypersonic flow in the viscous layer regime .
the conpressibility transformation and the turbulent boundary layer equations . the compressibility transformation first introduced by dorod-nitzyn has been applied in this paper to the equations of the turbulent boundary layer on a flat plate, considering heat transfer and arbitrary prandtl numbers . assuming the shear distribution to be invariant under the transformation, the stream function and the momentum equation take the proper form for incompressible flow, allowing the use of incompressible velocity profiles in the transformed coordinates . application of crocco’s method to the transformed energy equation permits integration of the energy equation resulting in a formulism remarkably similar to that proposed by eckert . finally, the reference condition was chosen to correspond to the edge of the sublayer from considerations of the assumptions made regarding the shear-stress distribution . with this choice, the reference enthalpy is in good agreement with eckert’s formula over the ordinary range of test conditions . in view of these results, the analysis may be considered to provide a theoretical basis for the reference-enthalpy method .
local heat transfer to a yawed, infite, circular cylinder in laminar compressible flow . this note presents a simplification of a method for calculating the ratio of local to stagnation-line heat-transfer coefficients on a yawed, infinite, circular cylinder in laminar compressible flow . a brief description of the method of ref. 1 is presented, followed by a discussion of the assumptions and mathematical procedure which lead to a considerable simplification .
use of local similarity concepts in hypersonic viscous interaction problems . the problem of predicting the characteristics of a hypersonic laminar boundary layer that interacts with the external flow field is approached using the tangent wedge formulation for the inviscid flow field and the method of similar solutions for the viscous flow . it is shown that the concept of local similarity which allows the pressure gradient parameter to vary in the streamwise direction leads to an explicit relation between the viscous and inviscid flows for all values of the hypersonic interaction parameter . the conditions of /strong/ and limits of the general relations . the present theory is compared with three independent experimental investigations . in each case, the agreement is found to be excellent over the range of investigated . it is shown, using asymptotic solutions to the exact boundary layer equations, that the present theory is applicable to a wide variety of viscous interaction problems . a large number of solutions to the laminar boundary layer similarity equations for a perfect gas with cross flow and surface mass transfer are given . these numerical results, when combined with the solutions of previous authors, are sufficient to describe the range of conditions with high precision .
similitude of hypersonic flows over slender bodies in non-equilibrium dissociated gases . this paper is concerned with the similitude laws governing inviscid, nonequilibrium gas flows around blunt or sharp-nosed slender bodies at zero angle of attack, based on the hypersonic small disturbance flow theory . some related features of the interaction between the effects of nose bluntness and nonequilibrium dissociation and vibration and the influence of a dissociated freestream are also discussed . the hypersonic equivalence principle and the related similitude for affinely related bodies are set forth for nonequilibrium flows in either diatomic gases or a gas mixture such as air . for a family of diatomic gases, as opposed to a given gas such as air, a generalized ambient gas state scaling condition is obtained, whereby the ambient density and temperature need not be simulated . a detailed discussion is given of blunted cylinders and slabs or sharp-nosed cones and wedges, including example nonequilibrium flow field correlations of numerical solutions available in the literature . low density nonequilibrium flows with a negligible shock layer atom recombination rate are also examined ,. as expected, a less restrictive small disturbance similitude law is obtained in this case .
biot’s variational principle in heat conduction . biot’s variational principle is applied to a number of different one-dimensional heat conduction problems . these problems show the applicability of the variational principle to problems involving prescribed heat flux boundary conditions and to those with temperature-dependent material properties . a method is introduced for including boundary conditions when these are expressed as prescribed heat fluxes . the idea behind this is overall energy balance within the body, which is a constraint condition to be satisfied by the time histories of the generalized coordinates . the variational principle is then applied to the well-known problem of constant surface heat flux in order to present the technique and provide a basis for the remaining sections . the equivalence of the result obtained in applying the variational principle for a prescribed surface temperature history to that obtained for a prescribed heat flux is also pointed out . radiation cooling due to fourth power radiation from semi-infinite solids and finite slabs together with radiation according to newton’s law of cooling is then treated . finally, the introduction of temperature-dependent material properties is discussed and the determination of the temperature distribution in a semi-infinite solid with variable properties is investigated .
the stacking of compressor stage characteristics to give an overall compressor performance map . a method of calculation is developed to compute the overall performance of a multi-stage axial compressor, from a knowledge of the individual stage characteristics, by a /stacking/ technique . compressor models are designed and their overall performance calculated . these results are compared to show, qualitatively, the effect of alterations in design and stage performance on overall performance and to find how compressors should be designed for optimum performance .
a theoretical and experimental study of oscillating wedge shaped aerofoils in hypersonic flow . aerodynamic stiffness and damping derivatives have been measured in a /hypersonic gun/ wind tunnel for sharp and blunt-nosed two dimensional single wedge shapes oscillating in the pitching mode in hypersonic flow . the results, which have been compared with theoretical prediction, modified to account for leading edge bluntness, show that this may increase the damping by up to 50 percent for certain axis positions . details of the experimental technique designed to measure the derivatives in the short running times available are described .
calculation of sideslip derivatives and pressure distribution in asymmetric flight conditions on a slender wing-fin configuration . the flow around slender wing-fin configurations having curved leading edges, whose shape is defined by polynomials, is considered . a general expression for the pressure distribution on such a configuration in asymmetric flow is derived and the derivatives due to the particular case of sideslipping motion are also given . no numerical results are given for wing-fin load distribution, but the sideslip derivatives have been evaluated in a number of cases for gothic and ogee wings .
measurements of aerodynamic heating on a 15 cone of graded wall thickness at a mach number of 6. 8. this note describes transient wall temperature measurements made on a in an airstream of mach number 6.8 . the skin of the model was sufficiently thin to allow it to reach zero heat transfer conditions within a running time of one minute . in order to reduce effects of longitudinal heat conduction during a run the electroformed-nickel skin of the model was made with graded thickness, and as a result fairly uniform temperature distributions along the surface were obtained at all times in both the laminar and turbulent regions . values of heat transfer, calculated from the wall temperature-time histories using the thin-wall temperature are compared to theoretical estimates using the intermediate enthalpy method 10, 11 .
boundary layer characteristics of caret wings . the theory of laminar boundary layers along flat surfaces has been used in conjunction with eckert’s approximations to the displacement thickness, skin friction and temperature profiles on the undersurface of a caret wing configuration . to a first approximation it has been assumed that parallel flow exits behind the shock outside the boundary layer, and the displacement of the shock by the boundary layer near the leading edge is neglected . conduction of heat within the body and along the surface is neglected but radiation is included, so that are found . examples are given for various altitudes and configurations and the effect of the skin friction on the lift drag ratio calculated, assuming the undersurfaces to be plane .
the contraction of satellite orbits under the influence of air drag . pt .iv with scale height dependent on altitude . the effect of air drag on satellite orbits of small eccentricity e was studied in part i (tech. note gw 533), on the assumption that atmospheric density varies exponentially with distance r from the earth’s centre, so that the ‘density scale height’ h, defined as, is constant . in practice h varies with height in an approximately linear manner, and in the present note the theory is developed for an atmosphere in which h varies linearly with r . equations are derived which show how perigee distance and orbital period vary with eccentricity, and how eccentricity varies with time . expressions are also obtained for the life-time and air density at perigee in terms of the rate of change of orbital period . the results are also presented graphically . the results are formulated in two ways . the first is to specify the extra terms to be added to the constant-h equations of part i . the second the best constant value of h for use with the equations of part i . for example, it is found that the constant-h equations connecting perigee distance (or orbital period) and eccentricity can be used unchanged without loss in accuracy, if h is taken as the value of the variable h at a height above the mean perigee height during the time interval being considered, where, and decreases from to 0 as e decreases from 0.02 to 0 . similarly the constant-h equations for air density at perigee can still be used if h is evaluated at a height above perigee, where, and decreases to zero as e decreases from constant-h equations can still be used if h is evaluated at the scale height below the initial height . variation of h with altitude has a small effect on the lifetime – about 3 – and on the e-versus-time curve .
experimental study of the velocity and temperature distribution in a high-velocity vortex-type flow . the vortex tube represents a simple device in which a particular type of vortex motion may be studied in the laboratory in an attempt to obtain a better understanding of such flows . such an investigation has been pursued in the heat transfer laboratory of the university of minnesota . the present paper summarizes the major results of this vortex-tube investigation .
laminar heat transfer in tubes under slip-flow conditions . the effects of low-density phenomena on the fully developed heat-transfer characteristics for laminar flow in tubes has been studied analytically . consideration is given to the slip-flow regime wherein the major rarefaction effects are manifested as velocity and temperature jumps at the tube wall . the analysis is carried out for both uniform wall temperature and uniform wall heat flux . in both cases, the slip-flow nusselt numbers are lower than those for continuum flow and decrease with increasing mean free path . extension of the results is made to include the effects of shear work at the wall, temperature jump modifications for a moving fluid, and thermal creep .
analysis of a loaded cantilever plate by finite difference methods . the various difference patterns necessary for finite difference solution of rectangular plate problems, with various boundary conditions and under various transverse loads, are developed . the solution of one particular problem on deuce is also described .
chemical kinetics of high temperature air . when a hypersonic object enters earth’s atmosphere, a shock wave is formed in front of it, and the air passing through this shock wave is heated to high temperatures . the shock heated molecules equilibrate their translational and rotational degrees of freedom within a distance of a few mean free paths . to achieve equilibrium, it is necessary to excite vibration, dissociate molecules, produce new molecules and produce ions and electrons . the problem is complex, since all these phenomena occur simultaneously and because the reaction rates depend on the temperature, density and composition which are changing during the relaxation toward equilibrium . the experimental techniques used to investigate these reactions are briefly discussed along with the resulting rate expressions obtained by the various investigators . a compilation of the rate expressions for these reactions representing the author’s evaluation of all the available data is presented . several pertinent problems which are not yet completely understood and which still require theoretical and experimental investigation are outlined . computed concentration, temperature and density time histories are shown for three different shock speeds in air . the time rate of change of concentration for each chemical reaction is also shown and regimes of importance for the various processes are discussed .
ablation of glassy materials around blunt bodies of revolution . the steady-state equations of motion for a thin layer of an incompressible glassy material on the surface of an ablating and radiating blunt body are reduced to a first-order ordinary differential equation which is integrated numerically . this solution is coupled with the solution of the air boundary layer for both laminar and turbulent heat transfer with or without mass vaporization of the ablating material . the distribution of the effective energy of ablation around the body is thus obtained for a cone cylinder with a hemispherical cap that re-enters the atmosphere at hypersonic flight speeds, and has quartz as the ablating material . it is found that the ablation process from turbulent heating is more efficient than from the laminar case because of increased vaporization . this solution of the equations of motion at the stagnation point has been verified by are wind tunnel experiments . the present state of development of the are wind tunnel does not permit its use for experimental investigations of ablation around blunt bodies under turbulent heating .
generalized heat transfer formulas and graphs . utilizing the research results of previously reported investigations of the laminar, turbulent and radiative heat transfer in dissociated air, some generalized formulas for calculating heat transfer are given . graphs for determining the laminar heat transfer, momentum thickness reynolds number, and turbulent heat transfer distributions around an axisymmetric body are also given . these heat transfer correlations are valid for velocities between 6000 and 26,000 fps and for altitudes up to 250,000 ft . this range of velocities and altitudes covers the important re-entry regime of practical re-entry trajectories having interest today . in the last section of this report these generalized results are specialized for icbm nose cone re-entry applications . these formulas and graphs may be found useful for making rapid engineering estimates and preliminary design evaluations of the heating problems associated with re-entry into earth’s atmosphere .
closing reply to comments on generalized heat transfer formulas and graphs for nose cone re-entry into the atmosphere . in a recent paper (1), detra and hidalgo have shown that, when the boundary layer is turbulent, the heat flux per unit area at the sonic point of a nose cone may exceed the corresponding laminar heat flux per unit area at the stagnation point . the ratio of turbulent sonic-point to laminar stagnation-point heat flux per unit area has been estimated (2) to vary from about 1.0 to 10 for a hemispherical nose as the reynolds number (based on nose diameter) increases from 10 to 10 . since for an axisymmetric body the surface area in the vicinity of the sonic point greatly exceeds the area in the vicinity of the stagnation point, the ratio of turbulent to laminar heat fluxes to the entire body will be much greater than the above quoted ratios of heat fluxes per unit area .
numerical comparison between exact and approximate theories of hypersonic inviscid flow past slender blunt nosed bodies . this paper presents numerical results of exact calculations of the inviscid equilibrium flow about a long hemisphere-cylinder in motion at hypersonic velocity . a comparison is made with blast wave as well as free layer theories of hypersonic flow . as a result of the comparison, it is concluded that the second-order blast wave theory can be used for the purpose of finding the shock shape and the body pressure distribution . however, this procedure is definitely empirical and cannot be justified on rational or theoretical grounds . we show that the presently calculated radial distribution of energy is radically different than that given by blast wave theory . if body shapes other than those considered here are of interest, the only reliable approach at the present time is to carry out numerical calculations . it was found that for certain flight velocities the pressure on the body does not decay to free stream pressure monotonically but overexpands .
a numerical comparison between exact and approximate theories of hypersonic inviscid flow past slender blunt nosed bodies . this note refers to paper of same title by feldman in ars j. 30, validity of blast wave theory cannot be justified on rational or theoretical grounds because of different values of energy in cross flow field as calculated by this theory and by method of characteristics . present note questions this conclusion, shows reasonably good agreement when energy is calculated for points where shock location, streamline pattern, and velocity, temperature, and pressure profiles are adequately defined, and still better agreement when energy is calculated from flow quantities provided by- characteristics method . results are checked using data from independent source . conclusion is reached that blast wave theory is still valid .
experimental measurements of turbulent transition motion, statistics and gross radial growth behind hypervelocity object. the laminar-turbulent transition behind 0.500-in.-diameter spheres at 8500 ft sec and behind measured as a function of pressure . schlieren motion-picture techniques were used to analyze the turbulent motion and the results are described . autocorrelation functions of the density fluctuations of the turbulence have been measured . from these values has been calculated and the results are given for several positions in the turbulent trail at 30 mm hg downstream air pressure . in addition the authors’ previous measurements of the gross radial growth of the turbulent wake have been extended to pressures of 10 mm hg for the case of 0.500-in.-diameter spheres and to the trail behind
heat transfer at the forward stagnation point of blunt bodies . relations are presented for the calculation of heat transfer at the forward stagnation point of both two-dimensional and axially symmetric blunt bodies . the relations for the heat transfer, which were obtained from exact solutions to the equations of the laminar boundary layer, are presented in terms of the local velocity gradient at the stagnation point . these exact solutions include effects of variation of fluid properties, prandtl number, and transpiration cooling . examples illustrating the calculation procedure are also included .
a theoretical study of the effect of upstream transpiration-cooling on the heat transfer and skin friction characteristics of a compressible laminar boundary layer . an analysis is presented which predicts the skin-friction and heat-transfer characteristics of a compressible, laminar boundary layer on a solid flat plate preceded by a porous section that is transpiration cooled . the analysis is restricted to a prandtl number of unity and linear variation of viscosity with temperature . the local skin friction has been found to have a low value in the region of transpiration cooling and then to increase until it approaches the value for a completely nonporous surface asymptotically . the initial increase in local skin friction is rapid as half of the ultimate increase occurs in a distance beyond the porous region that is about 20 percent of the length of the porous region for all rates of injection . when the total coolant flow rate is kept constant and the porous length is varied, it is found that the average skin friction on a partially porous plate is slightly lower than that on a fully porous plate . the local heat transfer behaves in a manner similar to that of the local skin friction . it is found, in an example, that the temperature at the end of a partially porous plate could be maintained at about the same temperature as a fully porous plate by doubling the total rate of coolant flow .
a geometric problem related to the optimum distribution of lift on a planar wing in supersonic flow . the problem studied may be regarded as a problem of geometry . its simplest form (loosely stated) is then as follows .. a mountain rises up from the x-y plane . determine the exact shape of the mountain knowing only the cross-sectional area of every possible cut which can be made through the mountain with a vertical plane . in a more complicated version of the problem, the given information might be restricted to the cross-sectional area of every cut which can be made by a vertical plane inclined less than 45 to the y-axis . this latter case has direct applications to certain minimum drag problems in supersonic flow . the shape of the mountain corresponds to the (unknown) shape of the optimum lift distribution on a planar wing . the cross-sectional area of a cut is the integrated value of the lift along a straight line crossing the wing plan form . for a restricted range of line inclinations, these optimum integrated lift values can sometimes be determined directly . here it is assumed that they are given . the problem in its simplest form was originally solved by radon, who found solutions for a large class of such problems . the derivation presented here may perhaps be more readily understood .
concerning the effect of compressibility on laminar boundary layers and their separation . the theory of compressible flow in a laminar boundary layer has been developed for the case when the viscosity is assumed to be proportional to the absolute temperature and the prandtl number is unity . (these assumptions may be compared with the empirical relations suggested by cope .) it is shown that a transformation of the ordinate normal to the layer can lead to a simplified form of equation of motion very similar to the ordinary incompressible equation but modified by a multiplicative factor g in the pressure term . this factor is greater than unity at the boundary and tends to one at the outside of the layer . several particular solutions are considered including accelerated flow with a linearly increasing velocity and retarded flow along a flat plate with a linearly decreasing velocity . the general implications of the theory are discussed and qualitative conclusions are drawn when the mainstream velocity starts from a stagnation point, rises to a maximum and subsequently falls . it is concluded that for such a velocity distribution increasing compressibility will reduce the skin friction, increase the boundary layer thickness and cause earlier separation as compared with the incompressible flow with the same mainstream velocity distribution and the kinematic viscosity corresponding to conditions at the stagnation point .
the law of the wake in the turbulent boundary layer . after an extensive survey of mean-velocity profile measurements in various two-dimensional incompressible turbulent boundary-layer flows, it is proposed to represent the profile by a linear combination of two universal functions . one is the well-known law of the wall . the other, called the law of the wake, is characterized by the profile at a point of separation or reattachment . these functions are considered to be established empirically, by a study of the mean-velocity profile, without reference to any hypothetical mechanism of turbulence . using the resulting complete analytic representation for the mean-velocity field, the shearing-stress field for several flows is computed from the boundary-layer equations and compared with experimental data . the development of a turbulent boundary layer is ultimately interpreted in terms of an equivalent wake profile, which supposedly represents the large-eddy structure and is a consequence of the constraint provided by inertia . this equivalent wake profile is modified by the presence of a wall, at which a further constraint is provided by viscosity . the wall constraint, although it penetrates the entire boundary layer, is manifested chiefly in the sublayer flow and in the logarithmic profile near the wall . finally, it is suggested that yawed or three-dimensional flows may be usefully represented by the same two universal functions, considered as vector rather than scalar quantities . if the wall component is defined to be in the direction of the surface shearing stress, then the wake component, at least in the few cases studied, is found to be very nearly parallel to the gradient of the pressure .
local heat transfer and recovery temperature on a yawed cylinder at a mach number of 4. 15 and high reynolds numbers . local heat transfer, equilibrium temperatures, and wall static pressures have been measured on a circular cylinder at yaw angles of 0, 10, 20, 40, and 60 . the reynolds number range of the tests was from 1×10 to 4×10 based on cylinder diameter . increasing the yaw angle from 0 to 40 increased the stagnation-line heat-transfer coefficients by 100 to 180 percent . a further increase in yaw angle to heat-transfer coefficients . at zero yaw angle the boundary layer over the entire front half of the cylinder was laminar but at yaw angles of 40 and 60 it was evidently completely turbulent, including the stagnation line, as determined by comparison of local heat-transfer coefficients with theoretical predictions . the level of heating rates and the nature of the chordwise distribution of heat transfer indicated that a flow mechanism different from the conventional transitional boundary layer may have existed at the intermediate yaw angles of 10 and 20 . at all yaw angles the peak heat-transfer coefficient occurred at the stagnation line and the chordwise distribution of heat-transfer coefficient decreased monotonically from this peak . the average heat-transfer coefficients over the front half of the cylinder are in agreement with previous data for a comparable reynolds number range . the theoretical heat-transfer distributions for both laminar and turbulent boundary layers are calculated directly from simple quadrature formulas derived in the present report .
similar solutions for the compressible boundary layer on a yawed cylinder with transpiration cooling . heat-transfer and skin-friction parameters obtained from exact numerical solutions to the laminar compressible-boundary-layer equations for the infinite cylinder in yaw are presented . the chordwise flow in the transformed plane is of the falkner-skan type . solutions are given for chordwise stagnation flow with both a porous and a nonporous wall . the effect of a linear viscosity-temperature relation is compared with the effect of the sutherland viscosity-temperature relation at the stagnation line of the cylinder for a prandtl number of 0.7 . the effects of pressure gradient, mach number, yaw angle, and wall temperature are investigated for a linear viscosity-temperature relation and a prandtl number of 1.0 with a nonporous wall . the results indicate that compressibility effects become important at large mach numbers and yaw angles, with larger percentage effects on the skin friction than on the heat transfer . the use of the two different viscosity relations gives about the same results except when large changes in temperature occur across the boundary layer, as for a highly cooled wall . the present solutions predict that a larger amount of coolant would be required at a given large mach number and yaw angle than would be predicted from solutions of the corresponding incompressible-boundary-layer equations .
investigation of local heat transfer and pressure drag characteristics of a yawed circular cylinder at supersonic speeds . local heat-transfer coefficients, temperature recovery factors, and pressure distributions were measured on a circular cylinder at a nominal mach number of 3.9 over a range of free-stream reynolds numbers from from 0 to 44 . it was found that yawing the cylinder reduced the local heat-transfer coefficients, the average heat-transfer coefficients, and the pressure drag coefficients over the front side of the cylinder . for example, at is reduced by 34 percent and the pressure drag by 60 percent . the amount of reduction may be predicted by a theory presented herein . local temperature recovery factors were also reduced by yaw, but the amount of reduction is small compared to the reduction in heat-transfer coefficients . a comparison of these data with other data obtained under widely different conditions of body and stream temperature, mach number, and reynolds number indicates that these factors have little effect upon the dropoff of heat transfer due to yaw .
aerodynamic characteristics of a circular cylinder at mach number of 6. 86 and angles of attack up to 90 . pressure-distribution and force tests of a circular cylinder have been made in the langley 11-inch hypersonic tunnel at a mach number of based on diameter, and angles of attack up to 90 . the results are compared with the hypersonic approximation of grimminger, williams, and young and with a simple modification of the newtonian flow theory . the comparison of experimental results shows that either theory gives adequate general aerodynamic characteristics but that the modified newtonian theory gives a more accurate prediction of the pressure distribution . the calculated crossflow drag coefficients plotted as a function of crossflow mach number were found to be in reasonable agreement with similar results obtained from other investigations at lower supersonic mach numbers . comparison of the results of this investigation with data obtained at a lower mach number indicates that the drag coefficient of a cylinder normal to the flow is relatively constant for mach numbers above about 4 .
shock wave effects on the laminar skin friction of an insulated flat plate at hypersonic speeds . an approximate theory on the phenomena of interaction between the shock wave and the laminar boundary layer on an insulated flat plate at hypersonic speeds has been formulated . results on the rate of growth of the boundary-layer thickness and the rate of decay of the shock-wave strength have been found that hold for . a new set of formulas for the average skin-friction coefficient, over an insulated flat plate at hypersonic speeds has been obtained . calculations on the basis of the new formulas yield the data shown in figs. steady decrease in as increases, the present results indicate that may increase with at hypersonic mach numbers .
an experimental investigation of leading edge shock wave boundary layer interaction at mach 5.8 . the boundary layer on a slender body tends to be very thick at hypersonic speeds . it interacts with the external flow by producing larger flow deflections near the leading edge than those due to the body alone flow around the boundary layer gives rise to an induced pressure with a negative gradient which thins the boundary layer and increases the skin friction with respect to the zero pressure gradient value . experiments on a flat plate with a sharp leading edge have been performed in the galcit 5-dash by 5-dash in. mach 5.8 hypersonic wind tunnel . the induced pressure was measured by means of orifices in the plate surface . profiles of mach number, velocity, mass flow, pressure, and momentum deficiency were calculated from impact pressure surveys normal to the plate surface made at various distances from the leading edge . the results are as follows . /1/ the induced pressures are 25 per cent higher than the weak interaction theory . /2/ the boundary layer and the external flow are distinctly separate for as low as 6,000 . /3/ the shock wave location is in good agreement with that predicted by the friedrichs theory for a body shape equivalent to the observed boundary-layer displacement thickness . /4/ expansion waves reflected from the shock are weak . /5/ the average skin-friction coefficient tends toward and nearly matches the zero pressure gradient value downstream, but increases to approximately twice that value as the leading edge is approached .
on the boundary layer equations in hypersonic flow and their approximate solutions . analytical solutions of the prandtl boundary-layer equations are obtained for the problem of the /strong/ interaction between the leading-edge shock and the viscous layer over a flat plate at hypersonic velocities . as the mach number increases and the interaction region spreads downstream over the plate, the local skin-friction coefficient increases rapidly over its conventional value . the local heat-transfer coefficient at first remains practically unaffected but then also begins to increase with mach number .
heat transfer to flat plate in high temperature rarefied ultra-high mach number flow . an investigation was conducted in a hypersonic shock tunnel to determine the local heat transfer rates for a sharp leading edge flat plate . the free stream mach number range was 7.95 to 25.1 with stagnation temperatures of approximately 2550 and 6500 r . for these temperature and mach number conditions, the strong interaction parameter, varied from 2.35 to 826 . the corresponding knudsen numbers, based on the ratio of the free stream mean free path and the leading edge thickness, varied from 0.38 to 85.5 . for free stream mach numbers greater than 10, knudsen numbers of approximately unity, and perfect gas conditions, the calculated heat transfer coefficients were found to vary as as predicted by the noninsulated flat plate theory of li and nagamatsu . for the case of, the leading edge slip phenomenon drastically reduced the local heat transfer coefficients as compared to the theoretical values predicted with no slip at the surface . for the extreme case of and, the measured local heat transfer rate was an order of magnitude less than the analytical value . both the knudsen number and the free stream mach number are important physical parameters that determine the extent of the slip-flow region .
boundary layer displacement and leading edge bluntness effects in high temperature hypersonic flow . two important features of hypersonic flow over slender or thin bodies are the displacement effect of the boundary layer and the large down-stream influence of leading-edge bluntness . the present paper contributes new theoretical and experimental results on this problem . the interaction of the two effects is treated theoretically by extending the basic shock-layer concept . in the outer inviscid flow, a model consisting of a detached shock layer and an entropy layer is introduced to account for bluntness . in the boundary layer, the approximate solution is found to be governed by a local flat-plate similarity . under the assumption of a strong bow shock and a specific heat ratio close to unity, a theory is developed for an arbitrary thin body . for flat-plate afterbodies, the theory yields a solution agreeing with blast-wave theory at one limit and strong-interaction theory at the other . within the framework of the present theory, the problems involving angle of attack are also analyzed . complementary to the above study, a hypersonic similitude involving strong shocks, but not requiring close to one a natural comparison with experimental data correlated on the basis of this similitude . flat-plate experiments in air, conducted in the c.a.l. 11 x 15-dashin. hypersonic shock tunnel under cold-wall conditions, included measurement of surface heat-transfer distributions and schlieren studies for zero and nonzero angle of attack . steady laminar heat-transfer rates were measured by means of thin-film resistance thermometers at air test-flow mach numbers around 12, free-stream reynolds numbers from 1.4 x 10 to 1. for most of the experiments, airflow stagnation temperatures ranged from ratios of about 0.15 . the range of test conditions at this stagnation temperature encompassed the limiting cases of dominant bluntness and dominant viscous-interaction effects . heat-transfer distributions were also measured on a sharp plate for air stagnation temperatures ranging from 2,000degreek up to 4,000degreek . the experimental data are quite well correlated in terms of the foregoing theoretical similitude variables characterizing combined effects of boundary-layer displacement and bluntness . the correlations obtained suggest that for the present experimental conditions, at least, the hypersonic viscous similitude is valid even with leading-edge bluntness in the paper, is generally fair .
viscous hypersonic similitude . an extension of classical hypersonic similitude is developed which takes into account the interaction effect of the displacement thickness of the boundary layer . a basic result of this viscous similitude is that the total drag including frictional drag obeys the classical similarity law for the pressure drag . additional similarity conditions governing viscous effects must be imposed in this similitude . underlying the similitude is a new hypersonic boundary-layer independence principle . according to this principle, the principal part of a hypersonic boundary layer with given pressure and wall temperature distributions and free-stream total enthalpy is independent of the (high) external mach number distribution outside the boundary layer . various features of viscous hypersonic similitudes are discussed . it is found, for example, that it applies to three- dimensional boundary-layer interaction effects on flat bodies, provided the concepts of strip theory may be applied, and provided the aspect ratio is an invariant .
inviscid flow with nonequilibrium molecular dissociation for pressure distributions encountered in hypersonic flight . one-dimensional inviscid nonequilibrium flows of a two-component model gas are studied for prescribed pressure variations and an average reaction rate based on recent data for oxygen recombination . these flows are interpreted in relation to the flow along streamlines around blunt hypersonic bodies . assuming equilibrium conditions in the subsonic region, it is estimated that the flow in the initial supersonic expansion region, which is approximately of prandtl-meyer character, will be chemically frozen with respect to the molecular dissociation of the primary components under the hypersonic, high-altitude flight conditions considered . the flight conditions consist of flight velocities between furthermore, on bodies of small surface inclination beyond the nose, the flow will continue to be effectively frozen for at least 20 ft down-stream of the nose . these conclusions may lead to the simplification of procedures for theoretical calculation and testing . the problem of distinguishing a dimensionless length-reaction rate parameter, which characterizes the extent of departures from equilibrium or from frozen behavior in the flow fields of interest here, is discussed
atomic recombination in a hypersonic wind tunnel nozzle . the flow of an ideal dissociating gas through a nearly conical nozzle is considered . the equations of one-dimensional motion are solved numerically assuming a simple rate equation together with a number of different values for the rate constant . these calculations suggest that deviations from chemical equilibrium will occur in the nozzle if the rate constant lies within a very wide range of values, and that, once such a deviation has begun, the gas will very rapidly ‘freeze’ . the dissociation fraction will then remain almost constant if the flow is expanded further, or even if it passes through a constant area section . an approximate method of solution, making use of this property of sudden ‘freezing’ of the flow, has been developed and applied to the problem of estimating the deviations from equilibrium under a wide range of conditions . if all the assumptions made in this paper are accepted, then lack of chemical equilibrium may be expected in the working sections of hypersonic wind tunnels and shock tubes . the shape of an optimum nozzle is derived in order to minimize this departure from equilibrium . it is shown that, while the test section conditions are greatly affected by ‘freezing’, the flow behind a normal shock wave is only changed slightly . the heat transfer rate and drag of a blunt body are estimated to be reduced by only about 25 per cent even if complete freezing occurs . however, the shock wave shape is shown to be rather more sensitive to departures from equilibrium .
viscous and inviscid stagnation flow in a dissociated hypervelocity free stream . high reynolds number hypersonic stagnation flow over a blunt-nosed body in a nonequilibrium dissociated free stream is analyzed and compared to a similar flow in an initially undissociated ambient gas . free stream dissociation effects on various equilibrium stagnation flow properties in air are presented as a function of the ambient atom mass fraction and dissociation energy for velocities ranging from 15,000 to 25,000 fps . significant changes in the bow shock geometry, stagnation gas state, and boundary layer behavior are found when the free stream dissociation involves more than 10( of the total energy . it is observed that for large amounts of both atomic oxygen and nitrogen ahead of the body, the equilibrium shock layer properties converge toward those pertaining to chemically and vibrationally-frozen flow across the bow shock . moreover, under certain conditions, the ionization level can be increased by an order of magnitude and the usual reduction in frozen boundary layer heat transfer due to a highly-cooled noncatalytic surface can increase from stall of adjacent stages . the effects of compromises of stage matching to favor part-speed operation were also considered . this phase of the study indicated that such compromises would severely reduce the complete-compressor-stall margin . furthermore, the low-speed stage stall problem is transferred from the inlet stages to the middle stages, which are more susceptible to abrupt-stall characteristics . the analysis indicates that inlet stages having continuous performance characteristics at their stall points are desirable with respect to part-speed compressor performance . these characteristics must, however, be obtained when the stages are operating in the flow environment of the multistage compressor . alleviation of part-speed operational problems may also be obtained by improvement in either stage flow range or stage loading margin . the results of this analysis are only qualitative . the trends obtained, however, are in agreement with those obtained from experimental studies of high-pressure-ratio multistage axial-flow compressors, and the results are valuable in developing an understanding of the off-design problem . in addition to these stage-matching studies, a general discussion of variable-geometry features such as air bleed and adjustable gas model . numerical solutions of non-equilibrium airflows with fully coupled chemistry provide a preliminary verification of such scaling for benser, w.a. limit characteristics . the analysis indicated that all these problems could be attributed to discontinuities in the performance characteristics of the front stages . such discontinuities can be due to the type of stage stall or to a deterioration of stage performance resulting blades is included .
on hypersonic similitude . tsien in a recent paper (j. math. phys. mass. inst. tech. sonic flows around slender bodies and has pointed out that the product of mach number and fineness ratio is a basic similarity parameter . the author enlarges on this notion, indicating that the problem of hypersonic flow about a slender body in three dimensions is the same as that of a certain two-dimensional nonsteady flow (with time replacing the lengthwise spatial coordinate) characterized by essentially the same similarity parameter .
dissociation scaling for nonequilibrium blunt nose flows . stage-stacking study . the principal problems considered were poor low-speed efficiency, multiple-valued performance characteristics at intermediate speeds, and poor intermediate-speed compressor surge or stall-naca rm e56b03b, 1956 . chapter xiii compressor operation with one or more blade rows stalled . an analysis of the part-speed operating problems of high-pressure-ratio air .
further developments of new methods in heat flow analysis . lagrangian methods in heat-flow problems and transport phenomena were introduced by the writer in some previous work . the present paper develops further one particular aspect of the method,–i.e., the elimination of /ignorable coordinates ./ this is accomplished by a special choice of generalized coordinates, each of which is constituted by an arbitrary temperature distribution and an /associated flow field ./ the latter is a vector field which is derived from the corresponding scalar field by a variational method . the procedure is valid for a certain class of nonlinear problems, provided we replace the temperature by the heat content as the unknown . it is shown that for normal coordinates derivation of the associated flow field is immediate . the use of normal coordinates and their associated flow fields is illustrated by an example . introduction of dirac functions and associated flow fields yields a procedure which constitutes a generalization of the classical formulation by green’s functions and integral equations . this is illustrated by application to one-dimensional problems of heating of a homogeneous or composite slab and directly verified by classical methods in the appendix .
new thermo-mechanical reciprocity relations with application to thermal stress analysis . based on the variational formulation of linear thermodynamics as developed previously by the writer, thermomechanical reciprocity relations are discussed which lead to new methods of analysis of thermal stresses . these reciprocity relations are quite different from the usual ones derived from the analogy of thermal loading with a combination of surface and body-force distribution . the results are applicable to stationary and transient temperatures in elastic and viscoelastic structures . the methods are entirely variational and do not require the evaluation of the temperature field . the stresses at one point are expressed directly in terms of any arbitrary distribution temperatures applied externally, including the effect of surface heat-transfer layer . the concepts and procedures are illustrated on a simple example . the relation is pointed out between the reciprocity property and the generalization of castigliano’s principle to thermomechanics .
approximate formulas for thermal-stress analysis . the basis of any thermal-stress analysis is the determination of the temperature distributions in the structure . for arbitrary flight histories, the determination of such distributions is rather tedious and not completely general . this latter fact handicaps optimization studies in the project design stage when it is desirable to be able to express the thermal-stress distributions in a general manner . in this note, general expressions are derived for the thermal-stress distributions in a typical i-section using similar assumptions to those of biot .
the melting of finite slabs . an approximate method, known as the heat-balance integral, is used to determine the melting rate of a finite slab which is initially at a uniform temperature below the melting point . the slab is acted upon by a constant heat input at one face and has its other face either insulated or kept at its initial temperature . the first three terms of series solutions in an intrinsically small parameter are obtained for the time histories of melting and the temperature distribution in the slab .
influence coefficients for real gases . in the analysis of one-dimensional fluid-flow problems, it is often assumed that the behavior of the medium is that of a perfect gas . this assumption is justified, provided the pressure and temperature range of interest is small and near atmospheric . at higher pressures and temperatures various deviations are introduced thereby causing deviations from the results obtained by using the ideal fluid-flow equations . in this note, influence coefficients, similar to those developed by shapiro, are presented for the case of real gases . this analysis is based upon the use of various functions of the compressibility factor emmons . some of the assumptions made were as follows.. (1) the flow is one-dimensional and steady, (2) changes in the stream properties are continuous, and (3) the flow is comprised of imperfect gases
conduction of heat in a solid with a power law of heat transfer at its surface . the nonlinear boundary value problem, where and m are constants, is solved formally by first introducing power series in t for the unknown temperature and flux at the surface and then determining the coefficients in those series . in this manner the temperature function is determined as a series of repeated integrals of error functions . the convergence is rapid only for small values of t . the special cases and generalizations of the condition at the surface for which the same method applies, are noted . surface temperatures are also found by methods of difference equations, where t is not limited to small values . graphs of these temperatures corresponding to various laws of heat transfer at the surface are shown .
nonlinear heat transfer problem . a study has been made of the time-dependent heat conduction in a semi-infinite medium subject to a boundary condition which can involve the temperature in a nonlinear manner . a formulation for the determination of the surface temperature, which is often of greatest physical interest, leads to a nonlinear volterra integral equation . a simple iterative solution method, with an accuracy suitable for many practical purposes is presented . as an example, the problem of the time-dependent surface temperature of a body receiving heat according to the stefan-boltzmann law is treated . the analysis is also applicable to physical adsorption or chemisorption processes which occur at the boundary .
an approximate treatment of unsteady heat conduction in semi-infinite solids with variable thermal properties . this very short paper presents an approximate procedure for the calculation of unsteady heat conduction in semi-infinite solids with variable thermal properties . it is claimed to be an improvement over previous efforts in this area since it yields physically sensible results for cases where thermal properties have a large dependence on temperature . instead of using polynomials to represent an unsteady temperature profile an exponential form is used . good agreement is shown for several cases where the method of the paper is compared with exact solutions .
variational analysis of ablation . the variational and lagrangian thermodynamics developed in earlier publications are directly applicable to problems of heat conduction with melting boundaries . these techniques are used here in treating the problem of a half-space subjected to a constant rate of heat input at the melting surface (fig. 1) . the applicability of the lagrangian equations to this case follows from the fact that the basic variational principle is valid whether the boundaries are fixed or move as arbitrary functions of time . this can be seen if we remember that the equations govern only the instantaneous configuration of the flow rates for a given geometry and temperature field .
compressor operation with one or more blade rows stalled . an analysis of the part-speed operating problems of high-pressure-ratio ratio multistage axial-flow compressors was made by means of a simplified stage-stacking study . the principal problems considered were poor low-speed efficiency, multiple-valued performance characteristics at intermediate speeds, and poor intermediate-speed compressor surge or stall-limit characteristics . the analysis indicated that all these problems could be attributed to discontinuities in the performance characteristics of the front stages . such discontinuities can be due to the type of stage stall or to a deterioration of stage performance resulting from stall of adjacent stages . the effects of compromises of stage matching to favor part-speed operation were also considered . this phase of the study indicated that such compromises would severly reduce the complete-compressor-stall margin . furthermore, the low-speed stage stall problem is transferred from the inlet stages to the middle stages, which are more susceptible to abrupt-stall characteristics . the analysis indicates that inlet stages having continuous performance characteristics at their stall points are desirable with respect to part-speed compressor performance . these characteristics must, however, be obtained when the stages are operating in the flow environment of the multistage compressor . alleviation of part-speed operation problems may also be obtained by improvement in either stage flow range or stage- loading margin . the results of this analysis are only qualitative . the trends obtained, however, are in agreement with those obtained from experimental studies of high-pressure-ratio multistage axial-flow compressors, and the results are valuable in developing an understanding of the off-design problem . in addition to these stage-matching studies, a general discussion of variable-geometry features such as air bleed and adjustable blades is included .
some stall and surge phenomena in axial flow compressors . observations of rotating stall have shown that a wide variety of stall patterns is possible . hot-wire anemometer data on a multistage compressor have shown a progressive-type stall at low speeds . the amplitude of the flow fluctuations increases in magnitude through the first few stages and then diminishes rapidly to a small value in the latter stages . a stage-stacking analysis has shown that rotating stall will exist over a large portion of the compressor map at low speeds but will be instigated almost simultaneously with compressor surge at high speeds . blades failures attributable to resonant vibrations excited by rotating stall have been experienced in single and multistage compressors . in the stage-stacking analysis no deterioration of stage performance due to unsteady flow resulting from stall of adjacent stages was considered . in general, the pressure drop at the stall point is believed to be much larger than indicated by an analytical formulation of compressor performance . compressor surge is attributed to a limit cycle operation about the compressor stall point, and, as indicated in a few compressor tests and in jet-engine tests, a small compressor discharge receiver volume may result simply in stall of the compressor without the cyclic characteristics of compressor surge . in this event, engine operation will be limited because of the large drop in performance which accompanies compressor stall .
effects of stage characteristics and matching on axial flow compressor performance . the use of stage characteristics obtained from test data in the performance analysis and development of an axial-flow compressor is described relative stage matching as shown by an idealized example and also by test experience . factors governing major performance parameters are discussed and certain development problems and possible solutions are reviewed .
an approximate equation for the /choke line/ of a compressor . discussion of a similarity between the pressure-ratio versus inlet-mass-flow-coefficient characteristic of a stream or gas turbine and the analogous characteristic of an expansion /laval/ nozzle . this idea is extended to a compressor and a compression nozzle, and an approximate expression for the /choke line/ of the compressor is developed .
design of axial compressors . the main types of axial compressors are described, and the use of generalized design curves to make performance estimates is advocated . the different variables are weight, power, pressure ratio, temperature rise, mass flow, rotational speed, stage efficiency, blade bending stresses due to aerodynamic loading, and methods and materials of construction . air outlets, flow coefficients and different blade forms are also considered .
theoretical considerations of flutter at high mach number . some of the theories for two-dimensional oscillatory air forces which may be applied in flutter calculations at high mach numbers are discussed . these include linear theory, van dyke’s second-order theory, piston theory, landahl’s method, tangent-wedge and tangent-cone approximations, newtonian theory, and a new nonlinear-pressure method . a comparison of the theories is made by showing the results of flutter calculations for mach numbers up to 10, and the possibility of flutter at these higher mach numbers is pointed out . results of flutter calculations are shown to illustrate the various effects arising from a nonlinear thickness theory . the possibility of large flutter speed thickness effects which depend on frequency ratio is shown . the influence of airfoil shape is discussed and flutter speed trends with center of gravity and elastic axis locations are presented . some possible refinements of piston theory are discussed for use at very high mach numbers . these include the use of local flow conditions and the use of newtonian theory over the leading edge of a blunt-nosed airfoil .
wind tunnel techniques for the measurements of oscillatory derivatives . this paper discusses the basic principles employed in techniques for the measurement of oscillatory derivatives in wind tunnels, and gives some account of the associated instrumentation . the suitability of the various techniques for different test conditions is also discussed and brief reference is made to wind tunnel effects on the measurements .
the equilibrium piston technique for gun tunnel operation . a modified technique for the operation of a gun tunnel is suggested based on experimental results . if the piston mass and the initial barrel pressure are chosen correctly, then the peak pressures associated with the gun tunnel may be eliminated . under these conditions the piston is brought to rest with no overswing . some measurements of the piston motion using a microwave technique are reported which confirm this idea . the wave diagram associated with this mode of operation is shown, and some calculations of the stagnation pressure are given which show that during the suggested running time, the stagnation pressure may be considerably greater than the driving pressure if the driving chamber cross-sectional area is large compared with that of the driven section . for a uniform shock tube the stagnation pressure will always be less than the driving pressure . the use of air, helium and hydrogen as driving gases has been considered . experiments in a gun tunnel are reported which show that the equilibrium piston technique enables steady stagnation pressures to be achieved over a time of approximately 15 ms using air as the driving gas . the expansion caused by the piston acceleration is shown to interact with the stationary piston, but this is found to produce only a small drop in stagnation pressure .
the properties of crossed flexure pivots, and the influence of the point at which the strips cross . it is shown that the rotational stiffness of a crossed flexure pivot varies considerably when subjected to an applied force . the type of variation can be radically changed simply by moving the point at which the strips cross . the relation between torque and rotation for a given applied force is not exactly linear and the extent of the non-linearity is determined by taking into account the small movements of the centre of rotation of the pivot . finally, for design purposes, an analysis of the maximum stresses in the strips is given .
measurements of pitching moment derivatives for blunt-nose aerofoils oscillating in two-dimensional supersonic flow . direct pitching moment derivatives have been measured using the method of scruton, woodgate et al for two single wedge blunt-nosed aerofoils . these measurements were made at mach numbers of 1.75 and 2.47 and frequency parameters less than 0.02 . in general, nose blunting was found to have little effect on the derivatives although changes were observed for the thinner wedge at a mach number of 1.75 .
new test techniques for a hypervelocity wind tunnel . the measurement of rocket exhaust effects on vehicle stability and the measurement of aerodynamic damping were made in an arc-discharge type of hypervelocity wind tunnel . sample data are given to indicate the quality of data obtainable in this tunnel, and samples of self-luminous and shadowgraph photographs are also presented .
aerodynamic forces, moments and stability derivatives for slender bodies of general cross section . the problem of determining the total forces, moments, and stability derivatives for a slender body performing slow maneuvers in a compressible fluid is treated within the assumptions of slender-body theory . general expressions for the total forces (except drag) and moments are developed in terms of the geometry and motions of the airplane, and formulas for the stability derivatives are derived in terms of the mapping functions of the cross sections . all components of the motion are treated simultaneously and second derivatives as well as first are obtained, with respect to both the motion components and their time rates of change . coupling of the longitudinal and lateral motions is thus automatically included . a number of general relationships among the various stability derivatives are found which are independent of the configuration, so that, at most, only 35 of a total of 325 first and second derivatives need be calculated directly . calculations of stability derivatives are carried out for two triangular wings with camber and thickness, one with a blunt trailing edge, and for two wing-body combinations, one having a plane wing and vertical fin . the influence on the stability derivatives of the squared terms in the pressure relation is demonstrated, and the apparent mass concept as applied to slender-body theory is discussed at some length in the light of the present analysis . it is shown that the stability derivatives can be calculated by apparent mass although the general expressions for the total forces and moments involve additional terms .
the calculation of lateral stability derivatives of slender wings at incidence including fin effectiveness, and correlation with experiment . comparisons are made between low-speed experimental results and estimates based on attached-flow theory for the lateral stability derivatives of slender wings at incidence, and it is found that the flow separation has little effect on the sideslip derivatives . the reduction in due to part-span anhedral is evaluated, and a semi-empirical formula is derived to account for important second-order terms . for the rotary derivatives, an attempt is made to estimate the effect of the leading edge vortices, but no satisfactory conclusions have been reached . the fin contributions to the derivatives are evaluated on the basis of treating the wing surface as a total reflection plate . good agreement with experiment is reached for the sideslip derivatives, and for the damping-in-yaw at moderate incidences . sidewash is found to have a large effect on the rolling derivatives, and further information on the strength and position of the leading edge vortices in non-symmetric flow is required before a complete calculation of the sidewash can be given .
calculation of the flow past slender delta wings with leading edge separation . the flow past a slender delta wing with a sharp leading edge, at incidence, usually separates along this edge, i.e. a vortex layer extends from the edge and rolls up to form a /core/ (a region of high vorticity) . a potential flow model of this is constructed in which the layer is replaced by a vortex sheet which is rolled up into a spiral in the region of the /core/ . this problem is reduced to a two-dimensional one by assuming a conical field and using slender wing theory . the shape and strength of the sheet are determined by the two conditions that it is a stream surface and sustains no pressure difference . use is made of results previously obtained for the core region and the remaining finite part of the sheet is dealt with by choosing certain functions for its shape and strength . the parameters in these functions are found by satisfying the two conditions stated above at isolated points . results are obtained for the pressure distribution, chord loading and norman force coefficient as functions of the ratio of the incidence to the apex angle . the lift for a given incidence is about 15 below that found by brown and michael . flow patterns are indicated in two typical cases . the effect of separation on the drag due to lift of a wing with small thickness is discussed .
the 7 x 7 in . hypersonic wind tunnel at rae farnborough, part 1, design, instrumentation and flow visualization techniques . this is the first of three parts of the calibration report on the r.a.e. some details of the design and lay-out of the plant are given, together with the calculated performance figures, and the major components of the facility are briefly described . the instrumentation provided for the wind-tunnel is described in some detail, including the optical and other methods of flow visualization used in the tunnel . later parts will describe the calibration of the flow in the working-section, including temperature measurements . a discussion of the heater performance will also be included as well as the results of tests to determine starting and running pressure ratios, blockage effects, model starting loads, and humidity of the air flow .
the 7 in. x 7 in. hypersonic wind tunnel at r.a.e. farnborough part ii. heater performance . tests on the storage heater, which is cylindrical in form and mounted horizontally, show that its performance is adequate for operation at m=6.8 and probably adequate for flows at m=8.2 with the existing nozzles . in its present state, the maximum design temperature of 680 degrees centigrade for operation at m=9 cannot be realised in the tunnel because of heat loss to the outlet attachments of the heater and quick-acting valve which form, in effect, a large heat sink . because of this heat loss there is rather poor response of stagnation temperature in the working section at the start of a run . it is hoped to cure this by preheating the heater outlet cone and the quick-acting valve . at pressures greater than about 100 p.s.i.g. free convection through the fibrous thermal insulation surrounding the heated core causes the top of the heater shell to become somewhat hotter than the bottom, which results in /hogging/ distortion of the shell . this free convection cools the heater core and a vertical temperature gradient is set up across it after only a few minutes at high pressure . modifications to be incorporated in the heater to improve its performance are described .
the 7 in. x 7 in. hypersonic wind tunnel at r.a.e., farnborough part iii – calibration of the flow in the working section . the fused silica nozzle to give m=7 in the 7 in. x 7 in. hypersonic wind tunnel produces a flow field with an average mach number of 6.85 along the centreline of the working section . the mach number gradually decreases towards the boundary layer, and over a core of approximately mach number . the nozzle heats up during a run but this has little effect on the mach number distribution . at one station the mach number was one-third per cent greater for a run of 1 minute than for a run of 10 seconds . the temperature field in the inviscid flow has an average variation of in temperature with time throughout a run .
pressure measurements on a cone-cylinder-flare configuration at small incidences for m 6.8 . pressure measurements were made on a slender cone-cylinder-flare configuration, slightly blunted at the nose, for 0, 3 and 6 degrees incidence at a free-stream mach number of 6.8 . it was found that the surface pressures obtained on the cone agreed with extrapolations to m equals 6.8 of theoretical values given in m.i.t. tables /kopal/for yawed cones, and that impact theory gave a good indication of the pressure level to be expected on all parts of the body where surface incidence was sufficiently large to merit its use . the semi-angles of the conical and flared parts of the model were both the pressure level on the flare rose in all cases to approximately that developed upstream on the cone surface . no evidence of a marked over-expansion to pressures below the free-stream value was noticed at the junction between cone and cylinder .
formulae and approximations for aerodynamic heating rates in high speed flight . this note gives formulae and approximations suitable for making preliminary estimates of aerodynamic heating rates in high speed flight . the formulae are based on the /intermediate enthalpy/ approximation which has given good agreement with theoretical and experimental evidence . in the general flight case they could be used in conjunction with an analogue computer or a step-by-step method of integration to predict the variations of heat flow and skin temperature with time . in the restricted case of flight at constant altitude and mach number, simple analytical methods and results are given which include the effects of radiation and can be applied to /thick/ as well as /thin/ skins where h is the aerodynamic heat transfer factor, and g, d and k are the heat capacity, thickness and thermal conductivity of the skin . if 0.1 the skin is approximately /thin/, i.e. temperature gradients across its thickness may be neglected .
duct flow in magnetohydrodynamics . this paper is an extension of the work of hartmann (2) and shercliff transverse magnetic fields — the simplest class of magnetohydrodynamic problems . we are concerned here mainly with the boundary value problems associated with flow in ducts with conducting walls .
aerodynamic noise in supersonic wind tunnels . hot-wire measurements in the free stream of a supersonic wind tunnel were made in the mach number range of 1.6 to 5.0 . it is shown that the mass-flow fluctuations increase very rapidly with increasing mach number . if the fluctuation field is assumed to consist of sound waves-dash an assumption that is consistent with the measurements-dashthe sound intensity is approximately proportional to m, within the range of the experiments . furthermore, the orientation of the field is found to be different from the mach line direction,. it corresponds to a sound-source velocity of approximately one-half the free-stream velocity for the higher mach numbers . it is shown that the turbulent boundary layer along the nozzle and the tunnel walls is responsible for this sound field .
on three dimensional bodies of delta planform which can support plane attached shock waves . this note collects together in one report available theoretical work on bodies which can support attached plane shock waves, discusses some of the possible merits of such shapes, and includes some calculations illustrating their properties . also, some preliminary results from wind tunnel tests are given, together with details of proposed future tests .
corner interference effects . the three-dimensional incompressible flow of fluid along the corner of two semi-infinite plates intersecting at right angles, especially the interference of the boundary layers of the two plates, is discussed . mainly, the more important case of turbulent boundary layer is treated by means of experimental studies carried out at the technical university of braunschweig . some theoretical results for laminar flow are also taken into account . in order to describe the interference effects in the boundary layer, an interference displacement thickness and an interference skin friction have been introduced . it is shown from experiments and also from theoretical considerations how these two quantities depend on reynolds number . furthermore, the influence of interference on the transition from laminar to turbulent flow is investigated . in addition, some preliminary results are given about the effect of the pressure gradient on the interference effects .
an approximate solution of the compressible laminar boundary layer on a flat plate . following a major assumption that enthalpy and velocity are dependent only on local conditions, an enthalpy-velocity relation is obtained for the laminar boundary layer on a flat plate where subscripts p refer to the plate, 1 to the free stream and e to the equilibrium temperature condition at the plate . when compared with general results, this relation (exact for prandtl number o = 1) gives a close approximation to crocco’s numerical results for o = 0.725 and 1.25, up to . using the above relation in conjunction with the approximate viscosity-temperature relation suggested by chapman and rubesin, and with young’s suggested first approximation for shearing stress it is shown that close approximations to displacement thickness and velocity distribution are given by and where and which serves to define c . these have the advantage of being algebraic in form whereas previous results have involved complex numerical integrations for individual cases .
pressure distributions and flow patterns at m=4 . on some delta wings of inverted ‘v’ cross section . wind tunnel tests have been made to measure pressure distributions and to study flow patterns on a series of delta wings of inverted ‘v’ cross-section . each of these wings was designed to have a plane shock wave in the plane of the leading edges at a chosen mach number and incidence . it was found that for a wide incidence range about the design point the shock wave remained virtually attached to the leading edges and at each incidence the pressure was approximately constant over the lower surface .
the contraction of satellite orbits under the influence of air drag part i . with spherically symmetrical atmosphere . the effect of air drag on satellite orbits of small eccentricity e/0.2/ is studied analytically by a perturbation method, on the assumption that the atmosphere is spherically symmetrical . equations are derived which show/1/how orbital period and perigee distance vary with eccentricity as the orbit contracts, and/2/how each of these quantities varies with time . the equations of type/1/are nearly independent of the oblateness of the atmosphere . in all the equations, terms of order e and higher are usually neglected . the results are also presented graphically, in a manner designed for practical use . the theory is to be extended to an oblate atmosphere in part ii, and will later be compared with observation .
the contraction of satellite orbits under the influence of air drag . part ii . with oblate atmosphere . the effect of air drag on satellite orbits of small eccentricity e/0.2/ was studied in part i/technical note no.g.w.533/on the assumption that the atmosphere was spherically symmetrical . here the theory is extended to an atmosphere in which the surfaces of constant density are spheroids of arbitrary small ellipticity . equations are derived which show how perigee distance and orbital period vary with eccentricity, and how eccentricity is related to time . expressions are also obtained which give lifetime and air density at perigee in terms of the rate of change of period . in most of the equations, terms of order e and higher are neglected . the results take different forms according as the eccentricity is greater or less than about 0.025, while circular orbits are dealt with in a separate section . the results are also presented graphically in a manner designed for practical application, and examples of the theory in use are given . the influence of atmospheric oblateness is difficult to summarize fairly simultaneously assume their’worst’values, some of the spherical-atmosphere results can be altered by up to 30( as a result of oblateness and 5-10( would be a more representative figure .
the contraction of satellite orbits under the influence of air drag . part iii . high eccentricity orbits . /0.2 e 1/ . the effect of air drag on satellite orbits of eccentricity e less than between 0.2 and 1 is presented . equations are derived which show how perigee distance and orbital period vary with eccentricity during the satellite’s life, and how eccentricity is related to time,.and formulae are obtained for the lifetime and the air density at perigee, in terms of the rate of change of period . the results are also presented graphically and their implications and limitations are discussed .
determination of upper-atmosphere air density and scale height from satellite observations . a solution is obtained for the rate of change of semi-major axis and perigee distance of a satellite orbit with time due to the resistance of the atmosphere . the logarithm of air density is assumed to vary quadratically with height, and the oblateness of the atmosphere is taken into account . the calculation of perigee air density in terms of the rate of change of satellite period is dealt with,. and the method is applied to data at present available on six different satellites . the variation of air density with height is obtained as in p-28.59/0.15/-h-200//46/5/0.028/0.013//h-200///46/ for h in the range of approximately 170 to 700 km, where p is in grams/c m, h is in kilometres and standard deviations are given in brackets .
determination of upper-atmosphere air density profile from satellite observations . the theory previously developed for the changes in the perigee distance and semi-major axis of a satellite orbit due to air drag is extended to enable the air-density profile/i.e.its relative variation with height/to be derived from the motion of the orbit’s perigee . the solution is first obtained in terms of the change in perigee distance and then in terms of the change in the radius of the earth at the sub-perigee point the scale height in the 180 and 220 km altitude regions .
orbit decay and prediction of the motion of artificial satellites . the rate of decay of elliptic satellite orbits, due to atmospheric drag, is investigated through variation of parameters and through use of an atmospheric model involving a power function between density and altitude . this model is shown to fit actual conditions better than an exponential function . the effects of the equatorial belt and the rotation of the earth are investigated . the conclusion is reached that through these anomalies atmospheric drag substantially affects the orbit elements, especially those defining the orbit plane . an alternate approach of variation of parameters is presented, by which a direct relation between period decay and instantaneous density conditions is established . this approach, by itself specifically adequate for prediction work, also opens an avenue for systematic and unified evaluation of observed decay .
density of the upper atmosphere from analysis of satellite orbits .. further results . the method previously described has been refined by taking into account atmospheric rotation . further results are given from satellites of latitude and season and day-to-night changes are reported .
earth satellite observations and the upper atmosphere . atmospheric densities have been derived from artificial satellites in altitudes 200-700 km. and from rockets up to about 200 km. to consolidate the two sets of data, h.k. kallmann suggested a model with a exact form of this curve has now been derived . corrections for the is excellent . very close correlation between atmospheric density variations/h180 km./ and the solar 20-cm. radiation implies that the origin of the’solar effect’may lie in the absorption of solar ultra-violet radiation . the atmospheric density curve between 180 and 200 km. shows a temperature inversion in the fl-layer . it is not yet possible to decide whether solar ultra-violet radiation as well as the solar he line and solar x-ray radiation contribute to the heating of the fl-layer . diurnal and seasonal density variations at altitudes 210, 562 and 660 km. have been derived from variations in acceleration of three satellites/sputnik 3, vanguard 1 and 2/ . group averages of diurnal variations are taken from different dates within the period may 15, 1958-october 1, 1959 . physcal conditions in the upper atmosphere are briefly summarized ..the’solar effect’originates in the fl-layer as a result of heating by the solar he line at 304 a. diurnal density variation at 210 km. is only a few per cent . absorption of solar electromagnetic radiation in the f2-layer, and large heat conductivity cause intense diurnal density and temperature variations above
latitude and diurnal variations of air densities from 190 to 280 km. as derived from the orbits of discoverer satellites . variations in air density between day and night in the region 190 to 280 km are found to be small/less than about 25(/ . the presence of a possible region of local heating at about 220 km which disappears at night . the night-time density profile conforms with a constant scale height of 35/2/km.no definite variation of air density with latitude is evident apart from a possible increase of about 60(, which is indicated by rather limited polar-region data . for other latitudes and seasons a variation of less than about 20( is indicated .
scale height in the upper atmosphere, derived from changes in satellite orbits . the’density scale height’h in the upper atmosphere is a measure of the rate at which air density p varies with height y, being given by h-p//dp/dy/ . the value of h, although important because/with the molecular weight of the air/it determines the air temperature, has not as yet been well determined at heights above 200 km . this note develops methods for finding h from the decrease in a satellite’s perigee height and from the decrease in the orbital period of a satellite in a small-eccentricity orbit . these methods are then applied to all the 14 satellites found suitable for the purpose . the 44 values of h obtained, for heights of 200-450 km, represent an average over day and night and probably have errors/s.d./of 5-10( . it is found that, as solar activity declined between 1957 and 1961, h decreased greatly ..e.g.at height 275 km, h decreased from 60 km in early 1958 to height becomes much less rapid above 350 km, and are consistent with the supposition that h had low values, near 35 km, at heights near 250 km, for 1959-61 . the results could be greatly extended in scope and improved in accuracy if more accurate orbits were available for short-lifetime satellites .
on the coupling between heat and mass transfer . in mixtures of two different gases or liquids, one constituent will migrate spontaneously toward the warmer parts, and the other toward the colder parts . this phenomenon, known as the soret effect, and its converse the dufour effect, were discovered as early as 1856 and 1873 respectively . the two effects can also be considered as a simultaneous transport of mass and heat, or as a coupling between heat and mass transfer . the effects of this coupling have been neglected in all investigations of heat transfer in multicomponent flow systems so far, on the a priori assumption that they are small . in a recent publication however, it was shown that they can be large in laminar-boundary-layer-type flows with helium injection . turbulent-boundary-layer measurements and an analysis conducted at the heat transfer laboratory clearly showed significant effects of the coupling on heat transfer and adiabatic wall temperature . from additional measurements, the results of which are presented below, it is possible to separate the heat flux at the model wall into one part depending on the temperature gradient and a second part caused by the coupling . it is shown that the latter exceeds the former, and hence the coupling may not be neglected a priori without careful consideration .
cruise performance of channel-flow ground effect machines . the performance theory for high-speed air-cushion vehicles operating in close proximity to the ground is developed . the analysis is restricted to cruise flight of vehicles of rectangular planform employing an air pressure seal between the ground and the vehicle along the two streamwise sides . the variation of the optimum rearward deflection angle of the side jet pressure seal with speed for minimum overall power expenditure and maximum range is found . it is concluded that a mixed propulsion system (jet deflection plus propeller(s)) is required . volume flow and the corresponding fan pressure rise needed are also calculated . the maximum lift drag ratio is determined . the maximum thickness ratios of the vehicles are considered to be large compared with the ground-height vehicle-length ratio . two-dimensional airfoil theory is employed to show that close to stagnation conditions exist below the vehicles . the lower-surface lift, pitching moment, and aerodynamic-center location are determined . the flow over the upper surface is identified with flow over mounds . upper-surface lift coefficients are determined for typical mound shapes . it is shown that high total lift coefficients are theoretically obtainable with almost zero induced drag . the conventional induced-drag power penalty is replaced by a sealing-air power expenditure, which is shown not to be excessive .
viscous and inviscid nonequilibrium gas flows . the condition of immediate freezing of the mass fraction of dissociated species of air at the equilibrium value behind the shock envelope prevails over a major portion of the flight spectrum associated with lifting re-entry vehicles . this is observed by means of order-of-magnitude considerations within the limits of the present knowledge of chemical reaction rates for the constituents of air . accordingly, investigations of the viscous and inviscid hypersonic flow about blunt and sharp leading edge slender bodies are made . the investigations are generalized to consider an arbitrary degree of dissociation in the ambient free stream . this condition is included in order to allow comparison with the flow field about a model in the test section of a hypersonic facility with dissociated air species present in the free stream . inviscid frozen flow investigations are made for blunt and sharp leading edge slender body power-law geometries . the results indicate that the influence of a finite leading edge, in inducing a pressure field far downstream (/blast-wave/ analogy), is considerably diminished for this model . this conclusion is verified numerically by a characteristics solution for the hypersonic flow about a /sonic-wedge/ slab . the viscous investigations consider the boundary-layer interaction problem with a frozen degree of dissociation . in this case, as in the inviscid analysis, the governing parameter is observed to be the ratio of the dissociation energy to the free-stream kinetic energy . the influence of this parameter on the boundary-layer interaction mechanism for a highly cooled, noncatalytic wall is presented . the influence of a frozen flow field on skin friction and heat transfer is also discussed . finally, since higher mach number gas flows may be generated in wind tunnel nozzles where dissociation nonequilibrium effects are present, the possibility of employing expansions with a controlled degree of dissociation as a technique for aerodynamic simulation is presented .
some features of supersonic and hypersonic flow about blunted cones . for a family of cones of various semiapex angles blunted by spherical caps, shock shapes and surface pressure distributions have been obtained from both the belotserkovskii method and experiment . these results are used to study convergence to conical flow . conditions leading to both overexpansion and underexpansion on the surface with respect to the asymptotic conical pressures are described as well as conditions leading to bow shock inflection points . conditions also exist for which a second shock may occur, or for which the sonic line cannot touch the body surface . the implications of these conditions for various blunt-body methods are discussed . for cones blunted in such a manner as to keep the flow entirely supersonic, the flow field is found to exhibit certain similarities with that for genuine blunting . this is related to the fact that the surface entropy layer for blunt bodies can be most influential, in determining surface pressure, in the interior of the flow field rather than near the surface .
flutter analysis of circular panels . the flutter problem of flat circular panels with edges elastically restrained against rotation has been formulated in terms of small-deflection plate theory . the panel is subjected to uniform all-round tension or compression in its middle plane, in addition to the supersonic compressible flow passing over its upper surface with still air below . linear piston theory is employed to predict the aerodynamic load on the vibrating panel . the problem is investigated by a rayleigh-type analysis involving chosen modes of the panel as degrees of freedom . in order to investigate the convergence of the solution, the flutter-mode shape of the clamped-edge panel has been expressed in a series form in powers of r cos o . the results of three-, four-, and five-term approximations have displayed oscillatory behavior with apparently rapid convergence of the solution .
thermal effects on a transpiration cooled hemisphere . an approximate method is used to obtain the injection distribution which would exist on an isothermal, transpiration-cooled hemisphere in a supersonic stream . this distribution is the same for both air and helium injection, and is independent of the blowing level . a model having this distribution was tested in the naval supersonic laboratory wind tunnel at a mach number of 3.53 . it is concluded that the design technique is reasonably accurate . data taken near the nose are compared with the theories for air and helium injection . the agreement in the case of the reduction in heat-transfer coefficient is good . the values of insulated wall temperature obtained near the nose with helium injection are 8 percent above the local stagnation temperature, and largely independent of injection rate . it is believed that this phenomenon may be attributed to the thermal diffusion of the helium within the boundary layer . air injection causes a slight reduction in the insulated wall temperature . it is shown that injection of either air or helium at the hemisphere nose considerably reduces the heat flux at the surface . the additional reduction in heat flux resulting from helium injection as opposed to air injection, and predicted by existing theory, is largely absent .
second-order effects in laminar boundary layers . second-order boundary layer disturbances are due to the displacement of the main flow by the boundary layer, surface curvature, freestream vorticity, and slip . a procedure for finding these is given for compressible flow of a perfect gas having a classically similar boundary layer . solutions are given for the flat plate and circular cylinder and for the hypersonic axisymmetric stagnation point . for the latter flow, the dominant effect is that of vorticity, which increases both shear and heat flux . for the plate or cylinder, the same conclusion tends to hold for high speed flow . the vorticity effect is governed by the entire outer flow–not just the wall vorticity .
stagnation region in rarefied high mach number flow . paper describes results of numerical solution of the viscous shock-layer equations for axisymmetric stagnation region, using the viscosity-temperature law with w=0.65, pr=0.71 and y=1.25 . purpose is to establish applicability of the simple approximation of w=1 (obtained earlier) to air at low reynolds numbers and low ratios of wall temperature to stagnation temperature . using a reference temperature (closely equal to eckert’s) to interpret the linear results, excellent agreement is found, in the limit of, over a wide range of reynolds numbers, covering fully merged shock layers as well as boundary layers with and without vorticity interaction . agreement with recent experiments of ferri et al is as good as to be expected from shock-layer approximation . paper provides valuable extension of the applicability of the reference temperature concept .
low speed wind tunnel tests on a two dimensional aerofoil with split flap near the ground . pressure distributions have been measured on a 10 thick two-dimensional aerofoil of r.a.e.101 section fitted with split flaps deflected at 15 and 55 . measurements were made at two distances above a ground plate, and also without the ground plate . the results have been integrated to give the sectional lift, drag and pitching-moment coefficients .
calculated lift distributions in incompressible flow on some sweptback wings . in the course of a larger survey of some aerodynamic characteristics of a family of sweptback wings, the low-speed lift distributions were calculated . the 35 planforms considered cover a range of leading-edge sweep angles from 55 to 70, and aspect ratios from 2 to 3.9 . the results are given here, together with a comparison with other calculations and with experimental results on one particular wing .
an extension of the method of generalised conical flows for lifting wings in supersonic flow . the method of generalised conical flows has previously been developed subject to the condition that the upwash divided by the streamwise co-ordinate to the power k, where k is the order of the conical flow, must have vanishing (k+1)th derivative with respect to the conical co-ordinate . in the present note this restriction is removed . the results are also used to discuss the effect of the application of the leading edge attachment condition on the wing pressure and geometry .
effects of leading edge bluntness on flutter characteristics of some square- planform double-wedge airfoils at a mach number of 15 .4. results are presented from a wind-tunnel investigation in helium flow at a mach number of 15.4 . the models were square-planform, double-wedge, shaft-mounted airfoils with leading- and trailing-edge radii of 0, 1, 3, and 6 percent chord . in general, the tests indicate that bluntness effects on the model flutter characteristics are stabilizing as the leading-edge radius is increased from 0 to destabilizing with further increase in bluntness . results of flutter calculations made by using newtonian theory aerodynamics and a combination of newtonian theory and piston theory aerodynamics in conjunction with an uncoupled two-mode analysis are compared with experimental results . the piston-theory results accurately predicted flutter speeds for the models with
heat transfer and pressure distributions on a hemisphere-cylinder and a bluff-afterbody model in methane-air combustion products and in air . an experimental investigation has been made to indicate the validity of using methane-air combustion products as the test medium for aerodynamic heating and loading tests . tests were conducted on a hemisphere-cylinder and on a bluff-afterbody model, both in methane-air combustion products and in air alone, and covered a range of mach numbers from 6 to the data showed that the nondimensional heating-rate distribution along a hemisphere-cylinder as obtained in combustion products was in good agreement with that obtained in air, and the results were in reasonable agreement with theory . the stagnation-point heating rates in air and in combustion products over the hemisphere-cylinder agreed within 10 percent of the theoretical values . the pressure distributions around a hemisphere-cylinder obtained from tests in combution products were in good agreement with those obtained in air and could be predicted by newtonian flow theory . the tests in combustion products of a bluff-afterbody model produced nondimensional heat-transfer coefficients which were in fair agreement with results obtained in air .
pressure distribution induced on a flat plate at a free-stream mach number of 1.39 by rockets exhausting upstream and downstream . an experimental investigation was made of the pressures induced on a flat plate at a free-stream mach number of 1.39 by a supersonic rocket jet exhausting upstream and downstream . measurements of the pressure distribution on a flat plate were made at zero angle of attack for 11 different locations of the jet exhaust nozzle beneath the wing . measurements were made at ratios of rocket-exit total pressure to free-stream static pressure from 6 to 60 and at a reynolds number per foot of approximately 10 times 10 to the power of 6 . the rocket when exhausted upstream produced a strong shock that moved further upstream with increasing rocket-exit total-pressure ratio . positive incremental normal-force coefficients were obtained at all test positions . data at 11 test positions are tabulated for rocket-on and rocket-off pressure coefficients as well as for incremental pressure coefficients for the 48 orifices of the flat plate for the range of ratio of rocket-exit total pressure to free-stream static pressure of the investigation . changing the location of the model with respect to the plate had a negligible effect when the rocket was varied in the chordwise direction, but the pressure coefficients were reduced as the rocket was lowered away from the flat-plate wing .
an integral equation relating the general time-dependent lift and downwash distributions on finite wings in subsonic flow . an integral equation for obtaining the unsteady air forces on finite wings in subsonic compressible flow is presented . this equation is applicable for any arbitrary time-dependent motion and can be utilized for flexible as well as rigid wings . the approach involves the derivation of an integral equation relating the unknown pressure the form of the equation is such that it should lend itself readily to modern high-speed computers for obtaining pressure distributions . special cases of the integral equation are treated for two-dimensional incompressible flow and are presented in an appendix .
longitudinal aerodynamic characteristics at low subsonic speeds of a highly swept wing utilizing nose deflection for control . an investigation has been conducted in the langley 7- by 10-foot transonic tunnel at low subsonic speeds to determine the longitudinal aerodynamic characteristics associated with deflection of the nose section of a highly swept delta wing having an aspect ratio of 1.33 . in order to illustrate the effectiveness of this forward control, the longitudinal control characteristics are also presented for the wing with upper-and lower-surface split flaps located at the trailing edge . comparison between the longitudinal aerodynamic characteristics of the wing utilizing the nose control and those of the wing utilizing the upper-surface split flap located at the trailing edge indicated similar control effectiveness for high control deflections (15) and similar values of trimmed lift-drag ratio with increasing lift coefficient . use of the nose control, however, indicated a lower value of trimmed angle of attack for a given value of trimmed lift coefficient than that realized from use of the upper-surface split flap . further reductions in trimmed angle of attack for a given value of trimmed lift coefficient may be realized from deflection of the lower-surface split flap at the wing trailing edge in combination with the nose control and would be accompanied by large reductions in lift-drag ratio .
analytical study of the tumbling motions of vehicles entering planetary atmospheres . the tumbling motion of vehicles entering planetary atmospheres is analyzed . a differential equation governing the tumbling motion, its arrest, and the subsequent oscillatory motion is obtained and identified as the equation for the fifth painleve transcendant . an approximate analytical solution for the transcendant is derived . comparisons with results obtained from numerical integration of the exact equations of motion indicate that the solution for the angle-of-attack history is sufficiently accurate to be of practical use .
the design of structures to resist jet noise fatigue . the design of structures to resist jet noise fatigue demands a knowledge of a wide range of subjects from pure acoustics at one hand to metal physics at the other . at the present time the various aspects of the problem are not sufficiently well know quantitatively for a purely theoretical design study to be made . never- the-less a knowledge of the behaviour of typical forms of construction in noise environments can be used with a limited amount of theoretical work to indicate tne most efficient types of structure . this approach to the problem is adopted in this lecture as it seems to be the most promising one available at the moment . it must be emphasized, however, that although some progress has been made in dicsovering the behaviour of a structure subjected to noise it is not possible to estimate the life of any component at the drawing board stage . some prototype strain measurements and proof testing are therefore essential if one is to prove the integrity of the design . within the structural limits of single skin construction set in this lecture the main conclusion to be reached is that no reasonable estimate of fatigue life can yet be made in the drawing board stage of a structure . nevertheless, a study of the form of behaviour of typical structures has led to a theoretical simplification of the problem of skin vibration . from this it has been possible to suggest an optimum deisgn for a skin stiffened by stringers . a suggestion for an optimum design of skin and rib for control surfaces to minimise stresses at the rib-skin intersection is put forward but no experience can check this yet . the most resonable basis for the future estimation of fatigue life of a component appears to be the /random/ s-n curve and consierable effort should be made to obtain the necessary test data . the life expectation of a new design will be uncertain and some proof testing is essential if the integrity of structure in high noise levels (150 db) is to be guaranteed .
reduction of the clamped plate to two membrane problems with an application to uniformly loaded sectors . the clamped plate problem in the classical theory for the small deflection bending of flat plates is reduced to the solution by variational methods of two successive membrane problems . the first requires the least square minimisation of the average curvature of the deflected surface while the second problem concerns the integral of the gaussian curvature . there is a similar reduction for extensional problems where the boundary tractions are specified . the method is demonstrated by giving three distinct solutions to the problem of the clamped sector under a uniformly distributed load . one solution is of special interest because it is derived from a single membrane problem . numerical data are given .
the buckling strength of a uniform circular cylinder loaded in axial compression . the theoretical estimation of the buckling strength of a cylinder loaded in axial compression is improved by the use of a more representative deflected form for the buckled cylinder than has previously been used . kempner’s buckling strength for dead weight loading is reduced by 18 . the presentation of the magnitude and distribution of the constraint system required to maintain the mode is novel and instructive .
an investigation of wing-aileron flutter using ground launched rocket models . control surface flutter of the wing torsion-control rotation type has been investigated for an unswept wing with an under-massbalanced, half span, outboard aileron . thirteen pairs of wings were tested, using ground launched rocket driven vehicles, and a range of values of aileron natural frequency was covered . the test results showed considerable scatter, but enabled upper and lower limits of a flutter boundary to be determined approximately . it was established that aileron flutter could be eliminated on the models tested provided the aileron frequency exceeded the wing torsional frequency by 20 per cent or more . in this condition the models were also free from single degree of freedom flutter
a study of the cantilever square plate subjected to a uniform loading . plate problems involving free edges have been historically difficult to solve, particularly when two free edges are adjacent, resulting in a free corner . the cantilevered square plate subjected to a transverse loading is one such problem for which an exact solution has not been achieved . in the present paper results obtained by various approximate methods are presented for this problem for the case of a uniform loading . solutions obtained by the authors using the technique of point matching and the rayleigh-ritz method are compared with previously published finite-difference and experimental results and with bernoulli-euler beam and plane-strain approaches . numerical results for deflections, slope components, bending and twisting moments, and transverse distributed shears are presented for a relatively fine gridwork of points on the plate boundary and within the interior . the antielastic curvature is exhibited by all methods except beam theory . all methods present the interesting conclusion that the free edge deflection is greater when the plate is treated as a plate rather than a beam .
thermodynamic coupling in boundary layers . experimental results gathered in recent years for binary mixture mass transfer models are shown to yield consistent evidence of discrepancies with analytic considerations . specifically, measured recovery temperatures are appreciably higher than those predicted ,. while heat transfer coefficients are satisfactorily reproduced . it is shown on the basis of both approximate and exact solutions for plates and stagnation points that the discrepancies in previous results are related to thermal diffusion effects, a major influence being apparent in application of the surface boundary condition for an adiabatic wall . as a result, some reexamination is necessary of past criteria for mass addition effects as they pertain to specific injected media . a prime example is the /equivalence/ of helium and air as coolants despite the heretofore suggested preference for low density injectants on a perfect gas basis . ref. 16 .
thermal diffusion effects on energy transfer in a turbulent boundary layer with helium injection . a circular cylinder with two-inch diameter and with a porous wall fabricated out of woven wire material was aligned with its axis parallel to an air stream with approximately 100 ft sec velocity . helium gas was injected into the turbulent boundary layer through the cylinder walls at a uniform rate in the range 1.55 x 10 to 1.08 x 10 of the free stream mass velocity . the local energy transfer along the cylinder was measured at various values of the wall temperature level for the situation that the energy flows from the cylinder to the boundary layer and vice versa . the results showed clearly that the wall temperature for zero energy transfer – the adiabatic wall temperature – was larger than the free stream temperature by up to about 40 f, although viscous dissipation effects are negligible . this temperature excess increases with increasing injection rate and is independent of reynolds number . an analysis in which the laminar sublayer is treated as couette flow with helium injection and which includes thermal diffusion in this layer is formulated . the results show appreciable thermal diffusion effects on adiabatic wall temperature, increasing it over its value for zero injection by amounts of the same order of magnitude as found by measurements . thermal diffusion however has negligible effects on the heat transfer coefficient . its effects on the concentration and temperature distribution are discussed and are shown to produce appreciable modifications in the latter .
bending of a uniformly loaded rectangular plate with two adjacent edges and the others either simply supported or free . the distribution of deflection and bending moment in a uniformly loaded rectangular plate having two adjacent edges clamped and the others either simply supported or free, are obtained by a method of superposition . numerical values are given for square plates and, in one case, the results are compared with those obtained by another method .
the approximate analysis of certain boundary value problems . a simple method is given which is suitable for the approximate analysis of certain boundary-value problems, including, for example, the small deflections of clamped plates and the torsion of prismatic bars . the analysis is particularly simple and lends itself well to the use of the digital computer . the method is applied here to four problems, the uniformly loaded, clamped square, and equilateral-triangle plates, and the torsion of bars of square and hexagonal cross section . the results agree well with the exact solutions, where these are known .
the hovercraft – a new concept in maritime transport . the hovercraft is the first operational british project in the ground-effect machine field . although there has, for a number of years, been a tentative searching after the principles underlying such machines, it is only now that their possibilities as commercial transport and service craft are beginning to be developed . since the hovercraft is a new vehicle, the appearance of the saunders-roe sr-n1, a manned experimental craft, excited considerable public attention and there have been a number of descriptive articles in the press . papers of a more technical type, on ground-effect machines, are now beginning to appear and it is to be expected that these will rapidly increase in number, especially since american interest in both the commercial and defence fields is expanding fast . the authors of the present paper have, therefore, concentrated attention upon features about which they had something personal to say, and which they consider to be of particular significance for assessing the possibility of the hovercraft becoming important in maritime transport . these features are ..- the hovercraft as a fundamentally new principle in the transport field . the powering requirements and resistance characteristics . the likely operating costs of hovercraft in comparison with other forms of maritime transport . in addition, relatively brief descriptions of the history and the current work being undertaken on the ground-effect machine and of the design, construction, and testing of the saunders-roe sr-n1 are provided . the final section discusses outstanding problems and some future possibilities .
some design problems of hovercraft . analysis of the influence various aerodynamic parameters have on the performance of a simple peripheral jet system . power weight ratio, lift drag ratio, and effect of jet angles and thickness are each considered . structural requirements, optimum cushion pressure, and dynamic stability over waves are examined and then related to the economics of ground-effect machine operation .
heat transfer to separated and reattached subsonic turbulen flows obtained downstream of a surface step . local heat-transfer coefficients and recovery factors are presented for separated and reattached turbulent flows as obtained by a downward step in an otherwise flat surface in a two- dimensional, subsonic, air flow . the region downstream of the step, the focus of this investigation, contained a region of separated flow with reattachment at about five step heights downstream, followed by a section of reattached flow . the salient feature of the results is the maximum in the local heat-transfer coefficient at the reattachment point, with values thereof diminishing in the separated region and also in the reattached region, where they tend toward values characteristic of turbulent boundary-layer flow . it is found that for most of the region the heat-transfer coefficient depends on the velocity to about the 0.8 power, though a decreased dependence may exist in the separated region . recovery factors have the characteristically low values associated with separated flows, and do not attain values typical of turbulent boundary-layer flows within the downstream lengths available .
pressure distribution on two dimensional wings near the ground . a simple method of calculating the pressure distribution in incompressible flow on two-dimensional aerofoils of arbitrary section at moderate distances from the ground is developed . comparisons with an /exact/ potential flow solution, and with measurements on a 10 thick aerofoil of rae.101 section, provide a satisfactory verification of the adequacy of the method ,. but it is shown that it is necessary to take account of the boundary layer on the aerofoil in the calculations .
transient magnetohydrodynamic duct flow . parallel flow of an electrically conducting viscous incompressible fluid in a rectangular duct with transverse magnetic field is considered . the walls of the duct which are parallel and perpendicular to the imposed magnetic field are taken to be nonconducting and perfectly conducting, respectively . assuming the fluid to be at rest at the initial moment, exact solutions for the velocity and magnetic field components are obtained in the form of convolution integrals taking the longitudinal pressure gradient as an arbitrary given function of time . later, taking a step function for the pressure gradient, these expressions are integrated . for this case, the effect of the strength of the imposed magnetic field on the development behavior of the flow is studied . it is found that except for very large magnetic fields, the flows are over damped .
on the propagation and structure of the blast wave . part 1. as a continuation of part 1 (j. phys. soc. japan 8 (1953) 662), the second approximation for the propagation and structure of a blast wave is now discussed . the solution for r=1.4 is obtained by a numerical method, using the results of the first approximation obtained in part 1 . by use of this solution, u-r curves, distance-time curves and the changing feature of distributions of velocity, pressure and density behind the shock front are discussed . further, the approximate solution of the equation is discussed by a refinement of the wkb method due to imai .
effects of boundary layer displacement and leading edge bluntness on pressure distribution, skin friction, and heat transfer of bodies at hypersonic speeds . results are presented of an investigation to determine the effect of boundary-layer displacement and leading-edge bluntness on surfaces in hypersonic flow . the presence of the boundary layer and the blunt leading edge induce pressure gradients which in turn affect the skin friction and heat transfer to the surface . methods for predicting these phenomena on two-dimensional surfaces are given and a brief review of recent three-dimensional results is presented .
departure from dissociation equilibrium in a hypersonic nozzle . the equations of motion for the flow of an ideal dissociating gas through a nearly conical nozzle have been solved numerically, assuming a simple equation for the rate of dissociation, and a number of different values of the rate constant . the results of these calculations suggest that deviations from dissociation equilibrium will occur in the nozzle if the rate constant lies within a very wide range of values . they also suggest that once such a deviation has begun the gas will very rapidly/freeze/, so that the dissociation fraction will remain almost constant if the flow is expanded further, or even if it passes through a constant area test section . an approximate method of solution, making use of this property of sudden/freezing/of the flow, has been developed and applied to the problem of estimating the deviations from equilibrium under a wide range of conditions . if all the assumptions made in this report are accepted, then lack of dissociation equilibrium may be expected in the working sections of hypersonic wind tunnels and hypersonic shock tubes . it is shown, however, that the flow behind a normal shock wave in such a wind tunnel will not be greatly affected by any freezing that may take place in the nozzle upstream of the shock wave . even so, the stand-off distance of a shock wave in front of a blunt model may be quite sensitive to deviations from equilibrium .
interferometric studies of supersonic flows about truncated cones . fringe shifts on interferograms of flows at m=2.45 about variously truncated 15 (half-angle) cone cylinders in free flight in a pressurized range have been examined for similarity of the flow fields, occurrence of scale effects, and convergence to conical flow . it was found that flows over similar objects with equal tip reynolds numbers were similar and that convergence to conical flow occurred before the disturbance at the tip had been reflected the second time along characteristics to the body . density distributions have been determined, and a number of comparisons have been made with theoretical predictions .
review of panel flutter and effects of aerodynamic noise part i.. panel flutter . with the development of high-speed aircraft and missiles, vibration of panels has become a problem of practical significance . many of the failures of the early german rockets after attaining supersonic speed have been attributed to the development of such panel oscillations . it appears this phenomenon is not of much concern in the subsonic speed range., however, in the supersonic speed range panels may develop oscillations which cause instability of the structure . this effect has been exhibited experimentally under controlled laboratory conditions motion is limited and buckling may not be a serious design problem . in these cases panel flutter is still of importance because of its effect on the fatigue life and the allowable stresses for design of the panel material . the oscillations of panels may be due either to aerodynamic force induced by the motion of the panel, or to aerodynamic noise, or buffeting (irregular motion induced by turbulence in the flow) . the interaction between aerodynamic forces and panel motions, usually referred to as /panel flutter,/ has been investigated by several workers in recent years . since the problem is too complex to be dealt with in its entirety, simplifying assumptions have been made in these investigations . the literature is marked by a certain degree of controversy over the validity of these assumptions and the applicability of the results obtained . a brief review of the literature with reference to several of the approximations made and the results obtained follows .
nonuniform shear flow past cylinders . a general method is described whereby an approximation of any desired degree of accuracy to the stream functions for two types of variable shear flows past finite cylinders can be obtained . the two shear distributions in the free stream can be approximated to the linear shear distribution and the shear present in an unretarded incompressible boundary layer respectively . in every case the stagnation streamline is displaced from the position opposite the line of symmetry of the cylinder, and general expressions are obtained for this displacement . the line of symmetry may be in the direction of or perpendicular to the direction of flow . the two particular examples cited are those of a general elliptic cylinder and cylinders of the form where and being the polar coordinates, and 2p the maximum width of the cylinder .
the fundamental solution for small steady three dimensional disturbances to a two dimensional parallel shear flow . after a brief review of methods of calculating the flow fields produced by disturbances in rotational basic flows, the author points out a fundamental difficulty in the treated as a perturbation of the disturbance field that would occur if the basic flow were uniform) .. slow attenuation of the secondary-flow disturbance with distance from the obstacle . the author conjectured (same j. 1 the trouble was caused by nonuniform validity of the approximation sequence in the region far from the obstacle . the analogy with /stokes’ and whitehead’s paradoxes/ is mentioned, and a solution analogous to oseen’s is suggested, one in which disturbances, but not the shear, are assumed to be small . in this paper, such a solution is found, and is shown to overlap with the small-shear, secondary-flow solution . the basic flow is a parallel, steady, inviscid, two-dimensional shear flow . the / fundamental solution/ due to a weak source is sought . the method of fourier transforms is used . simple solutions are found for a uniformly sheared basic flow (where the result coincides with the secondary-flow solution) and for an exponential basic-flow profile . in the general case it is assumed that the parallel basic flow becomes uniform at, where the x-axis lies in the flow direction . the character of the solution is determined by studying its hankel transform, especially for the class of flows where the total variation of the basic stream speed v(y) is small . an interpretation in terms of images, due to m. b. glauert, is given, and finally the relationship of the present work to theories of the displacement of the stagnation streamline (displacement effect of pitot tubes) is discussed .
summary of laminar boundary layer solutions for wedge-type flow over convection and transpiration cooled surfaces . a summary of exact solutions of the laminar-boundary-layer equations for wedge-type flow, useful in estimating heat transfer to such arbitrarily shaped bodies as turbine blades, is presented . the solutions are determined for small mach numbers and a prandtl number at the wall of 0.7 ,. ranges of mainstream pressure gradients and rates of coolant flow through a porous wall are considered for the following cases .. (1) small temperature changes in the boundary layer along a constant- and along a variable-temperature wall, and (2) large temperature changes in the boundary layer along a constant-temperature wall . dimensionless forms of heat-transfer and friction parameters and boundary-layer thicknesses are tabulated . the results indicate that coolant emission and increased stream-to-wall temperature ratios diminished the friction and heat transfer for a constant wall temperature . for a variable wall temperature with small temperature differences in the boundary layer, the friction was unaffected, but the heat transfer was greatly increased for increased wall-temperature gradient . heat-transfer results in the literature reveal that transpiration cooling is much more effective for prandtl numbers of the order of 5.0 than for 0.7 .
theoretical and experimental investigation of aerodynamic-heating and isothermal heat transfer parameters on a hemisphere nose with laminar boundary layer at supersonic mach numbers . the effect of a strong, negative pressure gradient upon the local rate of heat transfer through a laminar boundary layer on the isothermal surface of an electrically heated, cylindrical body of revolution with a hemispherical nose was determined from wind-tunnel tests at a mach number of 1.97 . the investigation indicated that the local heat-transfer parameter, based on flow conditions just outside the boundary layer, decreased from a value of 0.65 0.10 at the stagnation point of the hemisphere to a value of 0.43 0.05 at the junction with the cylindrical afterbody . because measurements of the static pressure distribution over the hemisphere indicated that the local flow pattern tended to become stationary as the free-stream mach number was increased to 3.8, this distribution of heat-transfer parameter is believed representative of all mach numbers greater than 1.97 and of temperatures less than that of dissociation . the local heat-transfer parameter was independent of reynolds number based on body diameter in the range from 0.6×10 to 2.3×10 . the measured distribution of heat-transfer parameter agreed within theoretical distribution calculated with foreknowledge only of the pressure distribution about the body . this method, applicable to any body of revolution with an isothermal surface, combines the mangler transformation, stewartson transformation, and thermal solutions to the falkner-skan wedge-flow problem, and thus evaluates the heat-transfer rate in axisymmetric compressible flow in terms of the known heat-transfer rate in an approximately equivalent two-dimensional incompressible flow . measurements of recovery-temperature distributions at mach numbers of 1.97 and 3.04 yielded local recovery factors having an average value of 0.823 0.012 on the hemisphere which increased abruptly at the shoulder to an average value of 0.840 0.012 on the cylindrical afterbody . this result suggests that the usual representation of the laminar recovery factor as the square root of the prandtl number is conservative in the presence of a strong, accelerating pressure gradient .
viscous flow along a flat plate moving at high speeds . by the distortion of coordinates, it is shown that, in the case of supersonic viscous flow past a flat plate, the boundary-layer and simple wave theories can be combined to give a complete representation of the velocity and pressure fields . consistent first-order solutions are considered . an expression for the induced pressure on the plate, correct to the second order, is obtained . at high mach numbers the important parameter satisfies the hypersonic similarity law ,. and for arbitrary mach and reynolds numbers and for different gases, the theoretical curve correlates closely the experimental data . asymptotic shock curve and skin-friction coefficient are also deduced, but the experimental verifications are yet to be made .
the boundary layer on a flat plate in a stream with uniform shear . the incompressible laminar boundary layer on a semi-infinite flat plate is considered, when the main stream has uniform shear . a solution is obtained for the first two terms of an asymptotic solution for small viscosity . it is shown that one of the principal effects of free-stream vorticity is to introduce a modified pressure field outside the boundary-layer region .
on the theory of hypersonic gas flow with a power law shock wave . plane and axisymmetric hypersonic gas flows are considered with shock waves of very great intensity that have a power-law form . on the basis of an investigation of the portions of the flow with high entropy adjoining the surface of the body (not necessarily for a shock wave of the given form) it is shown that the use in the flow problem of the exact solution for the corresponding unsteady self-similar gas motion requires a supplementary refinement of the thickness of the high entropy layer . a method is shown for introducing such a correction and constructing the shape of the body contour, on which is to be applied the pressure distribution obtained on the basis of the theory of small disturbances .
blunt body heat transfer at hypersonic speed and low reynolds numbers . an analytical method for the determination of effect of shock curvature on heat transfer in the region of the nose has been developed . it is shown that for practical body shape the viscous terms in the navier-stokes equations are not important in the region of the flow far from the wall, and the displacement thickness can be neglected . then the flow can be approximately represented by an inviscid-flow solution having as boundary conditions the body shape, which is not affected by the reynolds number, and by a boundary-layer type of flow near the wall, having appropriate boundary conditions . this approach permits us to determine the heat transfer in the region of the nose even at very low reynolds numbers . experimental results are presented . the experimental results agree with the values given by the analysis .
hypersonic shock layer theory of the stagnation region at low reynolds number . cheng, h.k. hypersonic flow at low reynolds number is studied utilizing the shock-layer concept . the present formulation takes into account the salient features of the transport processes within the shock layer in a manner consistent with the shock-layer approximation . the rankine-hugoniot shock relations are modified to include contributions due to heat conduction and viscous effects immediately behind the shock . the specific problem of an axisymmetric stagnation region is treated . the flow regimes for this problem can be classified according to whether or not the transport effects are important immediately behind the shock . in one regime where the ordinary rankine-hugoniot relations hold across the shock, the vorticity-interaction theory based on the boundary-layer approximation is shown to be sufficient . in the other regime where the rankine-hugoniot relations have to be modified but the continuum-flow model applies, an approximate, an analytical solution is obtained . this solution reveals a substantial reduction of the temperature behind the shock and of the shock stand-off distance in the presence of strong surface cooling . the present study is intended to provide a knowledge to bridge the gap between the free-molecule flow regime and that of the boundary layer via the continuum theory . in this respect, the solution obtained appears to be satisfactory in that it yields the correct free-molecule limits for the skin friction and surface-heat transfer rate .
measurements of stagnation point heat transfer at low reynolds number . measurements of stagnation point heat transfer are presented in the reynolds number range between the free molecular flow and the range where modified boundary layer theory still applies . the measurements are compared with the analytical methods set forth by ferri, zakkay, and ting . the results show smooth transition between the two regions and indicate that the predicted reynolds number for which the modified boundary layer theory can be used is in agreement with experiments . in the lower range of reynolds number the ratio of decreases and reaches a value of 1 at a reynolds number of 40 .
subsonic potential flow past a sphere inside a cylindrical duct . the subsonic potential flow of a compressible fluid past a sphere in an infinite medium was first determined by rayleigh . subsequently, caplan and tamada extended the solution to include the fourth power of the mach number . to the author’s knowledge, no solution for subsonic flow past a sphere in a finite medium has been published . it is the purpose of this note to present a solution for subsonic potential flow past a sphere inside a circular cylindrical duct .
on blunt-body heat transfer at hypersonic speed and low reynolds number . a discussion of differences arising between experimental and analytical results, in particular those due to inconsistencies introduced in the presentation of data and the way the comparison is made .
pressure and boundary-layer measurements on a two dimensional wing at low speed . results are given of pressure measurements and boundary-layer traverses on a two-dimensional wing with 10 per cent rae 101 section at reynolds numbers of 1.6×10 and 3.2×10 . these results which have been integrated to give lift, drag and aerodynamic-centre characteristics, are used to check some calculation methods for the growth of the turbulent boundary layer and for the effect of a known boundary layer on the pressure distribution . it is concluded that the calculation of the boundary layer still needs a little refinement before it is accurate enough to predict viscosity effects on pressure distribution, lift, drag and aerodynamic center,.but that these effects can be calculated if the actual boundary-layer characteristics are known .
tunnel interference effects . the problems of solid blockage, wake blockage, lift effect, and the influence of boundary constraint at high mach number are considered in detail . corrections are given for various open and closed tunnels, rectangular, circular and octagonal, and different speeds, two and three dimensional flows, with several aerofoils and wings . other interferences include the wall boundary layer, gradient of static pressure and problems with the working fluid used .
investigation of full scale split trailing edge wing flaps with various chords and hinge locations . an investigation was conducted in the n. a. c. a. full-scale wind tunnel on a small parasol monoplane equipped with three different split trailing-edge wing flaps . the object of the investigation was to determine and correlate data on the characteristics of the airplane and flaps as affected by variation in flap chord, flap deflection, and flap location along the wing chord . the chords of the flaps were 10, 20, and 30 percent of the wing chord and each flap was tested at deflections from 0 to 75 when located successively at 68, 80, and 88.8 percent of the wing chord aft of the leading edge . the investigation included force tests, pressure-distribution tests, and downwash surveys . the results give the lift, the drag, and the pitching-moment characteristics of the airplane, the flap forces and moments, the pressure distribution over the flaps and wing at one section, and the downwash characteristics of the flap and wing combinations . an increase in flap chord or distance of the flap from the leading edge of the wing increased the lift of the airplane but had an adverse effect on the wing pitching moment . the ld ratio of the airplane decreased with increase in flap deflection or flap chord . flap normal-force coefficients were primarily a function of flap deflection and were relatively independent of flap chord, hinge-axis location, and airplane attitude . the location of the flap center of pressure in percentage of flap chord aft of the hinge axis remained practically constant irrespective of airplane attitude and of flap deflection, chord, or location . flap hinge-moment coefficients varied with a power of flap chord greater than the square so that with regard to hinge moments narrow flaps were the most efficient in producing a given increase in lift . split trailing-edge flaps materially affected the magnitude and distribution of pressures over the entire wing profile . at low angles of attack the predominant effect of the flaps was to increase positively the lower-surface pressures ,. at high angles of attack, to increase negatively the upper-surface pressures . downwash surveys indicated that horizontal tail planes located above the wing chord line would be more effective than those below the chord in counteracting the increased diving moment of the airplane with flaps deflected .
the shapes and lift-dependent drags on some sweptback wings designed for m= 1. 2. the camber and twist distributions needed to produce a constant span-wise -distribution and certain linear chordwise load distributions have been calculated by linearised supersonic theory at for a set of 34 thin sweptback wings . the wing planforms cover a range of aspect ratios from 2.0 to 3.5 and leading-edge sweep angles from 55 to 70 . both leading and trailing edges are subsonic at the design mach number, and the slenderness parameter is between 0.19 and 0.40 . the lift-dependent vortex and wave drags associated with these loadings have also been calculated, and appear not to be excessive in almost all the cases considered .
pressure distribution and surface flow on 5( and 9( thick wings with curved tip and 60degree sweepback . extensive tables are given of pressure coefficients measured at reynolds numbers from 1.3×10 to 3.9×10 on two half-models of identical planform with 5( rae 101 and 9( rae 101 streamwise sections . the planform of aspect ratio 3.899 has a straight trailing edge with 60degree of sweepback, constant chord over most of the span and a parabolic outer portion of the leading edge curving to a pointed tip . the overall wing characteristics are obtained from integrated normal pressures and are compared with lifting-surface theory . the low-speed experimental pressure distributions and surface oil-flow patterns are analysed and discussed in relation to the onset of separation and the distinct vortex flows that develop at high incidence . series of contrasting upper-surface isobars illustrate some features of the different stalling processes of the two wings . the direct influence of the main vortex on local surface pressures is assessed in general terms . a fuller appraisal of secondary surface flow is obtained from the oil patterns, observations in water and measurements of high suction near the trailing edge . studies of the extent of leading-edge stall and location of part-span vortices, in particular two simultaneous leading-edge vortices on the thinner wing, follow from further analysis of local surface pressures . after a detailed discussion of the effect of reynolds number and the distinct types of separated flow, a few results with leading-edge roughness are considered in relation to scale effect on separation and the extensive influence of part-span roughness .
a simple method for calculating the span and chordwise loading on straight and swept wings of any aspect ratio at subsonic speeds . the methods of the classical aerofoil theory are used to derive a general theory for wings of any given planform . the load over the whole surface of a given wing can be calculated at a given subcritical mach number, and the procedure is as simple and rapid as that of the classical aerofoil theory . the calculated results are confirmed by experiments .
methods for calculating the lift distribution of wings /subsonic lifting surface theory/ . this report contains some fairly simple and economic methods for calculating the load distribution on wings of any plan form based on the conceptions of lifting surface theory . the computer work required is only a small fraction of that of existing methods with comparable accuracy . this is achieved by a very careful choice of the positions of pivotal points, by plotting once for all those parts of the downwash integral which occur frequently and by a consequent application of approximate integration methods similar to those devised by the author for lifting line problems . the basis of the method is to calculate the local lift and pitching moment at a number of chordwise sections from a set of linear equations satisfying the downwash conditions at two pivotal points in each section . interpolation functions of trigonometrical form are used for spanwise integration both in setting up the downwash equations and in getting the resultant forces on the wing from the local forces . the preliminary chordwise integrations for the downwash are predigested in a series of charts/figs.1-6/,.it is these which make the method a practical computing proposition . the theory is outlined in sections 2-5,.section 6 deals with the solution of the linear equation and section 7 with the resultant forces on the wing . some examples are worked out in section 8 to compare with other methods,. one solution is given in full detail in tables 8-30 as a guide for computers . appendices i-vi discuss more carefully some salient points of the mathematical theory, and appendix vii is intended to instruct the computer how to carry out the steps of the calculation .
the effect of end plates on swept wings . existing methods of calculating the effect of endplates on straight wings are modified so as to apply to swept wings . the changes in overall lift and drag, and also the spanwise distribution of the additional load, can be calculated . the theoretical results are compared with experimental results obtained on swept wings, including new measurements of lift, drag and pitching moment, made on an untapered the method of calculation is also extended to cover the effect of the tip vortex which is formed on wings without endplates .
low speed tests on 45 sweptback wings . this report contains the results of pressure measurements on three and aspect ratio 5, over an incidence range up to 10 . chordwise and spanwise lift distributions are given, mostly near the centre where, on two of the wings, modifications had been made to the section shape . it was found that altering the thickness distribution in the centre did not affect the loading but that approximately straight isobars could be obtained at values of below about 0.1 . by the incorporation of twist and camber in the central part the distortion of the lift distribution in the centre could be avoided at one particular incidence, and thus the same chordwise distribution obtained over most of the span . twist and camber alone do not improve the isobar pattern and therefore a thickness modification would be needed to give the desired lift distribution and isobar pattern at one particular incidence . the results of experimental investigations of the boundary layer and of the effect of aspect ratio will be given in a later report .
generalized conical flow fields in supersonic wing theory . linearized, compressible-flow analysis is applied to the study of quasi-conical supersonic wing theory . single-integral equations are derived which relate either the loading to the shape of a lifting surface or the thickness of a symmetrical wing to the pressure distribution for triangular wings with subsonic leading edges . the forms of these equations and their inversions are simplified through the introduction of the finite part and the generalized principal part of an integral . applications of the theory, in the lifting case, include previously known results . in the nonlifting case, it is shown that for a specified pressure distribution the theory does not always predict a unique thickness distribution . this is demonstrated for a triangular plan form having a constant pressure gradient in the stream direction .
integrals and integral equations in linearized wing theory . the formulas of subsonic and supersonic wing theory for source, doublet, and vortex distributions are reviewed, and a systematic presentation is provided which relates these distributions to the pressure and to the vertical induced velocity in the plane of the wing . it is shown that care must be used in treating the singularities involved in the analysis and that the order of integration is not always reversible . concepts suggested by the irreversibility of order of integration are shown to be useful in the inversion of singular integral equations when operational techniques are used . a number of examples are given to illustrate the methods presented, attention being directed to supersonic flight speeds .
the lift of twisted and cambered wings in supersonic flow . a generalised conical flow theory is used to deduce an integral equation relating the velocity potential on a delta wing/with subsonic leading edges/to the given downwash distribution over the wing . the complete solution of this integral equation is derived . this complete solution is composed of two parts, one being symmetric and the other antisymmetric with respect to the spanwise co-ordinate,. each part represents a velocity potential . for example, if y is the spanwise co-ordinate and x is measured in the free stream direction, then a downwash of the form w-a ux/y/is symmetric and will give rise to a symmetric potential, whereas w-a ux/y/sgn y is anti-symmetric and gives rise to an anti-symmetric potential . the velocity potentials of such flows are given in the form of tables for all downwashes up to and including homogenous cubics in the spanwise and streamwise co-ordinates . table iii gives similar formulae in the limiting case were used over a cycle of the tumbling motion . the analytical expression was in good agreement with numerical solutions of the complete non-linear equations of motion .
the use of conical camber to produce flow attachment at the leading edge of a delta wing and to minimize the lift-dependent drag at sonic and supersonic speeds . in an attempt to avoid flow separation at the leading edge of a thin delta wing with subsonic leading edges, an attachment line is prescribed there . this is done by requiring the load, as predicted by attached flow theory, to vanish along the leading edge at the design lift coefficient . for sonic speed, a complete account of this flow is given in terms of slender wing theory and the load distributions corresponding to arbitrary conical camber are calculated . for supersonic speeds load distributions arising in the slender wing theory are considered and the corresponding conical camber distributions are found by linearized theory . the lift-dependent drag for a given lift is then minimized with respect to the coefficients of a linear combination of these load distributions . it is found that the lift-dependent drag factor for these conically cambered wings approaches the value it takes for the attached flow/in which leading edge suction occurs/past the uncambered wing at the same mach number, as more terms are included in the linear combination . however, when the leading edge is almost sonic an appreciable reduction is predicted . the corresponding load distributions and wing shapes are calculated and drawn . the optimum shapes for a fixed number of terms resemble flat plates drooped downwards near their edges, so that the localised leading edge suction is replaced by a distributed force on a forward-facing surface, producing an effect of similar magnitude .
tables of complete elliptic integrals . the present paper contains a set of tables of complete elliptic integrals computed and collected especially for applications to certain dynamical problems . the tabulated functions are four in number and are denoted by f/a/, g/a/, e/a/, and/a,b/respectively . the definitions of these functions and their connections with the functions of legendre will be discussed in the following .
aerodynamic effects of some configuration variables on the aeroelastic characteristics of lifting surfaces at mach numbers from 0. 7 to 6. 86 . results of flutter tests on some simple all-movable-control-type models are given . one set of models, which had a square planform with double-wedge airfoils with four different values of leading- and trailing-edge radii from 0 to 6 percent chord and airfoil thicknesses of 9, 11, at mach numbers from 0.7 to 6.86 . the bending-to-torsion frequency ratio was about 0.33 . the other set of models, which had a tapered planform with single-wedge and double-wedge airfoils with thicknesses of 3, 6, 9, and 12 percent chord, was tested at mach numbers from 0.7 to 3.98 and a frequency ratio of about 0.42 . the tests indicate that, in general, increasing thickness has a destabilizing effect at the higher mach numbers but is stabilizing at subsonic and transonic mach numbers . double-wedge airfoils are more prone to flutter than single-wedge airfoils at comparable stiffness levels . increasing airfoil bluntness has a stabilizing effect on the flutter boundary at supersonic speeds but has a negligible effect at subsonic speeds . however, increasing bluntness may also lead to divergence at supersonic speeds . results of calculations using second-order piston-theory aerodynamics in conjunction with a coupled-mode analysis and an uncoupled-mode analysis are compared with the experimental results for the sharp-edge airfoils at supersonic speeds . the uncoupled-mode analysis more accurately predicted the flutter characteristics of the tapered-planform models, whereas the coupled-mode analysis was somewhat better for the square-planform models . for both the uncoupled- and coupled-mode analyses, agreement with the experimental results improved with increasing mach number . in general, both methods of analysis gave unconservative results with respect to the experimental flutter boundaries .
flutter tests of some simple models at a mach number of 7. 2 in helium flow . results of hypersonic flutter tests on some simple models are presented . the models had rectangular plan forms of panel aspect ratio 1.0, no sweepback, and bending-to-torsion frequency ratios of about . two airfoil sections were included in the tests ,. double wedges of 5-, 10-, and 15-percent thickness and flat plates with straight, parallel sides and beveled leading and trailing edges . the models were supported by a cantilevered shaft . the double-wedge wings were tested in helium at a mach number of 7.2 . an effect of airfoil thickness on flutter speed was found, thicker wings requiring more stiffness to avoid flutter . a few tests in air at a mach number of 6.9 showed the same thickness effect and also indicated that tests in helium would predict conservative flutter boundaries in air . the data in air and helium seemed to be correlated by piston-theory calculations . piston-theory calculations agreed well with experiment for the thinner models but began to deviate as the thickness parameter approached and exceeded 1.0 . a few tests on flat-plate models with various elastic-axis locations were made . piston-theory calculations would not satisfactorily predict the flutter of these models, probably because of their blunt leading edges .
oscillating airfoils at high mach number . a simple formula is given for the pressure distribution on an oscillating airfoil in two-dimensional flow at high mach number . the formula is expected to be reasonably accurate if the pressure on the surface remains within the range 0.2 to 3.5 times the mainstream pressure . to illustrate the application of the formula, some results for symmetrical airfoils performing pitching oscillations are obtained and compared with results obtained from existing theories in the case of high mach number .
tables of aerodynamic coefficients obtained from developed newtonian expressions for complete and partial conic and spheric bodies at combined angles of attack and sideslip with some comparisons with hypersonic experimental data . closed-form expressions and tables composed from these expressions are presented for complete and partial conic and spheric bodies at combined angles of attack and sideslip in newtonian flow . aerodynamic coefficients of these bodies are tabulated for various body segments over a range of angles of attack from 1degree to 85degree and angles of sideslip from 0degree to 15degree . some comparisons between newtonian predictions and hypersonic experimental aerodynamic characteristics were made for conic bodies having various surface slopes, nose bluntnesses, and body cross sections to indicate the range of validity of the theory . in general, the theory is shown to agree quite well with experimental results for sharp-nose complete cones and for configurations having large blunted noses and steep surface slopes . however, agreement between theory and experiment generally is poor for the more slender, slightly blunted complete or half conic bodies and also for sharp-nose half conic bodies where real-flow phenomena such as forebody interference, viscous forces, leeward surface contributions, or leading-edge pressure reductions may have significant effect . the agreement between theory and experiment for the bodies considered can be improved by using the stagnation pressure coefficient behind a normal shock rather than 2 as the newtonian coefficient, although for the sharp-nose half conic bodies there is no theoretical justification for this modification .
investigation of the laminar aerodynamics heat transfer characteristics of a hemisphere cylinder in the langley 11-inch hypersonic tunnel at a mach number of 6. 8. a program to investigate the aerodynamic heat transfer of a nonisothermal hemisphere-cylinder has been conducted in the langley 11-inch hypersonic tunnel at a mach number of 6.8 and a reynolds number from approximately 0.14×10 to experimental heat-transfer coefficients were slightly less over the whole body than those predicted by the theory of stine and wanlass (naca technical note 3344) for an isothermal surface . for stations within 45 of the stagnation point the heat-transfer coefficients could be correlated by a single relation between local stanton number and local reynolds number . pitot pressure profiles taken at a mach number of 6.8 on a hemisphere-cylinder have verified that the local mach number or velocity outside the boundary layer required in the theories may be computed from the surface pressures by using isentropic flow relations and conditions immediately behind a normal shock . the experimental pressure distribution at a mach number of velocity gradients calculated at the stagnation point by using the modified newtonian theory vary with mach number and are in good agreement with those obtained from measured pressures for mach numbers from 1.2 to 6.8 . at the stagnation point the theory of sibulkin, in which the diameter and conditions behind the normal shock were used, was in good agreement with the experiment when the velocity gradient at the stagnation point appropriate to the free-stream mach number was used .
investigaion of the flow over a spiked-nose hemisphere cylinder at a mach number of 6. 8. the shape and nature of the flow over a spiked-nose hemisphere-cylinder was studied in detail at a nominal mach number of 6.8 and in a reynolds number range (based on diameter and stream conditions ahead of the model) of 0.12 x 10 to 1.5 x 10 . schlieren photographs showed the effect of varying the spike length and reynolds number upon the shape of the separated boundary and upon the location of transition . the heat transfer and pressure distribution over the body were then correlated with the location of the start of separation, the location of reattachment, and the location of the start of transition .
calculation procedure for thermodynamic transport, and flow properties of the combustion products of a hydrocarbon fuel mixture burned in air with results for ethylene-air and methane-air mixtures . a procedure is presented whereby the composition, thermodynamic properties, and transport properties of the dissociated combustion products of a fuel consisting of a mixed hydrocarbon compound burned in air may be calculated . equations and procedures for determining supersonic nozzle ordinates and flow properties for the dissociated combustion products are presented in an appendix . results are presented for the respective hydrocarbon fuels, methane and ethylene, at the equivalence ratios of 1.0, 0.9, 0.8, and 0.7 for pressures varying between 10 and 8 x 10 atmospheres and temperatures from 200degree k to
investigation of the jet effects on a flat surface downstream of the exit of a simulated turbojet nacelle at a free-stream mach number of 2.02 . an investigation at a free-stream mach number of 2.02 was made to determine the effects of a propulsive jet on a wing surface located in the vicinity of a choked convergent nozzle . static-pressure surveys were made on a flat surface that was located in the vicinity of the propulsive jet . the nozzle was operated over a range of exit pressure ratios at different fixed vertical distances from the flat surface . within the scope of this investigation, it was found that shock waves, formed in the external flow because of the presence of the propulsive jet, impinged on the flat surface and greatly altered the pressure distribution . an integration of this pressure distribution, with the location of the propulsive jet exit varied from 1.450 propulsive-jet exit diameters to 3.392 propulsive-jet exit diameters below the wing, resulted in an incremental lift for all jet locations that was equal to the gross thrust at an exit pressure ratio of 2.86 . this incremental lift increased with increase in exit pressure ratio, but not so rapidly as the thrust increased, and was approximately constant at any given exit pressure ratio .
investigation of jet effects on a flat surface downstream of the exit of a simulated turbojet nacelle at a free-stream mach number of 1.39 . an investigation at a free-stream mach number of 1.39 utilizing a blowdown-type tunnel was made to determine the effects of a propulsive jet on a zero angle-of-attack wing surface located in the vicinity of both a choked convergent nozzle and a convergent-divergent nozzle . staticpressure surveys were made on a flat surface that was located in the vicinity of the propulsive jet . the nozzles were operated over a varied range of both exit static- and total-pressure ratios at different within the scope of this investigation, it was found that shock waves, formed in the external flow because of the presence of the jet exhaust, impinged on the flat surface and greatly altered the pressure distribution . an integration of this pressure distribution for the choked convergent nozzle, with the location of the propulsive-jet exit varied from 1.747 jet-exit diameters to 4.981 jet-exit diameters below the wing surface, resulted in a positive incremental normal force on the wing at all positions .
pressure distribution induced on a flat plate by a supersonic and sonic jet exhaust at a free-stream mach number of 1.80 . as a continuation of previous research at mach numbers of 2.02 and 1.39, an experimental investigation was made of the pressures induced on a flat plate by a propulsive jet exhausting from sonic and supersonic nozzles at a free-stream mach number of 1.80 . measurements of the pressure distribution on a flat-plate wing were made at zero angle of attack for four different locations of the jet exhaust nozzle beneath the wing . both a choked convergent nozzle and a convergent-divergent nozzle on the nacelle were used . the nozzles were operated at nacelle-exit total-pressure ratios from 2 to 16 and the reynolds number per foot was approximately 13 x 10 . two distinct shock waves impinged on the wing surface and greatly altered the pressure distribution at all nozzle positions . positive incremental normal force resulted on the wing at all positions . comparisons are presented for two free-stream mach numbers .
some experiments relating to the problem of simulation of hot jet engines in studies of jet effects on adjacent surfaces at a free-stream mach number of 1.80 . an investigation at a free-stream mach number of 1.80 in a blowdown type tunnel was made to study the effect on the pressure distribution of a zero angle of attack wing surface when certain exhaust parameters of a hot turbojet engine are varied . static-pressure surveys were made on a wing surface that was located in the vicinity of a small-scale propulsive jet . this propulsive jet was operated with four types of jet exhausts . these jet exhausts were a hot jet /hydrogen burned in air/, a cold air jet, a cold helium jet, and a jet composed of a mixture of two cold gases /hydrogen and carbon dioxide/ . the hot jet, because of its high exhaust temperature /3,300degreer/ and because combustion was performed in air, was believed reasonably able to simulate the exhaust parameters of an actual afterburning turbojet engine . the cold jets used were selected in order that the effects of a variation in the exhaust parameters of jet-exit static-pressure ratio, ratio of specific heats, density, and velocity, could be obtained by comparing each cold jet with the hot jet or with another cold jet . the tests were made over a range of jet-exit staticpressure ratios from 1 to 9 with values of the ratio of specific heats of 1.27, 1.40, and 1.66 and at variations in density and velocity of the order of approximately 8 and 3 times, respectively . within the scope of this investigation, it was found that jet-exit static-pressure ratio and the ratio of specific heats affected the pressure distribution on the wing associated with jet interference while a variation in exit velocity and density did not . the jet-exit staticpressure ratio affected the wing pressure distribution in a major way while the ratio of specific heats had only a minor effect . the addition of temperature in the propulsive jet exhaust at a jet-exit staticpressure ratio of 4 had little or no effect on the pressure distribution associated with jet interference on the wing .
pressure loads produced on a flat-plate wing by rocket jets exhausting in a spanwise direction below the wing and perpendicular to a free-stream flow of mach number 2.0 . an investigation at a reynolds number per foot of 14.4 x 10 was made to determine the pressure loads produced on a flat-plate wing by rocket jets exhausting in a spanwise direction beneath the wing and perpendicular to a free-stream flow of mach number 2.0 . the ranges of the variables involved were /1/ nozzle types – one sonic /jet mach number of two-dimensional supersonic /jet mach number of 1.71/,. /2/ vertical nozzle positions beneath the wing of 4, 8, and 12 nozzle-throat diameters,. and /3/ ratios of rocket-chamber total pressure to free-stream static pressure from 0 to 130 . the incremental normal force due to jet interference on the wing varied from one to two times the rocket thrust and generally decreased as the pressure ratio increased . the chordwise coordinate of the incremental-normal-force center of pressure remained upstream of the nozzle center line for the nozzle positions and pressure ratios of the investigation . the chordwise coordinate approached zero as the jet vertical distance beneath the wing increased . in the spanwise direction there was little change due to varying rocket-jet position and pressure ratio . some boundary-layer flow separation on the wing was observed for the rocket jets close to the wing and at the higher pressure ratios . the magnitude of the chordwise and spanwise pressure distributions due to jet interference was greatest for rocket jets close to the wing and decreased as the jet was displaced farther from the wing . the design procedure for the rockets used is given in the appendix .
effects on adjacent surfaces from the firing of rocket jets . this paper is a preliminary and brief account of some research currently being conducted to determine the jet effects on adjacent surfaces from the firing of rocket jets . measurements of jet-effect pressures on a flat plate as well as shadowgraphs are presented that were obtained when a rocket jet at a mach number of 3 was exhausted downstream and upstream into free-stream flow at a mach number of 2 located from 2 to 4.7 rocket-jet-exit diameters from the plate . the jet effects on the flat plate with the rocket jet exhausting downstream are of the same order of magnitude as those previously obtained from sonic exits with a total pressure 10 times lower . a maximum pressure coefficient on the plate of rocket-jet-exit diameters below the plate, and an integration of the measured jet-effect pressures at this position resulted in a normal force on the plate equal to 2.3 times the thrust output of the rocket jet .
the unsteady lift of a wing of finite aspect ratio . unsteady-lift functions for wings of finite aspect ratio have been calculated by correcting the aerodynamic inertia and the angle of attack of the infinite wing . the calculations are based on the operational method . the starting lift of the finite wing is found to be only slightly less than that of the infinite wing,. whereas the final lift may be considerably less . the theory indicates that the initial distribution of lift is similar to the final distribution . curves showing the variation of lift after a sudden unit change in angle of attack, during penetration of a sharpedge gust, and during a continuous oscillation are given . operational equivalents of these functions have been devised to facilitate the calculation of lift under various conditions of motion . as an application of these formulas, the vertical acceleration of a loaded wing caused by penetrating a gust has been calculated .
approximate indical lift functions for several wings of finite span in incompressible flow as obtained from oscillatory lift coefficients . the unsteady-lift functions for a wing undergoing a sudden change in sinking speed have been presented for delta wings having aspect ratios of 0, 2, and 4 and for rectangular and elliptical wings having aspect ratios of 0, 3, and 6 . for the elliptical and rectangular wings the spanwise lift distributions were also presented . these functions were calculated from the lift coefficients associated with a wing oscillating harmonically in pure translational motion, as obtained from several sources . the results of these calculations indicate that the normalized unsteady-lift functions are substantially independent of the shape of the plan form for elliptical, rectangular, or moderately tapered wings,. however, for delta wings the increase of lift toward the steady-state value is much more rapid than that for the aforementioned wings of the same aspect ratio . these results also corroborate the results of other investigations in that the rate of growth of lift tends to increase with a decrease in aspect ratio . the shape of the spanwise distributions of the indicial lift seems to be, for all practical purposes, independent of time for rectangular and elliptical wings .
two and three-dimensional unsteady lift problems in high speed flight . the problem of transient lift on two- and three-dimensional wings flying at high speeds is discussed as a boundary-value problem for the classical wave equation . kirchhoffs formula is applied so that the analysis is reduced, just as in the steady state, to an investigation of sources and doublets . the applications include the evaluation of indicial lift and pitchingmoment curves for two-dimensional sinking and pitching wings flying at mach numbers equal to 0, 0.8, 1.0, 1.2, and triangular wings in both forward and reversed flow are presented and compared with the two-dimensional values .
numerical determination of indical lift of a two-dimensional sinking airfoil at subsonic mach numbers from oscillatory lift coefficients with calculations for mach number 0.7 . the reciprocal equations for relating the incompressible circulatory indicial lift to the lift due to harmonic oscillations have been modified to include the noncirculatory lift associated with apparent-mass effects . although the apparent-mass effects are impulsive in nature in incompressible flow, the lift due to apparent-mass effects in compressible flow is a time-dependent function . the corresponding reciprocal equations for the total compressible lift are given . by use of the reciprocal equations for compressible flow, the indicial lift and moment functions due to an airfoil’s experiencing a sudden acquisition of vertical velocity are determined numerically for mach number 0.7 . lack of sufficient flutter coefficients prevents the calculation of these functions at other mach numbers . although the indicial lift and moment functions due to penetration of a sharp-edge gust may be obtained from the oscillatory tab or aileron coefficients by a similar analysis, sufficient coefficients are not available at the present . however, an approximate method is shown for determining a portion of this unsteady-lift function . when a comparison is made of the indicial lift functions at mach numbers appears to be less rapid for the compressible case than for the incompressible case . consequently, the calculation of the gust load factor at high subsonic mach numbers utilizing the two-dimensional incompressible indicial lift functions and an over-all correction for compressibility such as the prandtl-glauert factor might be conservative .
numerical determination of indical lift and moment functions for a two dimensional sinking and pitching airfoil at mach numbers 0.5 and 0.6 . the indicial lift and moment functions are determined approximately for sinking and pitching motion at mach numbers m of 0.5 and 0.6 . these functions are determined from a knowledge of the existing oscillatory coefficients at the low reduced frequencies and from approximate expressions of these coefficients at the high reduced frequencies . the beginning portion of the indicial lift function associated with an airfoil penetrating a sharp-edge gust in subsonic flow is evaluated by use of an exact method . by use of an approximate method for determining the remaining portion, the complete indicial gust function is determined for m 0.5, m 0.6, and m 0.7 . all the indicial lift and moment functions are approximated by an exponential series,. the coefficients which appear in the exponential approximations for each indicial function are tabulated for m 0.5, m 0.6, and m 0.7 .
general airfoil theory . on the assumption of infinitely small disturbances the author develops a generalized integral equation of airfoil theory which is applicable to any motion and compressible fluid . successive specializations yield various simpler integral equations, such as possio’s, birnbaum’s, and prandtl’s integral equations, as well as new ones for the wing of infinite span with periodic downwash distribution and for the oscillating wing with high aspect ratio . lastly, several solutions and methods for solving these integral equations are given .
a systematic kernel function procedure for determining aerodynamic forces on oscillating or steady finite wings at subsonic speeds . a detailed description is given of a method of approximating solutions to the integral equation that relates oscillatory or steady lift and downwash distributions on finite wings in subsonic flow . the method of solution is applicable to general plan forms with either curved or straight leading and trailing edges . moreover, it is directly applicable to control surfaces such as all-movable tails but modifications are needed to apply it to controls in general . applications of the method involve evaluations of numerous integrals that must be handled by numerical procedures but systematic schemes of evaluations have been adopted that are well suited to the routines of automatic digital computing machines . these schemes of evaluation have been incorporated in a program for an ibm 704 electronic data processing machine . with this machine, a pressure distribution together with such quantities as section or total lift and moment coefficients or generalized forces can be determined for a given value of frequency and mach number and for several /four or five/ modes of oscillation in about 4 minutes of machine time . in the case of steady downwash conditions corresponding quantities can be obtained in about 2 minutes of machine time . in order to illustrate applications of the method, results of several calculations are presented . in these illustrations total forces and moments are compared /1/ with results of analytic procedures for a circular plan form with steady downwash conditions, /2/ with results of other theories and with experiment for a rectangular plan form of aspect ratio 1 at a uniform angle of attack, and /3/ with some experimental results for a rectangular plan form of aspect ratio 2 undergoing pitching and flapping oscillations . also included in the illustrations are results of flutter calculations compared with experimental results for an allmovable control surface of aspect ratio 3.50 and for a cantilevered rectangular plan form of aspect ratio 5.04 .
on the kernel function of the integral equation relating the lift and downwash distributions of oscillating finite wings in subsonic flow . this report treats the kernel function of an integral equation that relates a known or prescribed downwash distribution to an unknown lift distribution for a harmonically oscillating finite wing in compressible subsonic flow . the kernel function is reduced to a form that can be accurately evaluated by separating the kernel function into two parts .. a part in which the singularities are isolated and analytically expressed and a nonsingular part which may be tabulated . the form of the kernel function for the sonic case /mach number of 1/ is treated separately . in addition, results for the special cases of mach number of o /incompressible case/ and frequency of o /steady case/ are given . the derivation of the integral equation which involves this kernel function, originally performed elsewhere /see, for example, naca technical memorandum 979/, is reproduced as an appendix . another appendix gives the reduction of the form of the kernel function obtained herein for the three-dimensional case to a known result of possio for two-dimensional flow . a third appendix contains some remarks on the evaluation of the kernel function, and a fourth appendix presents an alternate form of expression for the kernel function .
on som reciprocal relations in the theory of nonstationary flows . in the theory of nonstationary flows about airfoils, the /indicial lift/ function k /s/ of wagner and the /alternating lift/ function c /k/ of theodorsen have fundamental significance . this paper reports on some interesting relations of the nature of fourier transforms that exist between these functions . general problems in transient flows about airfoils may be given a unified broad treatment when these functions are employed . certain approximate results also are reported which are of notable simplicity, and an analogy with transient electrical flows is drawn .
thermal analysis of stagnation regions with emphasis on heat-sustaining nose shapes at hypersonic speeds . the leading edges and noses of hypersonic vehicles are subjected to severe aerodynamic heating and must be cooled in some manner-dash e.g., internal convection, transpiration, or radiation . it is this latter mode of handling the problem that is discussed in this paper . neglecting conduction in the leading-edge region, the maximum temperature for long-range hypersonic gliders is of the same order as the melting point of refractory materials, with a corresponding large temperature gradient away from the leading edge . inclusion of conduction in the aft direction reduces the maximum temperature and distributes the heat to a location that will radiate it out from the surface . for either steady-state or transient conditions, the temperature at the leading edge is reduced by conduction, while the temperature aft of the leading-edge shoulder is increased, thus setting up a heat transmission balance between the convective influx of heat, the redistribution of heat by conduction, and the radiation of heat from the surface . the feasibility of such a mechanism can be enhanced by suitably choosing leading-edge shapes and materials . the philosophy behind the choice of leading-edge shapes is discussed and the effects of varying parameters, such as shape, diameter, emissivity, conductivity, thickness, etc., are shown .
aerodynamic characteristics of two winged reentry vehicles at supersonic and hypersonic speeds . tests were conducted at the langley research center on two winged lifting hypersonic reentry glider configurations . performance, stability, and control data are presented at mach numbers of 1.62 and 2.91 for angles of attack up to 15degree and at mach numbers of 6.8 and 9.6 for angles of attack up to 25degree .
static longitudinal aerodynamic characteristics at transonic speeds and angles of attack up to 99degree of a reentry glider having folding wingtip panels . data are presented which were obtained from a transonic wind-tunnel investigation of a reentry glider having folding wing-tip panels . the tests were conducted at angles of attack from -4degrees to 99degrees . the reynolds number based on the mean geometric chord of the fixed planform varied from 2.35 x 10 to 2.99 x 10 . the maximum lift-drag ratio for the model with the folding wing-tip panels fully extended decreased from a maximum value of 7.8 at a mach number of 0.60 to about 3.4 at mach numbers from 1.03 to 1.20 . the model with the folding wing panels fully extended was stable for values of the lift coefficient from 0 up to at least 0.8 . above this lift coefficient pitch-up tendencies were observed, followed by an unstable or neutrally stable region which extended up to values of angle of attack of 50degrees or 60degrees . deflecting the folding wing panels between ducing a significant change in the trim angle of attack or in any of the force or moment coefficients in the angle-of-attack range from 49degree to 99degree .
the smallest height of roughness capable of affecting boundary-layer transition . an investigation was made to determine the smallest size of isolated roughness that will affect transition in a laminar-boundary layer . critical heights for three types of roughness were found in a low-speed wind tunnel . the types were /1/ two-dimensional spanwise wires, /2/ three-dimensional discs, and /3/ a sandpaper type . in addition to type of roughness, test variables included the location of roughness, pressure distribution, degree of tunnel turbulence, and length of natural laminar flow . the most satisfactory correlation parameter was found to be the roughness reynolds number, based on the height of roughness and flow properties at this height . the value of this critical reynolds number was found to be substantially independent of all test variables except the shape of roughness . this parameter also correlates well other published data on critical roughness in low-speed flow . the value of the roughness reynolds number necessary to move transition forward to the roughness itself was also determined for the three types of roughness and was found to be approximately constant for a given type of roughness . an investigation of the limited amount of available data on critical roughness in supersonic flow indicates that the effects of roughness may still be correlated by the roughness reynolds number . the value of this reynolds number depends primarily on the mach number at the top of the roughness . when this mach number is greater than 1.0, the roughness reynolds number based on conditions behind a shock is probably the characteristic parameter .
an investigation at subsonic speeds of aerodynamic characteristics at angles of attack from -dash 4degrees to 100degrees of a delta-wing reentry configuration having folding wingtip panels . an investigation was made at subsonic speeds in the langley highspeed lifting reentry configuration having folding wingtip panels . the configuration is of the type used in a high angle-of-attack /near 90degree/ reentry to minimize aerodynamic heating . by unfolding the wingtip panels into the airstream, a moderate angle-of-attack glide is used for a controlled landing . the basic configuration tested utilized a whose area was 25 percent of the total wing area . the effects of varying the plan form and size of the wingtip panels was studied as well as the effects of unfolding the wingtip panels in a high angle- of-attack attitude . tests were made at mach numbers of 0.40, 0.60, and
low-speed longitudinal aerodynamic characteristics associated with a series of low-aspect ratio wings having variations in leading-edge contour . an investigation has been conducted at various reynolds numbers and low subsonic speeds to determine the longitudinal aerodynamic characteristics associated with a series of low-aspect-ratio wings having variations in leading-edge contours . the planforms included a highly swept triangular wing, a rectangular wing, and intermediate wings including planforms having elliptic and parabolic leading-edge contours, all having an aspect ratio of 1.33 . the effects of changing aspect ratio for a given leading-edge contour were investigated for two of the wings presented,. also included are the longitudinal characteristics associated with various fuselage sizes . an effort has been made to estimate the lift variation with angle of attack for the wing planforms of the present investigation . improvements in the lifting capabilities at low subsonic speeds associated with a basic triangular planform of low aspect ratio are possible by slight alterations in leading-edge design, which should still conform to possible design requirements at hypersonic speeds . these changes in planform resulted in increases in lift-curve slope, lift at high angles of attack, and in the maximum untrimmed lift-drag ratio, provided the fuselage was sufficiently small . the longitudinal stability characteristics of the majority of planforms indicate more desirable stability characteristics at high lifts than either a triangular wing or rectangular wing of the same aspect ratio . the effects of increasing reynolds number for each of the planforms investigated generally resulted in slight reductions in the lift at high angles of attack . a method is presented for estimating the subsonic-lift variation with angle of attack for the low-aspect-ratio wings of the present investigation and indicated good agreement with experimental data throughout the angle-of-attack range of this investigation .
static longitudinal stability characteristics of a blunted glider re-entry configuration having 79.5degree sweepback and 45degree dihedral at a mach number of 6.2 and angles of attack up to 20degree . an experimental investigation was conducted at a mach number of 6.2 to determine the static longitudinal stability characteristics of a model of a blunted glider reentry configuration having 79.5degree sweepback and 45degree dihedral . the free-stream reynolds number for the investigation was 3.0 x 10 based on the basic model length of 7.5 inches . tests were made through an angle-of-attack range from 0degrees to investigation showed that incorporating 10degree nose incidence in the basic model resulted in a lower lift-curve slope, a lower lift-drag ratio, a higher value of trim lift coefficient, and a decrease in static longitudinal stability . in comparison, the effect of extending the configuration length and incorporating 10degrees and 20degrees boattail angles resulted in smaller changes in the longitudinal stability characteristics of the model .
blockage corrections for three-dimensional flow closed throat wind tunnels, with considerations of the effect of compressibility . theoretical blockage corrections are presented for a body of revolution and for a three-dimensional unswept wing in a circular or rectangular wind tunnel . the theory takes account of the effects of the wake and of the compressibility of the fluid, and is based on the assumption that the dimensions of the model are small in comparison with those of the tunnel throat . formulas are given for correcting a number of the quantities, such as dynamic pressure and mach number, measured in wind-tunnel tests . the report presents a summary and unification of the existing literature on the subject .
motion of a ballistic missile angularly misaligned with the flight path upon entering the atmosphere and its effect upon aerodynamic heating, aerodynamic loads and miss distance . an analysis is given of the oscillating motion of a ballistic missile which upon entering the atmosphere is angularly misaligned with respect to the flight path . the history of the motion for some example missiles is discussed from the point of view of the effect of the motion on the aerodynamic heating and loading . the miss distance at the target due to misalignment and to small accidental trim angles is treated . the stability problem is also discussed for the case where the missile is tumbling prior to atmospheric entry .
study of the oscillatory motion of manned vehicles entering the earth’s atmosphere . an analysis is made of the oscillatory motion of vehicles which traverse arbitrarily prescribed trajectories through the atmosphere . expressions for the oscillatory motion are derived as continuous functions of the properties of the trajectory . results are applied to a study of the oscillatory behavior of re-entry vehicles which have decelerations that remain within limits of human tolerance . it is found that a deficiency of aerodynamic damping for such vehicles may have more serious consequences than it does for comparable ballistic missiles .
motions of a short 10degree blunted cone entering a martian atmosphere at arbitrary angles of attack and arbitrary pitching rates . the dynamic behavior of two probe vehicles entering a martian atmosphere in a passive manner with arbitrary initial angles of attack and pitching rates to 12degree per second has been determined . results for an entry velocity of 21,700 feet per second and an entry angle of -40degree were obtained from machine calculated solutions of the six-degree- of-freedom rigid-body equations of motion using experimental aerodynamic characteristics for the vehicles . one of the vehicles had a flat base and was statically stable in two attitudes /nose forward and base forward/ . the other vehicle, derived from the first by adding a conical afterbody, was statically stable in only one attitude /nose forward/ . a 10-rpm vehicle spin rate, believed ample for the purpose of distributing solar and aerodynamic heating over the vehicle surface, and model atmospheres encompassing the probable extremes for the planet were also considered . it was found that while the motion of the flat-based vehicle could be oscillatory about either the nose-forward or base-forward stable trim attitudes when aerodynamic heating rates were high, the range of initial angles of attack resulting in base-forward orientation was reduced by more than a factor of 3. when initial pitch rates were increased from body having only nose-forward stability showed that oscillatory angles of attack at maximum heating-rate conditions probably would not exceed about 25degrees although angles of attack when heating rates were 50 percent of maximum could be as high as 40degree . values of these upper bound angles of attack were essentially independent of initial pitch rates for the range considered . furthermore, the envelope of maximum probable angles of attack was increased only slightly when the vehicle was given a 10-rpm spin rate . the relationship between maximum amplitudes of oscillation and heating rates through high heating portions of the trajectories was preserved when model atmospheres believed to encompass the extreme possibilities for mars were used in the calculations
means and examples of aeronautical research in france at onera . cosmonautics is currently very much to the forefront in the news . it embraces and extends aeronautics, and i would like to propose including both, at least on certain occasions, under a general denomination of /aerocosmonautics/ . in your country, the sciences and technology of space are subjects which have been backed by initial advances and abundantly treated . since france has not yet launched any artificial satellite or built any circumlunar space vehicle, i propose to confine myself here to the field of aeronautics, where there is still so much progress of manifest utility to accomplish . i shall accordingly content myself with presenting some examples of aeronautical research and experiments undertaken in my country by onera, a body whose mission is akin to that of the illustrious naca, now nasa, but bearing in mind the considerable difference between the scales of the respective resources .
tumbling bodies entering the atmosphere . the equations of motion of a tumbling flat plate entering an exponential atmosphere were linearized and solved analytically to obtain a simple expression for the altitude at which tumbling would cease and libration would commence . the plate had only three degrees of freedom, and aerodynamic forces were derived from newtonian impact theory . in the linear analysis, mean values of the drag and pitch damping coefficients so that flutter occurs in the range of a low-speed wind tunnel . a particular type of construction for supersonic flutter models is described in detail . methods of vibration testing, static testing, and flutter testing are discussed . particular emphasis is placed on the technique of varying flow parameters rather than model parameters to precipitate flutter . the tool for varying flow parameters is the variable mach number supersonic test section of the massachusetts institute of technology blowdown wind tunnel . the aerodynamic features of the supersonic test section are presented .
a note on the use of sandwich structures in severe acoustic environments . this paper reviews some of the experience to date of using sandwich type structures in severe acoustic pressure environments . the methods used for testing sandwich structures for acoustic fatigue are described and their limitations considered . experimental and theoretical work relating to the damping and mode-frequency relationships of certain sandwich configurations is also reviewed . special attention is given to the estimation of the stress in the bond of a honeycomb sandwich panel subjected to sudden pressure fluctuations . a /uni-modal/ theory is presented, relating the mean-square bond-stress to the random exciting pressure and panel dynamic characteristics . this theory indicates that tensile bond stresses may be encountered of up to six times the local r.m.s. exciting pressure . these must be combined with bending and shear stresses to obtain the principal stresses which precipitate bond fatigue failures . finally, an outline is given of some of the lines of future research which should lead to the achieving of the maximum possible fatigue resistance from sandwich configurations .
near noise field of a jet engine exhaust . aircraft structures located in the near noise field of a jet engine are subjected to extremely high fluctuating pressures that may cause structural fatigue . studies of such structures have been limited by lack of knowledge of the loadings involved . the acoustic near field produced by the exhaust of a stationary turbojet engine having a high pressure ratio was measured for a single operating condition without afterburning . the maximum over-all sound pressure without afterburning was found to be about 42 pounds per square foot along the jet boundary in the region immediately downstream of the jet-nozzle exit . with afterburning the maximum sound pressure was increased by 50 percent . the largest sound pressures without afterburning were obtained on a constant percentage band width basis in the frequency range from 350 to 700 cps . additional tests were made at a few points to find the effect of jet velocity on near-field sound pressures and to determine the difference in value between sound-pressure levels at rigid surfaces and corresponding free-field values . near the jet nozzle, over-all sound pressures were found to vary as a low power (approx. unity) of the jet velocity . over-all sound-pressure levels considerably greater than the corresponding free-field levels were recorded at the surface of a rigid plate placed along the jet boundary . the downstream locations of the maximum sound pressure at any given frequency along the jet-engine-exhaust boundary and the longitudinal turbulent-velocity maximum of the same frequency along a small cold-air jet at 1 nozzle-exit radius from the jet axis were found to be nearly the same when compared on a dimensionless basis . also, the strouhal number of the corresponding spectra maximums was found to be nearly equal at similar distances downstream . in addition to the magnitude and frequency distribution of the acoustic pressures, it is necessary to know the cross correlation of the pressure over the surface area . cross-correlation measurements with microphones were made for a range of jet velocities at locations along the jet and at a distance from the jet . free-field correlations of the over-all sound pressure and of the sound pressure in frequency bands from 100 to 1000 cps were obtained both longitudinally and laterally . in addition, correlations were obtained with microphones mounted at the surface of a rigid plate that was large compared with the distance over which a positive correlation existed . the region of positive correlation was generally found to increase with distance downstream of the engine to 6.5 nozzle-exit diameters, but remained nearly constant thereafter . in general, little change in the correlation curves was found as a function of jet velocity or frequency-band width . the distance from unity correlation to the first zero correlation was greater for lateral than for longitudinal correlations for the same conditions and locations . the correlation curves obtained in free space and on the surface of the plate were generally similar . the results are interpreted in terms of pressure loads on surfaces .
random excitation of a tailplane section by jet noise . the response of a section of tailplane structure to both discrete and random noise pressures has been studied in detail . initially the specimen was mounted behind a jet engine and the induced strains were analysed with the object of determining both the resonant frequencies and the corresponding modes of vibration . during these tests a survey was made of the spectrum and correlation pattern of the jet noise on the surface of the model . secondly the specimen was mounted in front of a loudspeaker in an acoustics laboratory and the structural resonances were excited by means of discrete frequency sound . the mode shapes were studied in detail with the aid of a stroboscope . it is concluded that the tailplane skin on this particular piece of structure only responds to any significant degree in one structural mode . although reasonable comparison has been obtained between the random and discrete tests, it was not possible to calculate the induced stresses using the observed mode shapes and measured pressure excitation .
on the fatigue failure of structures due to vibrations excited by random pressure fields . on the assumption that the forced modes of vibration of a structure, subjected to pressure fluctuations random in time and space, can be approximated by the composition of the motions of the uncoupled natural modes, a general analysis is made using the ideas of vibration theory and spectrum analysis . the power spectrum, and hence the rms value, of any quantity depending linearly upon structural distortions is derived and it involves a quantity (called the /joint acceptance/) concerning the spacewise structure of the pressure field and of the geometry of the modes of vibration . it is shown how this result may be used (on assuming /normal/ randomness) to estimate the fatigue life on the hypothesis of cumulative damage .
structural acoustic proof testing . with the introduction of high-powered propulsion systems, and paralleling their continued development, an accompanying increase in acoustical problems has arisen . of these acoustical problems, that of acoustical fatigue failures has become paramount in the eyes of the structural engineer . aircraft designed to normal strength requirements have been known literally to fall apart under acoustical loading . this problem has required much endeavour to produce a solution, and considerable structural research, based upon results of siren or other testing, have proved inadequate . this failure to find a satisfactory solution has resulted in the conviction that the final proof of a design can be found only in proof testing . proof testing, in the acoustic fatigue sense, is the testing of a design structure in a simulated acoustical environment for a period of time long enough to assure equality with design life .
the response of a typical aircraft structure to jet noise . an analysis is made of experimentally determined mode shapes excited on the rear structure of a modern airliner by jet noise from a pod-mounted turbojet engine . power spectra of stresses determined from strain-gage measurements are obtained and cross correlated . extensive measurements were made on skin panels of the fuselage and elevator and limited ones were made on fuselage stringers and frames . the skin-panel results are compared with theoretical predictions . reviewer believes that this paper is of considerable value for those concerned with response of aircraft-type structures to jet-induced noise .
on structural fatigue under random loading . experience has shown that the fluctuating loads induced by a jet may cause fatigue failure of aircraft structural components . in order to throw some light on this and similar problems, the stress spectrum and the /equivalent fatigue stress/ of an elastic structure subjected to random loading are studied . the analysis is simplified by assuming the structure to have only a single degree of freedom and by using the concept of cumulative damage, the results being expressed in terms of quantities that can be directly measured . as an example, a similarity expression for the probable value of the equivalent fatigue stress of a panel subjected to jet buffeting is derived .
a study of the acoustic fatigue characteristics of some flat and curved aluminium panels exposed to random and discrete noise . a study was made of the fatigue life of simple 2024-t3 aluminum-alloy panels measuring 11 by 13 inches and exposed to both discrete-frequency noise from a siren and random noise from an air jet . noise levels varied from approximately panel variables included thickness, edge conditions, curvature, and static-pressure differential . no significant differences were noted in the nature of failures experienced for the two types of loadings . at a given root-mean-square stress level, the failure times were generally shorter for the random loading than for the discrete-frequency loading . these differences in failure times were noted to be a function of stress level, the larger differences occurring at the lower stress levels . increases in time to failure were obtained as a result of increased panel thickness, increased panel curvature, and particularly for increased static-pressure differential across curved panels . for the discrete-type loading, the location of weak points in these simplified structural designs can be satisfactorily accomplished but quantitative predictions of fatigue life are much more difficult .
free vibrations of continuous skin stringer panels . the determination of the natural frequencies and normal modes of vibration for continuous panels, representing more or less typical fuselage skin-panel construction for modern airplanes, is discussed in this paper are considered . a numerical example is presented, and analytical results for a particular structural configuration agree favorably with available experimental measurements .
stresses in continuous skin stiffener panels under random loading . theoretical aspects involved in the prediction of stress levels for continuous skin-stiffener panels subjected to a random pressure field are considered in the light of powell’s general theory for statistical superposition of modal response . the choice of structural model is dictated by the prevalence of skin-stiffener construction in modern flight vehicle design . the present study clearly demonstrates that any truly adequate prediction of stress levels in actual aircraft structures requires a much better representation of structural characteristics than can be provided by single panel idealizations . in an example considering fuselage panels exposed to jet engine noise, essential agreement is shown with experimental data, although better correlation is shown for rms stress than for power spectrum . it is shown that reduction of stress level by increasing damping is effective only in the higher frequency range .
on the bending of a clamped plate . the present paper contains an application of a recently developed variational method to the boundary value problem of the bending of a clamped plate of arbitrary shape . it will be shown that this problem can be linked to the simpler problem of the equilibrium of a membrane by a chain of intermediate problems, which can be solved explicitly and in finite form in terms of the membrane problem . in the intermediate problems, the deflection converges uniformly in the domain of the plate of the clamped plate, and the derivatives of all orders of the deflection converge uniformly in every domain completely interior to the plate . (in the ritz method, not even the convergence of the slopes can be guaranteed .) the method yields numerical results for plates of all shapes for which the membrane problem (which we shall call the base problem) admits an explicit solution . as an example we shall consider a clamped square plate under a uniform load . this problem has been the object of numerous investigations, some of which are theoretical, while others are purely numerical, use infinite simple and double series, and operate with an infinite number of linear equations and an infinite number of unknowns . an inspection of the general formulae derived in the present paper, formulae which become simple in numerical applications, would show how some of the numerical methods might be rendered rigorous . the convergence of higher derivatives is of great practical interest for the approximate computation of the stresses .
upper and lower bounds for the solution of the first biharmonic boundary value problem . let w(x,y) be a solution of the boundary value problem where r is a plane domain with the boundary c . the authors obtain upper and lower bounds for, the value of w at a point in r, by a method which is applicable to many other problems . if u is a function satisfying the boundary conditions and v is a function satisfying the partial differential equation, then the authors obtain by applying green’s classical identity and schwarz’s inequality a pair of inequalities of the form where . together with the function w the authors consider a function the solution of the boundary value problem on c, and in analogy with the functions u and v associated with the function w a pair of functions and associated with the function . in the expression for derived from green’s classical identity appears an unknown line integral containing the values of w and on c . but the same line integral appears also in the expressions for to which the above inequalities are applicable . in this way the authors obtain two inequalities of the form where b and b’, respectively, are approximate values of . in order to improve these bounds one may add to u a linear set of functions and to v a linear set of functions and then minimize h(u-v) in order to determine the coefficients of the best linear combinations . if the sequences and are complete in a certain sense defined by the authors the approximations will converge to the value .
on the analogues relating flexure and extension of flat plates . the displacement of a flat plate bent by transverse loading, and the extensional or in ‘plane stress’, are governed by equations of identical form ,. and the boundary conditions have identical form when edge-displacements are specified in the flexural, edge-tractions in the extensional problem, so mathematically, in these circumstances, only a single problem is presented . this, the ‘first analogue’ relating flexure and extension, is well known . a ‘second analogue’, relating the flexural problem when edge-tractions with the extensional problem when edge-displacements are specified, is believed to have been first propounded in 1941 . by introducing two quantities u and v, analogous with the components u and v of extensional displacement, it permits a treatment of the flexural problem by any method–e.g. which yields extensional solutions of this second type . in this paper both analogues are combined in an inclusive statement covering the perforated (multiply connected) plates which were discussed in 1948 . reasons are stated for believing that ‘two-diagram technique’ is preferable in problems governed by ‘mixed’ boundary conditions .
the bending of a sectorial plate . the problem of evaluating the bending moments, existing in a uniformly loaded clamped plate having the form of a sector of a ring, is one which arises in connection with the stress analysis of reinforced piston heads and in other design problems . in this paper, expressions are derived for the bending moments along the edges of such a plate . similar problems, i.e., those of the clamped rectangular plate under uniform pressure, under a central concentrated load, and that of the simply supported sector of a disk under uniform pressure, have been discussed by previous authors . the general approach used in the foregoing problems is adopted in the present case ,. a considerable reduction in the computational work is achieved, however, by the use of an integral-equation method of solving the boundary-condition equations . numerical results are obtained for plates of various dimensions, and the edge moment distributions are plotted for these cases . curves are also plotted which indicate the relationship existing between the maximum bending moments derived for sectorial plates and those previously obtained for clamped rectangular plates of similar size .
the bending of uniformly loaded clamped plate in the form of a circular sector . the deflexion of a uniformly loaded plate in the form of a semicircle clamped along its boundary is obtained by a method due to weinstein . this problem requires the solution of the biharmonic equation where z is given, subject to the conditions that w = 0 and on the boundary, n being the direction of the outward normal . the solution is expressed in the form where, writing is found by solving (in succession) two harmonic equations of the forms where z may be zero, and where f and have to satisfy certain boundary conditions . the constants are then determined to satisfy the boundary condition . numerical calculations show that five or six terms of the series give a good approximation to the accurate value as judged by the closeness with which the approximate solution satisfies the boundary condition . the procedure to be adopted in the case of the general circular sector and for non-uniform loading is indicated briefly . the connexion between the deflexion problem and that of plane strain in which the stress function satisfies the equation, where and have given values on the boundary, is discussed as a preliminary to the further consideration of the latter problem by a method of the same type .
the bending of uniformly loaded sectorial plates with clamped edges . this paper analyzes the bending of a sectorial plate, clamped on all edges and subjected to uniformly distributed load, by using two different methods of superposition on the elementary solution for a uniformly loaded circular plate with a clamped edge .
the bending of a wedge shaped plate . a general method of solution is given in this paper for the problem of bending of a wedge-shaped thin elastic plate with arbitrary boundary conditions on the radial edges in the case of a single load . the solution is carried out for a plate with clamped edges and a single load on the bisector radius of the plate . stress distribution along the edges is shown and the behavior of the solution near the corner point is discussed for several opening angles of the plate .
on the analysis of elastic plates of variable thickness . the extensional and flexural equations governing the elastic behaviour of a plate of variable thickness are expressed in terms of the laplacian operator . temperature variations in the plane of the plate and across the thickness of the plate are taken into account . general solutions are given for a rectangular plate whose thickness varies exponentially along the length, and for a circular, or annular, plate whose thickness varies as a power of the radius . the large-deflexion equations, including effects of initial irregularities, are also discussed .
finding zero’s of arbitrary functions . a method for finding real and complex roots of polynomial equations, due to d. muller, is applied to finding roots of general equations of the form f(z) = 0, where f(z) is analytic in the neighborhood of the roots . the procedure does not depend on any prior knowledge of the location of the roots nor on any special starting process . all that is required is the ability to evaluate f(z) for any desired value of z . multiple roots can also be obtained . a general purpose program, prepared for the univac scientific 1103 and 1103a, is described and numerical results are presented for the following applications .. finding eigenvalues of differential operators ,. finding eigenvalues of arbitrary matrices ,. finding zeros of the generalized eigenvalue problem ,. finding roots of a number of transcendental equations .
the buckling of thin cylindrical shells under axial compression . in two previous papers the authors have discussed in detail the inadequacy of the classical theory of thin shells in explaining the buckling phenomenon of cylindrical and spherical shells . it was shown that not only the calculated buckling load is 3 to 5 times higher than that found by experiments, but the observed wave pattern of the buckled shell is also different from that predicted . furthermore, it was pointed out that the different explanations for this discrepancy advanced by l. h. donnell and w. flugge are untenable when certain conclusions drawn from these explanations are compared with the experimental facts . by a theoretical investigation on spherical shells the authors were led to the belief that in general the buckling phenomenon of curved shells can only be explained by means of a non-linear large deflection theory . this point of view was substantiated by model experiments on slender columns with non-linear elastic support . the non-linear characteristics of such structures cause the load necessary to keep the shell in equilibrium to drop very rapidly with increase in wave amplitude once the structure started to buckle . thus, first of all, a part of the elastic energy stored in the shell is released once the buckling has started,. this explains the observed rapidity of the buckling process . furthermore, as it was shown in one of the previous papers the buckling load itself can be materially reduced by slight imperfections in the test specimen and vibrations during the testing process . in this paper, the same ideas are applied to the case of a thin uniform cylindrical shell under axial compression . first it is shown by an approximate calculation that again the load sustained by the shell drops with increasing deflection . then the results of this calculation are used for a more detailed discussion of the buckling process as observed in an actual testing machine .
the behaviour of a cylindrical shell under axial compression when the buckling load has been exceeded . the value of the compressive stress at which a thin circular cylindrical shell becomes unstable has been worked out theoretically by southwell (1914) . subsequent experimental results, however, have indicated that this value is appreciably too high and that the form of distortion which occurs in practice differs from that assumed in theory . in recent years much work has been done on this problem in america . lundquist (1933) and donnell (1934) have concluded that the buckling of a cylindrical shell is greatly influenced by initial irregularities,. von karman and tsien (1941) have indicated that a thin cylindrical shell can be maintained in a buckled state by a compressive load considerably smaller than that previously predicted by theory . the present paper is an extension of the work of von karman and tsien . it shows that the smallest load which will keep a thin cylindrical shell in a buckled condition is about one-third of that given by southwell, a result in very fair agreement with experiment, and that once the cylinder has buckled, and so long as the stresses remain within the elastic range of the material, the cylinder has only about one-quarter of its original stiffness .
the behaviour of thin cylindrical shells after buckling under axial compression . the fundamental investigations of von karman and tsien on the buckling of cylindrical shells under axial compression are continued . the energy expression is simplified and minimized with respect to the axial and circumferential wave-length parameters . solution of the equations obtained yields curves of the reduced average stress and of the wave dimensions plotted against the reduced average strain . they illustrate the behavior of the cylinder during the buckling process . the minimum buckling stress is found to be 0.195e(tr) .
post-buckling behaviour of axially compressed circular cylinder shells . the postbuckling characteristics of an axially compressed thin-walled circular cylindrical shell loaded either by dead weights or by a rigid testing machine are determined . it is shown that for either loading condition the minimum applied stress in the postbuckling region is 0.182(er) and that the region of stable equilibrium corresponding to loading by the rigid testing machine includes and extends beyond that obtained with dead weight loading . the work here described is a continuation of work done earlier by von karman and tsien, by michielsen, and by leggett and jones .
new developments in the nonlinear theories of the buckling of thin cylindrical shells . in the present paper a short survey will be given first of the buckling and postbuckling behavior of isotropic cylindrical shells subjected to different loading conditions as obtained by the nonlinear theory of finite deflections of shells during the last twenty years . next a report will be given on new investigations carried out in the structures department of the dvl concerning the elastic stability of isotropic and orthotropic cylindrical shells loaded in axial compression and internal pressure . these studies are based on the nonlinear theory of finite deformations . the theoretical rsults will be compared with new experimental results obtained with a series of axially loaded pressurized isotropic and orthotropic cylindrical shells .
lower buckling load in the non-linear buckling theory of thin shells . for thin shells the relation between the load p and the deflection beyond the classical buckling load is very often non-linear . for instance, when a uniform thin circular cylinder is loaded in the axial direction, the load p when plotted against the end-shortening has the characteristic shown in fig. 1 . if the strain energy s and the total potential are calculated, their behavior can be represented by the curves shown in figs. 2 and 3 . it can be demonstrated that the branches oc and ab corresponds to stable equilibrium configurations and the branch bc to unstable equilibrium configurations . the point b is then the point of transition from stable to unstable equilibrium configurations .
an automatic method for finding the greatest or least value function . the greatest or least value of a function of several variables is to be found when the variables are restricted to a given region . a method is developed for dealing with this problem and is compared with possible alternatives . the method can be used on a digital computer, and is incorporated in a program for mercury .
aeroelastic problems in connection with high speed flight . a review is given of developments in the field of aeroelasticity during the past ten years . the effect of steadily increasing mach number has been two-fold .. on the one hand the aerodynamic derivatives have changed, and in some cases brought new problems, and on the other hand the design for higher mach numbers has led to thinner aerofoils and more slender fuselages for which the required stiffness is more difficult to provide . both these aspects are discussed, and various methods of attack on the problems are considered . the relative merits of stiffness, damping and massbalance for the prevention of control surface flutter are discussed . a brief mention is made of the recent problems of damage from jet efflux and of the possible aeroelastic effects of kinetic heating .
bodt freedom flutter of ground launched rocket models at supersonic and high subsonic speeds . a theoretical investigation of symmetric body freedom flutter of a rocket model is described . the results confirm that structural failures of models were caused by this type of flutter, and an extension of the investigation indicates the parameters that are of importance . a high ratio of body to wing mass and a well forward position of the overall centre of gravity are conditions under which flutter may occur . increase of body pitching radius of gyration and tailplane volume are beneficial . it is concluded that this type of flutter may be significant in some aircraft designs, and that the canard has no advantage in this respect over the conventional lay-out of wing and tailplane .
subsonic aerodynamic flutter derivatives for wings and control surfaces, /compressible and incompressible flow/ . this report gives tables of the two-dimensional subsonic flutter derivatives,. where possible the values given are based on the published work of various authors, but some have been specially calculated for this report . wing derivatives are given for mach numbers 0, 0.5, 0.6 and 0.7 for the frequency parameter range 0 /0.04/ 0.2 /0.2/ 1.6 and mach numbers 0 and 0.7 for frequency parameter 5.0 . control surface derivatives are given for mach numbers 0 and 0.7 for control surface/ wing chord ratios 0.02 /0.02/ 0.10 /0.05/ 0.50 and frequency parameters are also given for mach numbers 0, 0.5, 0.6 and 0.7 for frequency parameter 0 /0.04/ 0.2 /0.2/ 1.4 . control surface-tab derivatives are given for some particular values of the variables and methods of obtaining approximate values of these derivatives for other values of the variables are suggested . control surface and tab derivatives are in all cases for no aerodynamic balance .
the aerodynamic effects of aspect ratio and sweepback on wing flutter . the report describes tests to obtain direct measurements of the aerodynamic effects of aspect ratio and sweepback on wing flutter . the tests were made on rigid wings with root flexibilities . it is shown that measured effects of aspect ratio and sweepback on the flutter of these wings can be represented quite closely in flutter calculations based on two-dimensional flow theory by multiplying the two-dimensional aerodynamic coefficients by appropriate factors . the effect of sweepback is represented by multiplying all aerodynamic coefficients by cos, where is the wing leading-edge sweepback, and the effect of aspect ratio is represented by multiplying the aerodynamic damping coefficients by 1f(a) and, the stiffness coefficients by 1(f(a)) where a is the aspect ratio . for the wings tested an average value for f(a) is f(a) = (1 + (0.8a)) .
transonic flow in two dimensional and axially symmetrical nozzles . by means of suitable expansions in inverse powers of r, the radius of curvature of the nozzle profile at the throat measured in throat half-heights, the velocity components in the throat region of a convergent-divergent nozzle can be calculated . the first three terms of the series solution have been obtained both for two-dimensional and for axially-symmetric nozzles . the numerical accuracy of the solution is confirmed by comparison with the known exact solution along the branchline .
a note on the use of end plates to prevent three dimensional flow at the ends of bluff cylinders . the results are given of some observations of the effects of end plates on the three-dimensional separated flow at the ends of cylindrical models . while these are by no means exhaustive, it is felt that they are of sufficient interest to merit putting on record .
slender not-so-thin wing theory . a method for making an approximate thickness correction to slender thin-wing theory is presented . the method is tested by applying it to cones with rhombic cross-sections and the agreement is found to be good if the cones are not too thick . it is then suggested that the thickness correction to slender thin-wing theory may be applied unchanged to linear thin-wing theory . this suggestion is compared with some experiments on delta wings and it is found that there is considerable improvement over thin-wing theory near the centre line, but that this improvement is not maintained as the wing tips are approached .
development of a quasi-steady approach to flutter and correlation with kernel-function results . the quasi-steady approach to flutter utilizes experimental or theoretical steady-state aerodynamic data to arrive at increased understanding of the flutter mechanism, and also, in many cases, acceptably accurate quantitative flutter predictions . circulation lag effects are neglected, but aerodynamic damping is included in the evaluation of the air forces . situations requiring the inclusion of rate aerodynamics for accurate flutter estimation are specified . a quasi-unsteady approach is also discussed, in which the approximate magnitude of the circulation lag function at flutter is included in simple modifications of quasi-steady parameters . closed-form solutions are derived for the flutter characteristics of a typical section with and without rate aerodynamics . application is then made to the rational flutter analysis of three-dimensional multi-degree-of-freedom lifting surfaces . a specific planform is evaluated in the mach-number range from zero to two . quasi-steady, quasi-unsteady, and kernel-function results are compared subsonically . quasi-steady results are utilized supersonically . primary applications of the quasi-steady approach are in the areas of preliminary design and parameter-variation studies, modification of more sophisticated flutter theories to force compatibility with available steady-state data, and flutter evaluation of complex configurations which can be rationally analyzed by steady-state aerodynamic theories, but for which no complete unsteady aerodynamic theories are presently available .
heat transfer through laminar boundary layers on semi-infinite cylinders of arbitrary cross section . this paper shows how to calculate the rate of heat transfer through a laminar boundary layer on a semi-infinite cylinder of arbitrary cross section . the cylinder is placed in a stream of incompressible fluid, the flow at infinity being parallel to the generators, and is maintained at a uniform temperature . a series solution for small downstream distances and an asymptotic formula for large downstream distances are given . to cover the intermediate range an approximate pohlhausen solution is obtained,. a correction of the error involved in the pohlhausen solution is suggested which, it is believed, will lead to final errors of at most 2 percent . the calculations are applied to elliptic cylinders, and illustrate the effect on the local rate of heat transfer of varying the ratio of the major and minor axes of cross section, the length of perimeter being held fixed .
oscillatory derivative measurements on sting-mounted wind tunnel models method of test and results for pitch and yaw on a cambered ogee wing at mach numbers up to 2. 6. this report describes a method which has been developed for measuring oscillatory derivatives on sting-mounted models in the 8 ft by 8 ft supersonic tunnel at r.a.e. bedford . direct and cross derivatives with respect to angular displacements and velocities in pitch and yaw have been measured satisfactorily, and results are given of tests on a cambered ogee wing at six mach numbers from 0.2 to 2.6 . some tests were made on this model in the course of the preliminary development work in the 13 ft by 9 ft low speed wind tunnel, and results of these are included .
further comments on the inversion of large structural matrices . in a recent note, klein referred to a paper co-authored by the writer, and to ref. 3 . regarding the subject of inversion of large-order matrices, klein stated that he would show ‘that the situation is not as hopeless as the anove-mentioned authors intimate’ . the purpose of this note is not to take exception to klein/s conclusions, but rather to disagree with his implication that the authors of ref. 2 were pessimistic with respect to large-matrix inversions . two general methods of analysis were treated.. the method of consistent distortion and the method of transfer matrices . the first method leads directly to a relatively large matrix of structural coefficients of both internal forces and displacements . this matrix must be inverted to solve the problem . the second method ultimately produces a relatively small matrix requiring inversion., however, to arrive at this point one must perform a number of matrix multiplications .
an investigation of the flow about a plane half-wing of cropped delta planform and 6( symmetrical section at stream mach numbers between 0. 8 and 1. 41 . a study has been made of the flow development over the wing as the incidence and stream mach number vary and this is illustrated by surface pressure distributions and oil-flow patterns . the growth and movement of the two main surface shocks (the rear and forward shocks) is discussed, and conditions for flow separation through these shocks are considered . for the rear shock, which has little sweep, these conditions are similar to those for shock-induced separation on two-dimensional aerofoils . the forward shock is comparatively highly swept and separation seems to correspond to two rather different but simultaneously-attained conditions, one related to the component mach number normal to the shock front and the other to the position of the reattachment line . the flow in the region between the leading edge and the forward shock is shown to have certain characteristics analogous to those found upstream of the shock on two-dimensional aerofoils . to the rear of the forward shock, but ahead of the rear shock, the flow at low supersonic speeds resembles in some respects that about a simple cone . the general flow development is related in the text to the wing lift and pitching moment, and the drag . the first two are most affected by the aft movement of the rear shock, which also stimulates the transonic drag rise . the lift-dependent drag is shown to be influenced by the appearance of leading-edge separation and possibly also by some stage in the development of the forward shock . the flow over the cropped-delta planform is noteworthy for the absence of the strong outboard shock and this is attributed partly to the cropped tip and partly to the unswept trailing edge . a comparison is made with results obtained during preliminary tests in which the wing planform closely resembled that of a true delta .
the lower bound of attainable sonic-boom over-pressure and design methods of approaching this limit . from a study of existing sonic-boom theory it has been possible to establish an approximate lower bound of attainable sonic-boom overpressure, which depends only on the airplane length, weight, and volume and on the flight conditions . this lower bound may be approached over a narrow range of flight conditions through the application of appropriate design considerations . in general, for intermediate values of lift coefficient the major portion of the lift generating surfaces must be located aft of the maximum cross-sectional area, whereas for higher values of lift coefficient the maximum area must be well forward and or the lift-producing surfaces must extend well toward the airplane nose .
stability investigation of a blunted cone and a blunted ogive with a flared cylinder afterbody at mach numbers from 0. 30 to 2. 85 . a cone with a blunt nose tip and a blunt nose tip and a 20 flared cylinder afterbody have been tested in free flight over a mach number range from 0.30 to 2.85 and a reynolds number range from 1 x 10 to 23 x 10 . time histories, cross plots of force and moment coefficients, and plots of the longitudinal-force coefficient, rolling velocity, aerodynamic center, normal-force-curve slope, and dynamic stability are presented . with the center-of-gravity location at about models were both statically and dynamically stable throughout the mach number range . for the cone, the average aerodynamic center moved slightly forward with decreasing speeds and the normal-force-curve slope was fairly constant throughout the speed range . for the ogive, the average aerodynamic center remained practically constant and the normal-force-curve slope remained practically constant to a mach number of approximately 1.6 where a rising trend was noted . maximum drag coefficient for the cone, with reference to the base area, was approximately 0.6, and for the ogive, with reference to the area of the cylindrical portion, was approximately 2.1 .
inelastic buckling of initially imperfect cylindrical shells subject to axial compression . an analytical and experimental study is made for inelastic instability of initially imperfect cylindrical shells subject to axial compression . donnell’s equations and the principle of virtual work are adapted to determine the effects of initial imperfections on the buckling modes and the critical buckling stresses . the deformation theory and the incremental theory of plastic stress-strain relationships are both considered . the experimental results of ten tests on specimens made of aluminum alloy 3003-0 are presented . comparison of experimental with theoretical results indicates that the application of the deformation theory provides a fairly accurate prediction of buckling strength, but fails in this case to yield a correct description of post-buckling behavior . on the other hand, the application of the incremental theory, which is mathematically and physically more rigorous, leads to an overestimation of buckling strength, even though initial imperfections are considered . this paradox has existed for years, and remains to be resolved .
buckling of sandwich under normal pressure . a theoretical study is made of the buckling of a sandwich sphere comprised of a core layer of low-modulus material and two thin facing layers of higher modulus material . the solution for the buckling resistance of the sphere under normal external pressure is obtained by linearized theory, and is reducible to the classical solution for monocoque spherical shells . critical buckling pressures are calculated for various radius-thickness ratios and sphere materials .
allowable axial loads and bending moments for inelastic structures under nonuniform temperature distribution . a strain-analysis method is derived and demonstrated for the calculation of design allowable load-strain curves for the cross section of a structure supporting axial loads and bending moments . the temperature effects of thermal stresses and changed material properties and all inelastic effects are included in the calculations so that the final curve is a design curve for the applied stresses as calculated by room-temperature elastic procedures . the method allows for sequence application and removal of load and temperature, as well as cycling of load and/or temperature . applications are shown for a rectangular bar under temperature cycling with axial loads and/or bending moments and for a box beam with one bending-moment temperature cycle . interaction curves beyween axial load and bending moment with inelastic effects included are given, the calculations being done on a digital computer . a procedure is given for using the method to construct design curves .
effects of internal pressure on the buckling of circular-cylindrical shells under bending. the effect of internal pressure on the small-deflection buckling of thin-walled cylinders under bending is investigated by means of a modified donnell equation . the results indicate that the maximum critical stress due to bending increases with internal pressure, unlike the case of pressurized cylinders under compression . these results represent the moment at which significant deformations appear in the cylinder, rather than the maximum moment able to be carried, but may be a good approximation to the latter for metal cylinders .
breathing vibrations of a circular shell with an internal liquid . resonant breathing frequencies and mode shapes are determined experimentally for a thin-walled, circular cylindrical shell containing a nonviscous incompressible liquid . the resonant frequencies determined for the full shell are in good agreement with those predicted by reissner’s shallow-shell vibration theory with the inclusion of an apparent-mass term for the liquid . the effect of the internal liquid on the shell mode shapes is significant only for the partially full shell . in this case the circumferential node lines tend to shift toward the bottom or filled portion of the shell . excitation of low-frequency liquid-sloshing motion by high-frequency forced oscillation of a partially filled shell occurred in many cases . this low-frequency liquid response is tentatively explained as being excited by a beat frequency in the forced oscillation . a similar type of response has been reported by yarymovych in axially excited rigid tanks .
clamped short oval cylindrical shells under hydrostatic pressure . the principle of the minimum of the total potential is employed to obtain stresses and displacements for clamped, short, oval cylindrical shells under hydrostatic pressure . classical shell theory, in which buckling effects are not considered, was used . a fourier series is assumed for the deflections in the closed circumferential direction so that the partial differential equations of equilibrium are replaced by a set of ordinary differential equations . the energy solution is compared with a simplified approximation which can be considered an equivalent circular cylinder solution . graphs of the significant stresses and displacements are presented for oval cylinders having major to minor axis ratios of 1.10, 1.30, and 1.50 . it is shown that the maximum stresses and displacements increase significantly as the major to minor axis ratio is increased .
experimental investigation at mach number of 3. 0 of effects of thermal stress and buckling on flutter characteristics of flat single-bay panels of length-width ratio 0. 96 . flat, single-bay, skin stiffener panels with length-width ratios of 0.96 were tested at a mach number of 3.0, at dynamic pressures ranging from 1,500 to stagnation temperatures from 300 f to effects of thermal stress and buckling on the flutter of such panels . the panel supporting structure allowed partial thermal expansion of the skins in both the longitudinal and lateral directions . panel skin material and skin thickness were varied . a boundary faired through the experimental flutter points consisted of a flat-panel portion, a buckled-panel portion, and a transition point, at the intersection of the two boundaries, where a panel is most susceptible to flutter . the flutter region consisted of two fairly distinct sections, a large-amplitude flutter region and a small-amplitude flutter region . the results show that an increase in panel skin temperature flutter . the flutter trend for buckled panels is reversed . use of a modified temperature parameter, which approximately accounts for the effects of differential pressure and variations in panel skin material and skin thickness, reduced the scatter in the data which resulted when these effects were neglected . the results are compared with an exact theory for clamped panels for the condition of zero midplane stress . in addition, a two-mode /transtability/ solution for clamped panels is compared with the experimentally determined transition point .
mathematical techniques applying to the thermal fatigue behaviour of high temperature alloys . during thermal fatigue testing of a specimen with a thin edge, or during rapid temperature changes in the gas flow past turbine blades, the thin edges are deformed plastically in compression during heating and subsequently creep in tension as the bulk of the specimen or blade heats up . the plastic deformation is determined from temperature distributions, which are calculated by biot’s variational method . the creep deformation is determined as a function of time by a differential equation, which expresses the balance between increasing elastic stress and reduction of stress due to creep relaxation, and which is solved to a riccati equation soluble in terms of bessel functions, or (iii) by transformation to a second-order differential equation with a periodic coefficient . using the thermal stresses obtained from the solution of the differential equation, the theoretical thermal fatigue endurance is determined from cyclic (mechanical) stress endurance data . agreement between theoretical and experimental thermal fatigue endurances is obtained, over ranges of temperature, strain, and strain rate, or equivalently, over ranges of temperature-edge radius and heat transfer coefficient . this agreement supports the use of the theoretical methods in wider contexts . the accuracy of the temperature distributions is better than the accuracy of other factors entering into the correlation between theoretical and experimental endurances . improvement in the interpretation of experimental results requires consideration of the alteration of the stress cycles during the course of thermal fatigue testing . this requirement is catered for partially by the various solutions of the differential equation for thermal stress .
formulae for use with the fatigue load meter in the assessment of wing fatigue life . this note gives a method for the derivation of suitable constants which, when multiplied by the readings recorded at each appropriate acceleration level on a fatigue load meter and then added together, give directly the proportion of fatigue life used up in the wing . it is suggested that when the estimated proportion is of order 80, then a more detailed assessment of fatigue life should be made .
local circumferential buckling of thin circular cylindrical shells . the problem of circumferential buckling of a thin circular cylindrical shell due to compressive hoop stresses which vary in the axial direction is examined . for extremely localised compressive hoop stress distributions resulting from thermal discontinuity effects, or from a uniform, radial line loading, the buckle pattern should also be localised . simplified analyses into these two types of problem are considered which show that only a limited number of buckle deflection modes needs to be assumed .
the flow of a compressible fluid past a sphere . the flow of a compressible fluid past a sphere fixed in a uniform stream is calculated to the third order of approximation by means of the janzen-rayleigh method . the velocity and the pressure distributions over the surface of the sphere are computed and the terms involving the fourth power of the mach number, neglected in rayleigh’s calculation, are shown to be of considerable importance as the local velocity of sound is approached on the sphere . the critical mach number, that is, the value of the mach number at which the maximum velocity of the fluid past the sphere is just equal to the local velocity of sound, is calculated for both the second and the third approximations and is found to be, respectively, and .
on the flow of a compressible fluid past a sphere . it was shown by raleigh (philos. mag. 32, 1 (1916)) that the velocity potential for the subsonic flow of a compressible fluid past a sphere can be expressed as a power series in terms of mach’s number m (which is the ratio of the undisturbed velocity u, divided by the velocity of sound for the undisturbed flow) . the equation in question is and boundary conditions are prescribed for raleigh himself computed the first two terms of this series,. the author finds the third term . he gives some graphs showing numerical differences between raleigh’s and his approximation .
an experimental study of jet-flap compressor blades . the results of a preliminary experimental investigation to determine the feasibility of using the jet flap to improve the section characteristics of an axial-flow compressor blade are presented and discussed trailing edge . internal design of the blade is described and details of the resulting jet flow are given . also included are wind-tunnel design and test procedures for the two-dimensional cascade used in the test . test results are presented in the form of the measured turning angle, pressure rise, and lift coefficient . they are examined with particular reference to the prevention of rotating stall .
q . app . math . 7, 1950, 381 . experiments on porous-wall cooling and flow separation control in a supersonic nozzle . control of flow separation by fluid injection at one diverging boundary of a two-dimensional, transparent-walled de laval nozzle was investigated by spark schlieren photography of dry nitrogen flows expanded from two stagnation temperatures injection conditions at the permeable boundary were varied by the use of three grades of porous stainless steel with nominal pore diameters of 10, 20, and 30 microns, through which nitrogen was forced by coolant reservoir pressures of 25, 50, and 100 psig, in addition to the case of no forced injection . pressure distribution measurements were made along the nonpermeable diverging boundary . it was found that flow separation at expansion ratios approaching the optimum value for maximum thrust coefficient could be induced at the porous wall by a local injection mass velocity of the order of a few per cent of the local main-stream mass velocity . separation at the solid boundary was not noticeably influenced by injection at the opposite wall, and the asymmetrical separation thus effected jet deflections of up to 10 degrees at the lower stagnation-pressure levels . variation of the wall heat-transfer condition by changing the stagnation temperature did not significantly influence separation behavior . temperature measurements at the reservoir face of the porous section, together with use of published correlations and of the rube-sin analysis for estimation of stream-side stanton numbers under noninjection and injection conditions, respectively, permitted heat-transfer calculations which indicated that the effectiveness of the transpiration technique in controlling nozzle wall temperatures derives primarily from intimate fluid-solid contact in a porous material of high specific surface .
general characteristics of the flow through nozzles at near critical speeds . the characteristics of the position and form of the transition surface through the critical velocity are computed for flow through flat and round nozzles from subsonic to supersonic velocity . corresponding considerations were carried out for the flow about profiles in the vicinity of sonic velocity .
studies on two dimensional flows of compressible fluid. it is well-known that when the flow is everywhere subsonic in a field of flow, the nature of the two-dimensional isentropic flow of a compressible perfect fluid differs only slightly from that of the corresponding flow of an incompressible perfect fluid . thus, in such a case, we can calculate the field of flow by any of the well-known methods of approximation . on the other hand, if the flow is supersonic throughout the field, we can determine the flow pattern by the method of characteristics .
force measurements on square and dodecagonal sectional cylinders at high reynolds numbers . results are given of measurements in the compressed air tunnel of the forces on two cylinders, one of square cross-section and the other dodecagonal . the tests were carried out at various reynolds numbers ranging from approximately 0.1 x 10 to 5.5 x 10, based on the distance between parallel faces .
a technique for rendering approximate solutions to physical problems uniformly valid . a method is described for treating some of the characteristically non-linear problems of physics, in particular those involving a non-linear partial differential equation for which an approximate linearization is permissible everywhere except in a limited region, such as the neighbourhood of (5) a singular characteristic of the approximate solution, or of approximation is valueless . the method involves a transformation of an independent variable, which is determined progressively with successive approximations to the solution .. only one step being necessary if a first approximation valid uniformly be obtained . the method is most easily understood in its application to simple first order ordinary differential equations, which are studied in detail in 2 and 3 as a preparation for the extension to more complicated problems in 4, 5 and 6 . physically, the longest section, 6, concerns the /spread/ of a progressive wave at infinity, an important and essentially non-linear process .
an integral related to the radiation integrals . the author points out the relation of the integral to problems in astrophysics and quantum mechanics, and because of its importance supplies a table of values . the table gives rl(x) to seven places of decimals from x=0 to x=6.00 at intervals of 0.01 . second order central differences are tabulated to assist in interpolation .
calculation and compilation of the unsteady lift functions for a rigid wing subjected to sinusoidal gusts and to sinusoidal sinking oscillations . the total lift responses of wings to sinusoidal gusts and to sinusoidal vertical oscillations are calculated from the response to gust penetration and to a sudden change in sinking velocity through use of the well-established reciprocal relations for unsteady flow . the cases considered are two-dimensional wings in incompressible, subsonic compressible, sonic, and supersonic flow,. elliptical and rectangular wings in incompressible flow,. wide rectangular and delta wings in supersonic flow,. and delta wings of vanishingly low aspect ratio in incompressible and compressible flow . for most of the cases considered, closed-form expressions are given and the final results are presented in the form of plots of the square of the modulus of the lift coefficient for wings in a sinusoidally oscillating gust and in the form of the real and imaginary parts of the lift component for wings undergoing sinusoidal sinking oscillations . a summary table is presented as a guide to the scope and results of this paper,. this table contains the figure and equation numbers for the types of flow and plan forms considered .
the transonic characteristics of 38 cambered rectangular wings of varying aspect ratio and thickness as determined by transonic-bump technique . an investigation to determine the effects of camber on the aerodynamic characteristics of a series of rectangular wings having various aspect ratios and thickness-to-chord ratios was conducted in the ames the transonic-bump method . the mach number range of the investigation was from 0.6 to 1.12, with a corresponding reynolds number range of 1.7 to 2.2 million . the lift, drag, and pitching-moment data are presented for wings having aspect ratios of 4, 3, 2, 1.5, and 1, and naca 63a2xx and 63a4xx sections with thickness-to-chord ratios of
use of subsonic kernel function in an influence-coefficient method of aeroelastic analysis and some comparisons with experiment . this paper illustrates the development and application of an influence-coefficient method of analysis for calculating the response of a flexible wing in an airstream to an oscillating disturbing force and for treating such aeroelastic instabilities as flutter and divergence . aerodynamic coefficients are derived on the basis of lifting-surface theory for subsonic compressible flow by use of the method presented in nasa technical report r-48 . application of the analysis is made to a uniform cantilever wing-tip tank configuration for which responses to a sinusoidal disturbing force and flutter speeds were measured over a range of subsonic mach numbers and densities . calculated responses and flutter speeds based on flexibility influence coefficients measured at nine stations are in good agreement with experiment, provided the aerodynamic load is distributed over the wing so that local centers of pressure very nearly coincide with these nine influence stations . the use of experimental values of bending and torsional structural damping coefficients in the analysis generally improved the agreement between calculated and experimental responses . some calculations were made to study the effects of density on responses near the flutter conditions, and linear response trends were obtained over a wide range of densities .
calculated subsonic span loads and resulting stability derivatives of unswept and 45degree sweptback tail surfaces in sideslip and steady roll . subsonic span loads and the resulting stability derivatives have been calculated for a systematic series of vertical- and horizontal-tail combinations in sideslip and in steady roll in order to provide information embracing a wide range of probable tail configurations . all calculations were made by application of the discrete-horseshoe-vortex method to the problem of estimating loads on intersecting surfaces . the investigation covered variations in vertical-tail aspect ratio, the ratio of horizontal-tail aspect ratio to vertical-tail aspect ratio, the effects of horizontal-tail dihedral angle /for the sideslip case/, and the effects of vertical position of the horizontal tail for surfaces having their quarter-chord lines swept back 0degrees and 45degrees . the results of the investigation are presented in charts from which the span loads for the various conditions can be obtained . the resulting stability derivatives are presented as vertical- and horizontal-tail contributions as well as total-tail-assembly derivatives . the results of this investigation, which was made for a wider range of geometric variables than previous studies, showed trends which were in general agreement with the results of previous investigations . also presented in this paper and used in the computations is an extensive table of values of sidewash due to a rectangular vortex .
a method for calculating the subsonic steady-state loading on an airplane with a wing of arbitrary planform and stiffness . a method for computing the steady-state span load distribution on an elastic airplane wing for specified airplane weights and load factors is given . the method is based on a modification of the weissinger l-method and applies at subcritical mach numbers . it includes the effects of external stores and fuselage on the spanwise loading . modifications are outlined for treating tail-boom and tailless airplane configurations and for calculating the divergence dynamic pressure of a swept wing with a large external store . a method is also outlined for reducing wind-tunnel data to obtain effective aerodynamic coefficients which are free of model flexibility effects . the effects of mach number can readily be evaluated from the aerodynamic coefficients thus obtained .
heat transfer through the laminar boundary layer on a circular cylinder in axial incompressible flow . this paper presents a method of calculating the distribution of rate of heat transfer into a laminar incompressible boundary layer from the exterior surface of a long thin circular cylinder, when the surface of the cylinder is maintained at a constant temperature and the flow is parallel to the cylinder axis,. the temperature difference between the surface and the main stream is taken to be small enough to neglect buoyancy effects . a series solution, valid for small downstream distances from the nose, has been obtained already by seban, bond, and kelly . this is now extended by deriving an asymptotic series solution, valid at large downstream distances, and bridging the gap between these two series solutions by an approximate solution, based on the method used recently by davies and bourne to calculate heat transfer from a flat plate . the calculation is used to demonstrate the effect of curvature and of prandtl number on the local rate of heat transfer at various downstream distances by comparing with the corresponding flat plate results .
the flow of fluid along cylinders . the boundary layer equations for uniform flow parallel to the generators of any cylinder without corners are put into the form of a series of linear third-order differential equations . the first three of these are the same as those obtained by seban and bond /1/ for a circular cylinder and solved by kelly /2/ . the rest have additional terms depending on the radius of curvature of the cylinder and its derivatives . the problem is also attacked by a pohlhausen method as far as four terms of the series . for large distances from the front, rayleigh’s method, as given by hasimoto /3/, gives the first two terms of an asymptotic expansion for the drag . explicit calculations are made of the drag of an elliptic cylinder of eccentricity 1/2 3 . there is evidence that the drag is everywhere less than that of a circular cylinder of the same perimeter .
the skin friction on infinite cylinders moving parallell to their length . the frictional force on a cylinder moving steadily parallel to its length through a viscous liquid which is initially at rest is determined with reasonable accuracy over the whole range of values of the duration of the motion and for a wide variety of shapes of the cylinder cross-section . when the time t is small, the first approximation gives a force per unit area which is the same as that for a flat plate of infinite width . the second approximation takes the shape of the cylinder into account and the force on unit length of cylinder is determined in terms of the number of corners, and their angles, in the cylinder cross linder is the same, to this approximation, as that on a circular cylinder of the same perimeter . for large values of t the determination of the frictional force is reducible to that of a potential problem, the solution of which is known for a number of different shapes . the approximations for small and large values of t for any one cylinder do not overlap but can be joined without much ambiguity . for no value of t do the forces on cylinders of different shape /excluding those whose curvature is not everywhere inwards/ differ by more than about 25 per cent .
rayleigh’s problem for a cylinder of arbitrary shape . the motion of an incompressible viscous fluid generated by a cylinder of arbitrary cross-sectional form which is started to move suddenly from rest with uniform velocity in the direction of its length is considered formulae in powers of are derived for the velocity distribution /valid in the vicinity of the cylinder/ and for the frictional drag on the cylinder, correct to the order of a, where a is the characteristic length of the cross section, v is the kinematic viscosity, and t is the time . these formulae are given in terms of only the analytic function which maps conformally the region outside the cross section of the cylinder onto the region outside the unit circle, and of certain integrals e which are common to any arbitrary cylinder . in particular, when a is sufficiently small, the total frictional drag on the cylinder per unit length is expressed as, irrespective of the cross-sectional form, where b 2 and y 0.5772…/euler’s constant/ .
an approximate boundary layer theory for semi-infinite cylinders of arbitrary cross-section . an estimate is given of the distribution of skin frictional force per unit length, and of displacement area, on the outside of a semi-infinite cylinder, of arbitrary cross-section, moving steadily in a direction parallel to its generators . a pohlhausen method is employed with a velocity distribution chosen to yield zero viscous retarding force on the boundary layer approximations . /the smallness of the fluid acceleration far from the leading edge has been pointed out by batchelor reasonable results atlarge distances from the leading edge . however, for a large class of cross-sections, which includes all convex cross-sections and locally concave cross-sections with re-entrant angles greater than 1/2, the method yields the expected square root growth of the boundary layer at the leading-edge, with a fairly close approximation to the coefficient, and it is supposed that the skin-frictional force and displacement area are given with reasonable accuracy along the whole length of the cylinder . results for the elliptic cylinder and the finite flat plate are given in closed form, valid for the whole length of the cylinder, and are expected to be in error by at most 20 per cent . in addition, some estimate is given of the effect of corners on skin frictional force and displacement area .
a further note on the calculation of heat transfer through the axisymmetrical laminar boundary layer on a circular cylinder . by using a karman-pohlhausen method the distribution of local rate of heat transfer is ovaluated for the case of air flow in an axisymmetrical laminar boundary layer on a heated circular cylinder, the temperature of the cylinder being independent of downstream distance . this calculation serves to link the numerical values obtained by seban, bond, and kelly for small downstream distances to those obtained by bourne and davies for large downstream distances .
a wind-tunnel test technique for measuring the dynamic rotary stability derivatives at subsonic and supersonic speeds . a method is described for measuring the dynamic stability derivatives of a model airplane in a wind tunnel . the characteristic features of this system are that single-degree-of-freedom oscillations were used to obtain combinations of rolling, yawing and pitching motions., that the oscillations were excited and controlled by velocity feedback which permitted operation under conditions unfavorable for more conventional types of oscillatory testing., and that data processing was greatly simplified by using analog computer elements in the strain-gage circuitry . the system described is primarily for measurement of the damping derivatives damping in roll damping in pitch, damping in yaw, and the cross derivatives rolling moment due to yawing and yawing moment due to rolling . the method of testing also permits measurement under oscillatory conditions of the static derivatives rolling moment due to sideslip, yawing moment due to sideslip, and pitching moment due to angle of attack . all these derivatives are of particular importance in estimating the short-period oscillatory motions of a rigid airplane . a small number of experimental data are included to illustrate the general scope of results obtainable with this system .
measurements at mach numbers up to 2. 8 of the longitudinal characteristics of one plane and three cambered slender ‘ogee’ wings . measurements have been made of the longitudinal characteristics of one plane and three cambered slender ogee wings (p = 0.45,) at two subsonic and eight supersonic mach numbers up to 2.8 . the tests also included measurements of the zero-lift pressure drag and support interference of the plane wing . the results have been analysed to give data for estimating the performance of supersonic transport aircraft .
some low speed problems of high speed aircraft . the first part of the paper deals with the low speed aerodynamics of aircraft shapes suggested by kuchemann, at the second international congress in aeronautical sciences at zurich in 1960, as suitable for achieving a required range at supersonic speeds, namely wingbody arrangements with sweepback angles of 55degrees or 60degrees and streamwise thickness-chord ratio of about 5 per cent suitable for low supersonic speed, and slender near-triangular wings with sharp leading edges suitable for mach numbers of about 2 or more . no attention is given to /slewed/ wings, powered lift or variable geometry . in dealing briefly with swept wings, the need for avoiding separation of flow from the leading edge is demonstrated, with the conclusion that it is desirable to use leading edge flaps with blowing or suction at the knee together with blown trailing edge flaps . wind tunnel tests are described on a simplified model with these boundary layer control methods applied . mention is made of the possibility of adverse ground effect on maximum lift . more attention is given to the case of slender wings because their use involves a new type of flow with separation from all edges . this flow and its steadiness are therefore discussed from the point of view of the possibility of buffeting,. the effect of plan form on static longitudinal stability and pitch-up is analysed,. and a short summary of available results on damping in pitch is given . large rolling moments due to sideslip are shown to give rise to serious problems of control, and the present state of knowledge of static lateral and directional stability and rolling and yawing rotary derivatives is discussed . finally the effects of proximity to the ground are summarised . the second part of the paper is concerned with work aimed at clarifying some of the requirements for handling qualities of future aircraft . it is not so much concerned with forecasts of the dynamic behaviour of these future aircraft as with determining what the pilot wants . two aspects of control in the vertical plane are discussed in some detail namely speed control and glide path holding . flight tests on an avro 707a aircraft, with artificially worsened characteristics, are described, and it is shown that substantially constant performance in the piloting task can be achieved at the expense of increased pilot effort . some tentative conclusions on desirable levels of speed stability and phugoid damping are, nevertheless, drawn . a brief review of the present status of lateral/directional handling requirements, using mainly american data, is also included .
the flow pattern on a tapered sweptback wing at mach numbers between 0.6 and 1.6 . the development of the flow pattern on a swept wing with incidence and stream mach number is described . the wing, of aspect ratio 2 828, taper ratio 0 333 and leading-edge sweep 53 5 deg, was tested at mach numbers between 0 6 and 1 6 at incidences up to about 12 deg . the test reynolds number varied with mach number, being typically 2 3 x 10 at m leading edge . it is shown that the flow pattern at moderate incidences develops smoothly from a subsonic type involving leading-edge separation to a supersonic type where the flow is attached near the leading edge and with shock-induced separation further aft . the formation and movement of the shock-wave system and the vortices near the wing surface are briefly discussed .
experiments with a tapered swept-back wing of warren 12 planform at mach numbers between 0.6 and 1.6 . the development of the flow pattern on a wing of aspect ratio 2 828, taper ratio 0 333, leading-edge sweepback 53 5 deg and 6 per cent thickness/chord ratio in the streamwise direction has been described in part 1, which discussed oil-flow patterns obtained on the surface of the wing . the complete programme of tests also included pressure plotting at four spanwise stations and force measurements . these are discussed in relation to the flow development in this part of the report . the wing was tested at mach numbers between 0 6 and 1 6 for incidences up to about 14 deg . the tunnel stagnation pressure was held constant at a value near atmospheric pressure during the tests, so that the reynolds number varied with mach number .. at m 1 0 it was 2 3 x 10 based on the mean aerodynamic chord . boundary-layer transition was fixed by a roughness band at the leading edge . a detailed analysis has been made of the pressure distributions on the surface of the wing and the chordwise distributions integrated to determine the spanwise loading . the overall lift and pitching moment of the wing were also obtained from these data, as well as from direct measurements using a strain-gauge balance, by means of which the wing drag was also determined . these results are considered in some detail to illustrate the effects of mach number and incidence on the flow about the model . a preliminary analysis is also made of the conditions for boundary-layer separation due to shock waves on the wing surface . the principal factor appears to be the component of mach number normal to the shock front .
the operation of the npl 18in x 14in. wind tunnel in the transonic speed range . a brief description of the slotted liners used is given together with the power requirements and some flow surveys . some observations are made on wall interference on a half-model of a swept wing tested in the wind tunnel .
an investigation at transonic speeds of the performance of various distributed roughness bands used to cause boundary layer transition near the leading edge of a cropped delta half-wing . distributed roughness bands of no.320 and no.500 carborundum were found to be effective in causing boundary-layer transition if they extended over the first 5( and 10( respectively of the local chord . use of larger grain sizes, or increases in the band width for a given grain size resulted in a drag penalty . with very large particle sizes /about between the particles . the drag penalty was constant over the test mach number range /0.80 to 1.15/ and decreased slowly with incidence . the wing lift and pitching moment were only slightly modified by the presence of any of the roughness bands tested, but this result would not of course necessarily apply to wings of other planforms or section shapes . the test reynolds number was about 2.7 million . in the appendix, the structure of the roughness bands is discussed, as well as the details of the materials used and the techniques used to apply the band .
a study of the effect of leading-edge modifications on the flow over a 50degree sweptback wing at transonic speeds . summary . an investigation has been made in the n.p.l. 18 in. x 14 in . tunnel of the effects of leading-edge modifications on the flow and forces on an untapered wing of 50 deg leading-edge sweep, at stream mach numbers between 0 60 and 1 20 . seven leading-edge profiles were tested, ranging from a drooped extension of 18 per cent of the chord of the basic sharp-nosed section to a round-nosed section with a leading-edge radius of 1 0 per cent of the basic chord . leading-edge droop was found to increase the wing drag near zero lift but to reduce appreciably the lift-dependent drag component, except at the highest test mach numbers . droop also increased the lift coefficient at which leading-edge separation occurred on the upper surface at moderate subsonic speeds, but in addition reduced the mach number for transonic flow attachment . the appearance of the forward shock /but not the rear shock/ is considerably delayed when the leading edge is drooped . with the undrooped sections an increase in leading-edge radius was accompanied by successively earlier appearances of the forward shock, and hence the outboard shock with its attendant separation . the conditions at which the rear shock first appeared changed only slowly as the section was changed . the variations in wing flow pattern as the leading edge is modified are discussed and related to measured changes in the wing lift and drag . an attempt is also made to estimate the local mach numbers on some parts of the wing from the oil-flow patterns,. this material is used to assess the flow conditions appropriate to shock-induced separation . the main section of the report concludes with a tentative discussion of the significance of the present results to the design of swept wings . in an appendix results obtained with the wing in a sweptforward configuration are briefly considered .
interaction between shock waves and boundary layers, with a note on the effects of the interaction of the performance of supersonic intakes . the interaction between shock waves and boundary layers has important effects in many problems of high-speed flow . this paper has been written as a guide to the literature on the subject, and as a critical review of the present state of knowledge concerning both the underlying physical processes and the practical applications . it will be clear to the reader that, although substantial progress has been made, our knowledge is still far from complete and that more work both of a fundamental nature and on specific applications is needed before the problem is understood sufficiently well for design purposes . part i of the paper describes experiments on comparatively simple types of flow designed to provide fundamental information and to assist in the development of the theory . these experiments show that the interaction depends mainly on the mach and reynolds numbers and on the strength of the shock wave . in particular, the interaction of a shock wave with a laminar boundary layer is shown to produce much larger effects than if the boundary layer is turbulent . for most cases where the effects of the interaction are large enough to have serious practical consequences it is found that the boundary layer separates from the surface, and the difference between the interaction with laminar and turbulent layers arises mainly because the laminar layer separates much more readily in an adverse pressure gradient . the details of the interaction downstream of the separation point thus depend critically on the behaviour of the separated layer, and on the conditions under which it reattaches to the surface . many of the features found in the fundamental experiments appear also in practical applications and these are considered in parts ii and iii of the paper . although the emphasis hero is on the performance of aerfoils and wings moving at high subsonic speeds, the importance of the interaction in other examples such as at supersonic trailing edges and in supersonic intakes is also discussed briefly . the differences between the interaction with laminar and turbulent boundary layers are often a source of serious discrepancy between model experiments and full-scale conditions . for small-scale models it is, therefore, frequently essential to make the boundary layer turbulent by artificial means . some of the difficulties involved in doing this, and certain of the more promising methods are briefly discussed . it is shown that experiments on models with transition fixed can be used to explain a number of aerodynamic effects encountered in transonic flight, and connected with the occurrence of shock-induced separation of the turbulent boundary layers . for both two-dimensional aerofoils and straight and sweptback wings, turbulent separation occurs for shocks above a certain strength which applies for both model and full-scale conditions full-scale conditions,. differences in magnitude would be expected if the pressure recovery along the separated layer between the shock and the trailing edge is affected by reynolds number, but little information is at present available on this point . most of the repercussions of turbulent separation on the steady-motion characteristics of aerofoils and wings can be traced to the associated reduction in the pressure recovery over the roar of the surface . this is because the pressure at the trailing edge controls the inter-relation between the two surfaces /so long as the flow at the trailing edge remains subsonic/, and in particular the relative movements of the shock waves and the extents of the local regions of supersonic flow . certain unsteady-flow characteristics such as buffeting and control surface separation . some evidence is presented on the influence of section shape on the occurrence and effects of separation, but in this, as in many other respects, information relevant to turbulent boundary layers is scarce . some notes on the further work which is required are given in part iv of the paper .
some effects of wind-tunnel interference observed in tests on two-dimensional aerofoils at high subsonic and transonic speeds . in the high-speed research on two-dimensional aerofoils at the national physical laboratory the need to keep model size above a certain minimum, in order to reproduce correctly the boundary layer separation effects experienced at full scale, has been considered paramount even at the risk of incurring significant tunnel interference effects . this report discusses the interference effects for the slotted working sections now in use . the magnitudes of the blockage and lift effect corrections are deduced for the ratio of model chord to tunnel height normally used . it is shown that a simple adaptation to reduce the open area of the walls would reduce both corrections to insignificant proportions simultaneously, but would give a reduced choking mach number separated flows . the observed trends in the variation of the blockage effects for other ratios of model chord to tunnel height differ from those predicted theoretically, and so the results cannot be applied more generally until these trends have been checked by further investigations . it is suggested that wake interference effects can significantly influence the manner in which separated flows develop with increasing incidence or mach number, particularly for walls of small open area . examples are also given of effects of distortions in the local supersonic flow, which are most noticeable for walls with relatively large open areas .
wall interference at transonic speeds on a hemisphere cylinder model . tests have been made in three n.p.l. wind tunnels on a pressure-plotting model consisting of a long cylinder with a hemispherical nose . the surface pressure distributions were measured for stream mach numbers between 0.7 and 1.1 at zero model incidence, and schlieren photographs were taken . the blockage ratios were 0.211(, 0.117( and 0.120( . the principal feature of the flow is the effect of working section size on the rate at which the terminal shock wave moves back along the model with increasing stream mach number . this is thought to depend mainly on the distance from the model to the slotted walls of the tunnel, and not necessarily on the blockage ratio . the distance of the solid sidewall is important in influencing the local mach number ahead of the terminal shock, by reflecting the expansion-wave system originating near the model nose .
experimental study of the equivalence of transonic flow about slender cone-cylinders of circular and elliptic cross section . this report describes an experimental investigation of the equivalence relationship and the related theory for lifting forces proposed by transonic slender-body theory . the models chosen for this study are a flat, winglike, elliptic cone-cylinder and its equivalent body of revolution, a circular cone-cylinder . it is determined that the flows about the two models are closely related in the manner predicted by the theory, the relationship persisting over a mach number range of 0.92 to cone-cylinder vary linearly only over the small angle-of-attack range of approximately 1 and that the aerodynamic loading at sonic speed compares favorably with jones’ slender-wing theory . the results of the investigation suggest that at transonic speeds and at small angles of attack the calculation of all aerodynamic characteristics of slender, three-dimensional shapes can be made by use of transonic slender-body theory when the pressures on the equivalent body of revolution are known, either by experiment, or by an adequate nonlinear theory . from transonic slender-body theory it is deduced that the slenderness required for this application is the same as that required for the successful application of the transonic area rule .
the behaviour of supersonic flow past a body of revolution far from the axis . a theory is developed of the supersonic flow past a body of revolution at large distances from the axis, where a linearized approximation is valueless owing to the divergence of the characteristics at infinity . it is used to find the asymptotic forms of the equations of the shocks which are formed from the neighbourhoods of the nose and tail . in the special case of a slender pointed body, the general theory at large distances is used to modify the linearized approximation to give a theory which is uniformly valid at all distances from the axis . the results which are of physical importance are summarized in the conclusion (9) and compared with the results of experimental observations .
the shock pattern of a wing-body combination far from the flight path . the position and strength of the front shock wave at large distances from a wing-body combination, are deduced from the linear theory for the combination, using a method developed by whitham . the combination consists of a body of revolution and a wing which has thickness and is lifting . the effects of interference between the flow over the body and the flow over the wing are included . in any direction the flow far from the wing-body combination is equivalent to the flow past a body of revolution determined from the configuration of the combination . the modified formulae for unsteady flow are given and some results are evaluated for the combination of a body of revolution and a delta wing with subsonic leading edges .
a flight test investigation of the sonic boom . the /sonic boom/ as it is now popularly called, has become the center of considerable interest during the past few years because of widespread public disturbance and possible damage that can result from it . in the hopes of minimizing this disturbance and to extend the general knowledge of the shock waves which produce the booming noise, the aeronautical research laboratory, wright air development center, has initiated an extensive research program to study the sonic boom phenomenon . this report presents the results of flight tests undertaken as one phase of this program . the tests had as their objective the determination and measurement of the shock wave pressure pattern surrounding an f-100 aircraft in level supersonic flight . the flight tests were conducted at the air force flight test center, edwards air force base, california, under the authority of air research and development command test directive no. 5524-f1 .
ground measurements of the shock wave noise from airplanes in level flight at mach numbers to 1. 4 and at altitudes to 45,000 feet . time histories of noise pressures near ground level were measured during flight tests of fighter-type airplanes over fairly flat, partly wooded terrain in the mach number range between 1.13 and 1.4 and at altitudes from 25,000 to 45,000 feet . atmospheric soundings and radar-tracking studies were made for correlation with the measured noise data . the measured and calculated values of the pressure rise across the shock wave were generally in good agreement . there is a tendency for the theory to overestimate the pressure at locations remote from the track and to underestimate the pressures for conditions of high tailwind at altitude . the measured values of ground-reflection factor averaged about 1.8 for the surfaces tested as compared to a theoretical value of 2.0 . two booms were measured in all cases . the observers also generally reported two booms,. although, in some cases, only one boom was reported . the shock-wave noise associated with some of the flight tests was judged to be objectionable by ground observers, and in one case the cracking of a plate-glass store window was correlated in time with the passage of the airplane at an altitude of 25,000 feet .
ground measurements of airplane shock wave noise at mach numbers to 2, and at altitudes of 60,000 feet . the intensity of shock-wave noise at the ground resulting from flights at mach numbers to 2.0 and altitudes to 60,000 feet was measured . measurements near the ground track for flights of a supersonic fighter and one flight of a supersonic bomber are presented . level cruising flight at an altitude of 60,000 feet and a mach number of 2.0 produced sonic booms which were considered to be tolerable, and it is reasonable to expect that cruising flight at higher altitudes will produce booms of tolerable intensity for airplanes of the size and weight of the test airplanes . the measured variation of sonic-boom intensity with altitude was in good agreement with the variation calculated by an equation given in nasa technical note d-48 . the effect of mach number on the ground overpressure is small between mach numbers of 1.4 and 2.0, a result in agreement with the theory . no amplification of the shock-wave overpressures due to refraction effects was apparent near the cutoff mach number . a method for estimating the effect of flight-path angle on cutoff mach number is shown . experimental results indicate agreement with the method, since a climb maneuver produced booms of a much decreased intensity as compared with the intensity of those measured in level flight at about the same altitude and mach number . comparison of sound pressure levels for the fighter and bomber airplanes indicated little effect of either airplane size or weight at an altitude of 40,000 feet .
ground measurements of the shock wave noise from supersonic bomber airplanes in the altitude range from 30,000 to 50,000 feet . shock-wave ground-pressure measurements have been made for supersonic bomber airplanes in the mach number range from 1.24 to 1.52, for altitudes from about 30,000 to 50,000 feet, and for a gross-weight range from about 83,000 to 120,000 pounds . the measured overpressures were generally higher than would be predicted by the theory which accounts only for volume effects . there is thus a suggestion that lift effects on sonic-boom intensity may be significant for this type of airplane for the altitude range of the present tests .
an investigation of some aspects of the sonic boom by means of wind tunnel measurements of pressures about several bodies at a mach number of 2. 01 . an investigation of some aspects of the sonic boom has been made with the aid of wind-tunnel measurements of the pressure distributions about bodies of various shapes . the tests were made in the langley at a mach number of 2.01 and at a reynolds number per foot of 2.5 x 10 . measurements of the pressure field were made at orifices in the surface of a boundary-layer bypass plate . the models which represented both fuselage and wing types of thickness distributions were small enough to allow measurements as far away as 8 body lengths or 64 chords . the results are compared with estimates made using existing theory . to the first order, the boom-producing pressure rise across the bow shock is dependent on the longitudinal development of body area and not on local details . nonaxisymmetrical shapes may be replaced by equivalent bodies of revolution to obtain satisfactory theoretical estimates of the far-field pressures .
an investigation of the influence of lift on sonic-boom intensity by means of wind tunnel measurements of the pressure fields of several wing-body combinations at a mach number of 2. 01 . an investigation of the effect of lift on sonic-boom intensity has been performed by means of wind-tunnel measurements of the pressure fields surrounding small wing-body combinations . the tests were conducted in the langley 4- by 4-foot supersonic pressure tunnel at a mach number of per foot . effects of lift were found to be real and significant . measured bow-shock intensities agreed fairly well with, but were consistently less than, shock intensities estimated by theoretical methods . available flight data were examined for correlation with wind-tunnel test results .
the shock wave noise problem of supersonic aircraft in steady flight . data are presented which provide an insight into the nature of the shock-wave noise problem, the significant variables involved, and the manner in which airplane operation may be affected . flight-test data are also given, and a comparison with the available theory is made . an attempt is also made to correlate the subjective reactions of observers and some associated physical phenomena with the pressure amplitudes during full-scale flight . it is indicated that for the proposed supersonic transport airplanes of the future, booms on the ground will most probably be experienced during the major portion of the flight plan . the boom pressures will be most severe during the climb and descent phases of the flight plan . during the cruise phase of the flight, the boom pressures are of much lesser intensity but are spread laterally for many miles . the manner in which the airplane is operated appears to be significant,. for example, the boom pressures during the climb, cruise, and descent phases can be minimized by operating the airplane at its maximum altitude consistent with its performance capabilities .
an investigation of lifting effects on the intensity of sonic booms . this paper is a brief summary of an investigation made to check the effect of lift on the shock noise of aircraft flying at supersonic speeds . the method of hayes has been combined with the theory of whitham to predict the asymptotic shock strength of wings carrying lift and of combinations of bodies and lifting wings . (a similar, but not quite as general, method was derived by walkden in ref. 6 .) whitham’s formula, including only the volume term, has been used extensively to predict the boom intensity of aeroplane type bodies and the agreement with experiment has, so far, been quite reasonable . the test data obtained to date extends only up to about 40,000 ft. altitude and the calculations of this paper show that under those conditions the shock noise of the aircraft tested so far will, in most cases, be dominated by the volume term . it is shown that at higher altitudes lifting effects will dominate for even the small fighter and they will dominate over most of the altitude range for large bomber and supersonic transport aircraft . the boom intensity due to lift decreases with altitude as which compares to in the volume case (=pressure at altitude h) . it is insensitive to mach number, wing loading, wing plan shape and lift distribution . a simple rule for calculating the shock noise due to combined volume and lifting effects is proposed which is applicable to configurations with wings located towards the rear of the fuselage . the rule states that the shock noise of an aircraft carrying lift is equal to the shock noise due to volume (neglecting lift) or the shock noise due to lift (neglecting volume), whichever is the greater . a chart is presented from which rapid estimates can be made of the shock noise of lifting wing-body combinations .
investigation of two bluff shapes in axial free flight over a mach number range from 0. 35 to 2. 15 . a fineness-ratio-2.71 right circular cylinder and a fineness- ratio-been tested in free flight over a mach number range of 0.35 to 2.15 and a reynolds number range of 1 x 10 to 12 x 10 . time histories, cross plots of force coefficients, rolling velocity, and longitudinal-force coefficient are presented for both cylinders . in addition, cross plots of moment coefficients and plots of the normal-force curve slope and the aerodynamic center are presented for the fineness-ratio-2.71 cylinder . the average aerodynamic center of the right circular cylinder moved rearward with decreasing speeds until at the subsonic mach numbers it remained approximately constant and comparisons of the drag data of this test with wind tunnel and other free-flight data show good agreement . an appreciable decrease in drag was observed when the data of the present test of the rounded nose cylinder were compared with data of a right circular cylinder of a similar configuration .
the motion of rolling symmetrical missiles referred to a body-axis system . the linearized equations of motion have been derived for a rolling missile having slight aerodynamic asymmetries . time histories of rolling-missile motions referred to a body-axis system have been prepared to show the types of missile motions that can be encountered . the motions resulting from a trim change and a pulse-rocket disturbance are shown to be determined mainly by the ratio of rolling velocity to pitching frequency . finally, the derived equations are used in establishing a technique for the reduction of rolling-missile oscillation data . it is shown that the aerodynamic derivatives can be obtained from flight data if four accelerations are measured . the method is applied to the results obtained from a flight test of a missile configuration .
stability derivatives of cones at supersonic speeds . the aerodynamic stability derivatives due to pitching velocity and vertical acceleration are calculated by use of potential theory for circular cones traveling at supersonic speeds . the analysis is based on two theoretical techniques used successfully previously in application to the case of uniform axial and inclined flow . in the first, potential solutions for axial flow and crossflow are derived from the first-order wave equation but in application to calculations for the forces no approximations are made either to the tangency condition or to the isentropic pressure relation . the second method consists in combining the first-order crossflow potential with an axial-flow potential correct to second order . closed-form solutions by both methods are found for a cone, and numerical results for the stability derivatives are presented as a function of mach number for cones having semivertex angles of 10 and 20 . in addition, expressions for the forces, moments, and stability derivatives of arbitrary bodies of revolution are obtained using newtonian impact theory . numerical results for cones compare well with those obtained from the combined first- and second-order potential theory at the highest mach number for which the latter theory is applicable .
investigation of several blunt bodies to determine trans- onic aerodynamic characteristics including effects of spinning and of extendible afterbody flaps and some measurements of unsteady base pressures . several blunt bodies having shapes that may be suitable for atmospheric reentry vehicles were tested to determine the aerodynamic characteristics of such shapes for angles of attack up to 34 . the tests were conducted through the transonic mach number range and at reynolds numbers from 1.74 x 10 to 2.78 x 10, based on body diameter . a full-skirted rather than a short-skirted type of shape developed the greatest amount of static stability and the largest lift-curve slopes . the angle of attack for maximum lift for such bodies appears to be subject to mach number effects . spinning a full-skirted body about its longitudinal axis generally increased the lift and reduced the pitching moment at angles of attack and reduced the aerodynamic static stability parameter through the transonic mach number range . the extension of segmented clamshell-shaped flaps from the afterbody of a short-skirted model served to increase the lift and static stability only if the flaps extended into the airstream . some evidence was found of oscillatory base pressures on two dissimilar shapes at certain high angles of attack and the highest mach number in these tests . there is doubt, however, that these pressures can induce any significant oscillatory motion for a reentry vehicle because of their small amplitude and phasing .
experimental investigation at a mach number of 3. 11 of the lift, drag and pitching moment characteristics of a number of blunt low-fineness-ratio bodies . a number of blunt bodies having shapes that may be suitable for atmospheric reentry were tested to determine the lift, drag, and pitching-moment characteristics at a mach number of 3.11 and a reynolds number of 6 x 10 based on maximum body diameter of 2 inches . the results of the tests showed that all the bodies were statically stable about a point located one-third of the body length from the nose . the results also showed that high-drag bodies which have a large portion of their afterbodies negatively sloped (decrease in cross-sectional area from nose to base) may have a negative lift-curve slope . this negative slope results from the large negative lift component of the axial force obtained with those bodies and the fact that with negatively sloped afterbodies only small normal forces are developed .
loading paths and the incremental stress law . this paper will be concerned with some properties of the stress-strain law for work hardening materials introduced by w. prager incremental strain or plastic flow law by which it is meant that the differentials of strain are expressed as functions of the stresses, strains, and differentials of the stress . we shall also have occasion to refer to total strain or plastic deformation laws, in which the strains are given directly as functions of the stress .
a quantitative comparison of flow and deformation theories of plasticity . the stresses and displacements in a partly plastic, infinitely long, hollow cylinder are obtained according to the flow type of stress-strain law of prandtl-reuss and to the deformation law of hencky . in both cases the mises yield condition is used and the compressibility of the material is taken into account . it is shown that under these assumptions the two theories yield substantially the same results for this particular problem, but that one theory or the other may be preferable for computing purposes in certain cases . the results are compared with those of other investigations in which different combinations of stress-strain law, yield condition, compressibility, and end loading were assumed .
stresses in the plastic range around a normally loaded circular hole in an infinite sheet . the stresses in the plastic range around a normally loaded circular hole in an infinite sheet are found numerically on the basis of both deformation and incremental theories . the results of deformation theory are quantitatively assessed in the light of a criterion, recently developed by budiansky, for the acceptability of deformation theories . the criterion is completely satisfied . moreover, the results obtained by using these two different theories of plasticity do not differ greatly despite the fact that the stress paths are far from being radial .
theories of plastic buckling . the theory for the plastic buckling of columns which appears finally to have achieved a satisfactory form, rests upon the well-established uniaxial stress-strain relation . the development of a correspondingly satisfactory theory for the plastic buckling of plates has been hampered by the nonexistence of an established polyaxial stress-strain relation in the plastic range . present theories for the polyaxial stress-strain relation beyond the elastic range can be divided into two types, often called flow and deformation theories . theories of plastic buckling based on deformation theories are in better agreement with experiment than those based on flow theories . on the other hand, tests in which a material is compressed into the plastic range and then subjected to shear at constant compressive stress are in better agreement with flow than with deformation theories . legitimate doubt therefore has existed as to the validity of any theory for the plastic buckling of plates . as a result of studying these apparent contradictions, a new theory of plasticity has been developed which is of neither the flow nor the deformation type . it is based upon the concept of slip, and its formulation was guided more by physical, and less by mathematical, considerations than previous theories . experimental evidence of limited scope but of crucial character is in better agreement with the new theory than with either flow or deformation theories . the new theory accounts for the apparent contradictions previously alluded to and justifies the use of deformation theory in the analysis of the plastic buckling of plates .
inelastic column theory . the action of a column in the plastic range is analyzed on the basis that bending may proceed simultaneously with increasing axial load . this leads to a new column formula that includes both the tangent-modulus (engesser) and the reduced-modulus the tangent-modulus load and that the column load increases with increasing lateral deflection, approaching the reduced-modulus load as a limit if the tangent modulus is assumed to remain constant .
effects of imperfections on buckling of thin cylinders and columns under axial compression . von karman and tsien have shown that under elastic conditions the resistance of perfect thin cylinders subjected to axial compression drops precipitously after buckling . it is considered that this indicates that this type of buckling is very sensitive to imperfections or disturbances . in this paper the effects of certain imperfections of shape turbances combined) are studied by the large-deflection shell theory developed in a previous paper (2) . it is found that two types of buckling failure may occur . one is of a purely elastic type which occurs when the peak of the average stress versus average strain curve is reached, while the other type is precipitated by yielding, which for thicker cylinders or lower-yield material may occur before such a peak is reached . curves are derived giving the dependence of each type of failure upon the dimensions and elastic and yield properties of the specimen and also upon an /unevenness factor/ u which determines the magnitude of the initial imperfections and is assumed to depend on the method of fabrication . the relations derived are in line with test results, and similar studies of the buckling of struts indicate that the magnitude of the initial imperfections which have to be assumed to explain test strengths are reasonable .
plastic torsional buckling strength of cylinders including the effects of imperfections . the torsional buckling strength of a cylinder in the plastic range has been determined . an energy solution and a more exact solution, both based on a plastic stress-strain relationship given by the simple deformation theory, are presented . close agreement between the two solutions is found . the effects of large deflections and imperfections on buckling strength are analyzed . for two groups of experimental results used for comparison, the effects of geometrical imperfections in the plastic range are negligible . the theoretical results are found to be in good agreement with the experimental results .
on the concept of stability of inelastic systems . simple models are employed to bring out the large and important differences between buckling in the plastic range and classical elastic instability . static and kinetic criteria are compared and their interrelation discussed . nonlinear behavior in particular is often found to be the key to the physically valid solution . the nonconservative nature of plastic deformation in itself or in combination with the nonlinearity requires concepts not found in classical approaches . conversely, the classical linearized condition of neutral equilibrium is really not relevant in inelastic buckling . plastic buckling loads are not uniquely defined but cover a range of values and are often more properly thought of as maximum loads for some reasonable initial imperfection in geometry or dynamic disturbance . the models indicate that basically the same information is obtained from essentially static systems by assuming initial imperfection in geometric forms as by assuming dynamic disturbances . one approach complements the other and both are helpful in obtaining an understanding of the physical phenomena .
inelastic instability and incremental theories of plasticity . a most troublesome paradox has existed for a number of years with respect to buckling in the plastic range . theoretical considerations and all direct experimental evidence show conclusively that an incremental or flow type of mathematical theory of plasticity is valid . however, the results of plastic buckling tests are well correlated by a simple total or deformation theory and bear no resemblance to published predictions of incremental theory . the suggestion was made that initial imperfections of shape or loading might well explain this most peculiar result . however, subsequent investigations by several authors seem to have given the impression that excessively large imperfections would be needed and that the answer would be overly sensitive to the magnitude of such imperfections . it is the purpose of this paper to demonstrate that extremely small, and therefore unavoidable, imperfections of shape do account for the paradox in a simple manner . the buckling load is shown to be extremely insensitive to the amount of imperfection . the example chosen is a simplified version of the long rectangular plate hinged along one edge and free on the other under uniform compressive stress at the ends . this is the equivalent of the case of the cruciform column, which has been so disturbing in the past because incremental theory applied to a perfect cruciform column did lead to an entirely incorrect result .
small bending and stretching of sandwich type shells . a theory has been developed for small bending and stretching of sandwich-type shells . this theory is an extension of the known theory of homogeneous thin elastic shells . it was found that two effects are important in the present problem, which have not been considered previously in the theory of curved shells .. (1) the effect of transverse shear deformation and (2) the effect of transverse normal stress deformation . the first of these two effects has been known to be of importance in the theory of plates and beams . the second effect was found to occur in a manner which is typical for shells and has no counterpart in flat-plate theory . the general results of this report have been applied to the solution of problems concerning flat plates, circular rings, circular cylindrical shells, and spherical shells . in each case numerical examples have been given, illustrating the magnitude of the effects of transverse shear and normal stress deformation . the results of this investigation indicate the necessity of taking account of transverse shear and normal stress in sandwich-type shells, as soon as there is an order-of-magnitude difference between the elastic constants of the core layer and of the face layers of the composite shell . it was found that the changes due to transverse shear and normal stress deformation in the core may be so large as to be no mere corrections to the results of the theory without transverse core flexibility . the actual magnitude of the changes is greatly dependent on the geometry and loading condition of the structure under consideration so that no general rules may be given which indicate for which elastic modulus ratio the changes begin to be significant . solutions of problems in the present theory may in general be obtained by mathematical methods which are similar to those employed in the theory of plates and shells without the effect of transverse shear and normal stress deformation included . the present work does not include consideration of buckling and finite deflection effects .
a nonlinear theory of bending and buckling of thin elastic shallow spherical shells . a shallow spherical dome subjected to lateral pressure is a structure for which the deformation departs appreciably from the linear theory at relatively small values of the deflection amplitude . it is also one for which the buckling process is characterized by a rapid decrease in the equilibrium load once the buckling load has been surpassed . for structures having this type of buckling characteristics the question arises as to whether the proper buckling criterion to apply is the classical criterion, which considers equilibrium with respect to infinitesimal displacements, or the finite-displacement /energy criterion/ proposed by tsien . in this paper the problem of the finite displacement and buckling of a shallow spherical dome is investigated both theoretically and experimentally . in the theoretical approach the nonlinear equations are converted into a sequence of linear equations by expanding all of the variables in powers of the center deflection and then equating the coefficients of equal powers . the basic parameter for the shallow dome is proportional to the ratio of the central height of the dome h to its thickness t . for small values of this ratio the expansions converge rapidly and enough terms are computed to determine the buckling load according to the classical criterion . for higher values of h t, convergence deteriorates rapidly and it was not possible to determine the buckling load with the number of terms which were computed . however even for these higher values of h t the deflection shapes are determined for deflection amplitudes below the amplitude at which buckling occurs . these deflection shapes are characterized by their rapid change as h t increases and by the fact that, over most of the range of h t studied, the maximum deflection does not occur at the center of the dome . experimental results seem to indicate that the classical criterion of buckling is applicable to very shallow spherical domes for which the theoretical calculation was made . a transition to energy criterion for higher domes is also indicated .
stresses and small displacements of shallow spherical shells . the purpose of the present paper is to derive a system of equations which can be used for the analysis of shallow segments of thin, elastic, spherical shells . a segment will be called shallow if the ratio of its height to base diameter is less than, say . the results obtained on the basis of this assumption will often also be applicable to shells which are not shallow, namely then, when the loads are such that the stresses are effectively restricted to shallow zones . the problem of the spherical elastic shell has been the subject of numerous researches . for the rotationally symmetric case the fundamental results were obtained in 1912 (1) and have been the starting point of many applications . while it is possible to deduce from these results approximate equations equivalent to part of what follows, it is believed that the present approach to the problem of the shallow shell may be of some interest even for rotationally symmetric cases . a number of investigations have been concerned with the shell loaded in a non-rotationally symmetric manner (2,3,4) . in its general form this problem is quite difficult and the results so far obtained are not easy to apply . restricting attention to the shallow shell in the manner of the present paper brings with it a very considerable simplification of the analysis .
stability of thin-walled tubes under torsion . in this paper a theoretical solution is developed for the torsion on a round thin-walled tube for which the walls become unstable . the results of this theory are given by a few simple formulas and curves which cover all cases . the differential equations of equilibrium are derived in a simpler form than previously found, it being shown that many items can be neglected . the solution obtained is length ratio is zero and infinite, and is a good approximation for intermediate cases . the theory is compared with all available experiments, including about 50 tests made by the author . the experimental-failure torque is always smaller than the theoretical-buckling torque, averaging about 75 percent of it, with a minimum of 60 percent . as the form of the deflection checks closely with that predicted by theory and the experiments cover a great range of shapes and materials, this discrepancy can reasonably be ascribed largely to initial eccentricities in actual tubes .
nonlinear deflections of shallow spherical shells . the equations obtained by chien for the nonlinear deflection of shallow spherical shells under uniform external pressure are solved by means of power series expansions, following procedures introduced by friedrichs and stoker in their treatment of buckling of circular plates . these equations depend upon two parameters . one of these parameters is related to the external pressure, while the other depends upon the dimensions of the shell . the equations are solved for several ranges of the parameters under boundary conditions corresponding to a fixed edge . the solution, carried out numerically on the aec univac at new york university, yields a complete description of the stresses and deflections as functions of the polar angle over a wide range of values of the loading parameter and the dimensional parameter . prediction of the upper buckling load is then made by means of a numerical criterion based on the load vs. deflection curve . for some cases, the postbuckling behavior is investigated . the results agree well with existing experimental and theoretical studies and cover a wide range of cases not previously treated .
buckling of shallow shells under external pressure . a formula for the initial buckling loads for clamped, shallow spherical shells under uniform external pressure is obtained by combining the solutions of two linearized versions of the original nonlinear problem . one of these versions is a linear eigenvalue problem while the other is the bending problem for a shallow cap in the linear theory of elasticity . the formula, which is obtained in a simple manner, yields buckling loads that are in better agreement with experiments than previous approximate solutions to the nonlinear problem .
accelerating convergence of iteration processes . a technique is discussed which, when applied to an iterative procedure for the solution of an equation, accelerates the rate of convergence if the iteration converges and induces convergence if the iteration diverges . an illustrative example is given .
a simple method of matric structural analysis, part iv, non-linear problems . the method presented in the previous parts is employed to solve various kinds of nonlinear problems, such as problems concerning large deflections or buckling, or thermal creep, or inelastic stress redistribution involving thermal gradients, or design . the procedure used in each case is one of direct iteration–i. e., after one assumes a starting point all subsequent cycles are self-generating . simple numerical examples are worked out .
limit design for economical missile structures . a special safety factor alone won’t do the trick in the design of lightweight, high temperature missile structures . if you really want to end up with the most efficient structure you can get, an entirely new design approach is needed .
the problem of strain accumulation under thermal cycling . parkes and sprague and huang have shown that it is possible for strain growth to occur in a beam structure under temperature-load cycling . the various aspects of this problem as to criteria for convergence and divergence of the strain accumulation can be simply demonstrated by thermal cycling one element of a two-element structure .
analytical and experimental investigation of stress distributions in long flat plates subjected to lingitudinal loads and transverse temperature gradients . stress and strain distributions were studied in long flat plates in order to develop practical analytical procedures for the design analysis of aircraft structures at elevated temperatures . various load-temperature conditions are presented . these include .. methods of analysis for calculation of stress distributions under and plastic range,. ture . experimental verification of the analytical procedures is shown with comparisons between the use of constant room temperature or temperature dependent values of modulus of elasticity and coefficient of thermal expansion . the test specimen, equipment, instrumentation, and experimental program are discussed in detail . experimental data obtained from the specimen and associated material control coupon tests are presented .
inelastic behaviour of structures subjected to cyclic thermal and mechanical stressing conditions . a general analytical procedure is outlined for structures subjected to varying thermal and mechanical stressing conditions . consideration is given to the accumulation of time-independent plastic strains and creep strains . stress-strain-temperature-time relations for uniaxial and multiaxial stresses are defined, based on various material behavior assumptions . several of the assumptions are compared with a limited number of time-varying temperature and uniaxial stress tests . the procedure is illustrated by its application to uniaxial stress problems in which /planes originally plane remain plane/ and to plane stress plate problems . a solution, based on the influence coefficient approach to the plane stress plate problem, is obtained which is applicable to all plate plan forms, edge boundary conditions, and inplane thermal and mechanical loading conditions . from the predicted inelastic behavior of a three-bar structure subjected to cyclic thermal and mechanical loading conditions, it is shown that eventual failure could result from large permanent deformation accumulations, tensile rupture, or thermal-stress-fatigue . a sample plate with a centrally located hole was analyzed for two cycles of a time-varying temperature and edge stress condition . both plastic strain reversals and plastic strain growths were predicted at the hole . however, a test-theory comparison indicated failure by creep-rupture .
bending and compression tests of pressurised ring-stiffened cylinders . the results of tests on pressurized ring-stiffened cylinders subjected to compression and bending are presented and discussed . the results obtained at high values of internal pressure differ from those obtained by previous investigators in that the theoretical small-deflection compressive buckling coefficient of 0.6 was nearly achieved in each test . small amounts of internal pressure had a greater stabilizing effect in the bending tests than in the compression tests .
the bending stability of thin– walled unstiffened circular cylinders including the effects of internal pressure . in a recent paper, the authors presented a statistical, semiempirical design procedure for the determination of the buckling strength of unpressurized and pressurized cylinders under axial compression . this procedure has been extended in the present paper to the bending of unpressurized and pressurized cylindrical shells and allows the calculation of the critical bending stress with a knowledge of the cylinder geometry and the internal pressure only . because no published data could be found, an extensive series of bending tests of pressurized cylinders has been performed . these new data for pressurized cylinders are treated semiempirically together with all of the other known test data for unpressurized cylinders . best-fit curves are presented using applicable theoretical parameters . design curves for determining the critical buckling stress for unpressurized and pressurized cylinders in bending are then developed as 90 per cent probability curves from the test data .
analysis of partly wrinkled membrane . a theory is derived to predict the stresses and deformations of stretched-membrane structural components for loads under which part of the membrane wrinkles . rather than studying in detail the deformations in the wrinkled region, the present theory studies average displacements of the wrinkled material . specific solutions of problems in flat and curved membranes are presented . the results of these solutions show that membrane structures retain much of their stiffness at loads substantially above the load at which wrinkling first occurs .
on the bending of circular cylindrical shells under pure bending . the stability of circular cylindrical shells under pure bending is investigated by means of batdorf’s modified donnell’s equation and the galerkin method . the results of this investigation have shown that, contrary to the commonly accepted value, the maximum critical bending stress is for all practical purposes equal to the critical compressive stress .
an improvement on donnell’s approximation for thin-walled circular cylinders . donnell’s equation for thin-walled circular cylinders is replaced by where w is a non-dimensional form of the radial displacement and q is the distributed radial loading . this equation retains the essential simplicity of the original but, unlike donnell’s equation, the accuracy does not decrease as the wavelength of circumferential distortion increases .
a simplified method of elastic stability analysis for thin cylindrical shells . the equation for the equilibrium of cylindrical shells introduced by donnell in naca report no. 479 to find the critical stresses of cylinders in torsion is applied to find critical stresses for cylinders with simply supported edges under other loading conditions . it is shown that by this method solutions may be obtained very easily and the results in each case may be expressed in terms of two nondimensional parameters, one dependent on the critical stress and the other essentially determined by the geometry of the cylinder . the influence of boundary conditions related to edge displacements in the shell median surface is discussed . the accuracy of the solutions found is established by comparing them with previous theoretical solutions and with test results . the solutions to a number of problems concerned with buckling of cylinders with simply supported edges on the basis of a unified viewpoint are presented in a convenient form for practical use .
flexural vibrations of the walls of thin cylindrical shells having freely supported ends . the paper deals with the general equations for the vibration of thin cylinders and a theoretical and experimental investigation is made of the type of vibration usually associated with bells . the cylinders are supported in such a manner that the ends remain circular without directional restraint being imposed . it is found that the complexity of the mode of vibration bears little relation to the natural frequency,. for example, cylinders of very small thickness-diameter ratio, with length about equal to or less than the diameter, may have many of their higher frequencies associated with the simpler modes of vibration . the frequency equation which is derived by the energy method is based on strain relations given by timoshenko . in this approach, displacement equations are evolved which are comparable to those of love and flugge, though differences are evident due to the strain expressions used by each author . results are given for cylinders of various lengths, each with the same thickness-diameter ratio, and also for a very thin cylinder in which the simpler modes of vibration occur in the higher frequency range . it is shown that there are three possible natural frequencies for a particular nodal pattern, two of these normally occurring beyond the aural range .
the flexural vibrations of thin cylinders . the flexural vibrations of the walls of thin cylinders are considered . in this type of vibration many forms of nodal pattern may exist owing to the combination of circumferential and axial nodes . theoretical expressions are developed for the natural frequencies of cylinders with freely-supported and fixed ends and a comparison is made with the frequencies obtained experimentally . in practice, the ends of cylinders are subjected to a certain degree of fixing by end-plates, flanges, etc., and the natural frequencies thus lie between the corresponding values for freely-supported and fixed ends . to make possible the estimation of such frequencies, a method is devised in which an equivalent wavelength factor is used . this factor represents the wavelength of the freely-supported cylinder that would have the same frequency as the cylinder under consideration when vibrating in the same mode . the results of experimental investigations with various end thicknesses and flange dimensions are recorded, and from these the equivalent factors are derived . sets of curves calculated for cylinders with freely-supported ends and covering a range of cylinder thicknesses are given . from these it is possible to obtain close approximation to the frequencies of cylinders under other end conditions by the use of an appropriate factor . an example is given of frequency calculations for a large air-receiver for which two frequencies were identified by experiment .
on the vibration of thin cylindrical shells under internal pressure . the frequency spectra and vibration modes of thin-walled circular cylinders subjected to internal pressure are considered . it is shown that for very thin cylinders the internal pressure has a significant effect on the natural vibration characteristics . for these cylinders, particularly those having smaller length to diameter ratios, the mode associated with the lowest frequency is in general not the simplest mode . the exact number of circumferential nodes, n, which occur in the mode associated with the lowest frequency, depends on the internal pressure p . if this number n is large, it decreases rapidly with increasing p when p is small, and the /fundamental/ frequency–the lowest frequency at each p–increases rapidly with increasing internal pressure . at higher values of internal pressure the frequency spectrum tends to be arranged in the regular manner, the frequency increases with the increasing number of circumferential nodes, and the lowest frequency rises slowly with the internal pressure . experimental results on the frequency spectra, vibration modes, and structural damping of a series of thin-walled cylinders subjected to internal pressure are briefly described . these results show agreement with the features predicted by reissner’s the effect of slight deviation of the cylinder from perfect circular symmetry is discussed .
experimental study of the vibrations of a circular cylindrical shell . an apparatus is described which permits the mode shape of a vibrating circular cylindrical shell to be obtained quite easily . these measurements are made without contacting the cylinder and can be converted to actual lineal values . a representative number of results obtained with such a system are shown to illustrate the relationship between the nodal pattern and frequency in a cylinder as well as the effect of internal pressure on these frequencies . finally, comparisons are made between these results and timoshenko theory and an appropriate shell theory .
the effect of an internal compressible fluid column on the breathing vibrations of a thin pressurised cylindrical shell . the free oscillations of a thin pressurized cylindrical shell containing a compressible fluid are studied here . the use of an approximate set of shell equations (shallow shell theory) leads to a relatively simple formula for the natural frequencies of the coupled fluid-cylinder system . the results of some computations are presented .
a theory of imperfection for the vibrations of elastic bodies of revolution . various observations and preliminary experiments have shown that the effect of imperfections upon the vibrations of bodies of revolution cannot be neglected . owing to the possibility of applying the lagrange equation, the influence of the imperfections could be traced through the kinetic energy, the potential energy and the dissipation function . although the fundamental difficulty of the uncertainty of certain variables was not eliminated, this procedure permitted at least the making of general qualitative statements as to the behaviour of the system if imperfections are present .
remarks on donnell’s equations . flugge’s set of differential equations of equilibrium for circular cylindrical shells is expressed in a form analogous to the donnell equations . the results of solutions of the two sets of equations for a simply supported cylinder under a centrally applied, uniformly distributed radial line load over a generator segment, as well as under sinusoidally applied line loads, are in very good agreement for the particular geometry investigated .
energy expressions and differential equations for stress and displacement analysis of arbitrary cylindrical shells . energy expressions and the related equilibrium equations and natural boundary conditions for the determination of the stresses in and displacements of uniform, thin-walled cylinders of arbitrary cross section loaded in an arbitrary manner by surface and edge forces and moments are presented . the derivations are based upon the kirchhoff-love assumptions of the classical theory of shells and are performed to within a degree of accuracy employed by flugge in his derivation of the equilibrium equations applicable to circular cylindrical shells,. hence, in terms of stress resultants, the exact, small-deflection equilibrium equations are obtained . methods of simplification of the relations derived and of solution of the differential equations presented are indicated .
stress and displacement analysis of simply supported non- circular cylindrical shell under lateral pressure . this paper presents an analysis of the deflections of and stresses in a short noncircular cylindrical shell of uniform wall thickness whose median-surface cross section is described analytically by a simple expression corresponding to a family of doubly symmetric ovals . the cylinder is under a uniform lateral load and is simply supported at its edges . the small deflection analysis considered is based upon a series solution of appropriate differential equations of shell theory which leads ultimately to infinite sets of algebraic equations, truncated forms of which are considered . numerical values of the significant stresses and displacements for points of the oval cylinder, which are 5 percent of the axial length and 2.5 percent of the circumferential length apart, have been calculated for an oval cross section with a major-minor axis ratio of 1.10 .
the accuracy of donnell’s equations . solutions of donnell’s equations of the small deformations of the perfectly elastic thin-walled circular cylindrical shell are compared with those obtainable from flugge’s equations . the range of the basic parameters is found within which the two solutions are approximately equal .
boundary-value problems of the thin-walled circular cylinder . the homogeneous differential equations of donnell’s theory of thin cylindrical shells are integrated . expressions are obtained in closed form for the displacements, membrane stresses, moments, and shear forces .
simplified formulas for boundary value problems of the thin– walled circular cylinder . n. j. hoff has presented formulas which can be used in the solution of boundary-value problems of circular cylinders . the purpose of this note is to express these results in exact simplified form,. a more detailed investigation appears elsewhere . the notation will be that of hoff unless otherwise stated .
some experimental studies of panel flutter at mach 1 .3. experimental studies of panel flutter were conducted at a mach number of 1.3 to verify the existence of this phenomenon and to study the effects of some structural parameters on the flutter characteristics . thin rectangular metal plates were used in these studies and were mounted as a section of the tunnel wall . most of the data were obtained by using aluminum-alloy panels, although a few steel, magnesium, and brass panels were also used . different materials with various thicknesses and lengths were used to determine the effect of these parameters on panel flutter . the experimental program consisted of three phases .. panels clamped front and rear, and (3) buckled panels clamped on all four edges . panel flutter was obtained under controlled laboratory conditions and it was found that, at the flow conditions of these tests, increasing tensile forces were effective in eliminating flutter, as were shortening the panels or increasing the bending stiffness . no apparent systematic trends in the flutter modes or frequencies could be observed, and it is significant that the panel flutter sometimes involved higher modes and frequencies . the presence of a pressure differential between the two surfaces of a panel was observed to have a stabilizing effect . initially buckled panels were more susceptible to flutter than panels without buckling . buckled panels with all four edges clamped were much less prone to flutter than buckled panels clamped front and rear .
experimental studies of flutter of buckled rectangular panels at mach numbers from 1. 2 to 3. 0 including effects of pressure differential and of panel width-length ratio . experimental panel flutter data have been obtained at mach numbers from 1.2 to 3.0 for buckled rectangular panels and the effect of a pressure differential has been determined . increasing the pressure differential was effective in eliminating flutter on most of the panels tested . the effects of the variables in the panel flutter parameter sure, e is young’s modulus, and t and l are the panel thickness and length, respectively) were investigated for buckled panels clamped on the front and rear edges and a critical value of this parameter of 0.44 is indicated at zero pressure differential when the panel width-length ratio is 0.69 . an estimated flutter boundary is presented for buckled panels clamped on four edges, with width-length ratios of 0.21 to 4.0 . this boundary shows that the panel width is more significant than the panel length when the ratio of width to length is less than approximately 0.5 . panels clamped on four edges and buckled in two half waves in the direction of flow were found to be particularly susceptible to flutter . the results of limited tests on panels with applied damping, curvature, and lengthwise stiffeners are also presented and discussed .
experimental investigation at mach numbers 3. 0 of the effects of thermal stress and buckling on the flutter of four-bay aluminium alloy panels with length-width ratios of 10 . skin-stiffener aluminum alloy panels consisting of four bays, each bay having a length-width ratio of 10, were tested at a mach number of 3.0 at dynamic pressures ranging from 1,500 psf to 5,000 psf and at stagnation temperatures from 300 f to 655 f . the panels were restrained by the supporting structure in such a manner that partial thermal expansion of the skins could occur in both the longitudinal and lateral directions . a boundary faired through the experimental flutter points consisted of a flat-panel portion, a buckled-panel portion, and a transition point at the intersection of the two boundaries . in the region where a panel must be flat when flutter occurs, an increase in panel skin temperature (or midplane compressive stress) makes the panel more susceptible to flutter . in the region where a panel must be buckled when flutter occurs, the flutter trend is reversed . this reversal in trend is attributed to the panel postbuckling behavior .
flutter of aerodynamically heated aluminium-alloy and stainless steel panels with length-width ratio of 10 at mach 3. 0. an investigation of the effects of aerodynamic heating on the flutter of multibay external-skin panels has been carried out at a mach number of 3.0 in the langley 9- by 6-foot thermal structures tunnel . both aluminum-alloy and 17-7 ph stainless-steel panels with a length-width ratio of 10 for each bay were tested at dynamic pressures between addition, a few tests were made on the lower vertical stabilizer of the x-15 airplane which has external-skin panels unsupported for a length all panels showed flutter boundaries characterized by an increase in panel thickness required to prevent flutter with increasing thermally induced stress prior to buckling . after buckling the panels showed flutter boundaries characterized by a decrease in thickness required to prevent flutter with further increases in thermal stress . the largest thickness required to prevent flutter in the presence of aerodynamic heating occurred at the transition between the flat-panel boundary and the buckled-panel boundary . this peak value (for aluminum-alloy panel) was as much as 60 percent greater than the extrapolated value for an unheated, unloaded panel . values of the modified-thickness-ratio flutter parameter for the unstressed panels (obtained by extrapolation) were in fair agreement for the aluminum, steel, and x-15 stabilizer panels . peak values at transition, however, showed large differences due to apparently minor changes in panel-support construction and or changes in panel-skin material .
test of an aerodynamically heated multi– web wing structure (mw-1) in a free jet at mach number 2. a multiweb wing structure, representing an airplane or missile wing, was tested under simulated supersonic flight conditions to determine the transient temperature distribution . the aerodynamic loads played an important and unanticipated role, however, in that the model experienced a dynamic failure near the end of the test . the test is discussed and the conclusion reached that the model failed as a result of the combined action of aerodynamic heating and loading . the temperature data collected are analyzed and are shown to be in reasonable agreement with calculated values .
charts adapted from van driest’s turbulent flat-plate theory for determining values of turbulent aerodynamic friction and heat transfer coefficients . a modified method of van driest’s flat-plate theory for turbulent boundary layer has been found to simplify the calculation of local skin-friction coefficients which, in turn, have made it possible to obtain through reynolds analogy theoretical turbulent heat-transfer coefficients in the form of stanton number . a general formula is given and charts are presented from which the modified method can be solved for mach numbers 1.0 to 12.0, temperature ratios 0.2 to 6.0, and reynolds numbers 0.2 x 10 to 200 x 10 .
the phenomenon of change in buckle pattern in elastic structures . a model is analyzed which exhibits the important properties associated with change in buckle pattern of plates . the analysis includes a rigorous study of stability in its various modes . a discussion of how the present results may be applied to plates and other elastic structures is given .
loads and deformations of buckled rectangular plates . the nonlinear large deflection equations of von karman for plates are converted into a set of linear equations by expanding the displacements into a power series in terms of an arbitrary parameter . the post-buckling behavior of simply supported rectangular plates subjected to longitudinal compression and to a uniform temperature rise is investigated in detail by solving the first few of the equations . experimental data are presented for the compression problem . comparisons are made for total shortening and local strains and deflections which indicate good agreement between experimental and theoretical results .
status of flutter of flat and curved panels . representative results are presented to show the current status of the panel flutter problem . the discussion includes flat panels with and without midplane stresses, buckled panels, and both unstiffened and stiffened infinitely long circular cylinders .
a study of the thermal fatigue behaviour of metals . the effect of test conditions on nickel-base high temperature alloys . an attempt has been made to identify the significant factors governing the thermal-fatigue behaviour of nickel-base high-temperature alloys, mainly by using a laboratory technique with hot and cold fluidized beds as the heating and cooling media . a succession of heating shocks is generally more damaging than a succession of cooling shocks between the same temperature limits . the duration of the heating shock and the upper temperature of the cycle are dominant factors . the thermal-fatigue cracks are initiated at the surface and are intercrystalline in origin and propagation . surface oxidation, which is intergranular in nature for nickel-base alloys, has a significant effect on thermal-fatigue life .
regularities in creep and hot fatigue data . published experimental results are assembled to support a previously-given theory of uniaxial deformation, and the theory is then used to analyse published data on the creep-rupture and hot-fatigue of engineering materials . the theory enables data for different times and temperatures to be classed together, thereby providing information over a much greater range of times than could practicably be covered by experiments at a single temperature . an underlying numerical pattern common to all the widely different group 8 materials considered then shows through the experimental scatter . data for further engineering materials is considered in these terms in part 2 .
low frequency fatigue of nimonic 90 . low frequency fatigue – a rheological approach . an alloy of nimonic 90 type has been tested under cyclic loads at temperatures of 800 deg., 850 deg. and 900 deg. cent . graham’s descriptive theory of deformation has been developed in detail for the tests reported here, and shown to provide a satisfactory description of the experimental results . the variation of cumulative strain with number of cycles can be described by the sum of powers of cycle number, n, n, n, and n . there is qualitative agreement between the observed and the calculated stress-strain loops . the effect of variation of maximum stress per cycle can be described by the sum of power terms with simple exponents . within the scatter of observation, the total time to fracture is independent of the frequency .
design and operation of the n. g. t. e. thermal shock analogue . a description is given of the n.g.t.e. thermal shock analogue which is suitable for estimating the temperature in a turbine blade section as a function of position and time when the blade is subjected to a step change in gas temperature . the method of operating the analogue and obtaining results has also been described . the limitations of the analogue have been stated, but they are considered a small penalty in view of the essential simplicity of the design .
the calculation of transient temperature in turbine blades and tapered discs using biot’s variational method . transient temperatures in aerofoil sections and tapered discs are calculated taking advantage of simplifications in heat flow analysis achieved in biot’s variational method . cross-sections are represented by a line of adjacent squares of various sizes suitable for the local dimensions, e.g. small squares near the leading and trailing edges . the potential, dissipation and surface dissipation functions of biot’s method are set up, and the lagrange equations lead, by automatic procedures, to an eigenvalue formulation in matrix form for the temperatures and their first time derivatives . solutions are sums of exponentials in time, and are evaluated by digital computer, requiring about five minutes for each cross-section and heat transfer coefficient . transient temperatures in a particular aerofoil section for variation of heat transfer coefficient and for external temperature depending exponentially on time agree with results obtained on an analogue computer . maximum transient temperature differences are evaluated for tapered discs by a simple electrical analogue) with variation of edge radius and heat transfer coefficient . peculiarities in the solution for cyclic temperature external to an aerofoil over a range of frequencies indicate limitations in the mathematical formulation . a successful solution for cyclic external temperature might enable eigenvalues to be separated out in experimental measurements using electronic equipment, and this might be extended to exponential external temperature if a relationship between cyclic and exponential external temperature could be established . eigenvalues and eigenvectors as discrete values arise fictitiously from the sub-division into squares and the possibility of an integral formulation is mentioned . there is a possible, but not immediate, extension to cooled blades, whose cross-sections are multiply-connected regions . transient stresses due to creep, and viscoelasticity might be included .
effect of rheological behaviour on thermal stresses . since the conventional elastic analysis of thermal stress problems coupled with limiting creep rates and time-dependent fracture stresses as (inelastic) design criteria, results in design procedures for thermal stresses (in heat exchangers, nuclear reactors, flight structures at supersonic speeds, etc.) of considerable unreality, the effect of various types of rheological behavior (viscoelastic, plastic, work hardening) on the level of thermal stresses is analyzed under simplified assumptions, such as uniaxial stress and polar or cylindrical symmetry . the effect on the thermal stress intensity of the rheological behavior of the material is shown to be very significant, particularly with respect to stress relaxation and the development of residual stresses .
steady-state creep through dislocation climb . a dislocation climb creep model is considered which does not require the production of immobile dislocations . the creep equation that results from the analysis is where a and b are constants, is the stress, q is the activation energy of creep and kt has its usual meaning . this equation is quite similar to one previously proposed .
fundamentals of boundary layer heat transfer with streamwise temperature variations . boundary-layer heat transfer is analyzed for the case of a sinusoidal distribution of temperature in the direction of flow . it is shown that for both laminar and turbulent flow the spatial distribution of heat transfer is generally out of phase with the wall temperature by an angle of 30 to 45 . this leads to the conclusion that in some areas the heat flow is opposite to the temperature difference as used in the definition of the heat-transfer coefficient, and points to the basic shortcomings of this concept . the physical explanation for this behavior is found to be the temperature-field distortion by the fluid motion . the distortion is measured by the peclet number . approximate equations representing a /conduction analogy/ were used in this analysis and the validity of these equations for unsteady flow is examined with reference to limitations in frequency and wavelength . a solution of these equations is given for the case of a velocity profile which is not a straight line . the use of previously developed variational principles for the evaluation of convective heat transfer including cases of three-dimensional unsteady flow, turbulence, and nonparallel streamlines is also discussed .
lagrangian thermodynamics of heat transfer in systems including fluid motion . the lagrangian thermodynamic equations of irreversible processes are extended to convective heat transfer . this generalization provides equations for the unified analysis of transient heat flow in complex systems comprising solid structures and moving fluids in either laminar or turbulent flow . the concept of a surface-heat-transfer coefficient is eliminated from the formulation . the theory is developed along two different lines . in one approach a new concept referred to as the /trailing function/ is introduced . it represents the surface-heat-transfer properties and may be evaluated by quite simple but remarkably accurate variational procedures . the method of /associated fields/ is also generalized to convective phenomena . the second line of approach extends to convective heat transfer the thermodynamic concept of entropy production for both laminar and turbulent flow . the theory amounts to an extension of the thermodynamics of irreversible processes to systems for which onsager’s relations are not valid .
the use of models for the determination of critical flutter speeds . the use of model tests in the prediction of full-scale critical flutter speeds is now well established, and the technique of such tests is therefore worthy of discussion . in order to obtain critical speeds for the model within the speed range of ordinary wind tunnels it is necessary that the model should differ in some respect from a mere small suggested by mckinnon wood the modification of the model consists in a reduction of its effective stiffnesses . this method has the defect /in most cases probably not serious/ that the model experiment is conducted at a reynolds number much below that for full-scale . in the present paper it is pointed out that an alternative method of reducing the critical speed is to increase the mass loading of the model and to make the flutter tests in compressed air . * it is then quite feasible to reach the full-scale reynolds number . this method of reducing the critical speeds by a proportionate increase of all effective densities may also be combined with a reduction of the elasticity of the model . the relation of model and full-scale stresses at the critical flutter speeds is considered . where the reduction in critical speed is effected by increase of density only, the model and full-scale stresses are equal . in a model of reduced elasticity the stresses in the wires are the same as for full-scale, whereas, the stresses in the spars are less than for full-scale . this is in accord with the usual experience that the wires of such a model are the first parts of the structure to fail in a flutter . lastly, the influence of gravity on flutter is considered . this is negligibly small for full-scale, but not necessarily so for the model . gravitational effects can sometimes be corrected by suitable orientation of the model .
models for aeroelastic investigation . this addendum provides a short note on two aspects omitted from the original paper, viz. gravitational effects and structural damping . a short list of references to earlier papers dealing with the subject is also added .
on flutter testing in high speed wind tunnels . the requirements for simulating in a wind tunnel flutter conditions appropriate to high-speed flight are discussed, and an assessment is made of the desirable features of a wind tunnel suitable for flutter testing at transonic and supersonic speeds . it is concluded that such a tunnel should have either the mach number or the stagnation pressure variable during the tunnel run, and that it is of considerable advantage, and for some purposes essential, for high stagnation pressures to be available . the stagnation pressure required to allow flight conditions to be simulated with a flutter model is considered to range from at least 2 atmospheres for transonic speeds to about 15 atmospheres for m = 4 . no attempt to simulate kinetic heating is envisaged, although its effect on stiffness should be allowed for in the design of the model . to minimise uncertainties due to the variation of the model stiffness with temperature it is desirable that means for controlling the stagnation temperature should be incorporated in the tunnel .
the influence of aerodynamic heating on the flexural rigidity of a thin wing . this report considers the loss of flexural rigidity of a thin wing due to the presence of middle-surface stresses resulting from aerodynamic heating . the spanwise properties of the wing are assumed constant but the wing section is arbitrary . the loss of flexural rigidity is comparable with the corresponding loss of torsional rigidity .
experimental model techniques and equipment for flutter investigations . an outline is given of the uses of flutter models as an aid to the designer in the avoidance of flutter . details are given of the different types and methods of construction that are used for flutter models and of the various test facilities that are available for high speed and low speed tests . the procedure followed in the u.k. for flutter clearance of the full scale aircraft is described, and the value of the electronic flutter simulator in this field is discussed .
flutter model testing at transonic speeds . flutter research on reflection plane models of straight, swept, and delta wings in a 3 x 4 foot transonic test facility . techniques of model construction and testing developed .
the design and testing of supersonic flutter models . the basic problems of flutter testing in the low supersonic speed range simulate full-scale airplanes when mach number is included as a parameter are reviewed and are compared with those where velocity is scaled when the leading edges become transonic,. these are compared with results given elsewhere and serve as a check on the results of tables i and ii .
cumulative damage in fatigue . the phenomenon of cumulative damage under repeated loads was assumed to be related to the net work absorbed by a specimen . the number of loading cycles applied expressed as a percentage of the number to failure at a given stress level would be the proportion of useful life expended . when the total damage, as defined by this concept, reached 100 per cent, the fatigue specimen should fail . experimental verification of this concept for an aluminum alloy, using different types of specimens, various stress ratios, and various combinations of loading cycles is presented . these data are also analyzed to provide information on different stress ratios when an s-n curve for any one ratio is known . results of a sample analysis based on experiments are given . it is concluded that a simple and conservative analysis is possible using the concept of cumulative fatigue damage .
the variation of gust frequency with gust velocity and altitude . information on atmospheric turbulence obtained from counting accelerometer records is examined and relations giving the variation of gust frequency with gust velocity and altitude are obtained . the results are summarized in a form convenient for use in estimating the fatigue life of an aircraft .
correlated fatigue data for aircraft structural joints . results of fatigue tests carried out at r.a.e. on typical aircraft wing structural joints are correlated to give an indication of general fatigue behaviour . the results are plotted in the form of s – log n curves, and these indicate that the mode of behaviour cannot be attributed to any single factor, such as the type of aluminium alloy, the ultimate tensile strength, or the mean stress of the fatigue cycle . the detailed method of design undoubtedly has a predominant influence on behaviour, but this quality is not revealed by a broad classification according to the proportion of load transmitted at holes .
the estimation of fatigue damage on structural elements . a method is presented for the estimation of fatigue damage to aircraft structural elements . the gust spectrum to which the aircraft is subjected is analysed in terms of infinitesimal loading intervals . gust data supplied by j. taylor for flying below 15,000 ft are used to study fatigue damage in a number of aircraft structural joints and one whole structure .
buckling of thin cylindrical shells under hoop stresses varying in axial direction . the buckling of a thin cylindrical shell simply supported along the perimeter of its end sections is analyzed under hoop compressive stresses varying in the axial direction . the thermal stresses arising from a uniform increase in the temperature of the cylinder are determined . it is found that such thermal stresses are not likely to cause elastic buckling . simple approximate formulas are developed for buckling stress and thermal stress .
thermal buckling of clamped cylindrical shells . the problem of thermal buckling of shells arises in connection with air-frame bodies subject to aerodynamic heating at supersonic speeds . the case of the shell with clamped edges is presented, as this case typifies all structures with a tubular shell stiffened at intervals with stiffening rings . the shell is assumed to be unrestrained longitudinally and fully restrained laterally at the edges .
buckling due to thermal stress of cylindrical shells subjected to axial temperature distributions . thermal stress distributions in uniform circular cylindrical shells due to axial temperature distributions are investigated . the discontinuity effect due to the presence of a cooler stiffening bulkhead is considered, and the possibility of thermal buckling of the shell due to the circumferential discontinuity stress is examined . the buckling analysis is based on donnell’s shell equation, and particular attention is given to shells having clamped edges . an experimental investigation of this buckling problem is discussed, and the results obtained are seen to agree reasonably well with theory .
combinations of temperature and axial compression required for buckling of a ring-stiffened cylinder . a theory is presented to predict the buckling temperature of an axially compressed, uniformly heated ring-stiffened cylinder . the cylinder buckles because of the interaction of the axial stress due to applied compressive loads and the circumferential stress resulting from restraint of thermal expansion by the rings . buckling charts covering a wide range of cylinder proportions are presented for both clamped and simply supported cylinders . the buckling temperature for a given axial loading is determined from a simple equation involving a coefficient given in the buckling charts and the radius-thickness ratio of the cylinder .
a simplified method of elastic stability analysis for thin cylindrical shells . this paper develops a new method for determining the buckling stresses of cylindrical shells under various loading conditions . for convenience of exposition, it is divided into two parts . in part 1, the equation for the equilibrium of cylindrical shells introduced by donnell in naca report no. 479 to find the critical stresses of cylinders in torsion is applied to find critical stresses for cylinders with simply supported edges under other loading conditions . it is shown that by this method solutions may be obtained very easily and the results in each case may be expressed in terms of two nondimensional parameters, one dependent on the critical stress and the other essentially determined by the geometry of the cylinder . the influence of boundary conditions related to edge displacements in the shell median surface is discussed . the accuracy of the solutions found is established by comparing them with previous theoretical solutions and with test results . the solutions to a number of problems concerned with buckling of cylinders with simply supported edges on the basis of a unified viewpoint are presented in a convenient form for practical use . in part 2, a modified form of donnell’s equation for the equilibrium of thin cylindrical shells is derived which is equivalent to donnell’s equation but has certain advantages in physical interpretation and in ease of solution, particularly in the case of shells having clamped edges . the solution of this modified equation by means of trigonometric series and its application to a number of problems concerned with the shear buckling stresses of cylindrical shells are discussed . the question of implicit boundary conditions also is considered .
comments on ‘thermal buckling of clamped cylindrical shells’ . in the recent paper by zuk, an expression was presented for the critical buckling temperature of a clamped cylindrical shell in terms of the material and geometrical properties of the shell . restraint at the edges of the shell was assumed to be provided by rigid frames experiencing no temperature rise . the circumferential stress induced in the shell when it experienced a temperature rise, t, may be approximated by the function . in other words, there is a compressive circumferential stress along the entire length, l, of the shell . it is well known, however, that the discontinuity stresses introduced at the junction of a shell and a rigid frame (or bulk-head) are extremely localized, and the circumferential stresses induced in the shell decrease rapidly away from the joint .
buckling of a finite length cylindrical shell under a circumferential band of pressure . this paper is concerned with buckling of a circular cylinder of finite length subjected to a symmetrical band of external pressure . both experimental and theoretical results are presented . the experimental data were obtained from tests of three thin-walled steel cylinders subjected to external pressure by a pneumatic tube encircling the test cylinder at mid-length . the theory is based on the principle of minimum potential energy, and the rayleigh-ritz procedure is used to expand the displacement components in trigonometric series . theoretical results are given in the form of graphs which show buckling pressure as a function of the following ratios .. cylinder radius thickness cylinder length radius pressure bandwidth cylinder length theoretical results are in close agreement with existing solutions to special cases in which (1) the pressure is applied over the entire lateral surface, and (2) the pressure is concentrated along a circumferential line . the theoretical results are also in agreement with the test results .
research on unsteady flow . this is a survey of certain recent advances made in the study of aerodynamic unsteady flow and of some of the new problems arising which require further investigation . no attempt is made to reproduce classical theory, but emphasis rather is laid on validity and general usefulness, particular attention being given to unsteady boundary-layer effects, especially when there is flow separation . coverage is broad and author thus provides a useful review for those interested in this field .
a new design of pitot-static tube with a discussion of pitot-static tubes and their calibration factors . the report describes experiments devised to investigate some of the previously unexplained peculiarities of normal types of pitot-static tube . in the process use was made of what was as nearly as could be a standard static pressure tube . the experiments led to a new alternative design of instrument having a nose of modified ellipsoidal shape and for which the main characteristics have been investigated . it has been found to be necessary as well as desirable to include a discussion of the implications of the term /calibration factor/ and (although not in this respect comprehensive) of the special features and limitations of various pitot-static tubes .
flutter of a two dimensional simply supported buckled panel with elastic restraint against edge displacement . the critical flutter speed is evaluated for a two-dimensional thin buckled panel with one surface exposed to a supersonic airstream and the other to still air at the same static pressure . the panel is simply supported along the leading and trailing edges by rigid edge members separated by an elastic member represented by a compression spring . the whole system is acted upon by a constant compressive force uniformly distributed along the edge members . the aerodynamic forces acting on the deflected panel are found from two-dimensional /quasi-steady/ theory, valid for slow oscillations where the downwash velocity is small compared with the speed of flow and provided that the mach number is sufficiently greater than . the elastic behaviour of the panel is given by von karman’s large deflection equations modified to cover initially curved plates . the solution of the equations is carried out by means of galerkin’s method, which has been shown to give valid results for a panel with a non-zero bending rigidity . the influence of the midplane compressive force carried by the panel itself, the initial buckle amplitude and the elastic restraint against edge displacements is investigated, and curves are presented giving the critical dynamic pressure ratio as a function of these variables .
the airforces on the low aspect ratio rectangular wing oscillating in sonic flow . approximate expressions for the generalised airforces acting on a rectangular wing of low aspect ratio oscillating harmonically in sonic flow at low frequencies are derived in this paper . the modes of oscillation considered are rigid modes and a small selection of flexible modes . results are presented as the first few terms of infinite expansions . a brief description of the modes of oscillation and of the generalised airforces is given towards the end of the paper so that the results may be used without the main text of the paper having to be read .
the calculation of loads on a supersonic weapon in the steady circling case . the economy of a design depends on the accuracy with which it is possible to solve the various structural problems as this has a direct bearing on structure weight . this paper describes the calculation of the bending moments on a specific weapon for the high-g steady circling case . a hybrid method is used to obtain the aerodynamic loads . the results presented show the effect of a number of parameters, such as – altitude, weight, acceleration and mach number – on the magnitude of the maximum bending moment .
some results on buckling and postbuckling of cylindrical shells . in this summary paper, the effects of initial deformations on the buckling and postbuckling characteristics of circular cylindrical shells under hydrostatic pressure is determined in an approximate manner . the influence of initial axisymmetric deformations is stressed . also, the classical buckling of an axially compressed, noncircular the results show that the major-minor axis ratio of the cross section has a marked effect on the critical load, and that use of the maximum radius of curvature in the formula for the classical buckling stress of a circular cylindrical shell leads to good results for moderate eccentricities .
a survey of buckling theory and experiment for circular conical shells of constant thickness . a survey of the state-of-the-art for the stability of thin-walled conical shells is presented . known theoretical results are summarized and compared with experiment . the shortcomings of present knowledge and recommended work for the future are discussed .
aerodynamic effects on boundary layer unsteadiness . with a view to the study of aerodynamic problems, a review is made of boundary layer theory for a flat plate moving with a time-dependent velocity . unsteady effects are shown to enter according to the magnitude of the ratio of time for diffusion to act throughout the boundary layer to the characteristic time of the imposed unsteadiness . it is concluded that a boundary layer may be considered quasi-steady even during extreme flight manocuvres . generation of acoustic noise purely by boundary layer unsteadiness is generally small . thermal and heat-transfer effects are cited . unsteady boundary layer considerations are important in damping or amplifying certain instabilities, such as flutter of panels and stalling flutter of aerofoils . in connection with the aerofoil problem, laminar separation concepts and the stagnation-point boundary layer are described for unsteady flow . an analysis of aerofoil lift hysteresis is described, using unsteady laminar boundary layer considerations, which leads to a prediction of counter-clockwise hysteresis at maximum lift .
some measurements in the vortex flow generated by a sharp leading edge having 65 sweep . the report is concerned with the vortex flow which arises when separation occurs at a highly swept leading edge . measurements were made in the flow over flat plates at 15 incidence each having a sharp leading edge of 65 sweep . the pressure and velocity distributions both along the axis of the vortex and for one cross section of the flow are presented together with a preliminary discussion of their significance .
long slender delta wings with leading edge separation . the slender-body approximation of linearized compressible flow is applied to the problem of a delta wing in which flow separation occurs at the leading edges . the vortex sheets found in the real flow are approximated by concentrated vortices with feeding lattices, and a plausible adaptation of kelvin’s theorem is applied to simulate the force-free nature of the vortex sheet . the computations show that leading-edge separation produces an increase in lift over that given by the jones slender-wing theory and that the lift does not vary linearly with angle of attack . computed pressure distributions and span loadings are presented and the theoretical lift results are compared with the results of simple force tests made at a mach number of 1.9 .
some current and proposed investigations into the flow for slender delta and other wings in unsteady motion . the practical need for research into the aerodynamics of slender delta wings in unsteady motion has been emphasized in a recent paper by zbrozek . two important aspects are .. – formation and presence of leading-edge vortices . with oscillatory or transient modes of longitudinal (or chordwise) bending . the first of the aspects above, has already been briefly discussed in ref. 2 . one feature of the flow with leading-edge vortices which seems to be of particular significance to the dynamic behaviour of a wing is the shedding of vorticity at the leading edge as well as at the trailing edge . any time-dependent motion, or distortion, of the wing leads to a change in the rate at which vorticity is shed . with more conventional types of flow, the free vorticity being shed only from the trailing edge has diminishing influence on the wing, but when the free vorticity is shed from the leading edge, in passing downstream, it remains close to the upper surface of the wing . it might be expected then, that, although the magnitudes of the unsteady forces may not be greatly affected for a slender delta, the time delays associated with the forces may be significantly different for the attached and separated regimes of leading-edge flow .
two dimensional transonic unsteady flow with shock waves . a study is made of the unsteady flow around an airfoil at transonic mach numbers, the situation being such that local supersonic regions terminated by shock-waves are present in the vicinity of the airfoil . for the unsteady part of the flow, small perturbations technique is employed and the interaction with the shock wave is taken into account . the case of an oscillating aileron is considered first, and a solution is derived for the pressure distribution on the aileron . it is found that the solution has a simple form when the shock-wave is well ahead of the hinge axis of the aileron . as the shock approaches the hinge-axis a correction must be added to the solution . an interpretation of these results is given . the results are compared with results of a theory which neglects the presence of the shock and it is found that both agree for m = 1 . for m 1, however, neglecting the presence of the shock waves introduces errors of the order of magnitude (1 – m), where m is the local mach number behind the shock . the theory is finally extended to include the case in which the whole airfoil oscillates, but only the solution for the subsonic region behind the shock is treated . the role of the unsteady shock-boundary layer interaction is discussed and it is shown that this mechanism can be included in the results of the present theory .
calibration of the standard pitot-static head used in the rae low speed wind tunnels . recent results of tests in the r.a.e. wind tunnels concerned with the measurement of pressure distributions have shown slight discrepancies between the readings of various static pressure tubes and calculated pressure distributions . as a consequence some doubt was felt concerning the calibrations of tunnel static pressure and upon the validity of the reading given by the standard pitot-static head . it was therefore decided to check the standard pitot-static head used in the r.a.e. wind tunnels, against an instrument similar to the measurements of static pressure were also made using a long tube where the interference from head and support is calculated to be small . this note gives the results of tests made in the 5 ft open jet wind tunnel and the no. 1 11 ft wind tunnel in order to determine the necessary correction to the reading of static pressure given by the r.a.e. pitot-static head . the tests were made during september and october, 1951 .
comparative tests of pitot-static tubes . comparative tests were made on seven conventional pitot-static tubes to determine their static, dynamic, and resultant errors . the effect of varying the dynamic opening, static openings, wall thickness, and inner-tube diameter was investigated . pressure-distribution measurements showing stem and tip effects were also made . a tentative design for a standard pitot-static tube for use in measuring air velocity is submitted . this report covers an investigation conducted under the auspices of the national research council .
review of the pitot tube . this paper is an attempt to bring together the important information regarding pitot tubes and their use,. to summarize the available data on the application of various types of impact and velocity probes for the guidance of engineers and research workers,. and to aid them in the design of flow instruments for specific applications .
cavitation and pressure distribution head forms at zero angle of yaw . early in the fall of 1943 the iowa institute of hydraulic research undertook the design and fabrication of a variable-pressure water tunnel . as the tunnel neared completion, however, its immediate use for the study of the pressure distribution around various body forms was requested . the original request for this investigation was a natural out-growth of the need for systematic data on the distribution of pressure in flow around various bodies, particularly under conditions leading to cavitation, information which is desirable for the design of a wide variety of navy equipment . ultimately the study is to include data for two- and three-dimensional head and tail forms at various angles of yaw . the first phase of the study, namely the investigation of three-dimensional head forms at zero angle of yaw, is described herein . three general geometric series have been studied.dash rounded, ellipsoidal, and conical.dash together with other related forms . the data obtained have been systematized to yield information for a wide variety of geometrical forms either directly or by interpolation . whenever possible, analytical methods have been used to corroborate the experimental data and to provide a reliable means of generalizing the results .
random vibration . random vibration is vibration which results from an excitation which is not well represented by any simple function (sinusoid, step, etc.) or any simple combination of such functions but which is satisfactorily modeled by a stochastic process . it is perhaps not too much of an exaggeration to say that /all vibration is random vibration ./ every vibration record contains /hash/ at some level . nevertheless, until recently, engineering vibration theory has been able to get along without including the consideration of random excitations . now in several fields simultaneously there has occurred a burst of activity in the application of random processes . the response of aircraft to buffeting from atmospheric turbulence and the response of ships to confused seas have been put on reasonably firm footing . possibly the most dramatic problems have been posed by the development of large jet and rocket engines which produce spectacular amounts of random vibrational energy . the high level of random vibration in a jet plane or a missile provides a severe environment with respect to fatigue failure of structural members and with respect to malfunctions of sensitive equipment .
the effect of jet noise on aircraft structures . the present state of knowledge on the problem of fatigue failure due to vibrations excited by jet noise is reviewed . it is concluded that it should currently be possible to make reasonable estimates of the stress levels set up in a structure by jet noise but, in general, the resultant fatigue life of the components cannot be estimated with any confidence .
natural frequencies of continuous beams of uniform span length . a simple graphical network is used to determine the natural frequencies of flexural vibration of continuous beams having any number of spans of uniform length . the network is based upon a relatively few calculated values .
experimental study of the random vibrations of an aircraft structure excited by jet noise . recordings have been made of the strains induced in a full scale rear fuselage test structure of the caravelle air-liner when one jet engine is running at maximum take-off thrust . the analysis has been concentrated on the strains in the centres of panels . correlation measurements indicate that the larger panel strains occur above resonance peak in each panel has been identified with the fundamental stringer-twisting mode but the mode-shapes for the two smaller peaks have not been completely determined . an attempt has been made to calculate the panel resonant frequencies theoretically .
the axisymmetric free-convection temperature field along a vertical thin cylinder . with a view to studying the effect of strong transverse curvature on boundary-layer problems, the axisymmetric free- convection problem along a vertical thin cylinder is investigated theoretically as well as experimentally . a theory is developed as an extension of the pohlhausen solution of a thick axisymmetric laminar boundary layer by mark and by glauert and lighthill . experiments consist of a thermocouple survey of the temperature field over an electrically-heated brass cylinder of diameter and 10 ft. height and an interferometric study of the density field over a bare tungsten wire of 0.02-in. diameter and 5 ft. height . the thermal-layer thicknesses are about five and fifty times the radii of the cylinders, respectively . experimental results of the local heat-transfer coefficient are in excellent agreement with the theory . this, in turn, justifies the theories of laminar boundary layer along a thin cylinder, at least indirectly .
vibrations of beams on many supports . the natural frequencies of a continuous beam resting on an arbitrary number of uniformly spaced supports are determined from a difference equation formulation . these frequencies fall in periodically spaced groups that are separated by spectral gaps of widths equal to approximately half the interval between the natural frequencies of a single beam on a square root frequency scale . these groups tend to uniform spectra as the number of supports tends to infinity, but the gaps remain, giving a band-pass character to the entire spectrum . wave propagation along an infinite, periodically supported beam is discussed and the phase and group velocities evaluated as functions of frequency .
transtability flutter of supersonic aircraft panels . for certain aero-elastic configurations it is possible to ascertain critical flutter conditions from static considerations alone . the idea is simply one of negation.. when the air speed exceeds a certain value statically stable equilibrium – and sometimes equilibrium itself take place . there are times when the dynamics of a situation are complex enough to defy a tractable analysis . the value of being able to indicate a flutter criterion from the simpler statics is clear . we will suppose flutter begins when some critical value of the air speed (or some parameter simply related the to) is exceeded . here we will show that there is a critical value which, when exceeded, precludes static equilibrium . underlying our work is the premise that these two critical values are the same . this assumption begs discussion . we will call the lowest value of our air speed parameter to preclude statically stable equilibrium of the system the transtability value . in some cases, excess of this value will ban all possibility of static equilibrium – stable or not., we will then call it a strong transtability value .
a buckled plate in a supersonic stream . the forcible buckling of an external skin plate from a guided missile, with a pinned end, is considered in a supersonic air flow . conidtions of dynamic stability orthogonality and expansion of the buckling mode, the degree of freedom and cases of small deflection are considered . without a 50 half angle conical afterbody in a pressurized ballistic of range at nominal mach numbers of 3.5 and of 90000 and 220000, respectively . it
the flow around oscillating low aspect ratio wings at transonic speeds . when certain conditions are fulfilled for thickness ratio, aspect ratio, and reduced frequency for a three-dimensional wing, it can be shown that the partial differential equation for the non-steady perturbation potential can be reduced to a comparatively simple linear equation . the solution is then obtained by applying a fourier transformation in the free-stream direction and then using an iterative process developed by adams and sears for steady flow . the method gives solutions valid for low combinations of aspect ratio and reduced frequency . the method is applied to a delta wing oscillating in some selected rigid and elastic modes . from the results it can be seen that the special non-steady forces in the potential equation, which are neglected in slender-body theory, are very important . stability derivatives can also be obtained by the method and it is seen that the damping in pitch may be negative at m 1 for delta wings of too high aspect ratio .
a method of calculating the short period longitudinal stability derivatives of a wing in linearised unsteady compressible flow . a method is developed for the calculation of the pressure distribution and the aerodynamic forces and moments on a wing performing harmonic pitching and heaving oscillations . the calculation is based on the assumption of inviscid potential flow without shock waves and is restricted to small incidence, so that the linearized theory is valid . in contrast to other work in the field the theory applies to all mach numbers . it is restricted to small values of the reduced frequency and should be valid for the usual range of short periods occurring at present in flight . the formal solution yields two integral equations for the parts of the load, which are in phase and go out of phase with the oscillation,. these are of the same form as the corresponding equation in steady flow . the way is thus opened for solutions over the whole mach number range at small frequencies, if the corresponding steady solutions can be found . the calculation is in fact easiest for m = 1 and has been done here for delta-wings to supplement a previous supersonic calculation, made on different frequency assumptions, which broke down near m = 1 . it appears from the two sets of results that the short period oscillation will be unstable near m = 1, if the apex angle of the delta wing is greater than about 60 . this confirms a now generally recognised trend . such results near m = 1 must of course be invalidated to an unknown extent by thickness viscosity and shock waves at their maximum effect . nevertheless it is unlikely that these factors will remove the critical nature of the transonic damping as calculated by this method . with all its obvious limitations this method, when extended to other planforms, should provide a useful tool in studying the effect of geometrical parameters on the stability of an aircraft at transonic speeds .
on the low aspect ratio oscillating rectangular wing in supersonic flow . the laplace transform of the lift distribution on an oscillating rectangular wing in a supersonic flow is obtained by separating the linearised equation for the velocity potential in elliptic (cylindrical) co-ordinates . the results for the case of no spanwise distortion are expanded in ascending powers of the aspect ratio in order to compare with the slender body theory, and the longitudinal stability derivatives are calculated . it is found that at either supersonic or transonic speeds single-degree- of-freedom instability in pitch is impossible insofar as the fourth power of the aspect ratio is neglected .
theoretical studies of unsteady transonic flow . part iii . the oscillating low aspect ratio rectangular wing . by expanding the velocity potential in an asymptotic series, the aerodynamic forces on an oscillating low aspect ratio rectangular wing are calculated . the approximate theory is valid for small values of ko /o semi-span-to-chord ratio,. k reduced frequency/ and complements an earlier low-aspect-ratio-wing theory by the author valid only for pointed wings like delta wings . the present report gives formulas for the calculation of generalized forces for any smooth, flexible or rigid mode of oscillation with spanwise symmetry . comparisons with the slender-wing theory show that, except for wings of very low aspect ratio, unsteady-flow effects are appreciable even at fairly low reduced frequencies . near the upper limit in ko for the applicability of the present theory good agreement is obtained with a recent theory for high aspect ratios .
supersonic flow over an inclined wing of zero aspect ratio . an asymptotic expression is found for the lift distribution on a long, narrow, laminar wing, at incidence in a supersonic stream . the approximations of the linearized potential theory are used .
slender-body theory-review and extension . the approximate theory of flow about slender bodies and wings originated by munk and jones is reviewed . it is presented here in a form that emphasizes the relation to the source-sink methods of von karman and others . the extension to noncircular bodies is made for subsonic flow, paralleling ward’s extension for supersonic flow . the calculation of pressures and forces and the extension of the theory to unsteady flows are reviewed, and some discrepancies in the published literature are explained . finally, interpreting the jones slender-wing result as the first term of an expansion in powers of a breadth parameter /e.g., aspect ratio/, it is shown how a more accurate theory can be developed by carrying additional terms for both subsonic and supersonic speeds . this theory of not-so-slender wings is applied to some practical wing problems, including direct problems of flow past given wings and problems of wing design for minimum drag . the accuracy of the new results is assessed by comparison with linearized supersonic-airfoil theory for the special case of a flat delta wing .
supersonic flow past slender bodies of revolution, slope of whose median section is discontinuous . the theory of supersonic flow around slender bodies of revolution, yawed or unyawed, with pointed or open bows, based on the linearized equation, is extended to the case when the meridian section of the outer surface has discontinuities in slope . expressions for the pressure distribution on the surface are obtained . it is found that the drag coefficient is no longer independent of mach number, and tends to zero more slowly than the square of the thickness of the body . the large pressure change behind a discontinuity is made up remarkably rapidly . the first approximation to the lift coefficient is unchanged .
methods for estimating lift interference of wing-body combinations at supersonic speeds . the modified slender-body method used by nielsen, katzen, and tang in rm a50f06, 1950, to predict the lift and moment interference of triangular wing-body combinations has been adapted to combinations with other than triangular wings . that part of the method for predicting the effect of the body on the wing has been retained, but a new method for predicting the effect of the wing on the body has been presented . these methods have been applied to the prediction of the lift-curve slopes of nearly 100 triangular, rectangular, and trapezoidal wing-body configurations . the estimated and experimental values for the lift-curve slopes agree for most of the cases within 10 percent . some of the higher-order effects that must be taken into account in a theory that is to give greater accuracy than the present one are discussed . a numerical example illustrating the method is included .
a method for calculating the lift and centre of pressure of wing-body-tail combinations at subsonic, transonic speeds . a method is presented for calculating the lift and pitching-moment characteristics of circular cylindrical bodies in combination with triangular, rectangular, or trapezoidal wings or tails through the subsonic, transonic, and supersonic speed ranges . the method covers unbanked wings, sweptback leading edges or sweptforward trailing edges, low angles of attack, and the effects of wing and tail incidence . the wing-body interference is handled by the method presented in naca rm’s a51j04 and a52b06, and the wing-tail interference is treated by assuming one completely rolled-up vortex per wing panel and evaluating the tail load by strip theory . a computing table and set of design charts are presented which reduce the calculations to routine operations . comparison is made between the estimated and experimental characteristics for a large number of wing-body and wing-body-tail combinations . generally speaking, the lifts were estimated to within 10 percent and the centers of pressure were estimated to within effect of wing deflection on wing-tail interference at supersonic speeds was not correctly predicted for triangular wings with supersonic leading edges .
factors affecting loads at hypersonic speeds . this paper gives a brief summary of current loads information at hypersonic speeds . several methods which the designer can employ in estimating the loads on various aircraft components are discussed . the paper deals with the characteristics of both slender and blunt configurations and touches upon the effects of boundary-layer and aerodynamic interference .
post buckling behaviour of circular cylinderical shells under hydrostatic pressure . the postbuckling behavior of initially perfect, thin-walled, circular cylindrical shells under hydrostatic pressure is investigated with the aid of the principle of stationary potential energy together with appropriate approximate deflection functions . calculations show that postbuckling equilibrium configurations exist for loads greater than as well as loads slightly less than the critical load calculated from small-deflection theory . loads less than the critical load are obtained only for a finite range of a parameter indicative of shell geometry . for loads corresponding to radial displacements of the order of the shell thickness, it is found that the number of circumferential waves remain essentially constant with increasing deflection and equal to the number of waves developed at buckling .
investigation of normal force distributions and wake vortex characteristics of bodies of revolution at supersonic speeds . the supersonic aerodynamic characteristics of inclined bodies of revolution at high angles of attack have been investigated in order to provide a more basic understanding of the body vortex wake flow and its relation to the problem of body-wing interference . the results of wind-tunnel tests, whereby the normal force, pitching moment, normal force distributions, and the local flow properties in the vicinity of the body were determined, are discussed and analyzed . comparisons of experimental normal force coefficient and center of pressure data with values calculated in accordance with theories which include methods for estimating the effects of viscosity show that the accuracy of these estimates is strongly dependent on the body fineness ratio and the angle of attack . further comparisons of the distributions of theoretical and experimentally derived cross-flow drag coefficients clearly show that, in general, the disagreement between experiment and existing theories is due to the inadequate prediction of the magnitude and distribution of the forces resulting from flow separation . the circulation strengths of the concentrated vortices and the circulation strengths of the vortex feeding sheets in the body vortex wake are determined by closed-contour velocity-perimeter integrations for paths enclosing the vortex or the feeding sheet . the values of vortex strength calculated in this manner are in close agreement with the values predicted by vortex strength formulas written for a simple theoretical model for which it is assumed that the cross-flow in any plane along the cylindrical portion of the body is represented by the steady incompressible potential flow about a cylinder, two symmetrical vortices of equal strength, and the attendant image vortices . however, in computing these strengths it is necessary to use the vortex locations and the viscous normal force distributions determined from experiment . the experimentally determined values of vortex strength are, in turn, used to calculate–by means of the aforementioned incompressible cross-flow potential–the local flow inclination angles which are in good agreement with the measured values, except in the vortex core, in the vicinity of the feeding sheet, and in regions for which transonic cross-flow velocities are expected . a consideration of these various regions with simple methods which account for the observed phenomena leads to substantial improvement in the agreement between theory and experiment . it is indicated that the complete vortex wake flow may be adequately predicted for a body of revolution (for conditions represented by the theoretical flow model), provided that the distribution of the viscous normal force and the vortex locations are accurately known .
a new theory for the buckling of thin cylinders under axial compression and bending . the results of experiments on axial loading of cylindrical shells (thin enough to buckle below the elastic limit and too short to buckle as euler columns) are not in good agreement with previous theories, which have been based on the assumptions of perfect initial shape and infinitesimal deflections . experimental failure stresses range from 0.6 to 0.15 of the theoretical . the discrepancy is apparently considerably greater for brass and mild-steel specimens than for duralumin and increases with the radius- thickness ratio . there is an equally great discrepancy between observed and predicted shapes of buckling deflections . in this paper an approximate large-deflection theory is developed, which permits initial eccentricities or deviations from cylindrical shape to be considered . true instability is, of course, impossible under such conditions,. the stress distribution is no longer uniform, and it is assumed that final failure takes place when the maximum stress reaches the yield point . the effect of initial eccentricities and of large deflections is much greater than for the case of simple struts . measurements of initial eccentricities in actual cylinders have not been made,. however, it is shown that most of these discrepancies can be explained if the initial deviations from cylindrical form are assumed to be resolved into a double harmonic series, and if certain reasonable assumptions are made as to the magnitudes of these components of the deviations . with these assumptions the failing stress is found to be a function of the yield point as well as of the modulus of elasticity and the radius-thickness ratio . on the basis of this a tentative design formula (5) is proposel, which involves relations suggested by the theory but is based on experimental data . it is shown that similar discrepancies between experiments and previous theories on the buckling of thin cylinders in pure bending can be reasonably explained on the same basis, and that the maximum bending stress can be taken as about 1.4 times the values given by equation buckling problems can probably be explained by similar considerations, and it is hoped that this discussion may help to open a new field in the study of buckling problems . the large-deflection theory developed in the paper should be useful in exploring this field, and may be used in other applications as well . the paper presents the results of about a hundred new tests of thin cylinders in axial compression and bending, which, together with numerous tests by lundquist, form the experimental evidence for the conclusions arrived at .
stability of the cylindrical shell of variable curvature . the report is a first attempt to devise a calculation method for representing the buckling behavior of cylindrical shells of variable curvature . the problem occurs, for instance, in dimensioning wing noses, the stability behavior of which is decisively influenced by the variability of curvature . the calculation is made possible by simplifying the stability equations (permissible for the shell of small curvature) and by assuming that the curvature as a function of the arc length s can be represented by a very few fourier terms . we evaluated the formulas for the special case of an ellipse-like half oval with an axis ratio under compression in longitudinal direction, shear, and a combination of shear and compression . however, the results can also be applied approximately to an unsymmetrical oval-shell segment under compression, shear, and bending so that the numerical values contained in the diagrams 10 to 12 represent directly dimensioning data for the wing nose .
general theory of large deflections of thin shells with special applications to conical shells . a general theory is developed for the case of large deflections but with rotations of the elements negligible compared to unity . the derivation is carried out in tensor form and therefore any coordinate system on the surface of the shell can be used . the effect of initial imperfections is included . it is shown that for shells of negligible gaussian curvature (shallow shells and developable surfaces), the problem can be reduced to the solution of two fourth-order partial differential equations in a stress function and the deflection normal to the shell . for shells forming a surface of revolution the results are indicated in terms of the equation of the generating curve . the differential equations for the conical shell are then listed .
stability equations for conical shells . the author rewrites v. s. vlasov’s equations for (linear) stability theory of shells (prikl. mat. meh. 8(1944), 109-placements . the result is a single eighth-order equation for normal deflection, and two fourth-order equations relating the displacement components in the shell middle surface to the normal displacement .
buckling of circular cones under axial compression . presented are the results of an experimental investigation to determine the buckling strength of right circular cones under axial compression . correlation of these data is made with existing theory and with previously published experimental data on circular cylinders,. thus a recommended procedure for predicting the buckling load of right circular cones under the foregoing loading condition is presented .
the characteristics of roughness from insects as observed for two-dimensional, incompressible flow past airfoils . advances in the practical development of boundary-layer control for the maintenance of extensive laminar flow have drawn attention to the problem of surface roughness, due not only to artificial irregularities such as rivet heads, lap joints, window panels, etc., but also to the kind generated in flight from impact with insects . this natural form of roughening, the effects of which have been noted, though not investigated previously, is the subject of the present paper . the phenomenon may be divided into two parts–namely, and (2) its effect upon the stability of the laminar boundary layer . wind-tunnel experiments with the fruit fly, drosophila, and the common housefly for the investigation of both (1) and airfoils are fully described . the former problem has also been treated mathematically in a separate paper, not yet published, agreement between theory and experiment being satisfactory in all essentials . the characteristics of the roughness profile consist principally of a pronounced peak near the leading edge, followed by an extensive area of surface over which there is a much reduced and gradually diminishing value of the excrescence height . further, it is shown that, if the severe leading-edge roughness, or its effect upon the boundary layer, can be eliminated, then the down-stream roughness causes no disturbance to the passage of a laminar layer–i.e., the surface, though roughened, is aerodynamically smooth . moreover, it appears that the conditions defining the upstream boundary to this region of insignificant roughness are fundamentally the same as those which determine the critical state for transition at an artificial disturbance of a three- dimensional character .
stability of cylindrical and conical shells of circular cross section, with simultaneous action of axial compression and external normal pressure . we consider in this report the determination of the upper limit of critical loads in the case of simultaneous action of a compressive force, uniformly distributed over plane cross sections, and of isotropic external normal pressure on cylindrical or conical shells of circular cross section . as a starting point we use the differential equations for neutral equilibrium of conical shells (ref. 1) which have been used for the solution of the problem of stability of conical shells under torsion and under axial compression (ref. 2),. upon solution of the problem it is possible to satisfy all boundary conditions, in contrast to the report (ref. 3) where no attention is paid to the fulfillment of the boundary conditions and to the report (ref. 4) where only part of the boundary conditions are satisfied by solution of the problem according to galerkin’s method . approximate formulas are used for the determination of the critical external normal pressure with simultaneous action of longitudinal compression . let us note that the formulas suggested in reference 5 are not well founded and may lead, in a number of cases, to a substantial mistake in the magnitude of the critical load .
buckling of thin single- and multi-layer conical and cylindrical shells with rotationally symmetric stresses . the buckling of simply supported, thin, single- and multi-layer conical shells under axially symmetrical loading is analyzed in this paper . the results are presented in a compact manner so that they may be easily used for design and/or experimental purposes . the results are compared with known experimental values .
a donnell-type theory for asymmetrical bending and buckling of thin conical shells . equations, somewhat more accurate than those recently presented by n. j. hoff, are derived for bending and buckling of thin circular conical shells under arbitrary loading . these equations reduce to donnell’s equations for thin cylindrical shells when the cone semivertex angle becomes very small and the minimum radius of curvature of the median surface approaches a constant value . at the other end of the scale the equations reduce to the well-known equations for flat circular plates when the cone semivertex angle approaches a right angle . in addition, for the entire range of cone semivertex angles the equations reduce to the known equations for axisymmetrical bending when variations of the displacements around the circumference vanish . the problem of bending is reduced to the solution of a single fourth-order partial differential equation with variable coefficients .
on the buckling of truncated conical shells in torsion . the problem of the buckling of thin circular conical frustums in pure torsion is solved in a manner similar to that employed previously by the author for buckling under uniform hydrostatic pressure . synthesis of the numerical results indicates that the critical torsion of a truncated cone is equal to that of an equivalent cylinder whose length and thickness are the axial length and wall thickness of the cone and whose radius is a function of the semivertex angle and the taper ratio of the cone . curves and equations to aid in the analysis of conical frustums are given . it is shown that a previous recommendation for the analysis of truncated cones in torsion may be seriously unconservative in some cases .
calculations for the stability of thin conical frustums subjected to external uniform hydrostatic pressure and axial loads . calculations are presented for the problem of the stability of conical shells subjected to combined external uniform hydrostatic pressure and axial tension or compression . stress interaction curves are found to vary only slightly as a function of the ratio of the end radii of the cone .
some explicit solutions for constant-temp . magnetogas dynamic channel flow . in order to simplify the process of estimating the aerodynamic loading on the after portions of slender vehicles, it is frequently assumed that there is no nose-tail interaction . it is the purpose of this note to show that, aside from boundary-layer effects, this assumption is not warranted when the nose hypersonic-similarity parameter, tan, is of the order of unity, or greater . physically speaking, the entropy change associated with a strong bow wave reduces the stagnation pressure down-stream of the shock, and hence, lowers the dynamic pressure in the vicinity of the tail .
of a turbulent free shear layer . the problem of predicting the mean velocity on streamlines through the pre-asymptotic turbulent free shear layer in two-dimensional incompressible flow is resolved into two parts . the linearized momentum equation in terms of a generalized axial co-ordinate is solved in the usual way . a relation between and the distance from the separation point is then established analytically in contrast to the previous use of empirical expressions . it is shown that except in the region close to separation the velocity on the streamlines can be predicted by the simple approximation proposed by kirk .
viscous compressible and incompressible flow in slender channels . an analytical study is made of viscous flow in slender channels . similar solutions to the approximate equations of motion, valid for flow at moderate or high reynolds numbers in slender channels, are found for incompressible two-dimensional and axisymmetric flows and for compressible flows through two-dimensional channels with adiabatic walls . a study of compressible flows in convergent-divergent channels yields results regarding the effect of viscosity on the location of the sonic line, on the pressure ratio at the geometric throat and on the discharge coefficient for such channels .
secondary gas injection in a conical rocket nozzle . data are presented on side forces generated by secondary gas injection in a 15 conical rocket exhaust nozzle . the side force was measured directly with a force transducer and the data examined in terms of an amplification factor, where is the measured effective specific impulse of injectant, and is the specific impulse of injectant for sonic flow into a vacuum . injection was normal to the axis of the nozzle through a single circular orifice at a fixed point in the diverging portion of the nozzle . a variety of ambient temperature gaseous injectants and orifice diameters were carefully studied . injectant flow rate was varied for each configuration . the main propellant was hot gas ( catalytically decomposed), and motor conditions were held essentially constant .
compressible free shear layer with finite initial thickness . the momentum equation was uncoupled from the other conservation equations for the case of a finite initial profile in a laminar free shear layer . the equation was solved numerically, in the crocco coordinate system, using an implicit finite difference method . profiles of velocity and shear function were obtained as a function of streamwise distance . the initial profiles as the flow separates from the rear of the body correspond to the blasius profile in transformed coordinates . for large distances downstream, the profiles approach the chapman distribution, corresponding to the case of zero initial free shear layer thickness . the effect of these results on calculations of base pressure and wake angle is discussed . a method for the calculation of finite chemical kinetic effects on the profiles of temperature and chemical composition in the free shear layer with finite initial thickness is outlined .
one dimensional heat conduction through the skin of a vehicle upon entering a planetary atmosphere at constant velocity and entry angle . closed-form solutions of the one-dimensional heat-conduction equations for the flow of heat into a plate with a laminar boundary layer have been obtained for a configuration entering a planetary atmosphere with constant velocity and negative entry angle . the atmospheric density was assumed to obey an exponential law and the temperature was assumed constant initially . the solution is in the form of a fourier series expansion which, for most practical applications, can be approximated by retaining only one term of the expression . the solution applies to the initial part of the entry before the maximum heating conditions are encountered .
method for design of pump impellers using a high speed digital computer . a method of designing pump impellers is derived from the equations of motion and continuity for incompressible nonviscous relative flow . the flow is assumed to follow a known stream surface (representing blade shape) that extends from hub to shroud . equations are also derived for approximate blade-surface velocities and pressures . a detailed numerical procedure and block diagram are given for use on a digital computer . a numerical example that illustrates limited use of the method is presented and further uses are indicated .
exploratory investigation of the effect of a forward facing jet on the bow shock of a blunt body in a mach number 6 free stream . the effect of a forward-facing jet on the bow shock of a blunt body in a mach 6 free stream was investigated experimentally . the models tested had forward-facing jets using air and helium exhausting at mach numbers from 1 to 10.3 and were run through a range of the ratio of jet total pressure to free-stream total pressure of 0.03 (jet off) to 2.5 . the ratio of body diameter to jet-exit diameter varied from 1.12 to 55.6 and the angle of attack was varied from 0 to 35 . the experimental results show that the main-stream shock can be affected by the jet in two significantly different ways . one way is simply to move the strong shock away from the body without altering its shape . the second and perhaps more interesting case occurs when the jet causes a large displacement of the main shock and considerably changes its shape . it was found that the ratio of jet total pressure to free-stream total pressure necessary to obtain the large displacements of the main-stream shock depended on the ratio of body diameter to jet-exit diameter and also on the jet-exit mach number . the maximum amount the shock could be displaced in percent of body diameter was seen to increase with increasing jet-exit mach number and also with decreasing ratio of body diameter to jet-exit diameter . for the models that were investigated through an angle-of-attack range, the displacement became very unsteady and fell off sharply as the angle of attack was increased . simplified theoretical considerations applied to the shock-displacement phenomena provide a possible explanation for the two different types of main-stream shock displacement . theoretical curves show the regions where these types of displacement would occur for different exit mach numbers and pressure ratios for a forward-facing jet in a mach 6 stream .
static aerodynamic characteristics of a short blunt 10 semi-vertex angle cone at a mach number of 15 in helium . axial force, normal force, pitching moment, and shock-wave shape were determined for a body of revolution consisting of a short blunt 10 semivertex angle cone with a flat base and also with a conical afterbody having a semi-vertex angle of 50 . measurements were made in helium at a free-stream mach number of 15 and a free-stream reynolds number of 2.25×10 based on maximum body diameter over an angle-of-attack range from the configuration with the conical afterbody was statically stable in the nose-forward attitude only, whereas the configuration with no afterbody was statically stable in both the nose-forward and base-forward attitudes . the force and moment data of both shapes were predicted reasonably well by modified newtonian theory at all angles of attack, except the pitching-moment coefficient for the model without afterbody near 180 angle of attack . in this region, measurements indicated static stability, whereas theory indicated static instability . the helium data agreed reasonably well with a limited amount of force and moment data obtained in a ballistic range at small angles of attack in air at a mach number of 15 and also with force and moment data obtained in air over a complete angle-of-attack range at a mach number of 5.5 . the value of axial-force coefficient and the shape of the bow shock wave at zero angle of attack for both models obtained from a numerical flow field calculation agreed very well with the data . the value of the axial force coefficient at 180 angle of attack for the model with afterbody agreed reasonably well with the theoretical value for a cone . the position and shape of the shock envelope near the stagnation point also could be predicted accurately by an approximate method over an angle-of-attack range from
panel flutter tests on full scale x-15 lower vertical stabilizer at mach number of 3. 0. panel flutter tests were conducted on two full-scale vertical stabilizers of the x-15 airplane at a mach number of 3.0 in the langley at dynamic pressures from 1,500 psf to 5,000 psf and stagnation temperatures from 300 f to 660 f . flutter boundaries were obtained for four of the five distinct types of panels which make up the vertical sides of the stabilizers . the boundaries consisted of a flat-panel boundary and a thermally buckled-panel boundary . the flat-panel boundaries were characterized by a reduction in dynamic pressure with increasing skin temperature,. whereas, after thermal buckling the trend was reversed . the minimum dynamic pressure for flutter occurred at the intersection of the flat-panel and buckled-panel boundaries and represented a large reduction in the dynamic pressure over the extrapolated, unstressed value . as a result of panel flutter, three of the five distinct types of panels were modified to provide the required flutter margin on the design flight dynamic pressure of the aircraft .
charts for equilibrium flow properties of air in hyper-velocity nozzles . for initial stagnation pressures up to 1,000 atmospheres and stagnation enthalpies up to 10,000 btu per pound, nozzle-flow properties for equilibrium air have been computed and plotted on charts . the work of nasa tn d-693 has been extended to include flow properties for closer intervals of specified stagnation enthalpies . properties which have been charted as a function of mach number are as follows .. temperature, pressure, density, velocity, area ratio, dynamic pressure, reynolds number, isentropic exponent, and molecular weight ratio . ratios of temperature, pressure, and density across normal shock waves are also charted, and weight-flow rate is plotted as a function of stagnation enthalpy .
comparison of theoretical and experimental creep buckling times of initially straight, centrally loaded columns . the creep-buckling times of initially straight, centrally loaded columns as predicted by the hypotheses of shanley, gerard, and rabotnov and shesterikov are compared with appropriate experimental data . it is found that the theoretical predictions are generally conservative, due possibly to the fact that the hypotheses predict initial instability times while the experiments normally record final collapse times . of the three hypotheses, that of gerard generally gives predictions which agree best with the experimental data .
a unified theory of creep buckling under normal loads . a general theory of creep buckling, with the initial imperfection as a parameter, is developed for the case of normal loading . a hyperbolic-sine law is used to describe the process of creep . the theory is believed to be applicable to, among other structures, columns, tubes, and possibly conical shells . the wall of the structure is idealized as a sandwich in order to simplify the integration of the equations . experimental data on columns and tubes, from two different sources, are compared with the predictions of the theory .
study of creep collapse of a long circular cylindrical shell under various distributed force systems . an analysis is presented for determining the collapse of circular rings and long cylinders subjected to primary and secondary creep conditions at elevated temperatures . the types of loading considered for the present investigation are dead loading and hydrostatic pressure-type forces . the method of solution is based on an application of the variational theorem for creep described in ref. 1 with some additional terms being introduced for the pressure-type loading case . the general results are reduced to a relatively simple form for the theoretical predictions of collapse time and are graphically illustrated for a typical sample material .
vibrations of infinitely long cylindrical shells under initial stress . the general bending theory of shells under the influence of initial stress presented recently by herrmann and armenakas is applied in this investigation to study the effect of initial uniform circumferential stress, uniform bending moment and uniform radial shear on the dynamic response of an infinitely long cylindrical shell .
analysis of stress at several junctions in pressurized shells . theoretical and experimental results are presented for the discontinuity stresses arising at a change of wall thickness in a cylinder, a cylinder-hemisphere junction, and a cone-spherical torus junction in pressure vessels . the effect of mismatch of nonconcurrence of the middle surfaces of two joined cylinders is considered . in addition, a cylinder with a special closure which has considerably reduced stresses is described, and curves with theoretical and experimental stresses are presented .
the membrane approach to bending instability of pressureized cylindrical shells . recent theoretical and experimental research is briefly described to trace the development of deformation and the occurrence of collapse in pressurized circular cylindrical membranes under applied moment loading . the collapse of pure membrane cylinders is then compared with instability of pressurized cylindrical shells . this approach leads to a better understanding of the behavior of pressurized cylinders under bending loads . the results suggest possibilities for further research utilizing the membrane approach .
elastic stability of simply supported corrugated core sand- wich cylinders . theoretical buckling coefficients are obtained for the general instability of simply supported, corrugated core sandwich circular cylinders under combined loads with the core oriented parallel to the longitudinal axis of the cylinder . buckling curves are presented for axial compression, external lateral pressure, torsion, and some typical interactions . the differential equations of equilibrium used to obtain the buckling equations were derived from the small deflection equations of stein and mayer which include the effect of deformation due to transverse shear . these equations are solved by galerkin’s equation . remarks are made concerning the probable validity of the results of the small deflection theory for sandwich shells .
axisymmetric snap buckling of conical shells . the authors give a brief account of some of their recent analytical and numerical studies of cone buckling, limiting the discussion to axisymmetric deformations . pertinent numerical results for the relaxation buckling of full cones subjected to uniform external pressure and belleville springs deformed by axial edge loads are presented . in addition, bifurcation buckling problems are discussed . for a specific case, the existence of friedrichs’ intermediate buckling load, as applied to cones, is established . upper and lower bounds for its value are given .
air scooping vehicle . a satellite vehicle is described which collects gases from the upper atmosphere and stores them in liquid form . such a vehicle could serve as a filling station in space, furnishing liquid oxygen or air to other spacecraft . the vehicle represents an alternative to launching these liquids into orbit from the surface of the earth . the two methods are compared on an economic basis, and it is shown that the proposed vehicle permits substantial savings when operated beyond about one year . the feasibility of developing such a system for long-time operation is investigated . several practical designs are discussed .
heat transfer in separated flows . results of an experimental heat-transfer investigation in regions of separated flow are presented and compared with the theoretical analysis of naca tn 3792 . the average heat transfer for both laminar and turbulent separated boundary layers was found to be from 35 to 50 per cent less than that for equivalent attached boundary layers . the overall scope of the measurements included mach numbers from 0.3 to 4.0 and reynolds numbers from 10 to 4 x 10 . the results for laminar boundary layers agree well with the analysis of tn 3792 . the results for turbulent boundary layers, however, disagree considerably . results of velocity and temperature surveys in the separated turbulent boundary layer are presented and partially explain the discrepancy between the experiments and analysis . the maximum local heat-transfer rates were found to occur in the reattachment region of the separated boundary layers investigated . the effect of transition on heat transfer in the separated laminar boundary layers is described and data showing effects of mach number and wall temperature on the transition reynolds number of separated laminar flows are also included .
investigation of free turbulent mixing . a discussion of the integral relations for flow of the boundary-layer type is presented . it is shown that the characteristic laws of spread of jets, wakes, and so forth, can be obtained directly for the laminar case and, with the help of dimensional reasoning, for the turbulent case as well . measurements of the mean velocity, the intensity and scale of the turbulent fluctuations, and of the turbulent shear in a two-dimensional mixing zone are presented . the results of these measurements are compared with the mixing-length theories . it is shown that both mixing length and exchange coefficient vary across the mixing zone . the theories based on the assumption of constant mixing length or exchange coefficient are thus in error . a discussion of the energy balance of the fluctuating motion is given and the triple correlation is estimated .
compressible two dimensional jet mixing at constant pressure . an analysis is made of turbulent constant pressure mixing for a compressible jet boundary, taking into consideration effects of the initial boundary layer . velocity profiles in the mixing region are represented in a transformed plans by one-parameter families of curves, with no specification for the mixing mechanism beyond that of an exchange coefficient concept being made . the exchange coefficient is represented by the bornel function of an integral transform for the x coordinate of an intrinsic system of coordinates . this intrinsic system and the physical coordinate system are related by means of a momentum integral . satisfactory correlation of theory and experimental low-speed data is obtained with a simple form of kernal function . an asymptotic solution, corresponding to a fully developed velocity profile in the jet boundary, allows the calculation of the mechanical energy level along the separating streamline in the jet boundary without the use of empirical information .
contributions to the theory of heat transfer through a laminar boundary layer . an approximation to the heat transfer rate across a laminar incompressible boundary layer, for arbitrary distribution of main stream velocity and of wall temperature, is obtained by using the energy equation in von mises’s form, and approximating the coefficients in a manner which is most closely correct near the surface . the heat transfer rate to a portion of surface of length l breadth is given as where k is the thermal conductivity of the fluid, o its prandtl number, p its density, u its viscosity, r(x) is the skin friction, and t(x) the excess of wall temperature over main stream temperature . a critical appraisement of the formula indicates that it should be very accurate for large, but that for of order 0.7 (for most gases) the constant should be replaced by 0.73, when the error should not exceed this yields a formula for nusselt number in terms of the reynolds number r and the mean square root of the skin friction coefficient c, in the case of uniform wall temperature . however, for the boundary layer with uniform main stream, the original formula is accurate to within 3 percent even for . by known transformations an expression is deducted for heat transfer to a surface, with arbitrary temperature distribution along it, and with a uniform stream outside it at arbitrary mach number (equation (42)) . from this the temperature distribution along such a surface is deduced in the case (of importance at high mach numbers) when heat transfer to it is balanced entirely by radiation from it . this calculation, which includes the solution of a non-linear integral equation, gives higher temperatures near the nose, and lower ones farther back (figure 2), than are found from a theory which assumes the wall temperature uniform and averages the heat transfer balance . this effect will be considerably mitigated for bodies of high thermal conductivity., the author is not in a position to say whether or not it will be appreciable for metal projectiles . but for stony meteorites at a certain stage of their flight through the atmosphere it indicates that melting at the nose and re-solidification farther back may occur, for which the shape and constitution of a few of them affords evidence . an appendix shows how the method for approximating and solving von mises’s equation could be used to determine the skin friction as well as heat transfer rate, but this line seems to have no advantage over established approximate methods .
a variational principle for convection of heat . authors extend variational principle of biot to various cases of heat transfer due to forced convection . numerical results are given for one-dimensional problems of fluid flowing between parallel walls with uniform or parabolic velocity profiles . agreement with exact solutions is excellent .
on the theory of discharge coefficients for round entrance flowmeters and venturis . a theory of rounded-entrance flowmeters, based on a consideration of the potential and boundary-layer flows in a converging nozzle, is constructed . curves are presented showing the discharge coefficient as a function of diameter reynolds number, with the /total equivalent length tional length-diameter ratio of the contraction section of the asme long-radius nozzle is presented . the theoretical curves of discharge coefficient versus diameter reynolds number are in good agreement with experiment over a range of reynolds number from 1 to 10 . the theory provides a rational framework for correlating and extrapolating experimental results,. it shows the effects of contraction shape and location of pressure taps,. it furnishes values of discharge coefficient for untested designs,. and it suggests precautions to be taken in design, installation, and operation .
analytic determination of discharge coefficients of flow nozzles . integration of the velocity profile at the throat of a flow nozzle yields the discharge coefficient as a function of the ratio of boundary solution of the approximate momentum equation for the boundary layer . the resulting expression for the discharge coefficient is then a function of the reynolds number based on nozzle diameter and of the geometry of the nozzle . good agreement is shown between this expression and published experimental data on flow nozzles for reynolds numbers between
on fully developed channel flows,. some solutions and limitations, and effects of compressibility, variable properties, and body forces . an examination of the effects of compressibility, variable properties, and body forces on fully developed laminar flows has indicated several limitations on such streams . in the absence of a pressure gradient, but presence of a body force liquid this follows also for the case of a constant streamwise pressure gradient . these motions are exact in the sense of a couette flow . in the liquid case two solutions /not a new result/ can occur for the same boundary conditions . an approximate analytic solution was found which agrees closely with machine calculations . in the case of approximately exact flows, it turns out that for large temperature variations across the channel the effects of convection /due to, say, a wall temperature gradient/ and frictional heating must be negligible . in such a case the energy and momentum equations are separated, and the solutions are readily obtained . if the temperature variations are small, then both convection effects and frictional heating can consistently be considered . this case becomes the constant-property incompressible case /or quasi-incompressible case for free-convection flows/ considered by many authors . finally, there is a brief discussion of cases wherein streamwise variations of all quantities are allowed but only in such form that the independent variables are separable . for the case where the streamwise velocity varies inversely as the square root of distance along the channel, a solution is given .
a study of laminar compressible viscous pipe flow accelerated by an axial body force, with application to magnetogasdynamics . a study is made of the steady laminar flow of a compressible viscous fluid in a circular pipe when the fluid is accelerated by an axial body force . the application of the theory to the magnetofluidmechanics of an electrically conducting gas accelerated by electric and magnetic fields is discussed . constant viscosity, thermal conductivity, and electrical conductivity are assumed . fully developed flow velocity and temperature profiles are shown, and detailed results of the accelerating flow development, including velocity and pressure as functions of distance, are given for the case where the axial body force is constant and for the case where it is a linear function of velocity . from these results are determined the pipe entry length and the pressure difference required .
rocket propulsion systems for interplanetary flight . a comparison is made of several different propulsion systems for interplanetary flight . liquid and solid propellant rockets, propulsion systems which use nuclear energy sources, are heating rockets, magneto-plasma devices, ion rocket propulsion, solar heating rockets, and solar sails are briefly described and their current status reviewed . engine performance requirements for different interplanetary missions are established . these several propulsion systems are then compared on the basis of several performance criteria, environmental characteristics, vehicle requirements, reliability, current status, growth potential, and efficiency . predictions on various propulsion system capabilities and an analysis of multiple rocket engine reliability is included . it is concluded that electrical rockets are superior for long-time inter-planetary flight applications, and that chemical rockets are satisfactory for most of the immediate applications in /near/ space . none of the several propulsion schemes discussed can be rejected until further technical work has been accomplished .
on the use of side-jets as control devices . wind tunnel experiments with side-jets, issuing laterally near the base of slender bodies in a supersonic stream, have suggested the existence of a sizable and usable interaction . with this interaction force, the use of jet reaction controls may be as attractive for flight within the atmosphere as it obviously is for flight outside the atmosphere . this note indicates the altitude regime of interest and the order of magnitude of the interaction bonus for a lateral control jet located near the base of a body of revolution .
loads induced on a flat plate wing by an air jet exhausting perpendicularly through the wing and normal to a free-stream flow of mach number 2.0 . measurements were made of loads induced on a flat-plate wing by an air jet exhausting perpendicularly through the wing and normal to the free-stream flow . the investigation was conducted at a free-stream mach number of 2.0 and a reynolds number per foot of 14.4 x 10 . an axially symmetric sonic nozzle and two supersonic nozzles were employed for the jets . the supersonic nozzles consisted of an axially symmetric nozzle with exit mach number of 3.44 and a two-dimensional nozzle with exit mach number of 1.76 . the ratio of nozzle total pressure to free-stream static pressure was varied from 20 to 110 . negative loads were induced on the flat-plate wing by all the jets . as the nozzle pressure ratio was increased the magnitude of interference loads due to jet thrust decreased . the chordwise center-of-pressure location generally moved toward the nozzle center line as the pressure ratio was increased .
surface pressure distributions with a sonic jet normal to adjacent flat surfaces at mach 2.92 to 6.4 . an investigation was made to determine the interference effects on surface pressure distributions caused by a sonic jet exiting normal to the surface . two configurations, a flat plate and an arrow-wing reentry-type vehicle, with sonic nozzles near the leading edge were tested over a range of pressure ratios and reynolds numbers for mach numbers from the data indicate that jet pressure ratio had considerable effect on the pressure levels and distributions on both configurations . also, for a constant jet pressure ratio, the free-stream mach number effect on the distributions and levels was quite large . over the limited range investigated, the effect of reynolds number at constant mach number and pressure ratio was small compared to the mach number and pressure ratio effect .
aerodynamic interaction effects ahead of a sonic jet exhausting perpendicularly form a flat plate into a mach number 6 free stream . an investigation of the effects of the interaction ahead of a two-dimensional sonic jet exhausting perpendicularly into a mach number were made at an angle of attack of 0degree at a reynolds number per foot of approximately 6 x 10 and with conditions of both transitional and turbulent separation on the flat plate . the ratio of jet stagnation pressure to free-stream static pressure was varied from 8 to 460 and the jet slot width was varied from 0.001 to 0.05 inch . the force ratio due to reaction of jet/, calculated ahead of the jet, was sizable and varied from 0.5 to 9 . in general, the ratio increased with increasing pressure ratio and decreasing slot width . for the turbulent boundary-layer separation tests it was found that the first peak pressure and the chordwise pressure distribution of the separated boundary layer ahead of the jet were similar to those for a separation caused by a forward-facing step at the same test conditions .
interaction effects produced by jet exhausting laterally near base of ogive-cylinder model in supersonic main stream . the experimentally determined interaction effects of a side jet exhausting near the base of an ogive-cylinder model are presented and discussed . the interaction force appears to be independent of main-stream mach number, boundary-layer condition /laminar or turbulent/, angle of attack, and forebody length . the ratio of interaction force to jet force is found to be inversely proportional to the square root of the product of jet stagnation-to-free-stream pressure ratio and jet-to- body diameter ratio .
approximate analysis of thrust vector control by fluid injection . a study has been made of the side force generated by injection of secondary material into the main stream of a rocket nozzle . two cases have been analyzed .. gas injection and liquid injection . for the gas injection case, it is assumed that the turbulent boundary layer ahead of the injection point separates from the wall . the pressure in the separated region and the extent of the separated region are determined by a consideration of turbulent boundary layer-shock wave interaction and the accommodation height of the injected gas stream . equations are derived for calculating the side force, and the side forces predicted by the theory are compared with experimental data . the agreement between theory and experiment is fair . for the case of liquid injection, it is assumed that the liquid flows along the nozzle wall and evaporates into the main stream . the resulting side force on the nozzle wall is determined on the basis of linearized theory, thus restricting the analysis to small rates of liquid injection . the effects of small rates of heat addition are also included in the analysis . a very simple equation for calculating the side force is obtained .
one dimensional flows of an imperfect diatomic gas . with the assumptions that berthelot’s equation of state accounts for molecular size and intermolecular force effects, and that changes in the vibrational heat capacities are given by a planck term, expressions are developed for analyzing one-dimensional flows of a diatomic gas . the special cases of flow through normal and oblique shocks in free air at sea level are investigated . it is found that up to a mach number of 10 the pressure ratio across a normal shock differs by less than 6 percent from its ideal gas value,. whereas at mach numbers above 4 the temperature rise is considerably below and hence the density rise is well above that predicted assuming ideal gas behavior . it is further shown that only the caloric imperfection in air has an appreciable effect on the pressures developed in the shock process considered . the effects of gaseous imperfections on oblique shock flows are studied from the standpoint of their influence on the lift and pressure drag of a flat plate operating at mach numbers of 10 and 20 . the influence is found to be small .
turbulent diffusion in the wake of a blunt nosed body at hypersonic speeds . at reynolds numbers greater than about 5 x 10 corresponding to altitudes below about 180,000 ft, the hot outer inviscid wake behind the bow shock wave produced by a blunt-nosed body at hypersonic speeds is cooled mainly by turbulent diffusion and conduction . turbulence originates in the inner wake formed by the coalescence of the free shear layers (or annulus) shed from the body surface when the boundary layer separates from the surface . as this turbulence spreads outward, it swallows enthalpy or momentum defect originally contained in the outer inviscid wake . if the turbulence is locally similar–i.e., if it behaves at each station like a slice of a low-speed /self-similar/ wake–then the turbulent diffusivity grows from a low initial value near the body to a value corresponding to the total drag of the body at about 300 body diameters downstream . at flight velocities of the order of 9,000-10,000 ft per sec. the growth of the turbulent inner wake predicted on the basis of locally similar turbulence is in good agreement with shadowgraph measurements of wake widths behind spheres obtained in ballistic ranges in the region from 200 to 4,000 body diameters downstream of the body . tentatively, one concludes that the turbulence mechanism in the wake with respect to a fixed observer is similar to the low-speed case, in spite of the large mean temperature gradients . in order to illustrate the behavior of an observable such as electron density in a turbulent wake behind a blunt body, the two limiting cases of thermodynamic equilibrium and pure diffusion (zero electron-ion recombination rate) are calculated for m = 22 at altitudes of 100,000 and 200,000 ft . even for the case of thermodynamic equilibrium, the predicted turbulent radar trail length is about 200 body diameters at l-band (1,300 mc) at 100,000-ft altitude and about 150 body diameters for uhf (400 mc) at 200,000 ft . one interesting result is that the width of the plasma cylinder corresponding to the plasma requency at l-band remains virtually constant at about 3.5 body diameters in the range 30 150 at 100,000-ft altitude . these results are sufficiently encouraging that one can consider including the effects of finite chemical and electron-ion recombination rates in the analysis in order to give a more complete picture of the wake at hypersonic speeds .
concerning some solutions of the boundary layer equations in hydrodynamics . the boundary-layer equations for a steady two-dimensional motion are solved for any given initial velocity distribution (distribution along a normal to the boundary wall, downstream of which the motion is to be calculated) . this initial velocity distribution is assumed expressible as a polynomial in the distance from the wall . three cases are considered .. first, when in the initial distribution the velocity vanishes at the wall, but its gradient along the normal does not,. second, when the velocity in the initial distribution does not vanish at the wall,. and third, when both the velocity and its normal gradient vanish at the wall (as at a point where the forward flow separates from the boundary) . the solution is found as a power series in some fractional power of the distance along the wall, whose coefficients are functions of the distance from the wall to be found from ordinary differential equations . some progress is made in the numerical calculation of these coefficients, especially in the first case . the main object was to find means for a step-by-step calculation of the velocity field in a boundary layer, and it is thought that such a procedure may possibly be successful even if laborious . the same mathematical method is used to calculate the flow behind a flat plate along a stream . the results are shown in curves in the original .
temperature profiles inafinite solid with moving boundary . a numerical solution is presented to the transient heat conduction equation for a cylinder of finite thickness with one moving boundary . the implicit method of solution is developed with conductivity as an arbitrary function of temperature . application is made to a sample case of re-entry heating encountered by aerodynamic bodies, with erosion by sublimation and combustion occurring at the body surface .
correlation of base pressure and wake structure of sharp and blunt-nose cones with reynolds number based on boundary layer momentum thickness . it has been established in the past that there is a certain relationship between base pressure and boundary-layer behavior . the base-pressure and wake-flow conditions were found to be dependent upon the local flow characteristics at the surface of supersonic vehicles directly upstream of the base or of the region of wake-flow separation . in order to use existing data on cones and other shapes to predict wake angle and base pressure on blunt bodies, an attempt was made recently at the naval ordnance laboratory to establish a unique relationship between given local flow conditions at the downstream end of sharp and blunt cones at supersonic speeds and the corresponding wake-flow conditions with zero heat transfer .
a method of computing the transient temperature of thick walls from arbitrary variation of adiabatic-wall temperature and heat-transfer coefficient . a method of calculating the temperature of thick walls has been developed in which are used relatively new concepts, such as the time series and the response to a unit triangle variation of surface temperature, together with essentially standard formulas for transient temperature and heat flow into thick walls . the method can be used without knowledge of the mathematical tools of its development . the method is particularly suitable for determining the wall temperature in one-dimensional thermal problems in aeronautics where there is a continuous variation of the heat-transfer coefficient and adiabatic-wall temperature . the method also offers a convenient means for solving the inverse problem of determining the heat-flow history when temperature history is known . a series of diversified problems were solved by exact analysis as well as by the new method . a comparison of the results shows the new method to be accurate . the labor involved is very modest in consideration of the nature of the thick-wall temperature problem . limiting solutions for the /infinitely thick/ wall and for walls so thin that thermal lag can be neglected were also obtained .
solutions to the heat-conduction equation with time dependent boundary conditions . design charts based on the analytical solution to the problem of one-dimensional heat flow in a solid body of constant material properties with time-dependent boundary conditions were presented by kaye and yeh . this solution dealt with aerodynamic heating at hypersonic speeds where the surface coefficient of heat transfer and the temperature potential were taken to be linear functions of time of flight . in order to make these charts of more general application, general solutions are presented which, together with the charts, enable rapid and reasonable estimates to be made of the transient temperature distributions in many practical cases .
the temperature history in a thick skin subjected to laminar heating during entry into the atmosphere . during high speed entry into the earth’s atmosphere, a vehicle can be afforded thermal protection for the short period of entry heating by a thick outer skin, sometimes called a /heat sink/ . the temperature distribution in such a heat sink has been found by integrating the product of the laminar aerodynamic heating rate and the appropriate green’s function for a finite-thickness wall over the generalized trajectory for a vehicle entering the earth’s atmosphere at high speeds dimensional heat conduction problem for laminar heating . the maximum surface temperature that occurs during the generalized entry trajectory for any combination of wall thickness and thermal properties is obtained from which the performance of any material can be found, provided that the average thermal properties may be used . as an example of the use of the solution, the performance of copper, graphite, molybdenum and tungsten are compared .
addendum to ‘heat transfer to satellite vehicles re-entering the atmosphere . the original paper gave a correlation formula for stagnation point heat transfer rate to a blunt body of revolution in hypersonic flow . this note gives a somewhat refined version, based on further calculation and shock tube data . its effect on the conclusions of the original paper is negligible except at surface temperature parameters of over 5000 r-ft1/8 . in other problems, where heat transfer rate itself is important, it can make a significant defference .
method of analysis for compressible flow through mixed-flow centrifugal impellers of arbitrary design . a method is presented for analysis of the compressible flow between the hub and the shroud of mixed-flow impellers of arbitrary design . axial symmetry was assumed, but the forces in the meridional /hub to shroud/ plane, which are derived from tangential pressure gradients, were taken into account . the method was applied to an experimental mixed-flow impeller . the analysis of the flow in the meridional plane of the impeller showed that the rotational forces, the blade curvature, and the hub-shroud profile can introduce severe velocity gradients along the hub and the shroud surfaces . choked flow at the impeller inlet as determined by the analysis was verified by experimental results .
a rapid approximate method for the design of hub shroud profiles of centrifugal impellers of given blade shape . a rapid approximate method for the design of centrifugal compressors of given blade shape with compressible nonviscous flow characteristics has been developed using techniques based upon stream-filament theory . axial symmetry is assumed, but meridional-plane forces derived from tangential pressure gradients are included . the method was applied to the design of an impeller in order to determine the approximate maximum meridional streamline spacing that could be used . three numerical solutions for different streamline spacings were made using the same hub profile, blade shape, and prescribed velocity distribution along the hub . the shroud profiles obtained from the three solutions, which utilized 3, 5, and 9 stream-tubes, were negligibly different . the approximate computing time required was 15 hours per streamtube .
design and test of mixed-flow impellers, viii – comparison of experimental results for three impellers with shroud redesigned by rapid approximate method . three centrifugal impellers with parabolic, circular, and skewed-parabolic blading were modified by a recently developed design procedure to reduce the velocity gradients along the hub from inlet to outlet . all original dimensions except the shroud contours were retained . experimental investigation showed that the modified impellers had better performance characteristics than the original impellers at all speeds investigated, the greatest gains occurring at speeds of 1300 feet per second and higher . these large gains probably resulted primarily from more favorable velocity gradients and from designing these impellers further away from the condition necessary for eddy formation . the modified impellers were thus able to operate over a wider range of weight flows at high speeds . the modified impellers were investigated over a range of equivalent speeds of 900 to 1500 feet per second and flow rates from maximum to the point of incipient surge . at 1300 feet per second, the peak pressure ratio and maximum adiabatic temperature-rise efficiency for the parabolic-bladed impeller were 3.07 and 0.825, respectively . for the same conditions, the circular-bladed impeller and the skewed-parabolic-bladed impeller had pressure ratios of 3.13 and 3.15 and efficiencies of 0.737 and 0.805, respectively . of the three, the parabolic-bladed impeller had the highest maximum efficiencies /0.854 to 0.800/ and the best weight-flow range over the speed range tested . on the basis of the parameters investigated, it appears that parabolic blading is superior to circular blading . the experimental results indicate that the design method of naca tn 3399 is a reliable method for use in designing centrifugal impellers .
a general theory of three dimensional flow in subsonic and supersonic turbo-machines of axial-radial-and mixed-flow types . a general theory of steady three-dimensional flow of a nonviscous fluid in subsonic and supersonic turbomachines having arbitrary hub and casing shapes and a finite number of blades is presented . the solution of the three-dimensional direct and inverse problem is obtained by investigating an appropriate combination of flows on relative stream surfaces whose intersections with a z-plane either upstream of or somewhere inside the blade row form a circular arc or a radial line . the equations obtained to describe the fluid flow on these stream surfaces show clearly the several approximations involved in ordinary two-dimensional treatments . they also lead to a solution of the three-dimensional problem in a mathematically two-dimensional manner through iteration . the equation of continuity is combined with the equation of motion in either the tangential or the radial direction through the use of a stream function defined on the surface, and the resulting equation is chosen as the principal equation for such flows . the character of this equation depends on the relative magnitude of the local velocity of sound and a certain combination of velocity components of the fluid . a general method to solve this equation by both hand and high-speed digital machine computations when the equation is elliptic or hyperbolic is described . the theory is applicable to both irrotational and rotational absolute flow at the inlet of the blade row and at both design and off-design operations .
nonviscous flow through a pump impeller on a blade-to-blade surface of revolution . the nonviscous incompressible flow through a typical pump impeller is analyzed on a blade-to-blade surface of revolution . solutions are obtained for a variety of inlet conditions including several with prewhirl of the assumed location of the rear stagnation point . comparison of results from two approximate methods of analysis showed good agreement for the zero-angle-of-attack case and reliable indication of the existence of an eddy on the driving face at a large positive angle of attack .
incompressible nonviscous blade-to-blade flow through a pump rotor with splitter vanes . the nonviscous flow through a mixed-flow pump impeller having one splitter vane between adjacent main blades has been analyzed on a blade-to-blade surface of revolution using a previously reported analysis method . solutions were obtained for a variety of flow conditions including several cases in which whirl is imparted to the flow upstream of the impeller . the velocity distributions on the main-blade surfaces and on the splitter-vane surfaces in the region of the splitter vane were strongly dependent on the assumed location of the rear stagnation points . solutions were obtained by assuming values of slip factor and of division of flow around the splitter in addition to assuming the location of the rear stagnation points . these solutions indicated that the velocity distributions in the splitter-vane region are largely determined by the division of flow around the splitter vane and that only the region in the immediate vicinity of the trailing edge is affected by the slip factor . blade surface velocities were obtained from two approximate methods by specifying flow division and slip factor, and these results are compared with the more exact solutions of the analysis .
a rapid approximate method for determining velocity distribution on impeller blades of centrifugal compressors . a rapid approximate method of analysis was developed for both compressible and incompressible, nonviscous flow through radial- or mixed-flow centrifugal compressors with arbitrary hub and shroud contours and with arbitrary blade shape . the method of analysis is used to determine approximately the velocities everywhere along the blade surfaces, but no information concerning the variation in velocity across the passage between blades is given . in eight numerical examples for two-dimensional flow, covering a fairly wide range of flow rate, impeller-tip speed, number of blades, and blade curvature, the velocity distribution along the blade surfaces was obtained by the approximate method of analysis and compared with the velocities obtained by relaxation methods . in all cases the agreement between the approximate solutions and the relaxation solutions was satisfactory except at the impeller tip where the velocities obtained by the approximate method did not, in general, become equal on both surfaces of the blade as required by the joukowski condition .
wing-flow study of pressure drag reduction at transonic speed by projecting a jet of air from the nose of a prolate spheroid of fineness ratio 6 . a study was made at transonic speeds by the naca wing-flow method of the pressure-drag reduction obtained by projecting a high-energy jet of air from the nose of a prolate spheroid . supplementary information was obtained by taking shadowgraphs of the model mounted in a small supersonic tunnel at a constant mach number of 1.5 . the high-velocity jet was observed to alter the pressure distribution over the body in such a way that the pressure drag of the body was reduced,. thus, in a restricted sense, the nose jet produced a thrust on the body . under the conditions investigated, the thrust produced by the nose jet was never so large as that which would be expected from a conventional rearward jet . for example, under the best conditions tested /mach number of 1.07/ the reduction in body pressure drag caused by the nose jet more than compensated for the negative thrust of the jet itself . however, the magnitude of the net reduction in drag /change in body pressure drag with jet on and jet off minus the adverse thrust of the jet/ was only about one-half of the thrust which would be produced by the same jet exhausting rearward . the appearance of such an unexpectedly large effect in the first trial indicated the phenomenon to be worth further study .
the effects of a small jet of air exhausting from the nose of a body of revolution in supersonic flow . an investigation has been made at a mach number of 1.62 to determine the effects of a small jet of air exhausting from the nose of an elliptical body of revolution upon boundary-layer transition and the viscous, pressure, and total drag of the forebody at three body stations body nose were also obtained . the tests were conducted at reynolds numbers of 2.13 x 10 and 7.66 x 10, based on body length . the maximum range of thrust coefficients for the small jet was from 0 to about at the lower test reynolds number, for which the boundary layer was laminar over the entire body in the jet-off condition, a very small flow from the jet moved the point of transition forward to the vicinity of the 20-percent-body station . as the jet flow was increased, the transition point moved abruptly to the nose at a thrust coefficient of about gardless of the type of boundary layer . at the higher test reynolds number for which the boundary layer was largely turbulent in the jet-off condition the total drag, including skin friction, was reduced somewhat by the action of the jet . although the forward-exhausting small jet was found to have the above favorable effects upon the drag, these findings are not believed too important since the question arises as to the benefits of the same small jet exhausting from the rear of the body in the conventional manner . no attempt was made to establish geometric optimums in the present investigation, yet, from a general consideration of the benefits indicated by the present results and the phenomena known to occur in the vicinity of rearward-exhausting jets, the benefits of a small jet exhausting rearward would appear to exceed those of the same small jet exhausting forward, particularly so when the flow over the body is laminar in the jet-off condition .
the extent of the jet interference flow fields . jet effects on cylindrical afterbodies housing sonic and supersonic nozzles which exhaust against a supersonic stream at angles of attack from 90degree to 180degree . an investigation has been made to determine jet effects on cylindrical afterbodies housing sonic and supersonic nozzles which exhaust against a supersonic stream at angles of attack from 90 to 180 . the tests were conducted at a free-stream mach number of 2.91 and at free-stream reynolds numbers, based on body diameter, of 0.15×106 and stream static pressure investigated was from jet off to about 400 . the data presented herein showed that, in general, variation of the ratio of jet total pressure to free-stream static pressure, jet-exit mach number, and ratio of jet-exit diameter to body diameter had large influences on the body pressures on the windward halves of the after-bodies and negligible influences on the leeward pressures . there was a negligible effect of reynolds number on the body pressures . the ratio of jet total pressure to free-stream static pressure also had a large influence on the base pressures at all angles of attack . schlieren studies showed details of the shock-wave structure caused by the jet and the extent of the jet interference flow fields .
investigation of a retrocket exhausting from the nose of a blunt body into a supersonic free stream . the pressure distribution and pressure drag of a blunt body with a supersonic jet issuing upstream from its center were determined at free-stream mach numbers of 1.60, 2.00, and 2.85 . the thrust of the jet issuing from the model nose was varied to study its effects on flow around the model and to determine variation of pressure distribution and pressure drag of the model with the thrust . at all mach numbers investigated, the pressure drag decreased with increasing retrorocket thrust until a minimum value was reached . further increases in retrorocket thrust resulted in increases in the pressure drag . the resultant drag /pressure drag plus retrorocket thrust but excluding base and skin-friction drag/ of the model was reduced by retrorocket operation below the drag for a jet-off condition, except at very low retrorocket thrust coefficients . the flow about the nose of the blunt body was very unstable throughout the range of mach numbers and retrorocket thrust coefficients investigated .
extension of boundary layer separation criteria to a m=6 .5 utilizing flat plates with forward-facing steps . an experimental investigation has been made of the separation phenomena on a flat plate to which forward-facing steps were attached to force separation . both laminar and turbulent flows were investigated over a mach number range of approximately distributions, shadowgraph and chemical film techniques, the pressure rise at separation, the laminar plateau pressure, and the turbulent peak pressure were determined . boundary-layer surveys were made on a smooth flat plate and on a flat plate with roughness to force transition . examinations of the separated flow showed that the predominant variable in the determination of the pressure distribution was the location of transition relative to the separation point and reattachment . pure laminar, transitional, and turbulent types of separation were found in this mach number range . the peak static-pressure-rise ratios for identical forward-facing steps at a mach number of 6.25 were approximately 1.5 and 5.0, respectively, for pure laminar and turbulent separation . the effect of reynolds number on the peak pressure rise for turbulent separation for the lower mach number range was found to be very minor provided the step height was of the order of the boundary-layer thickness . as the mach number is increased, the peak pressure coefficient for turbulent separation decreased from approximately 0.18 at a mach number of 4 to about 0.13 at a mach number of 6.25 . the pressure coefficient at the separation point for laminar separation decreases from approximately 0.014 at a mach number of value at a mach number of 6.5 . the results obtained with forward-facing steps agree with the trends predicted, based upon lower mach number studies .
experimental and theoretical studies of axisymmetric free jets . some experimental and theoretical studies have been made of axisymmetric free jets exhausting from sonic and supersonic nozzles into still air and into supersonic streams with a view toward problems associated with propulsive jets and the investigation of these problems . for jets exhausting into still air, consideration is given to the effects of jet mach number, nozzle divergence angle, and jet static-pressure ratio upon jet structure, jet wavelength, and the shape and curvature of the jet boundary . studies of the effects of the ratio of specific heats of the jets are included as are observations pertaining to jet noise and jet simulation . for jets exhausting into supersonic streams, an attempt has been made to present primarily theoretical curves of the type that may be useful in evaluating certain jet interference effects and in formulating experimental studies . the primary variables considered are jet mach number, free-stream mach number, jet static-pressure ratio, ratio of specific heats of the jet, nozzle exit angle, and boattail angle . the simulation problem and the case of a hypothetical hypersonic vehicle are examined
equations, tables and charts for compressible flow . this report, which is a revision and extension of naca tn useful in the analysis of high-speed flow of a compressible fluid . the equations provide relations for continuous one-dimensional flow, normal and oblique shock waves, and prandtl-meyer expansions for both perfect and imperfect gases . the tables present useful dimensionless ratios for continuous one- dimensional flow and for normal shock waves as functions of mach number for air considered as a perfect gas . one series of charts presents the characteristics of the flow of air (considered a perfect gas) for oblique shock waves and for cones in a supersonic air stream . a second series shows the effects of caloric imperfections on continuous one-dimensional flow and on the flow through normal and oblique shock waves .
static aerodynamic characteristics of short blunt cones with various nose and base cone angles at mach numbers of 0. 6 to 5. 5 and angles of attack to 180 . wind-tunnel tests have been performed at mach numbers from 0.6 to 5.5 to determine coefficients of normal force, axial force, and pitching moment for short blunt cones, as affected by changes in nose and base cone angles . models with nose half-angles of 10 and 20 were investigated . the 10 nose half-angle models were tested with a flat base and with base cones of 50 and 70 half-angle . the 20 nose half-angle model had a 50 half-angle base cone . reynolds numbers for the test ranged from about maximum diameter . variations in the base cone angle resulted in significant changes in the aerodynamic characteristics, with lesser effects resulting from changes in nose cone angle . in particular, the model with the 50 half-angle conical base had only one trim angle flat base and 70 half-angle conical base had two trim angles (a = 0 and a = 180) . estimated variations of the aerodynamic characteristics with angle of attack by means of a modified newtonian theory were in good agreement with the experimental results . the theory, however, failed to predict the trim point at a = 180 for the flat-based model .
free-flight measurements of the static and dynamic stability and drag of a 10 blunted cone at mach numbers 3. 5 and 8 .5 . tests were made of a short blunt-nosed without a 50 half-angle conical afterbody in a pressurized ballistic range at nominal mach numbers of 3.5 and of 90,000 and 220,000, respectively . it was found that the models were statically stable about the center- of-gravity location tested but exhibited neutral dynamic stability for flight at constant altitude . the static stability was not affected by the but was nonlinear with angle of attack and varied with mach number . the nonlinear variation of the pitching moment with angle of attack was accurately approximated by a cubic polynomial . the static stability was only qualitatively predicted by modified newtonian theory . the drag characteristics were in good agreement with values calculated by use of modified newtonian theory . calculations of the oscillatory behavior of the configurations flying an example entry trajectory through the martian atmosphere indicated the configurations to be dynamically satisfactory . pitching motions should converge to a small fraction of the amplitude at entry, provided the initial angle of attack and pitch rate are not large enough to cause tumbling .
wind tunnel investigation of the static and dynamic stability characteristics of a 10degree semivertex angle blunted cone . the static and dynamic stability characteristics of a blunted 10degree semivertex angle cone were studied . the cone which had a modified spherical segment nose was tested with a flat base and with a truncated conical base . all tests were performed in air at mach numbers from 0.65 to 2.20 with the angle-of-attack range from -4degree to +18degree . presented are measurements of the normal force, axial force, base pressure, and pitching moment from the static tests, and the damping-in-pitch moment from the dynamic tests . both models had satisfactory stability characteristics throughout the test mach number range but the addition of the conical afterbody had a large destabilizing effect .
preliminary investigations of spiked bodies at hypersonic speeds . generally accepted solutions for the problems of hypersonic flight appear, at the moment, to be centered around the use of blunt bodies to minimize the heat-transfer rates . there are, however, several other solutions to the problem, and, as part of an exploratory study of these solutions, a detailed examination has been made of the flow over blunt bodies equipped with a spike . these tests, carried out at a mach number of about 14 in the princeton helium hypersonic tunnel, have investigated the effect of varying spike lengths for flat-faced and hemispherically-nosed axially symmetric bodies . detailed pressure distributions have been obtained as well as heat-transfer rates . these exploratory studies have shown that the use of a spike protruding from a hemispherical-nosed cylinder at m 14 decreased the pressure level by an order of magnitude and the heat transfer to a fraction of that measured on a hemisphere without a spike . the general technique appears to hold considerable promise for hypersonic flight .
free-flight measurements of the static and dynamic . the real-gas hypersonic flow parameters for helium have been calculated for stagnation temperatures from 0 f to 600 f and stagnation pressures up to 6,000 pounds per square inch absolute . the results of these calculations are presented in the form of simple correction factors which must be applied to the tabulated ideal-gas parameters . it has been shown that the deviations from the ideal-gas law which exist at high pressures may cause a corresponding significant error in the hypersonic flow parameters when calculated as an ideal gas . for example, the ratio of the free-stream static to stagnation pressure as calculated from the thermodynamic properties of helium for a stagnation temperature of 80 f and pressure of 4,000 pounds per square inch absolute was found to be approximately 13 percent greater than that determined from the ideal-gas tabulation with a specific heat ratio of
free-flight measurements of the static and dynamic . the effects of contamination of helium by air upon static-pressure, total-pressure, heat-transfer, and temperature measurements have been investigated in the 2-inch helium tunnel at the langley research center . within the scope of the tests, even a small amount of air is shown to affect these measurements . the heat-transfer and temperature measurements were made on a 26.6 half-angle cone and demonstrated the effects of contamination qualitatively . the wall static and center-line pitot pressures show that if the contaminating air is held to less than about 0.2 percent by volume, the error in indicated mach number is less than 1 percent as calculated from the rayleigh pitot equation . the corresponding errors in wall static and center-line pitot pressures are about 1.7 and 0.4 percent, respectively .
free-flight measurements of the static and dynamic . equations based on newtonian impact theory have been derived and a computational procedure developed with the aid of several design-type charts which enable the determination of the aerodynamic forces and moments acting on arbitrary bodies of revolution undergoing either separate or combined angle-of-attack and pitching motions . bodies with axially increasing and decreasing cross-sectional area distributions are considered,. nose shapes may be sharp, blunt, or flat faced . the analysis considers variations in angle of attack from -90 to 90 and allows for both positive and negative pitching rates of arbitrary magnitude . the results are also directly applicable to bodies in either separate or combined sideslip and yawing maneuvers .
free-flight measurements of the static and dynamic . the inviscid flow of a perfect gas over blunt-nosed axisymmetric and two-dimensional bodies at zero angle of attack has been calculated numerically on an ibm 7090 computer . the computation consisted of the fuller blunt-body solution for the subsonic and transonic regions and the method of characteristics for the supersonic region . the flow fields about a number of blunt bodies were studied, and the calculated results showed good agreement with experimental shock-wave shapes, surface-pressure distributions, and flow-field surveys .
free-flight measurements of the static and dynamic . this report presents equations, tables, and figures for use in the analysis of helium flow at supersonic and hypersonic speeds . the contents of the report and presentation of the data parallel that of a similar reference work (naca rep. 1135) prepared for air flow . the perfect-gas relations for continuous one-dimensional flow, normal- and oblique-shock waves, and prandtl-meyer expansions are the same as for air but are presented here for completeness . the tables present the values of useful dimensionless ratios for continuous one-dimensional flow and for normal-shock waves as functions of mach number . the helium viscosity relation as a function of temperature, mass-flow rates as a function of mach number and temperature, and the reynolds number as a function of mach number and stagnation temperature are plotted . the oblique-shock characteristics of wedges and cones in helium at mach numbers of 12, 16, 20, and 24 are presented in a series of plots . throughout all the computations, helium is considered to be a perfect gas .
free-flight measurements of the static and dynamic . representative experimental results are presented to show the current status of the panel flutter problem . results are presented for unstiffened rectangular panels and for rectangular panels stiffened by corrugated backing . flutter boundaries are established for all types of panels when considered on the basis of equivalent isotropic plates . the effects of mach number, differential pressure, and aerodynamic heating on panel flutter are discussed . a flutter analysis of orthotropic panels is presented in the appendix .
free-flight measurements of the static and dynamic . charts of thermodynamic properties for equilibrium air are presented with sufficient accuracy to permit the calculation of flow parameters in hypersonic nozzles operating at stagnation temperatures up to 4,950 r and pressures up to 1,000 atm . flow parameters calculated from these charts are presented for a series of stagnation temperatures between use of these parameters, it is possible to calibrate a nozzle in the conventional way . a method is also presented from which the flow parameters for conditions other than those chosen herein may be calculated . real-gas effects on the calculation of a hypersonic nozzle contour are shown by an example calculation in which the nozzle contour for mach number 12 was determined by including real-gas effects, and this contour was compared with one calculated by ideal-gas considerations . also presented are the approximate limiting mach numbers at which equilibrium air will just condense for various combinations of stagnation temperatures and pressures .
free-flight measurements of the static and dynamic . air-flow properties in nozzles were calculated and charted for equilibrium flow and two types of frozen flows . in one type of frozen flow, air was assumed to be in equilibrium from the nozzle reservoir to arbitrary points where chemical reactions and molecular vibrations became frozen . in the other type, it was assumed that molecular vibrations were in equilibrium throughout the nozzle and that chemical reactions became frozen at arbitrary points . the calculations were made for a range of stagnation pressures up to 10,000 poinds per square inch absolute and stagnation enthalpies up to 24,500 btu per pound . the flow properties charted were temperature, pressure, density, velocity, dynamic pressure, mach number, reynolds number, molecular weight fraction, and mass flow . equilibrium flow properties through normal shock waves were also included .
free-flight measurements of the static and dynamic . charts have been prepared relating the thermodynamic properties of air in chemical equilibrium for temperatures to 15,000 k and for pressures from 10 to 10 atmospheres . also included are charts showing the composition of air, the isentropic exponent, and the speed of sound . these charts are based on thermodynamic data calculated by the national bureau of standards .
principles of creep buckling weight-strength analysis of aircraft structures . the possibility of a gradual instability failure of a column under compressive load has been recognized for some time . marin presented an analysis of creep buckling based on a theory of creep bending, but did not take into account the average stress due to axial loading . the theory also neglected the transient (nonlinear) portion of the creep curve . in efficient column design, the average stress should be relatively high in comparison with the bending stresses,. that is, the column should be as straight as possible and the slenderness ratio should not be too great . under these conditions marin’s theory is not directly applicable, although it gives good agreement with tests of columns having large slenderness ratios or large eccentricities .
principles of creep buckling weight-strength analysis . published work on creep buckling has implied that failure of columns after a critical time is caused by initial imperfections . such analyses are relatively complex and ultimately leave the choice of selecting the proper value of the initial imperfection to the designer . furthermore, recent test results on creep buckling of columns have indicated that there is a random and relatively unimportant effect of small initial imperfections on the critical time . to avoid the difficulties associated with initial imperfections, a formulation of the creep buckling phenomenon in terms of classical stability theory is presented . the theory permits the extension of known solutions for plastic buckling of certain thin plates and shells to creep buckling problems .
principles of creep buckling weight-strength analysis . a problem of creep stability of columns and plates is considered . in an analysis use is made of two forms of the creep theory based on the strain hardening hypothesis . for a uniformly compressed palte a comparison is made between the results according to the flow theory and strain theory .
principles of creep buckling weight-strength analysis . the determination of column deflections and column buckling loads has been considered for many years . the available theories, however, do not provide for materials which creep with time under constant loads . for the design of structural members made of these materials, a consideration of creep may be of practical importance . plastics, concrete, and some metals creep at normal temperatures while other metals creep only at high temperatures and at stress values beyond the yield point . a consideration of creep in the design of some structures appears appropriate in view of the modern developments in plastics and the presence of high stress values which are sometimes beyond the yield stress . this paper gives a rational theory for predicting creep deflections in columns . a special case using this theory is applied to the interpretation of some preliminary tests of an aluminum alloy .
principles of creep buckling weight-strength analysis . the relation of the time-dependent tangent-modulus load–as conceived by shanley–to actual column capacity is clarified . it may be interpreted as a limiting case of the conservative estimate . the time-dependent tangent-modulus load is, therefore, an approximation to a conservative estimate . the approximation, however, may be either conservative or nonconservative when applied to imperfect or real columns . typical cases are discussed and experimental results for two alloys are cited .
note on creep buckling of columns . it appears from librove’s interesting analysis that, for the case of creep buckling of columns, the initial imperfections contained in ordinary columns provide the mechanism by which failure due to creep occurs after a period of time . in fact, it can be concluded from this analysis that a theoretically perfect column that is initially loaded below the time-independent critical load will not buckle at all . this is an interesting contrast to the case of static buckling where small initial imperfections play an insignificant role, since the failing load of an initially imperfect column is substantially the same as that of a theoretically perfect column . it is of interest, therefore, to conjecture whether there is any possible mechanism by which a column containing no initial imperfections can fail as a result of creep when the initial load is less than the theoretical buckling load .
note on creep buckling of columns . the results of short-time elevated-temperature creep tests of objective of obtaining procedures for predicting column lifetime . semiempirical lifetime curves are obtained with the aid of a previously published column creep theory and are used for deriving column curves . the semiempirical lifetime curves are also used to study the effect of varying applied stress and out-of-straightness . in the range considered, small variations in out-of-straightness are found to be of little practical significance,. whereas, small stress variations change the column lifetime considerably . for the range of out-of-straightness encountered in the tests, the data can be presented in plots that do not explicitly include out-of-straightness, and plots of this type should be satisfactory for predicting column lifetime for design purposes .
note on creep buckling of columns . a method for estimating allowable load capacities of columns subject to creep is presented . the method, which utilizes approximate stress distributions derived from isochronous-stress-strain curves to estimate column load capacities, is shown to be conservative for the time for which the estimate is made . an application of the method is made to test data on as-received and on stabilized 24s-t4 aluminum alloy . a comparison of the computed column capacities with experimental capacities indicates that the method is satisfactory for estimating the decrease in capacity with increasing time . easily obtained, time-dependent tangent-modulus loads are discussed . they are interpreted as being approximations to allowable load-capacity estimates . a limited application is made to test data, and the results appear promising . it is concluded that if certain limitations are recognized, the method may prove to be useful because of its simplicity . a presentation of the results of an experimental investigation of the effects of column imperfection and column-material variation is made . it is found that column-capacity variations of the order of 10 per cent can result from column-imperfection differences and column-material variation . the results of an experimental study of the variation of column capacity with temperature of exposure are presented . they indicate that column efficiency, as measured by decrease in capacity, can be acceptable for very short times at the higher temperatures . the efficiency at these higher temperatures falls rapidly, however, with increasing time .
note on creep buckling of columns . the results of short-time creep-buckling and creep-bending tests of slenderness ratio 111 are presented . the tests were performed at 600 f, and strain measurements were taken with high-temperature electric-resistance strain gages . a description of the development of the gages is given in an appendix . the column tests show that the critical time decreases much more rapidly with increasing load than with increasing initial deviation from straightness . the bending tests indicate that the steady creep rate of the curvature is a simple power function of applied moment . these latter results, together with a previously derived creep-buckling theory, are used to develop a semiempirical formula suitable as a guide for the determination of the critical time for columns .
note on creep buckling of columns . this paper describes theory and tests of the creep collapse of long thin aluminum-alloy cylinders under external radial pressure . steady-state creep is assumed in the theoretical derivation . the test temperatures were between 300 and 500 f . the collapse time for each cylinder was calculated theoretically . agreement between theoretical and test results was fair .
note on creep buckling of columns . forty-three cylinders of 40-inch length and 16-inch diameter, made of 5052-0 aluminum-alloy sheets of thickness, were subjected to bending moments constant along the cylinder and in time in an oven which maintained a constant temperature of 500 f during the test . all the cylinders failed by buckling . the time that elapsed between load application and collapse was measured .
note on creep buckling of columns . a theory of creep buckling is presented in which the instantaneous elastic and plastic deformations following the application of a load, as well as the steady creep deformations, are considered in an approximate manner . equations are given from which the critical time, that is, the time elapsing between load application and the collapse of the column, can be computed .
note on creep buckling of columns . the general dynamic equation of creep bending of a beam loaded laterally and axially was derived for a linearly viscoelastic material whose mechanical properties can be characterized by four parameters . the material can exhibit instantaneous and retarded elasticity as well as pure flow . the equation derived was used to obtain the creep bending deflection of a beam in pure bending and of a column with initial sinusoidal deviation from straightness . as expected, the ratio of the creep deflections of the beam in pure bending and the deflections of a corresponding purely elastic structure is identical to the ratio of the creep strain and the corresponding elastic strain of a bar under simple tension or compression . the results of the analysis of the creep deflection of the column showed that the deflections increase continuously with time and become infinitely large only when the loading time is correspondingly large . however, large deflections are obtained in reasonably short periods of time if the applied load is near to the euler load of the column . the deflection-time curves obtained from a numerical example are of the same type as those determined by experiment with aluminum columns .
note on creep buckling of columns . the creep of a slightly crooked section column carrying a constant load is studied theoretically . the material of the column is characterized by a strain-time relationship, under constant uniaxial stress, of the form, where is the total strain, is the constant stress, is the time, and e,a,b, and k are material constants . this form was selected because it applies to at least two alloys–75s-t6 aluminum alloy at 600 f . and a low-alloy steel at 800 and 1,100 f . however, the analysis is intended for any material having creep properties of the same form and for which the material constants are known . a strain-time relationship under variable uniaxial stress, necessary for the column analysis, is formulated from the constant-stress properties with the aid of shanley’s engineering hypotheses of creep . the analysis leads to the conclusion that the lateral deflection approaches infinity–that is, the column collapses–in finite time . results are given showing the maximum length of time the column can support a given load before it collapses and the growth of stresses, strains, and deflections prior to collapse .
note on creep buckling of columns . this paper is concerned with the solution of the creep buckling of columns . instantaneous elastic and plastic deformations, as well as the transient and secondary creep, are considered . formulae for the critical time at which a column fails are presented for integral values of the exponents appearing in the creep law .
note on creep buckling of columns . a phenomenological relation between stress, strain rate, and temperature is suggested to account for the behavior of polycrystalline metals above the equicohesive temperature . the properties of the metal included in the relation are elasticity, linear thermal expansion, and viscosity . the relation may be integrated under various conditions to provide information on creep rates, creep rupture, stress-strain curves, and rapid-heating curves . it is shown that for one material – 7075-t6 aluminum-alloy sheet – the information yielded by the relation for these four applications agrees reasonably well with test data .
note on creep buckling of columns . the phenomenological theory previously proposed in naca technical note 4000 for the behavior of metals at elevated temperatures has been modified to yield transient creep curves by assuming that the metal consists of two phases, each with its own elasticity and viscosity . the extended theory satisfies the basic requirements for a theory of transient creep at elevated temperatures .. that the transient creep be closely connected with the subsequent steady creep, and that the apparent exponent of the time in the transient region be permitted wide variations between 0 and 1 . from this theory it is possible to construct nondimensional creep curves which extend continuously from the transient region into the steady-state region . the corresponding family of creep curves for any metal may be obtained from the nondimensional family by use of appropriate constants . the constants required are those obtained from steady creep measurements, together with two additional constants which represent the difference between the phases . the transient creep curves resulting from this theory are compared with the experimental curves for pure aluminum, gamma iron, lead, and agreement is found .
note on creep buckling of columns . a variational theorem is presented for a body undergoing creep . solutions to problems of the creep behavior of plates, columns, beams, and shells can be obtained by means of the direct methods of the calculus of variations in conjunction with the stated theorem . the application of the theorem is illustrated for plates and columns by the solution of two sample problems .
note on creep buckling of columns . some general topics in elastic stability are discussed . in particular, attention is given to the relationship between adjacent-equilibrium-position and energy techniques, to the effects of nonlinearity, and to the sensitivity of certain stability problems to the character of the loading .
note on creep buckling of columns . a general variational theory of elastic stability that was originated by e. trefftz (1) is applied to the problem of buckling of rings of rectangular cross section subjected to uniform external pressure . the theory is believed to be more rigorous than previous treatments of the problem, since it avoids conventional assumptions of curved-beam theory, such as the assumptions that plane sections remain plane and that radial stresses vanish . the classical result of levy (2) is confirmed for a ring of infinitesimal thickness . new results are obtained which show the effect of the finite thickness of a ring on the coefficients in the buckling formula .
on the conservativeness of various distributed force systems . the necessity of determining the conservativeness of force systems in instability problems is discussed in reference 1 . it is shown that, whereas kinetic methods are generally applicable for the determination of instability loads, the statical methods usually employed are valid only for conservative and nongyroscopic systems . small changes in the character of the loading could make an otherwise conservative system nonconservative and cause a large change in the magnitude of the buckling load . the buckling load of the cantilever column example in reference 1 is, for tangential end loading, eight times that for conservative, constant directional loading .
the design of tubes under uniform external pressure on the basis of assumed inaccuracies . since the failure of tubes under uniform external pressure depends very much upon the various kinds of imperfections in them, it seems logical to derive a design formula for such tubes in which the quantities depending on imperfections will appear explicitly . the most common imperfection in tubes is an initial ellipticity, the limiting value of which in each type of tube is usually well-known from numerous inspection measurements . the deviation of the shape of the tube from a perfect circular form can be defined by the initial radial deflections w’ .
note on creep buckling of columns . a long, thin-walled cylindrical shell is loaded by a uniform external pressure . equations are developed for the time behavior of the shape of the cross section under the following conditions .. formations expressible by a power creep law,. (b) the initial and subsequent mode shape of the deviations from circularity of any cross section is two-lobed,. and (c) the shell construction is of the sandwich type, with concentric cylindrical membranes taking normal stresses and an annular core supporting shear without deformation . explicit solutions are obtained for the particular case of the cubic creep law . it is shown that the nondimensional amplitude of the cross-sectional mode shape (briefly, shape factor) will become infinite in a finite time . curves of shape factor versus time and of collapse time versus initial value of the shape factor are presented . also given are an explicit expression for and a curve of the expected variation in collapse time owing to uncontrollable deviations from a nominal initial value of the shape factor . it is shown that the expected variation is small if the nominal initial shape factor value is sufficiently large .
note on creep buckling of columns . the stability of a compressed elastic ring has been studied by a method which can be extended to solve the problem of the stability of a flexible heavy structure spread by a system of hoops as in a crinoline skirt . the original work by levy, which was developed by timoshenko and love, cannot be generalized to problems in which the compressing forces are affected by the deformation of the ring . it is shown that the load at which a ring will buckle depends not only upon the magnitude of the load but also upon its first derivative relative to the radial distance . a positive derivative causes the ring to buckle at a higher load . when this result is applied to a cone of heavy and loosely draped fabric spread by a rigid hoop of radius and a larger and flexible hoop of radius below it, both hoops being in horizontal planes, then various modes of buckling other than oval are possible according to the relative magnitudes of and . it is found that oval buckling changes to three-wave buckling when three-wave changes to four-wave when, and as and approach nearer to equality the buckled form progressively changes to more waves . when applied to a structure spread by many horizontal hoops of which the top one is rigid and oval, it is found that all other hoops, if each is designed to the criterion, will have the same absolute deviation from circularity as the rigid hoop . if any one hoop is designed so that, then the oval shape of the rigid hoop is magnified on all flexible hoops .
on transverse vibrations of thin, shallow elastic shells . according to marguerre (proc. 5th internat. congress appl. mech., cambridge, mass., 1938, wiley, new york, shells are governed by three simultaneous differential equations in the three displacements . the author has considerably simplified this theory for the case of transverse vibrations by ignoring the longitudinal inertia terms, thus reducing the problem to that of solving two simultaneous differential equations in a stress function and one displacement component . this simplification is justified by an order-of-magnitude analysis, and illustrated by considering the vibrations of a paraboloidal shell with a rectangular boundary .
on transverse vibrations of thin, shallow elastic shells . it is shown that a perfectly straight bar, subjected to a state of uniform shear stress, will buckle, in a manner similar to a column under compression, if the shear stress exceeds a certain critical value . the buckling equations are obtained by the newtonian approach and also by the application of the principle of minimum potential energy . in order to provide additional insight into this buckling mechanism, a simple model is introduced and analyzed .
on transverse vibrations of thin, shallow elastic shells . the deformation and complete stress distribution are determined for each of the following edge-loaded thin shells .. (1) a right circular cylinder, (2) a frustum of a right circular cone, and (3) a portion of a sphere . the locations of maximum circumferential and meridional stresses are also found . equations are developed for discontinuity shear and moment at the following junctions .. circular cylinder, (2) axial change of thickness in a cone, (3) change of thickness in a portion of a sphere, cylinder and a portion of a sphere,
on transverse vibrations of thin, shallow elastic shells . an experimental investigation was made (1) to evaluate previously published theoretical procedures for the prediction of stress distribution for cases of radially symmetric abrupt change in wall thickness of thin-walled cylinders subject to internal pressure and (2) to investigate the significance of stresses attributable to the presence of thickness changes typical of design practice . one theory was adequate in itself for solution of the case of continuous middle surface,. use of the second theoretical procedure was required to determine the additional stresses arising from discontinuous middle surfaces at the change in thickness . comparisons were made between theoretical and experimental stress distributions for cases with continuous middle and continuous inner surfaces for radially symmetric changes in wall thickness of a cylinder subject to internal pressure for diameter to larger wall thickness ratios of 117 and 28 and for the case of a continuous outer surface for a ratio of 28 . in all tests the ratio of wall thicknesses at the change in wall thickness was 0.4 . there was reasonably good correlation between theoretical and experimental curves of stress distribution . on the basis of this correlation, it was concluded that the applicable theories were valid . it was shown that inclusion of the stresses arising from the condition of discontinuous middle surfaces at a change in thickness has an important effect on stress distribution . in the case of a cylinder with a continuous outer surface, the maximum mean effective stress was of sufficient magnitude to indicate that this geometry should be avoided in design if possible . the maximum mean effective stress was not increased to a significant degree by the presence of a change in wall thickness in the other cases .
on transverse vibrations of thin, shallow elastic shells . experimental measurements of pressures and heat-transfer rates over three blunt afterbodies of small fineness ratio in fully separated wakes are presented . the afterbodies are generally similar in shape but have different stepdown heights from the end of the forebody . tests were made by means of a new shroud technique over a range of reynolds numbers closely corresponding to typical flight conditions at mach numbers on the order of 20, considering models on the order of 5 ft. in diameter at about 120,000 ft. altitude . stagnation temperatures on the order of 1,300 r. to strictly speaking, the test flows correspond to prototype flows which would be created by a forebody consisting of a sharp or slightly blunted 54 half angle cone which turns cylindrical for a short distance and then connects with the afterbody . judiciously interpreted, the results may be considered to have a somewhat wider applicability for approximation purposes . the results are presented and compared with each other in terms of nondimensional variables based on flow conditions at the end of the forebody . the pressure distribution along an afterbody is seen to be roughly uniform in each run . for a given point on an afterbody, the ratio of pressure to the stagnation pressure at the forebody end (or exit) decreases with increasing stagnation pressure or reynolds number . the present pressures and pressure-reynolds number variations (fig. 8) are compared with values obtained from chapman’s mach 2 or 3 base-pressure data,. qualitative and some quantitative agreement is noted . in the reynolds number range comparable to those of the present tests, chapman’s exit boundary layers were considered to be laminar . an approximate check of the heat-transfer rate at the forebody end in the present tests also indicates a laminar rate . no information was obtained concerning the possible transition of the free-mixing separated boundary layer covering the wake . an adverse pressure gradient on the cylindrical end of the forebody figs. 7(a) and 7(c) was observed . heat-transfer rates are seen to be roughly uniform over an afterbody in each run, although some increase in the streamwise direction is noted . the afterbody nusselt number (n) varies with the reynolds number (r evaluated at the forebody end) roughly in the manner n r where generally (fig. 13) . heat rates on the rear faces of the afterbodies are almost twice the values on the sides . the heat rates on the large-step body are higher than those on the body of zero stepdown height . in an addendum, it is shown that the prandtl-meyer expansion angle of the flow leaving the afterbody increases with increasing test reynolds number, and that the corresponding local mach number square increases linearly with reynolds number . the effect is to keep the local wake reynolds numbers virtually constant with increasing test reynolds number while the afterbody heat rates increase sharply . the expansion angle on the afterbody of zero stepdown height is significantly smaller than on the stepped down bodies,. this may affect the decreased heat rates on this body .
analysis of stresses in the elements of shell structure . the love-meissner analysis for thin shells has previously been applied to cones of uniform wall thickness, and solutions for the stress resultants were given in terms of kelvin’s functions . since tabulation of these functions for large arguments is not practical, considerable computation was still required . in the present paper, the authors define special functions which eliminate the necessity of evaluating kelvin/s functions and which may be used with simple algebraic and trigonometric functions to compute the boundary forces and displacements for cones for various loading conditions . these special functions also make clear the magnitude of errors which result from geckeler/s and other approximate solutions .
on transverse vibrations of thin, shallow elastic shells . the report presents information on the stress problems in the analysis of pressurized cabins of high-altitude aircraft not met with in other fields of stress analysis relating to aircraft . the material may be roughly divided into shell problems and plate problems, the former being concerned with the curved walls of the cabin or pressure vessel and the latter being concerned with small rectangular panels of its walls, framed by stiffeners, but not necessarily plane .
on transverse vibrations of thin, shallow elastic shells . a numerical analysis is given for the solution of the general equations of thin shells of revolution subjected to rotationally symmetric pressure and temperature distributions . the basic differential equations are in a very general form, which permits the geometry of the shells considered, to be specified by discrete data points . the analysis determines elastic stresses, strains and displacements for multi-layer and multi-sectional shells of revolution . surface loads, temperatures, thicknesses and material properties may vary arbitrarily in the meridional direction . temperatures and material properties can also vary through the thickness . the solution is obtained by direct computation using a numerical method that employs two by two coefficient matrices,. and hence avoids the problems of slow convergence . the solution has been programmed in a semi-algebraic language which can be used on most high speed computers . comparisons of numerical solutions to known exact and approximate solutions of the thin shell equations are made to demonstrate the accuracy of this method .
on the theory of thin elastic shells . general equations for the symmetrical finite deflection of a rotationally symmetric thin shell are first obtained . for small deflections these equations are reduced to a pair of equations for the change of slope of the shell surface and the product of the undeformed radius of the shell to the radial stress . this choice of dependent variable is shown to be advantageous . two cases of shallow shells give particularly simple solutions .. parabolic shells of nth degree and uniform thickness, and parabolic shells of mth degree with thickness varying as mth power of the distance from the apex . for the first case, the solutions can be expressed in terms of cylinder functions,. for the second, in powers of the paper concludes with a discussion of the asymptotic solutions for small thickness .
the bending strength of pressurized cylinders . discussion of previously presented experimental data for the loading of pressurized cylinders, in terms of membrane theory .
the bending strength of pressurized cylinders . a theoretical solution is given for the critical stress of thin-walled cylinders loaded in torsion . the results are presented in terms of a few simple formulas and curves which are applicable to a wide range of cylinder dimensions from very short cylinders of large radius to long cylinders of small radius . theoretical results are found to be in somewhat better agreement with experimental results than previous theoretical work for the same range of cylinder dimensions .
the bending strength of pressurized cylinders . be described here is attributed to the russian investigator v. g. galerkin, whose original papers are inaccessible to the present writer . his knowledge of the method is derived from a description given in a paper by e. p. grossman . grossman states that the method was given by galerkin in his treatise p. 897), and that applications to oscillation problems were first made by v. p. lyskov . it is pointed out by grossman that galerkin’s process in applications to mechanics leads to the same results as lagrange’s principle of virtual work, but employs a special co-ordinate system . the method of galerkin belongs to the same general class as those of rayleigh and ritz, for it seeks to obtain an approximate solution of a differential equation with given boundary conditions by taking a function which satisfies these conditions exactly, and proceeds to specialise the function in such a manner as to secure approximate satisfaction of the differential equation . the selected function is a linear combination of n independent functions, and the coefficients are determined by a process of integration . the galerkin process can be considered from two points of view, (a) simply as a means for the approximate solution of differential equations, and treatment of problems concerning the statics and dynamics of elastic and other deformable bodies . these two aspects are treated separately in parts 1 and 2 of the paper respectively, and will now be briefly discussed . which satisfies the boundary conditions, in the differential equation be . since the result should be zero, is the error in the differential equation . then the galerkin process consists in choosing the n coefficients in the function in such a manner that n distinct weighted means of the error, taken throughout a certain range of representation, shall all be zero . as a generalised force, and the multipliers used to weight the errors are the virtual displacements corresponding to increments of each of the generalised co-ordinates in turn . thus the vanishing of the weighted mean is here interpreted as the vanishing of the virtual work in the appropriate displacement . the degree of accuracy attaindd can be increased indefinitely by increasing the number of independent functions employed, but this entails a great increase of labour . however, when the functions are well chosen, an excellent approximation can be obtained by the use of a very small number, as is sufficiently shown by the examples included in this paper .
a small deflection theory for curved sandwich plates . a small-deflection theory that takes into account deformations due to transverse shear is presented for the elastic-behavior analysis of orthotropic plates of constant cylindrical curvature with considerations of buckling included . the theory is applicable primarily to sandwich construction .
elastic constants for corrugated core sandwich plates . the sandwich plate consisting of corrugated sheet fastened between two face sheets is considered . application of existing theories to the analysis of such a sandwich plate requires the knowledge of certain elastic constants . formulas and charts are presented for the evaluation of these constants . the formulas for three of these constants were checked experimentally and found to give values in close agreement with the experimental values .
compressive buckling of simply supported curved plates and cylinders of sandwich construction . theoretical solutions are presented for the buckling in uniform axial compression of two types of simply supported curved sandwich plates .. the corrugated-core type and the isotropic-core type . the solutions are obtained from a theory for orthotropic curved plates in which deflections due to shear are taken into account . results are given in the form of equations and curves .
the stability of thin-walled unstiffened circular cylinders under axial compression including the effects of internal pressure . in the design of high-speed aircraft the importance of unpressurized and pressurized monocoque cylinders necessitates a reliable analysis procedure for the compressive buckling of cylindrical shells . analysis by the classical small-deflection theory has proved inadequate . recent large-deflection theoretical treatments of the problem have shown reasonable correlation with experiments but require a prior knowledge of the initial imperfections of the cylinder . developed in this paper is a semiempirical procedure which permits a compressive buckling analysis of cylindrical shells with a knowledge of the cylinder geometry only . this analysis is achieved by correlating experimental data statistically with theoretical parameters . in order to provide data not previously available, an extensive series of axial compression tests of pressurized cylinders has been performed . these data, together with all other known test data, are analyzed semiempirically . in the analysis best-fit curves are presented using theoretical parameters and shapes of curves where applicable . unpressurized and pressurized cylinder compressive buckling curves are then developed as 90 per cent probability curves from the test data . in general, these statistically defined design curves are significantly lower than previously available design curves .
recent advances in the buckling of thin shells . the importance of the field of shell analysis is evidenced by the fact that in august, 1959, the international union of theoretical and applied mechanics conducted a symposium on the theory of thin elastic shells in delft, holland . this special meeting was attened by approximately 65 scientists in this field from 14 countries . this symposium indicated that considerable interest currently exists in such relatively new topics as the buckling of bimetallic shells, pressurized shells, creep buckling, and dynamic buckling, as well as in the more traditional problems involving isotropic shells of various geometries .
spherical cap snapping . a nonlinear boundary value problem for the determination of the rotationally symmetric deformations of a clamped spherical cap under external pressure is solved by finite differences . the numerical solutions are obtained by employing a previously developed iteration procedure . a special case of the difference equations is solved explicitly and yields a justification of the iteration method as well as insight into the properties of the more accurate numerical solutions . buckled and unbuckled equilibrium states are obtained and the shape of the pressure-deflection curve which is usually assumed for these states is verified for a large class of caps . close estimates are given for the upper and lower buckling loads and an intermediate buckling load–i.e., the /dead-weight/ load . the stresses and deflections in the buckled and unbuckled states are examined and compared with an asymptotic solution valid in the interior of very thin shells . boundary layers are found to develop in the buckled states both as the loading increases and as the thickness of the shell decreases .
iterative solutions for the non-linear bending of circular plates . the authors study non-linear von karman equations for bending of a thin circular plate under uniform normal pressure . discussion is mainly concerned with plates clamped at the edges and with zero radial displacement, but analysis is valid for other edge conditions . solution is by an iterative procedure whose convergence properties are studied by means of integral equations . method is then applied to finite difference formulation of the differential equations in order to obtain numerical solutions . numerical results are compared with previous work by other authors and the advantages of the present method are indicated .
non-linear bending and buckling of circular plates . iterative solutions of finite difference approximations of the non-linear von karman plate equations are presented . results are obtained for circular plates under a variety of boundary conditions subjected to either uniform lateral pressure or uniform edge thrust . the solution, carried out numerically on the aec univac at new york university, yields a complete description of stresses and deflections for an apparently unlimited range of load parameters . in addition, boundary layer phenomena are discussed . for computing purposes, this iterative method proves to be superior to the previously used power series method and may be applicable to other non-linear problems .
axisymmetric large deflections of circular plates subjected to thermal and mechanical load . this paper is concerned with the nonlinear axisymmetric analysis of circular plates with in-plane edge restraint . both temperature and mechanical loads are accommodated as an extension of investigations performed for the isothermal mechanical loading problem . an exact mathematical formulation within the framework of the v. karman large strain-displacement relations is developed . the equilibrium equations and boundary conditions are then derived by utilizing the calculus of variations for arbitrary axisymmetrical temperatures and normal distributed loading . the satisfaction of equilibrium and compatibility equations requires the solution of two simultaneous nonlinear ordinary differential equations subject to the prescribed boundary conditions . analytical solutions of such equations are apparently not possible and therefore numerical procedures must be employed . a finite difference procedure utilizing /relaxed iterations,/ developed by h. keller and e. reiss, and employed by them for the solution of isothermal problems with apparently unlimited load parameter ranges, is used here for combined thermo-mechanical problems . numerical results are presented for the special case of a simply supported circular plate with radially immovable boundaries, subject to a uniform pressure and an arbitrary temperature variation through the thickness tained for a large range of temperature and load parameters . however, because of space limitations, only a limited amount of data are presented in this paper .
the uniform section disk spring . the authors point out in this paper that initially coned annular-disk springs of uniform cross section may be proportioned to give a wide variety of load-deflection curves not readily obtainable with the more conventional forms of springs, and that, although the versatility of this type spring has long been indicated, the formulas available have not been presented in a manner to disclose readily the effect of spring proportions on characteristics . therefore the authors have derived the formulas presented in this paper with the intention that the formulas will aid the designer in arriving at suitable characteristics by choice of spring geometry . these new formulas have been in use for several years at the general motors corporation research laboratories section, and their reliability has been checked by tests of springs used in a variety of special test equipment .
the conical disk spring . this paper presents approximate formulas to describe the behavior of the conical disk spring . it is assumed that the shallow conical shell remains conical when subjected to the axially symmetric edge loads . the principle of stationary potential energy is used to derive the relations between load and deflection . formulas relating the applied loads, stresses, and deflections are given for several types of edge constraint . the analysis is essentially a refinement and extension of the previous work of almen and laszlo .
the nonlinear conical spring . the large symmetric deformations of shallow conical shells are of interest in the design of nonlinear conical disk springs . in most applications a uniformly distributed axial load acts at the inner and outer edges,. these edges are otherwise free . several approximations have been proposed to describe the behavior of these springs . a first approximation (1) is based on the assumption that meridional strains are negligible . this requires that the shell remain conical after deformation and also that the extensional strain of meridional lines on the middle surface vanish . another approximation (2) retains only the assumption that the shell remains conical . the first assumption satisfies neither of the two boundary conditions at the free edges,. the latter violates the condition of vanishing moment at the free edges . recently the authors presented a series solution (3) for a special case, namely, the case of an annular plate under similar loading . numerical solutions for the shallow conical shell under these conditions of load have also been obtained (4) . an examination of these results indicates that the meridional bending stresses are of much smaller magnitude than the circumferential bending stresses . hence the present analysis is based on the neglect of the meridional bending moment .
buckled states of circular plates . authors discuss the thin elastic circular disk of constant thickness subjected to a constant compressive thrust applied at its edge . the analysis presented is based upon the nonlinear von karman equations of plate theory and is applied to disks with completely clamped and completely, simply supported edges .
turbulent mixing of a rocket exhaust jet with a supersonic stream including chemical reactions . the equations for the turbulent mixing of a two-dimensional supersonic jet issuing into an ambient supersonic stream are formulated . both streams consist of a mixture of chemically active and possibly reacting gases, therefore any heat release by chemical reaction is included,. the net mass rate of production of species is obtained on the assumption that the reaction rate constant is given by an expression reducible to the classical arrhenius law . the equations first given in terms of the x, and y coordinates, are expressed in dimensionless form and in terms of the x and coordinates, where is the stream function . the resulting expressions are all of the /heat conduction/ type,. they are put in a finite difference form by using the crank-nicolson method of substituting finite difference approximations for both the /time/ and /space/ derivatives . the mixture is assumed to consist of six species, namely h2o, h2, o2, co2, co, and n2, and the oxidation of h2 and co is assumed to take place according to a single-step chemical reaction . the solution of the problem is based on the simultaneous solution of 8n linear algebraic equations in 8n unknowns, n being the number of internal grid points at every step in the x-direction, and 8 the total number of unknowns at each grid point, namely velocity, temperature, and concentration for each of the six species . a method of obtaining initial and boundary conditions from available inviscid jet flow solutions is discussed . the equations are programed for calculation on an ibm-704 computer . finally, one typical case is considered, and plots of velocity, temperature, and concentration profiles are given for the initial stages of development of the mixing layer .
an experimental and theoretical investigation of second-order wing-body interference at high mach number . the second-order wing-body interference theory of landahl and beane is used in the theoretical calculation of the pressure distributions over the wing of a wing-body combination . results are compared with experimental values obtained from wind-tunnel tests, at a mach number of 7.35, on a cone-cylinder non-lifting body with a triangular wing of wedge section set at incidences of 0, 3, 6 and 10 . it is shown that interference effects can be very large and can be calculated theoretically with good accuracy .
on obtaining solutions to the navier-stokes equations with high speed digital computers . the purpose of this paper is to show how to obtain steady state solutions to the navier-stokes equations on a high-speed digital computer . first the relative merits of various finite difference formulae are discussed . thereafter the main part of the paper is concerned with the methods used to solve the finite difference equations and an investigation is made of all the simpler iterative methods .
propeller slipstream effects as determined from wing pressure distribution on a large-scale six-propeller vtol model at static thrust . during static-thrust tests of a large-scale general research model having a tilting wing and double-slotted flaps, static-pressure measurements were made on a wing segment behind one propeller to survey the effects of the slipstream . for the conditions of highest slipstream energy, the hovering end point of aerodynamic parameters for aircraft having vertical and short take-off and landing capability the tilt-wing configuration (zero flap deflection) was a 6 spanwise variation in effective angle of attack in a span of slightly less than 1 propeller diameter . effective changes in camber on the tilt-wing configuration as a result of slipstream rotation, the radial velocity gradient, and the resultant spanwise flow were negative and had a maximum magnitude of less than 2-percent chord . for the deflected-slipstream configuration (double-slotted flaps deflected), effects important to the hovering performance were found, including a 40-percent spanwise variation in effective thrust recovery and a 20 spanwise variation in effective thrust turning .
a free-flight investigation of ablation of a blunt body to a mach number of 13 .1. a five-stage rocket-propelled research-vehicle system was flown to a maximum mach number of 13.1 at an altitude of approximately 78,000 feet to determine ablation characteristics of teflon in free flight . continuous in-flight measurements were made using sensors developed by the national aeronautics and space administration . the sensors were located on the blunted face of a nose cone constructed from teflon, with one at the stagnation point and two others at a surface distance of 0.62 radius on opposite sides of the stagnation point . the ablated-length measurements were in close agreement with analytical predictions . the analytical predictions, upon inclusion of the pertinent material property values, should be applicable to other materials as well as teflon .
wind tunnel measurements of aerodynamic damping derivatives of a launch vehicle vibrating in free-free bending modes at mach numbers from 0. 70 to 2. 87 and comparisons with theory . the aerodynamic damping of a flexibly mounted aeroelastic model with a blunted conical nose and a cylindrical afterbody was measured at mach numbers from 0.70 to 1.20 at several levels of dynamic pressure and two weight conditions and at mach numbers from 1.76 to 2.87 at one weight condition . the first two free-free flexible modes of vibration were investigated . also investigated at mach numbers from 0.9 to 1.2 was the aerodynamic damping in the first free-free modes of a model which had a /hammerhead/ nose (the base diameter of the blunted cone was greater than the diameter of the afterbody which necessitated a reflex angle downstream from the cone base) . two basically different methods, the /electrical power-input/ and the /decaying oscillations/ methods were used to determine the damping and frequencies . the experimentally determined values are compared with some applicable theories . the results of the investigation indicate that the aerodynamic damping in the elastic modes of vibration was small for all configurations tested . the maximum aerodynamic damping measured in the first mode was on the order of damping . the aerodynamic damping was found to be even less for vibration modes higher than the first . reduced-frequency effects were found to be negligible for the range investigated . agreement of calculated aerodynamic damping derivatives with the experimental results was not good . generally, the experimentally determined derivatives were larger than those predicted by the various theories used . the bond-packard theory appeared to give the best agreement for the first free-free vibration mode but gave the worst agreement for the second mode . measurements made on the configuration that had a hammerhead nose indicated small negative aerodynamic damping in the mach number range from 0.95 to 1.00 . aerodynamic stiffness effects were found to be small and within the experimental scatter . (wind-on frequency determination was accurate only to approximately 1 percent .)
plastic stability theory of geometrically orthotropic plates and cylindrical shells . a linear eighth-order equilibrium differential equation for plastic buckling of geometrically orthotropic thin cylindrical shells is derived . this equation is used to obtain explicit solutions for long flat plates and wide columns under axial compression and moderate-length cylinders under external pressure, torsion, and axial compression .
instability analysis of cylindrical shells under hydrostatic pressure . to determine the elastic buckling pressure of simply supported cylindrical shells subjected to lateral and axial hydrostatic forces, various versions of linear bending theories have been employed in the past . for certain shell dimensions, however, the expressions commonly used may yield substantially differing results . in what follows, recent work on this problem by a. e. armenakas and the writer is briefly reviewed . this work consisted primarily in employing a general bending theory of circular cylindrical shells under the influence of initial stress, developed earlier by the same authors, to re-examine the problem mentioned, and compare the results with those of previous investigations . the outcome was the establishment of a simple but accurate expression for the buckling pressure applicable to a wide range of shell dimensions .
design and testing of honeycomb sandwich cylinders under axial compression . experimental results for 36 diameter honeycomb cylinders fabricated with thin (0.010) aluminum faces and cores prove that it is quite feasible to stabilize thin faces so they can be loaded beyond the yield point . the effect of initial imperfections and the various modes of failure are discussed .
buckling of orthotropic and stiffened conical shells . donnell type stability equations for thin circular orthotropic conical shells are presented and solved for external pressure, axial compression and combined loading . the solution is likewise applied to stiffened conical shells . correlation with equivalent cylindrical shells yields a simple approximate stability analysis for orthotropic or ring-stiffened conical shells under hydrostatic pressure . the general instability of stiffened conical shells under hydrostatic pressure is also analysed by a more accurate approach . preliminary experimental results for buckling of ring-stiffened conical shells under hydrostatic pressure are presented and discussed .
stability of thin torispherical shells under uniform internal pressure . the stability of the toroidal portion of a torispherical shell under internal pressure is considered from the point of view of the linear buckling theory . a detailed stress analysis of the prebuckled shell is made employing asymptotic integration . the change in potential energy of the shell is then minimized using a rayleigh-ritz procedure for actual computation of the critical pressure . numerical results reveal that elastic buckling may occur for very thin shells whose material has a relatively high value of the ratio of yield stress to elastic modulus .
ignition and combustion in a laminar mixing zone . the analytic investigation of laminar combustion processes which are essentially two- or three-dimensional present some mathematical difficulties . there are, however, several examples of two-dimensional flame propagation which involve transverse velocities that are small in comparison with that in the principal direction of flow . such examples occur in the problem of flame quenching by a cool surface, flame stabilization on a heated flat plate, combustion in laminar mixing zones, etc . in these cases the problem may be simplified by employing what is known in fluid mechanics as the boundary-layer approximation, since it was applied first by prandtl in his treatment of the viscous flow over a flat plate . physically it consists in recognizing that if the transverse velocity is small, the variations of flow properties along the direction of main flow are small in comparison with those in a direction normal to the main flow . the analytic description of the problem simplifies accordingly . the present analysis considers the ignition and combustion in the laminar mixing zone between two parallel moving gas streams . one stream consists of a cool combustible mixture, the second is hot combustion products . the two streams come into contact at a given point and a laminar mixing process follows in which the velocity distribution is modified by viscosity, and the temperature and composition distributions by conduction, diffusion, and chemical reaction . the decomposition of the combustible stream is assumed to follow first-order reaction kinetics with temperature dependence according to the arrhenius law . for a given initial velocity, composition, and temperature distribution, the questions to be answered are .. (1) does the combustible material ignite,. and (2) how far downstream of the initial contact point does the flame appear and what is the detailed process of development . since the hot stream is of infinite extent, it is found that ignition always takes place at some point of the stream . however, when the temperature of the hot stream drops below a certain value, the distance required for ignition increases so enormously that it essentially does not occur in a physical apparatus of finite dimension . the complete development of the laminar flame front is computed using an approximation similar to the integral technique introduced by von karman into boundary layer theory .
a practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type . three approximate methods for the solution of the nonlinear equation of heat flow in a medium where heat is being generated by a chemical reaction are compared . the equations are where subscripts indicate partial differentiations and q, k, a are
theoretical and experimental investigation of second-order supersonic wing-body interference . approximate second-order solutions for the supersonic flow around wing-body combinations are calculated, using two different theoretical models small and the wing sweep small in comparison with that of the mach cone are considered . the analysis is restricted to such high mach numbers that m-2 1, and an approximate formula common to the two models is then found for the second-order interference term . this formula can also be used to correct experimental pressure distributions for the effect of nonuniformities in the wind-tunnel flow . in order to test the theory, wind-tunnel experiments on non-lifting cone-cylinder bodies in combination with wings of simple shapes were performed . pressure distributions were measured at m 3 and m 4, both around the bodies and on the wings separately, as well as in combination, and it was found that the second-order interference was predicted reasonably well by the simplified theory .
an experimental and theoretical investigation of second-order supersonic wing-body interference, for a non-lifting body with wings at incidence . pressure distributions on the wing of two wing-body combinations are measured experimentally at mach numbers 3 and 4 with the wing at various incidences in the range 0degree to 10degree . the results are compared with theoretical results which include interference effects calculated according to the second-order supersonic wing-body interference theory due to landahl and beane /1/ . this theory, having been tested previously for non-lifting wing-body combinations, is thus tested also for wings at incidence . the agreement between theory and experiment is found to vary with mach number and wing sweepback . for the higher mach number and moderate sweepback the theory gives a good prediction of pressure distribution, but for the most adverse condition of low mach number and large sweepback the theory is found to overestimate the interference effects . this is expected as the theory assumes the sweepback of the wings is small compared with that of the mach line . an empirical guide to the limit of application of the interference theory is given . within this limit the agreement between theory and experiment is found to deteriorate only a little with increase of incidence, over the range tested .
an approximate method for determining the displacement effects and viscous drag of laminar boundary layers in two-dimensional hypersonic flow . a simplified approximate theory is presented by means of which the laminar boundary layer over an insulated two-dimensional surface may be calculated, a linear velocity profile being assumed, and an estimate made of its effect in changing the pressure distribution over the profile upon which the boundary layer is formed . skin friction is also determined . comparisons of results from this theory are made with experimental results at a mach number of 6.86 and a reynolds number of
a method of solution with tabulated results for the attached oblique shock wave system for surfaces at various angles of attack, sweep, and dihedral in an equilibrium real gas including the atmosphere . a new method is derived for solving the attached oblique shock-wave system for surfaces at various angles of attack, sweep, and dihedral in any real gas in equilibrium . results are tabulated for the following ranges .. angle of attack, 0degree to 65degree,. angle of sweep, 0degree to 75degree,. angle of dihedral, 0degree to 30degree,. mach number, 3 to 30,. and /effective specific-heat ratio/ parameter, 1.10 to 1.67 . both the method and tabulated solutions are easily adaptable to flight in any gas or in the atmosphere of any planet . an illustrative example is presented based on the ardc 1956 model atmosphere .
the steady flow of a viscous fluid past a circular cylinder at reynolds numbers 40 and 44 . this paper describes the numerical solution of the complete navier-stokes equations for the steady flow of an incompressible viscous fluid of unlimited extent past a circular cylinder at reynolds number 40 . a new device developed for the numerical solution is described . the results of the investigation are .. good agreement with experimental results . higher reynolds numbers even though they may not exist in nature . a solution has been obtained at reynolds number 44 but it has not been carried to the same accuracy as the solution at reynolds number 40 . portion of the cylinder continues to increase with reynolds number in such steady-state solutions up to a reynolds number 44 and no indication has been found that this process will not continue as the reynolds number is increased beyond 44 .
finite difference formulae for the square lattices . the paper gives approximate formulae for derivatives (including combinations like and ), and integrals, of a function of two independent variables, in terms of its values at nodes of a square lattice, primarily for use in the numerical solution of partial differential equations . consideration is given to the form, as well as to the magnitude, of the leading terms in the error, and what is believed to be for most purposes optimum combinations are thus selected for the simpler compact sets of nodes .
viscous flow round a sphere at low reynolds numbers . /l40/ . relaxation methods are outlined, and the present problem formulated in modified spherical polar co-ordinates . the results of calculations made for r 5, 10, 20, 40 are presented in the form of stream function and vorticity distributions,. and further results of pressure distributions, velocity distributions, and drag coefficients, calculated from them . these results are shown to compare favourably with experimental work, showing a steady trend from symmetrical stokes’s flow, towards boundary layer flow . the phenomenon of separation of the forward flow and development of a circulating wake, is explained and illustrated, the first formation of a wake being at r 17 .
numerical solution of the navier-stokes equations for the flow around a circular cylinder at reynolds number 40 . the steady two-dimensional flow around a circular cylinder submerged in a viscous fluid for the case r 40 is investigated, integrating numerically the exact navier-stokes equations . the main results are as follows . /i/ the steady flow solution exists even for the reynolds number as high as 40 . moreover, it seems that the solution goes over smoothly to the solution of the kirchhoff discontinuous flow theory which seems to be the limiting flow for the case r . /ii/ the flow pattern and the coefficients of pressure and drag are in good agreement with the experimental data .
the flow past pitot tube at low reynolds numbers, part 1-dash the numerical solution of the navier-stokes equations for steady viscous axisymmetric flow, part 2-dash the effects of viscosity and orifice size on a pitot tube at low reynolds numbers . in this report numerical methods used to solve the navier-stokes equations for steady viscous two-dimensional flow are extended to include the case of axial symmetry . the equations and their finite difference approximations are derived working in cylindrical polar co-ordinates with the stokes’ stream function and the vorticity as variables . a new method of dealing with the boundary conditions is given . the effects of viscosity and orlfice size on a blunt-nosed pitot tube have been theoretically investigated up to a reynolds number of ten, where the reynolds number has been based on the radius of the tube . results are expressed in terms of a pressure coefficient where p is the pressure measured in the tube, p the density of the fluid, and p and u the static pressure and velocity in an undisturbed flow at the position of the tube . the values of c for a blunt-nosed tube are found to be less than those for tubes with hemispheroidal heads, but always greater than unity in the range considered . the effect of the orifice size is to decrease c as the orifice size increases, this decrease is very small but increases with the reynolds number . at a reynolds number of ten the decrease is at most five per cent of the value of c when there is no orifice . it is suggested that the decrease of c below unity found in some experimental investigations at a higher reynolds number could be due to the effects of orifice size .
an investigation of fluid flow in two dimensions . flow of an inviscid fluid . -dash there are in existence several methods of obtaining numerical solutions to the two-dimensional flow of a perfect fluid for given boundary conditions . part 2 of the present paper gives a method of obtaining a numerical solution for viscous steady flow solution of the simpler problem, illustrating it with examples bearing on the experimental work described in part iv .
the flow past circular cylinders at low speeds . this paper deals chiefly with calculations and experiments on the flow past circular cylinders, but the arithmetical methods of solution of the equations of steady viscous flow proposed and used in section i, are applicable to other equations and may be of interest .
note on the convergence of numerical solutions of the navier-stokes equations . a criterion is given for the convergence of numerical solutions of the navier-stokes equations in two dimensions under steady conditions . the criterion applies to all cases of steady viscous flow in two dimensions and shows that if the local ‘mesh reynolds number’, based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge .
a note on the numerical solution of fourth order differential equations . an old numerical method of solving fourth order differential equations is put in relaxation form . the higher order correction terms are included and the technique is illustrated by an example . the method has the advantage of being more rapidly convergent than the usual relaxation procedure for fourth order equations . some comments are made on the numerical solution of the viscous flow equation .
convergence rates of iterative treatments of partial differential equations . the development of high-speed digital computers has made feasible the numerical solution by iterative methods of some partial differential equations . the convergence rates of several such iterative methods are estimated here . it is found that with the familiar elementary iterative methods some quite simple problems require prohibitive computational labor . the iterative methods here considered are related to the various forms of the southwell /relaxation method/ in that they involve successively applied local corrections to improve an approximate solution . however, these iterative methods are routinized in conformity with the requirements of automatic computers while the relaxation method is flexible and depends in an essential way on the skill of its practitioners .
iterative methods for solving partial difference equations of elliptic type . this paper considers linear systems /1/ where a includes matrices of a sort frequently occurring in the solution of elliptic partial differential equations by difference methods /in particular, a o/ . rewriting superscript is number of iteration cycle/ are used to compute u when u are used . also, one may /over-relax/ .. ser. a. 210, 307-357 /1910/ who suggested changing from time to time to speed up convergence . in the present paper over-relaxation /with fixed w/ is combined with immediate introduction of newly-computed u’s, a la gauss-seidel . various theorems on convergence are proved$. in particular, it is shown that there exists an ordering of the equations and an optimum value wb such that in general /3/ converges much more rapidly than the gauss-seidel method /w 1/ . means are suggested for estimating wb,. the sensitivity of the rate of convergence to the choice of w is studied . the paper concludes with a theoretical comparison of gauss-seidel and the method proposed, /successive over-relaxation/, for solving dirichlet’s difference problem over a square using a high-speed computing machine .
aerodynamic characteristics of propeller-driven vtol aircraft . this paper discusses the two major configurations that are usually considered for achieving vtol while keeping the fuselage essentially horizontal – that is, the tilt-wing and the deflected-slipstream configurations . because of the high turning losses incurred by deflected-slipstream configurations in hovering and because of the wing-stalling problem of the pure tilt-wing configurations during the transition, it appears that a combination of the two principles should be used . this tilt-wing and flap configuration should make use of a programed extensible-chord slotted flap together with a leading-edge high-lift device in order to avoid the performance and handling qualities problems associated with wing stalling during the transition while keeping the wing area as low as possible for efficiency in cruising flight .
pressure distribution and force measurements on a vtol tilting wing-propeller model . pt .ii, analysis of results . this report presents in graphical form the results presented in reference 1 from pressure distribution and force measurements on a half-wing model of a twin-engined tilt-wing vtol configuration . the profound influence of the slipstream on these results is discussed in some detail .
data from a static thrust investigation of a large scale general research vtol-stol model in ground effect . the model was tested at two different elevations with the wing pivot at 1.008 and 2.425 propeller diameters above the ground . the slipstream of the propellers was deflected by tilting the wing and propellers, by deflections of large-chord trailing-edge flaps, and by combinations of flap deflection and wing tilt . tests were conducted over a range of propeller disk loadings from 7.41 to 29.70 pounds per square foot . force data for the complete model and pressure distributions for the wing and flaps behind one propeller were recorded and are presented in tabular form without analysis .
wing-nacelle-propeller interference for wings of various spans . force and pressure distribution tests . an experimental investigation was made in the n. a. c. a. full-scale wind tunnel to determine the effect of wing span on nacelle-propeller characteristics and, reciprocally, the lateral extent of nacelle and propeller influence on a monoplane wing . the results provide a check on the validity of the previous research on nacelles and propellers with 15-foot-span wings tested in the the scale propeller and the n. a. c. a. cowling used in the former researches were tested in three typical tractor locations with respect to a thick wing of 5-foot chord and 30-foot span . the span was progressively reduced to 25,20, and 15 feet and the same characteristics were measured in each case . the efficiency factors–propulsive efficiency, nacelle drag efficiency, and net efficiency–were obtained for each wing length by means of force tests and the values are compared to determine the effect of span . pressure-distribution measurements show the lateral extent of the nacelle interference and the propeller-slipstream effect on the span loading for the various conditions . complete polar curves and curves showing the variation of nacelle drag with lift coefficient are also included . force and pressure-distribution tests concur in indicating that, for engineering purposes, the influence of a nacelle and of a propeller, in a usual combination, may be considered to extend laterally on a wing the same maximum distance, or about five nacelle diameters or two propeller diameters outboard of their common axes . all important effects of scale nacelle-propeller combinations may be measured within practical limits of accuracy by tests of a 15-foot-span wing .
induced interference effects on jet and buried-fan vtol configurations in transition . recent investigations of some jet and buried-fan configurations have indicated that in the transition speed range, configurations with considerable area surrounding the jet or buried fan can encounter large losses in lift and nose-up pitching moments due to the pressures induced on the lower surfaces by the interaction of the jet and free-stream flow . the obvious way of minimizing these effects is to reduce the surface area surrounding the jets or buried fans, that is, to consider these effects in the preliminary stages of the airplane design .
investigation of the effects of ground proximity and propeller position on the effectiveness of a wing with large chord slotted flaps in redirecting propeller slipstream downward for vertical take-off . an investigation of the effects of ground proximity and propeller position on the effectiveness of a wing equipped with large-chord slotted flaps in redirecting the slipstreams from large-diameter propellers downward for vertical take-off has been conducted in a static-thrust facility at the langley aeronautical laboratory . the results indicate that, with the propeller thrust axis on the wing chord plane, both the angle through which the slipstream is deflected and the ratio of resultant force to thrust are reduced as the ground is approached . at positions nearest the ground some of the loss in resultant force is regained . lowering the thrust axis below the wing chord plane reduces the adverse effects of the ground and also reduces the large diving moments associated with the slotted-flap arrangement . the static-thrust efficiency of the propellers is slightly reduced by the ground effect .
investigation of effectiveness of large-chord slotted flaps in deflecting propeller slipstreams downward for vertical take-off and low-speed flight . an investigation of the effectiveness of a wing equipped with large-chord slotted flaps in rotating the thrust vector of propellers through the angles required for vertical take-off and for flight at very low speeds has been conducted in the facilities of the langley 300 mph 7- by 10-foot tunnel . under conditions of static thrust and with zero incidence between the thrust axis and the wing chord plane, the slotted flaps were effective in rotating the thrust vector upward about than 10 percent of the thrust . when an auxiliary vane was added above the wing, the thrust vector was rotated upward configuration, vertical take-off could be achieved with an initial attitude of 16 and at airplane weights up to 90 percent of the total propeller thrust . the addition of 10 incidence between the thrust axis and the wing increased the upward rotation of the thrust vector about 10 . for the same turning angle, the diving moments associated with the slotted-flap configurations were approximately twice as large as the diving moments of the configurations with plain flaps and two auxiliary vanes .
qualitative measurements of the effective heats of ablation of several materials in supersonic air jets at stagnation temperature up to 11,000 f. the effective heats of ablation of a number of materials were derived from tests in supersonic air jets at stagnation temperatures ranging from 2,000 f to 11,000 f . the materials included the plastics teflon, nylon, lucite, and polystyrene,. the inorganic salts ammonium chloride and sodium carbonate, several phenolic resins of varied resin content and type of reinforcement,. and a melamine-fiber glass laminate .
experimental ablation cooling . this paper presents the results of an experimental investigation on the ablation of a number of promising materials for heating conditions comparable to those which may be encountered by unmanned reentry satellite vehicles, as well as for higher heating conditions comparable to those associated with reentry ballistic missiles . materials tested included the plastics teflon, nylon, and lucite,. the inorganic salts ammonium chloride and sodium carbonate,. graphite,. a phenolic resin and fiber glass composition,. and the commercial material haveg rocketon . results of these tests indicated heat-absorption capabilities which are several times greater than those of current metallic heat-sink materials . the results with teflon showed that for hemispherical noses there was no apparent effect of size or stagnation-point pressure on ablation rate for the range of variables covered in the tests . for flat-faced configurations, however, there was a definite increase in the ablation rate with increased stagnation-point pressure . the results for the several materials tested at heating rates associated with reentry ballistic missiles showed considerable increase in the effective heats of ablation over the results obtained at lower heating rates . this trend of increased effectiveness with increased heating potential is in agreement with the predictions of ablation theories . comparisons of the results for several materials tested at the higher heating rates showed graphite to have the lowest ablation rate of all materials tested .
an experimental investigation of ablating material at low and high enthalpy potentials . the ablation performance characteristics of a number of materials were derived from tests conducted in a mach number 2.0 ethylene-heated high-temperature air jet having a maximum stagnation enthalpy potential of approximately 1,200 btu lb . the tests were conducted with 6- inch-diameter blunt nose shapes . the surface of most of the materials after testing was generally smooth and the unablated portions of the specimens were in appearance the same as before testing . in all cases, the back or inside surface of the specimens exhibited no evidence of heating . an evaluation of the enthalpy potential effect was obtained by comparison of the present data with previous tests conducted, on the in a subsonic arc-heated air jet . the stagnation enthalpy potential of this facility was approximately 7,000 btu lb . for teflon, the effective heat of ablation increased from approximately 1,250 btu lb to enthalpy potential was increased from
a theoretical study of stagnation point ablation . a simplified analysis is made of the shielding mechanism which reduces the stagnation-point heat transfer when ablation takes place at the surface . the most significant result of the analysis is that the effective heat capacity of the ablation material increases linearly with stream enthalpy . the automatic shielding mechanism is discussed and the significant thermal properties of a good ablation material are given parameters .
an analytical investigation of ablation . an analytical procedure is described which enables the derivation of effective heat of ablation relationships for any type of boundary layer from transpiration cooling results . the procedure enables the inclusion of such effects as the ratio of wall temperature to local stream temperature, surface radiation, and surface combustion . the predicted effective heats of ablation for a three-dimensional laminar stagnation boundary layer for teflon material were in agreement with those derived from tests conducted at boundary-layer enthalpy potentials of 800 and approximately 7,000 btu/lb . the predicted equilibrium surface temperatures on nonablating surfaces behind an ablating material were in agreement with the values derived from tests conducted with inconel cylinders having teflon hemispherical nose pieces .
a sensor for obtaining ablation rates . a variable-capacitance ablation-rate sensor which allows continuous measurements of ablation rates for teflon and similar polymers has been developed and tested in an ethylene-heated high-temperature jet at stagnation temperatures ranging from 2,400degree to 3,800degree f . the data /length changes/ were measured by using the same telemeter equipment as that used in rocket-propelled flight vehicles . test results indicate measurement error to be a maximum of 4 percent between the telemetered length changes and the length changes that were obtained from photographic records of the test .
a five-stage solid fuel sounding rocket system . a five-stage solid-fuel sounding-rocket system which can boost a payload of 25 pounds to an altitude of 525 nautical miles and that of 100 pounds to 300 nautical miles is described . data obtained from a typical flight test of the system are discussed .
pressures, densities and temperatures in the upper atmosphere . averaged and internally consistent values of atmospheric pressure, density, and temperature from the ground to an altitude of 219 km have been determined and compiled by the united states groups active in upper-atmospheric research by rockets . additional relevant data by similar groups engaged in research on meteors and on the anomalous propagation of sound are also included, particularly in a brief discussion of variations with time and with place of these three atmospheric parameters .
aerodynamic heating of blunt nose shapes at mach numbers up to 14 . results are presented from recent investigations of the aerodynamic heating rates of blunt nose shapes at mach numbers up to 14 . data obtained in flight and wind-tunnel tests have shown that the flat-faced cylinder has about 50 percent the stagnation-point heating rates of the hemisphere over nearly the entire mach number range . tests made at a mach number of 2 on a series of bodies made up of hemispherical segments of varying radius of curvature showed that slight amounts of curvature can decrease the local rates at the edge of the flat-faced cylinders with only a slight increase in the stagnation rate . the total heat transfer to such slightly curved bodies is also somewhat smaller than the total heat transfer to flat-faced cylinders . comparison of several tests with theoretical heating-rate distributions showed that both laminar and turbulent local rates can be predicted by available theories /given the pressure distribution about the body/ reasonably well, although the scatter of the available data still leaves open the choice between the theories at the edge of the bodies, where they usually differ . tests on a flat-faced cylinder at a mach number of 2.49 and at angles of attack up to 15degree showed the movement of the apparent stagnation point from the center of the body to the 50 percent windward station at creased about 30 percent while that near the leeward edge decreased about 20 percent at 15degree angle of attack . preliminary results on a concave nose have indicated the possibility that this type of design may be developed to give heating rates significantly lower than even the flat-faced cylinder rates . the test results have also shown, however, the existence of an unsteady flow phenomenon which can increase the heating rates to extremely high values .
numerical solutions for supersonic flow of an ideal gas around blunt two-dimensional bodies . the method described is an inverse one,. the shock shape is chosen and the solution proceeds downstream to a body . bodies blunter than circular cylinders are readily accessible, and any adiabatic index can be chosen . the lower limit to the free-stream mach number available in any case is determined by the extent of the subsonic field, which in turn depends upon the body shape . some discussion of the stability of the numerical processes is given . a set of solutions for flows about circular cylinders at several mach numbers and several values of the adiabatic index is included .
free-flight observation of a separated turbulent flow including heat transfer up to mach 8. 5. a turbulent boundary layer separated by a forward-facing step was observed on the cylindrical portion of a hemisphere-cone-cylinder test vehicle . tip blunting, producing a shear flow, was found to induce higher pressures on the cylindrical portion than were predicted from ballistic tunnel data of unblunted projectiles . an approximate method for predicting this blunt-body pressure distribution was hypothesized . these findings, along with the hypothesis, were substantiated by a wind tunnel test of a similar body . the peak pressure ratios of the separation were smaller in magnitude than flat plate theory predicted because of the effect of the shear flow . the decrement in heating of the separated flow, relative to the corresponding attached flow, was found to compare well with the expected results .
the flow field over blunted flat plates and its effect on turbulent boundary growth and heat transfer at a mach number of 4. 7. surface pressures, impact and static pressure distributions in the flow field over the plate, and local heating rates were measured on a flat plate with various leading-edge diameters . the tests were conducted at a mach number of 4.7 and a free-stream reynolds number of 3.8×10 per foot . it was found that the shape of the shock wave indicated the existence of an outward deflection of the flow over the plate . the flow deflection caused an outward deflection of the shock-wave asymptote of approximately the shock-wave angle calculated including boundary-layer growth . the mach number distributions in the shear layer evaluated from pitot and static pressure surveys agreed with predictions based on shock-wave shape . the predicted turbulent heat-transfer coefficients for the blunted flat plates agreed with the measured heat-transfer coefficients . a comparison between the measured heat-transfer coefficients for the blunted flat plates and the calculated coefficients for a sharp leading-edged plate indicated that the coefficients were highest near the leading edge of the most blunted plate . the measured heat-transfer coefficients dropped to approximately 80 percent of the sharp-plate values at a considerable distance from the leading edge for all of the blunted flat plates .
a study of second-order supersonic flow theory . an attempt is made to develop a second approximation to the solution of problems of supersonic flow which can be solved by existing first-order theory . the method of attack adopted is an iteration process using the linearized solution as the first step . for plane flow it is found that a particular integral of the iteration equation can be written down at once in terms of the first-order solution . the second-order problem is thereby reduced to an equivalent first-order problem and can be readily solved . at the surface of an isolated body, the solution reduces to the well-known result of busemann . the plane case is considered in some detail insofar as it gives insight into the nature of the iteration process . again, for axially symmetric flow the problem is reduced to a first-order problem by the discovery of a particular integral . for smooth bodies, the second-order solution can then be calculated by the method of von karman and moore . bodies with corners are also treated by a slight modification of the method . the second-order solution for cones represents a considerable improvement over the linearized result . second-order theory also agrees well with several solutions for other bodies of revolution calculated by the numerical method of characteristics . for full three-dimensional flow, only a partial particular integral has been found . as an example of a more general problem, the solution is derived for an inclined cone . the possibility of treating other inclined bodies of revolution and three-dimensional wings is discussed briefly .
unsteady laminar compressible boundary layers on an infinite plate with suction or injection . this study deals with unsteady compressible laminar boundary layers on an infinitely extended porous plate . an integral solution based on two types of assumed velocity and temperature profiles is presented for the general case where the unsteady free-stream velocity and rate of surface suction or injection are both arbitrary . also indicated is an exact solution, applicable, however, only to certain specific unsteady free-stream and surface suction or injection variations . the reliability and range of validity of the integral solutions is then established on the basis of numerical results from the exact solution . finally, several general qualitative conclusions of the unsteady effects of free-stream velocity and surface suction or injection on laminar boundary-layer behavior are made .
on supersonic flow past a slightly yawing cone . this paper is concerned with the motion of a circular cone, of not too blunt an angle, through air at high speed . if the direction of motion of the cone coincides with its axis of symmetry, the resulting air flow is well known . here we consider the perturbation produced by a small /yaw/–i.e., the case in which the cone is moving not quite in the direction of its axis . the results are confirmed experimentally, and have applications to ballistics, though we are not concerned with the latter here,. they may also be useful as providing a check on various approximate methods of wider applicability . the square of the yaw is neglected–an approximation of which the validity is discussed . (similar methods can be applied to the second-order effects of the yaw, which are also of ballistic significance,. but the computations have not yet been completed .) it should be observed that, because of the lack of symmetry, the flow will be neither irrotational nor isentropic .
some research on high speed flutter . paper presents brief discussions of many topics currently of interest in the flutter field . these include /a/ the sonic speed case, /b/ oscillating pressure field of propellers, /c/ wing flutter with various configurations including effects of body modes, and /d/ propeller stall flutter .
unsteady aerodynamic forces on a slender body of revolution in supersonic flow . linearized slender-body theory is applied to the computation of aerodynamic forces on an oscillating, or deforming, body in supersonic flow . the undeformed body is a body of revolution and the deformed body is represented by movement of a line through the centers of the cross sections which are assumed to remain circular . the time dependence is based on sinusoidal motion . for a body of vanishing thickness the slender-body theory yields the apparent mass approximation as it is obtained for incompressible crossflow around a cylinder . both linearized slender-body theory and the apparent mass approximation are used to calculate the pitching-moment coefficients on a rigid slender body with a parabolic arc nose cone, and these coefficients are compared with some experimental results .
an electronic apparatus for automatic recording of the logarithmic decrement and frequency for oscillations in the audio and subaudio frequency range . an electronic apparatus for automatic evaluation of the damping of a harmonic oscillation has been designed and constructed . the apparatus is based on the idea of representing the harmonic damped oscillation by a rotating vector on the screen of a cathode-ray tube in such a way, that the rate of decrease of the length of the vector is a measure of the damping . the results are obtained simultaneously with the oscillation test as two numbers in decimal digits, which are inversely proportional to the logarithmic decrement and the frequency, respectively . the apparatus, which is named the /dampometer/, has been used for some time for free oscillation measurements of the dynamic stability derivatives of aeroplane models in windtunnels, and has proved to be very satisfactory . it gives results of usually higher accuracy than evaluation methods in common use, and permits a most considerable saving of time .
steady and fluctuating pressures at transonic speeds on two space-vehicle payload shapes . steady and fluctuating pressures have been measured at mach numbers which were varied from 0.6 to 1.2 on two bodies of revolution typical of two space-vehicle payload shapes, the centaur and the able v . the results of the investigation showed that significant fluctuations of pressure occurred on both bodies between mach numbers of 0.75 and 1.00 . the maximum fluctuations measured at any mach number and angle of attack occurred in the region of the normal shock wave as a result of shock-wave motion . large regions of unsteady pressure also occurred as a result of separation on the converging afterbody of the able-v model . the maximum pressure fluctuations occurring on the bodies increased with increasing angle of attack . for angles other than are indicated since pressure fluctuations were larger on the upper half of the bodies than on the lower half . no definite conclusions could be drawn regarding the form of the spectral densities of pressure fluctuations in the region of the shock wave . the spectral densities in regions of separation following the shock wave appeared flat except for some increase in energy level below due to slight model motions .
measurements of aerodynamic forces for various mean angles of attack on an airfoil oscillating in pitch and on two finite-span wings oscillating in bending with emphasis on damping in the stall . the oscillating air forces on a two-dimensional wing oscillating in pitch about the midchord have been measured at various mean angles of attack and at mach numbers of 0.35 and 0.7 . the magnitudes of normal-force and pitching-moment coefficients were much higher at high angles of attack than at low angles of attack for some conditions . large regions of negative damping in pitch were found, and it was shown that the effect of increasing the mach number from 0.35 to 0.7 was to decrease the initial angle of attack at which negative damping occurred . measurements of the aerodynamic damping of a 10- percent-thick and of a 3-percent-thick finite-span wing oscillating in the first bending mode indicate no regions of negative damping for this type of motion over the range of variables covered . the damping measured at high angles of attack was generally larger than that at low angles of attack .
general instability of stiffened cylinders . theoretical buckling stresses are determined in explicit form for circular cylinders with circumferential and axial stiffening . the loadings are axial compression, radial pressure, hydrostatic pressure, and torsion . analyses were confined to moderate-length and long cylinders . the investigation was based upon the use of a form of donnell’s equation derived by taylor which is applicable to orthotropic cylinders . the derivation of this equation is presented in this report .
stability of orthotropic cylindrical shells under combined loading . the increasing use of fiber and whisker reinforced materials makes necessary the availability of methods of analyzing cylinders and cones composed of an orthotropic material . this paper treats the buckling of such shells under a combination of axial compression and uniform external pressure . the differential equation approach of flugge is used, and the resulting buckling equation is presented in terms of axial and circumferential bending rigidities, shear rigidity, poisson’s ratio, geometry parameters and mode shapes . design curves are presented which allow quick determination of critical loads on cylinders, and, by using the equivalent cylinder concept, on conical shells of small included angle . the curves also lend themselves to /tailoring/ of materials to fit the load carrying requirements of the structure .
elastic stability of orthotropic shells . a small-deflection theory for general instability of orthotropic circular cylindrical shells has been derived for external pressure, torsion, and axial compression . for the first two types of loading, comparison of the theory with experimental data for orthotropic cylinders reveals agreement comparable with that obtained for isotropic shells . for axial compression, experimental data have been found to agree reasonably well with theory for orthotropic cylinders, in contrast to the agreement usually obtained for isotropic cylinders .
plastic stability theory of thin shells . considerable interest is currently centered on the role of deformation and flow types of plasticity theories in the solution of stability problems . for thin flat plates, deformation theory combined with classical stability theory appears to yield results which are in substantially good agreement with test data . on the other hand, flow or incremental theories appear to require the introduction of initial imperfections in order to obtain a satisfactory degree of correlation with tests . thus, in view of the current state of development of plastic stability theory, it appears fruitful to exploit the mathematical simplicity inherent in deformation theory in the investigation of the plastic stability of thin shells . although there may be theoretical objections to deformation theories as a class, test data on flat plates do suggest the predictive value of the results obtained from this theory . in this paper, a set of equilibrium differential equations for the plastic buckling of thin shells of constant unequal radii is derived . this set of three equations applies to flat plates, cylinders, and spheres under any loading system leading to buckling . for particular problems such as buckling of cylinders under axial compression, torsion or lateral pressure, and spheres under external pressure, the set of equations can be reduced to a single eighth-order partial differential equation of the donnell type in terms of the radial displacement only . these donnell-type equations are used to obtain solutions for plastic buckling of spheres under external pressure and long and moderate length cylinders under lateral pressure or torsion loads . the limiting cases of a simply supported flat plate under compression or shear, represent the solutions for short cylinders under lateral pressure or torsion, respectively .
a unified theory of plastic buckling of columns and plates . on the basis of modern plasticity considerations, a unified theory of plastic buckling applicable to both columns and plates has been developed . for uniform compression, the theory shows that long columns which bend without appreciable twisting require the tangent modulus and that long flanges which twist without appreciable bending require the secant modulus . structures that both bend and twist when they buckle require a modulus which is a combination of the secant modulus and the tangent modulus .
compressive and torsional buckling of thin-wall cylinders in the yield region . based on assumptions which have led to the best agreement between theory and test data on inelastic buckling of flat plates, a general set of equilibrium differential equations for the plastic buckling of cylinders has been derived . these equations have been used to obtain solutions for the compressive and torsional buckling of long cylinders in the yield region . test data are presented which indicate satisfactory agreement with the theoretical plasticity-reduction factors in most cases . where a difference in results exists, test data are in substantially better agreement with the results obtained by use of the maximum-shear law rather than the octahedral-shear law to transform axial stress-strain data to shear stress-strain data .
on the role of initial imperfections in plastic buckling of cylinders under axial compression . in a recent paper lee treated the complex problem of the plastic buckling and postbuckling behavior of an axially compressed cylindrical shell containing initial imperfections, representing an important step forward in our understanding of this common, yet perplexing, structural element . lee drew two major conclusions .. (a) even with initial imperfections the incremental theory of plasticity considerably overestimates the buckling strength as compared with the deformation theory, which is in substantially good agreement with experiments, and strength of cylindrical shells subject to axial compression are significant . it is the purpose of this note to discuss the second conclusion in terms of lee’s experimental and theoretical results, other experimental data on inelastic buckling of 7075-t6 aluminum-alloy cylinders, and recent theoretical results on the inelastic buckling of cylinders in the axisymmetric and circumferential modes . in particular, this writer does not believe that lee has proved that initial imperfections are important for the group of cylinders that he has tested . on the contrary, it is believed that initial imperfections are completely insignificant for this group of cylinders although of probable significance in other cases .
an extension of donnell’s equation for a circular cylindrical shell . in 1933, donnell succeeded in simplifying the equations of equilibrium for a circular cylindrical shell he obtained simple relations between the critical buckling shearing stress and the physical properties of a thin circular cylinder under pure torsion . his approach reduces the tedious computations involved in the classical solutions and is still in good agreement with them . furthermore, it is easy to show that the well-known classical solution for critical compressive stress of the cylinder under axial compression can readily be obtained from donnell’s equation .
design of missile bodies for minimum drag at very high speeds – thickness ratio, lift, and center of pressure given . newtonian flow theory has been used to develop a procedure for the design of minimum drag bodies of revolution having a given thickness ratio and center of pressure . it is shown that the optimum body shape is independent of lift . center of pressure location, however, exerts a powerful influence on both the shape of the body and on the drag coefficient at zero lift .
collapse by instability of thin cylindrical shells under external pressure . this paper discusses the collapse by instability of thin-walled cylindrical vessels subjected to external pressure . the most important of the theoretical and empirical formulas that apply to this subject are presented in a common notation . a new and simple instability formula is developed . three classes of tubes are considered .. tubes of infinite length,. tubes of finite length with uniform radial pressure only,. and tubes of finite length with both uniform radial and axial pressure . collapsing pressures calculated by the various formulas are presented in tabular form as a means of comparing the formulas . the formulas are discussed briefly and checked against the results of tests conducted at the u. s. experimental model basin for the bureau of construction and repair, navy department . this paper is a sequel to one previously published as a part of the work of the a.s.m.e. special research committee on the strength of vessels under external pressure .
an engineer’s conceptual approach to the buckling of cylindrical shell (axial loading) . by using the well known analogy between the bending of a beam on an elastic foundation and the axial symmetric displacement of a cylinder, a physical insight is obtained for the buckling of cylindrical shells under axial compression . the technique is equivalent to classical small deflection theory and provides good agreement with the more elaborate solutions for the buckling strength of various sandwich, multi-layered, and orthotropic cylinders, including the effects of internal pressure or an elastic core .
the buckling of sandwich type panels . fifty-one flat rectangular sandwich-type panels were tested in edgewise compression with the unloaded edges of the panels restrained by v-grooves . the sandwich consisted of papreg faces and a cellular cellulose acetate core . the thickness of the faces varied from 0.00675 to 0.02025 in.,. the core, from 0.066 to 0.741 in.,. the width of the panel, from 4 to 11 in . the length of the panel was always 10.5 in . the buckled shape consisted of a ripple of short wave length across the panel . it was either symmetric, the two faces bulging out symmetrically according to sine curves, or skew, the two faces deflecting in the same sense according to sine curves having a phase angle of 90 . a strain energy theory of buckling is presented for both the symmetric and the skew cases, and the buckling load in the symmetric case is also calculated by integration of the differential equation . the agreement between the theoretic and the experimental buckling stress is reasonable, that between the predicted and actual buckled shape good . a simple formula is developed which permits a choice of the most suitable core material when the mechanical properties of the face material are given .
face wrinkling and core strength in sandwich construction . the effect of initial waviness on the wrinkling of faces in sandwich construction is studied . formulae are derived to determine the failing stress when the faces wrinkle due to failure of the core in tension, compression or shear . the importance of core strength requirements in maintaining surface smoothness is noted . a comparison of theory with experiments is made, and the agreement between the two is found to be reasonably good . the strength of the core . williams has related the strength of the core in tension and shear to an arbitrarily assumed initial irregularity which, to ensure laminar flow in a wing is assumed to have a maximum admissible value (initial wave amplitude critical wavelength=0.0005 to 0.001) .
general instability of a ring stiffened circular cylindrical shell under hydrostatic pressure . the general instability load of a ring-stiffened, circular cylindrical shell under hydrostatic pressure is determined by analyzing an equivalent orthotropic shell . a set of differential equations for the stability of an orthotropic shell is derived and solved for the case of a shell with simple end supports . the solution is presented in terms of parameters of the ring-stiffened, isotropic shell, and a relatively simple expression for the general instability load is obtained . some numerical examples and graphs of results are presented . in addition, an energy-method solution to the problem is outlined, and the energy and displacement functions that could be used in carrying out a rayleigh-ritz approximation are indicated .
handbook of structural stability . pt .vi . strength of stiffened curved plates and shells . a comprehensive review of failure of stiffened curved plates and shells is presented . panel instability in stiffened curved plates and general instability of stiffened cylinders are discussed . the loadings considered for the plates are axial, shear, and the combination of the two . for the cylinders, bending, external pressure, torsion, transverse shear, and combinations of these loads are considered . general instability in stiffened cylinders was investigated . for bending and torsion loads, test data and theory were correlated . for external pressure several existing theories were compared . as a result of this investigation a unified theoretical approach to analysis of general instability in stiffened cylinders was developed .
the effect of axial constraint on the instability of thin conical shells under external pressure . author considers elastic axial restraint which may be (1) active from the beginning of loading and (2) active only at the onset of buckling . buckling loads for the two cases are related by a simple conversion factor . effect of the restraint on the axisymmetric type buckling is negligible, but the amplification of the critical load for the nonaxisymmetric type buckling may be very large for type (1) restraint . design curves are included for a range of l d.and restraint stiffness . results are of doubtful value considering the known inadequacy of the linear theory of buckling under axial compression .
general instability of ring stiffened cylindrical shells subject to external hydrostatic pressure – a comparison of theory and experiment . tests are described of a number of machined-stiffened cylinders subjected to external hydrostatic pressure, and the observed general instability strengths compared with predictions from theories of kendrick and nash . agreement with kendrick was found rather good . results also are presented from electrical strain gages which show in detail the growth of embryonic lobes and nonlinear characteristics of deformation at the threshold of buckling . weakening effects of imperfect circularity are discussed .
a simple extension of southwell’s method for determining the elastic general instability pressure of ring-stiffened cylinders subject to external hydrostatic pressure . a simple extension of southwell’s method is presented whereby the elastic general instability pressure of ring-stiffened cylindrical shells subject to external hydrostatic pressure may be determined . an actual application of the method is illustrated in the paper and the results of several other examples are summarized .
torispherical shells – a caution to designers . it has recently become apparent, through a rigorous stress analysis of a specific case that designing torispherical shells by the current edition of the asme code on unfired pressure vessels can lead to failure during proof-testing of the vessel . the purpose of the present paper is to show in what respects the code fails to give accurate results . as an illustrative example, a hypothetical pressure vessel with a torispherical head having a diameter-thickness ratio of 440 was selected . the supports of the vessel were considered to be either on the main cylinder or around the torus . the vessel was subjected to internal pressure and the elastic stresses in it were determined rigorously and by the code . a comparison of the two revealed that the code predicted stresses in the head which were less than one half of those actually occurring . furthermore, the code gave no indication of the presence of high compressive circumferential direct stresses which exceeded 30,000 psi for practically the entire torus . if the head had been fabricated using a steel with a yield point of would have failed or undergone large deformations, whereas the code would have predicted that it was safe . the code’s rules for torispherical heads are thus in need of revision for certain geometries . the implications of the foregoing results are currently being studied by the asme,. in the interim, however, designers should exercise care in applying the code to torispherical shells . it is also shown in the paper that the use of the membrane state as a particular solution of the differential equations is not a good approximation for toroidal shells of the type considered .
limit analysis of symmetrically loaded thin shells of revolution . the yield surface for a thin cylindrical shell is shown to be a very good approximation to the yield surface for any symmetrically loaded thin shell of revolution . hexagonal prism approximations to this yield surface, appropriate for pressure vessel analysis, are described and discussed in terms of limit analysis . procedures suitable for finding upper and lower bounds on the limit pressure for the complete vessel are developed and evaluated . they are applied for illustration to a portion of a toroidal zone or knuckle held rigidly at the two bounding planes . the combined end force and moment which can be carried by an unflanged cylinder also is discussed .
design of thin walled torispherical and toriconical pressure – vessel heads . the failure under hydrostatic test of a large storage vessel designed in accordance with current practice stimulated earlier analytical studies . this paper gives curves and a table useful for the design and analysis of the knuckle region of a thin torispherical or toriconical head of an unfired cylindrical vessel . a simple but surprisingly adequate approximate formula is presented for the limit pressure, np, at which appreciable plastic deformations occur .. where p is the design pressure, is the yield stress of the material, and n is the factor of safety . the thickness t of the knuckle region is assumed uniform . upper and lower bound calculations were made for ratios of knuckle radius r to cylinder diameter d of 0.06, 0.08, 0.10, 0.12, 0.14, and 0.16, and ratios of spherical cap radius l to d of 1.0, 0.9, 0.8, 0.7, and 0.6 . toriconic1a heads may be designed or analyzed closely enough by interpreting in table 1 as the complement of the half angle of the cone .
on the theory of thin elastic toroidal shells . the author obtains asymptotic solutions to the problem of rotationally symmetric small deflection of thin toroidal elastic shells . he first reduces the problem to that of integrating a single linear nonhomogeneous ordinary differential equation involving two parameters . asymptotic formulae for the complementary function are obtained by applying the general method of langer (trans.amer.math.soc.33, advantage of yielding results valid near the points where the tangent plane is perpendicular to the axis of revolution, where the methods of asymptotic integration customary in shell theory fail (see the preceding review) . for two problems in which only the complementary function is required, the author’s results are compared with those obtained by wissler (dissertation, zurich, 1916) by a method of power series expansion,. the agreement is within 4 or better . the author observes that the usual method of obtaining asymptotic expressions for a particular integral, being based on using as an approximation the complementary function obtained from the membrane theory, will fail near points where the tangent plane is perpendicular to the axis of revolution . he therefore introduces a new method, which he states was developed jointly with e. reissner . he applies his results to the cases of an joint loaded symmetrically and parallel to its axis, a corrugated pipe subject to axial load, and a corrugated cylinder subject to axial pressure . many numerical calculations are involved and there are two tables of functions occuring in the solutions .
asymptotic solutions of toroidal shell problems . method of asymptotic integration developed by e. reissner and author is refined, and solutions previously obtained for problems of bending of curved tube and of a toroidal expansion joint subject to an axial force are generalized and extended . results are compared to those obtained by l. beskin . for large values of a certain parameter, agreement is good .
the effect of entrance velocity on the flow of a rarefied gas, through a tube . the flow of a rarefied gas through a circular tube is considered . molecules entering the tube have a mass velocity directed down the tube, as well as a randomly directed thermal velocity . it is assumed that the conditions for free-molecule flow hold, and that molecules striking the tube wall are reflected diffusely . the mass velocity and tube dimensions are restricted only by the limitation to free-molecule flow . the theory is illustrated by an example of the effect of an entrance tube on the measurement of pressure and density by a gage placed on a satelite .
shock-standoff distance for spherical bodies at high mach numbers . usaf-sponsored development of a simple expression for shock-standoff distance, by consideration of an apropriate specific-heat ratio behind the shock . the theory gives predictions which are almost as accurate as those of the van dyke and hays methods .
the wake behind an oscillating vehicle . the incompressible laminar far wake behind an oscillating vehicle is analyzed with the use of the oseen linearization, and the assumption that the wake cross section is axisymmetric . time-wise drag variations are thus represented as changes in the wake diameter and velocity defect . it is shown that the unsteadiness in this flow can be predicted by a quasi-steady theory .
effect of wall divergence on sonic flows in solid wall tunnels . the effect of wall divergence (in excess of the normal value given for compensating the boundary layer growth on the tunnel walls), on the simulation of sonic flows in solid wall wind tunnels is presented in this note which is a condensed version of ref. 1 .
a one-foot hypervelocity shock tunnel in which high-enthalpy real gas flows can be generated with flow times of about 180 milliseconds . a shock tunnel is described in which high-enthalpy, real-gas air flows can be generated with flow times of about 180 milliseconds . this shock tunnel is operated with a combustion-heated driver gas and consists of a combustion chamber, shock tube, supersonic nozzle, test section, and vacuum tank . an essential feature of this shock tunnel is a means for achieving a constant-pressure air reservoir for the duration of the test . air streams with velocities in excess of achieved at a mach number of about 10 . the corresponding stream total enthalpy is about 4,500 btu lb and the stagnation pressure is 3.25 psia .
slipstream flow around several tilt-wing vtol aircraft models operating near the ground . a collection of data from a number of brief investigations made with three different models to determine the character of the slipstream flow along the ground is presented for multiple-propeller tilt-wing vtol aircraft configurations operating near the ground . in general, the tests involved tuft surveys and slipstream dynamic-pressure measurements for several tilt-wing vtol models . a more extensive series of tests, including some measurements of the erosion of gravel by the slipstream and some measurements of the unsteady rolling, yawing, and pitching moments, was also made on one of the models operating in the hovering condition near the ground . the results of the flow studies indicated the presence of a stronger and deeper slipstream flow along the center line of the aircraft, and to some extent along parallel planes between adjacent propellers (on one wing), than to the side of the aircraft . this effect is caused by an intensification of the individual slipstreams as they meet at the planes of flow symmetry . the intensified flow along the center line of the aircraft is amplified by the presence of the fuselage and causes the dynamic pressure to be greater in front of the aircraft than would be expected on the basis of the slipstream of the individual propellers . in the erosion tests it was found that gravel, if sufficiently small, was rapidly eroded by the slipstream and that this gravel could be thrown high into the air if it struck even very small fixed obstacles on the ground (obstacles with a height less than the diameter of the gravel) . results of the investigation of moment fluctuations indicated that there are large, erratic variations of rolling, yawing, and pitching moments and that the propellers, reacting to an erratic inflow from the recirculating slipstream, are the primary source of these moments .
buckling of core-stabilized cylinders under axisymmetric external loads . an equation is derived for the elastic stability of a circular cylindrical shell which is filled with a soft elastic core and is subjected to general axially-symmetric lateral pressure combined with a central axial force . numerical results are given for three lateral pressure distributions of interest in rocket motor case analysis.. uniform pressure, linearly varying pressure, and a circumferential band of pressure located at an arbitrary distance from one end of the cylinder . comparison is made with results of previous theoretical and experimental investigations, where available .
thermal buckling of cylinders . several theoretical and experimental investigations on the buckling of cylinders due to both axial and circumferential thermal stresses are reviewed . differences that exist among the various results are discussed and areas of future work are indicated .
heat transfer to bodies traveling at high speed in the upper atmosphere . a general method has been developed, using the methods of kinetic theory, whereby the surface temperatures of bodies can be calculated for steady flight at any speed in a rarefied gas . the particular solution was made for a flat plate., however, the calculations can be easily extended to bodies of arbitrary shape . it was found that the aerodynamic heating problem in the absence of solar radiation, that is, for the case of nocturnal flight, becomes of negligible importance at altitudes of 125 miles and higher and up to steady flight speeds of 36,000 feet per second . the effect of solar radiation, for the case of daytime flight, becomes increasingly important as the flight altitude is increased . at an altitude of 150 miles and higher, solar radiation is the predominating factor that determines skin temperature . owing to the strong effect of solar radiation on skin temperatures at high altitudes, the desirability of nocturnal flight is indicated in order to minimize skin temperatures . in order to maintain low skin temperatures, it was found that the angle of inclination of the body with respect to the flight path should be kept as small as possible . this may be accomplished in practice by designing the body to be finely tapered and by flying the body at small angles of attack . it is pointed out that skin temperatures may be reduced by insuring thermal contact between portions of the skin inclined at positive and negative angles with respect to the flight path . as much surface as possible should be inclined at negative angles . practically, this may be accomplished by boattailing the body . in the event that an internal skin-cooling system is employed, it is shown that the rate of internal cooling must be of the same order of magnitude or greater than the rate at which heat is lost naturally by emitted radiation . if the cooling rate is below the natural radiation rate, cooling has little effect upon skin temperatures . it is shown that, in the case of a missile designed to fly over a wide range of altitudes and speeds, it is desirable to make the emissivity of the skin as high as possible . this conclusion, however, is based upon a skin surface for which the emissivity is independent of the wave length of the emitted and absorbed radiant energy . a possible method of reducing surface temperatures is indicated by the decrease in skin temperature which accompanies a decrease in thermal accommodation coefficient . this phenomenon may be used to advantage if it is possible to decrease the accommodation coefficient by altering the surface characteristics of the skin .
knudsen flow through a circular capillary . the problem of knudsen flow through a circular capillary has been often discussed, usually by the momentum transfer method . however, p. clausing gave a rigorous formulation for the problem and obtained an integral equation for which he gave an approximate solution . from time to time the accuracy of clausing’s solution has been questioned and since clausing did not give a rigorous estimate of his error we have reinvestigated the problem .
similar temperature boundary layers . conditions for the existence of similar solutions are known for (a) two-dimensional, incompressible, steady and nonsteady laminar boundary layers and (b) three-dimensional, incompressible, steady, laminar boundary layers for a body of revolution rotating in a fluid at rest or a body of revolution in a rotating fluid flow . corresponding conditions for the existence of similar temperature boundary layers in both cases are given for constant and variable wall temperatures . the general conclusion is that, in all these cases, with or without viscous heating, and with constant wall temperature, conditions for the existence of similar velocity boundary layers are at the same time the conditions for the existence of similar temperature boundary layers . if the wall temperature is variable, the conditions for the existence of similar velocity boundary layers are at the same time the conditions for the existence of similar temperature boundary layers if the wall temperature varies as a power of the local free-stream velocity or surface velocity . numberical solutions are given for the nondimensional temprature distributions function and the nondimensional temperature gradient at the wall for several prandtl numbers in the case of a rotating flow over an infinite plate at rest .
preliminary results of density measurements from an air force satellite. atmospheric density was determined from a singly mounted ionization gauge flown on an air force satellite . included is a brief description of the experiment and theory as well as a discussion of some of the problems involved in performing these measurements . density data are given for the altitude range of 370 to 400 km during early morning hours for the two days 17 and 18 june 1961 . results are compared with those of the 1961 revised u.s. standard atmosphere .
experiments on supersonic blunt-body flows . recently, progress has been made in the theoretical calculation of the inviscid flow between the detached shock wave and the surface of a blunt body travelling at supersonic speed . detailed experimental data are needed for comparison . experiments have been made in the supersonic wind tunnels of the jet propulsion laboratory on spheres, disks, and blunted cones over the mach number range 1.8 to 5.0 in air . surface pressures, shock wave shape and detachment distance, certain velocity gradients, the sonic line location, and some streamline paths were determined . the sonic line is found as the trace of the termination point of a weak shock generated by a probe ahead of the model . for a sphere, good agreement between theory (of van dyke) and experiment is found .
on periodically oscillating wakes in the oseen approximation . studies in maths . and mechs ., the oscillating vortex wake behind an obstacle at reynolds numbers of order 10 is studied by means of the oseen approximation .
a study of the simulation of flow with free stream mach number 1 in a choked wind tunnel . the degree to which experimental results obtained under choking conditions in a wind tunnel with solid walls simulate those associated with an unbounded flow with free-stream mach number 1 is investigated for the cases of two-dimensional and axisymmetric flows . it is found that a close resemblance does indeed exist in the vicinity of the body, and that the results obtained in this way are generally at least as accurate as those obtained in a transonic wind tunnel with partly open test section . some of the results indicate, however, that substantial interference effects, particularly those of the wave reflection type, may be encountered under certain conditions, both in choked wind tunnels and in transonic wind tunnels, and that the reduction of these interference effects to acceptable limits may require the use of models of unusually small size .
on the influence of wall boundary layers in closed transonic test sections . the boundary layers at the test section walls of a transonic wind tunnel are known to reduce the wall interference . in the present paper this effect is studied by means of small perturbation theory, assuming viscosity to be negligible when perturbing a turbulent boundary layer . an approximation for thin boundary layers leads to a modified boundary condition at the wall of the test section, expressing the normal streamline slope induced by changes in mass flow density and crossflow within the boundary layer . this boundary condition is applied to the linearized equations of subsonic flow and to the non-linear transonic equations at choking, the cases of plane and circular test sections only being treated in detail . the results of linear theory show that all corrections except the three-dimensional angle-of-attack correction are considerably reduced by the presence of the boundary layers at mach numbers greater than 0.9, the essential part of their influence being due to the change of mass flow density with pressure . in the case of choking the analysis indicates that the presence of boundary layers will increase the maximum model size for which the flow can be interpreted as corresponding to mach number one in free flight . finally, the technique of using artificial thickening of the wall boundary layers for a reduction of wall interference is considered, though without reaching a definite conclusion as to its value as compared to other techniques .
some experimental investigations on the influence of wall boundary layers upon wind tunnel measurements at high subsonic speeds . pressure distribution measurements and drag determination by means of balance measurements have been carried out for a number of models at high subsonic velocity in wind tunnels, where the boundary layer of the walls has been varied . within the investigated range it appeared that a thickening of the boundary layer reduced the disturbing influence of the walls, which also caused an increase of the choking mach number . the phenomenon described should be of a certain importance from the point of view of wind tunnel technique, since it is possible to increase the choking velocity for a given model by means of thickening the boundary layer .
experimental investigation of attenuation of strong shock waves in a shock tube with hydrogen and helium as driver gases . an experimental investigation has been made of the attenuation of strong shock waves in air in a shock tube . time-history measurements were made of the static pressure at several stations in the wall of the tube . the internal diameter of the tube is 3.75 inches . shock- wave-velocity data were taken for a distance along the tube of about 120 feet . the range of the shock-wave mach number covered was from 5 to 10 and the initial pressure ahead of the shock wave varied from 5 to 100 millimeters of mercury . hydrogen and helium were used as driver gases . a helium-driven shock wave was found to decay only about one-half as rapidly as a hydrogen-driven shock wave . the pressure level had little effect on the attenuation rate of a shock wave of given strength for the pressure range investigated . the static-pressure measurements indicated that a severe pressure gradient existed in the latter portion of the air flow . this gradient limits the testing time useful for obtaining reliable aerodynamic data .
hypersonic shock tunnel . a hypersonic shock tunnel has been developed for obtaining fluid mechanic information at the high mach numbers and corresponding stagnation temperatures encountered in flight by long range ballistic vehicles and satellites . this report describes the hypersonic shock tunnel and presents some of the results obtained in the driven tube and in the nozzle helium is ignited in the driver to produce strong shock waves in air . a shock velocity in air as high as 55,000 fps with a calculated equilibrium temperature of 16,000 k has been produced in the driven tube . the effects of high stagnation temperatures upon the detached shock wave and the pressure distribution for blunt bodies have been observed in the nozzle test section . the detachment distance devreased greatly at high temperatures . the pressure distribution for the hemisphere was found to be less than that predicted by the modified newtonian theory . shock wave boundary layer interaction at the leading edge of a flat plate was observed, and the results agreed with the analytical prediction . a detached shock wave was observed for a blunt two-dimensional body at very low densities in the test section with a flow mach number of 19.6 .
the tailored-interface hypersonic shock tunnel . the /tailored-interface/ hypersonic shock tunnel provides a means for producing the high mach number, high stagnation temperature flow conditions encountered in hypersonic flight . various gasdynamic phenomena associated with shock tunnels are discussed, and experimental evidence of the successful application of this technique is presented . as an indication of its research application, the results of heat-transfer experiments on a hemisphere-cylinder model are presented and compared with theory .
experimental investigation of the effect of yaw on rates of heat transfer to transverse circular cylinders in a 6500-foot-per-second hypersonic air stream . a technique has been developed by which air can be shock-compressed by helium to 3660 degrees rankine to generate a 6500-foot-per-second air stream with a flow duration of 40 milliseconds . the resulting equipment is described . experiments were conducted to determine rates of heat transfer to transverse circular cylinders of 0.003-, greater than 100 . the cylinders were tested at a nominal mach number of 11 with a stagnation reynolds number (evaluated with free-stream mass flow and stagnation viscosity of 4.00 times 10 to the 4th power per foot .
recent advances in gaseous detonation . a review of recent work in gaseous detonation is presented . early work is briefly mentioned and treatises listed . theoretical calculations of chapman-jouguet detonations are reviewed, compared and the ambiguity concerning the speed of sound in a reacting gas mixture discussed . experimental chapman-jouguet measurements are reviewed . recent studies of the interior of a detonation wave are presented . standing detonation wave research, detonation limits, two-dimensional detonations, spectra, ionization and magnetohydrodynamic treatments are brought to the reader’s attention . a qualitative description of the development of a flame to a detonation is presented . experimental observations are examined and recent theoretical attempts to explain these observations are reviewed .
stagnation point heat transfer measurements in dissociated air . the results of an experimental investigation of the laminar heat transfer at the stagnation point of a blunt body in partially dissociated air are presented and are compared to the theoretical treatment of fay and riddell . heat-transfer results are presented for air temperatures as high as 8,000 k. where more than periments were performed in a shock tube and the new experimental techniques and principles are discussed briefly . simulation of flight stagnation conditions at velocities up to satellite velocity of 26,000 ft. sec. is shown to be possible in shock tubes and data has been obtained over a large altitude range at these velocities .
force-test investigation of the stability and control characteristics of a 1/8 scale model of a tilt-wing vertical take-off and landing airplane . a force-test investigation has been made to determine the aerodynamic characteristics of a scale model of a tilt-wing vertical-take-off-and-landing airplane in the short- and long-wing configurations . the model had two 6-blade dual-rotating propellers that were not interconnected mounted on a wing that could be tilted up to an incidence angle of about 90 for vertical take-off and landing . the investigation included measurements of both the longitudinal and lateral stability and control characteristics in the normal- forward-flight, transition, and hovering ranges . tests in the forward-flight and transition conditions were made at various wing incidences and power conditions . tests in the hovering condition were made in the presence of the ground . the data are presented without analysis .
force-test investigation of the stability and control characteristics of a scale model of a tilt-wing vertical take-off and landing aircraft . a wind-tunnel investigation has been made to determine the aerodynamic characteristics of a scale model of a tilt-wing vertical- take-off-and-landing aircraft . the model had two 3-blade single-rotation propellers with hinged (flapping) blades mounted on the wing, which could be tilted from an incidence of 4 for forward flight to 86 for hovering flight . the investigation included measurements of both the longitudinal and lateral stability and control characteristics in both the normal forward flight and the transition ranges . tests in the forward-flight condition were made for several values of thrust coefficient, and tests in the transition condition were made at several values of wing incidence with the power varied to cover a range of flight conditions from forward-acceleration (or climb) conditions to deceleration (or descent) conditions the control effectiveness of the all-movable horizontal tail, the ailerons and the differential propeller pitch control was also determined . the data are presented without analysis .
effect of ground proximity on the aerodynamic characteristics of a four- engined vertical take-off and landing transport airplane model with tilting wing and propellers . an investigation has been made to study the effect of ground proximity on the aerodynamic characteristics of a four-engine vertical- take-off-and-landing transport-airplane model with tilting wing and propellers . tests were made with the wing at an angle of incidence of 90, the position used for vertical take-off or landing . with the model at various heights above the ground, the lift, drag, and pitching moment were measured and tuft studies were made to determine the flow field caused by the propeller slipstream . data were obtained for the complete model, for the model with horizontal tail removed, and for the wing-propeller combination alone . the results of the investigation showed that, when the model was hovering near the ground, there was a strong upwash in the plane of symmetry and also an increase in lift of about 10 percent of the propeller thrust . about one-half of this lift resulted from an increase in propeller thrust and one-half resulted from an up load on the fuselage induced by the upwash . as the model approached the ground, it also experienced an increasing nose-down pitching moment that evidently resulted from the up load on the fuselage, the rear part of which was longer than the front part . the addition of the horizontal tail which was located about halfway up the vertical tail did not increase the nose-down pitching moment because the fuselage decreased the energy of the upwash before it reached the tail .
an investigation of the effect of downwash from a vtol aircraft and a helicopter in the ground environment . dynamic-pressure measurement, in ground effect, have been obtained about a single-rotor helicopter and a dual-propeller vtol aircraft . the results indicate that the slipstream dynamic pressure along the ground, some distance from the center of rotation, is not a function of disk loading but merely a function of the gross weight or thrust of the aircraft . furthermore, for a given gross weight the thickness of this outward flowing sheet of air is less for a small-diameter propeller (higher disk loading propeller) . the variation of the dynamic-pressure flow field for single and dual propellers or rotors is significantly different in the plane of symmetry between the two rotors than in a direction normal to this plane . the interaction of the two flows produces a region of upflow in this plane where the fuselage is located, and the decay of the maximum dynamic pressure with distance ahead of the fuselage is slower .
an investigation to determine conditions under which downwash from vtol aircraft will start surface erosion from various types of terrain . results of an investigation with small-scale equipment of the conditions under which the downwash from a hovering vertical-take-off- and-landing (vtol) aircraft will start surface erosion indicate that the onset of erosion depends only on the dynamic pressure of the outward flow of air near the surface . for a rotor or propeller at a height of about 1 slipstream diameter above the surface, this surface dynamic pressure was found to be equal to the disk loading . for the vtol aircraft supported by a ducted fan, the surface dynamic pressure with the ducted fan exit at a height of about one-half the exit-area loading . the surface dynamic pressure decreases rapidly with increasing height of the vtol device . erosion of sand and loose dirt started at surface dynamic pressures of 1 to 3 lb sq ft, which is in general agreement with helicopter experience . thoroughly soaking the sand and loose-dirt surfaces increased the resistance to erosion to surface dynamic pressures of 30 to 50 lb sq ft . spray from water started at surface dynamic pressures of 1.5 to pressures up to about 1,000 lb sq ft .
an experimental study of the effect of downwash from a twin-propeller vtol aircraft on several types of ground surfaces . a full-scale, twin-propeller vtol aircraft with a maximum gross weight of 3,400 pounds has been operated on the ground to study the effect of downwash on several types of ground surfaces . static operation over loose snow indicated a zone of obliterated vision ahead of the pilot in an arc of approximately 10 on each side of the plane of symmetry . an arc 10 to 45 each side of the center line was found to be an area of fair visibility while the arc from 45 to 90 was an area of poor visibility . static operation in the presence of loose surface material indicated that the downwash cleared the area near the aircraft of these particles without recirculation or damage to any components . short-time operation at moderate forward speed over loose gravel, with the thrust axis at an angle of in propeller-blade erosion and numerous small dents and fabric punctures in the sides of the fuselage . the propeller-blade erosion was superficial except for the leading edges where several layers of glass fiber were eroded .
damage incurred on a tilt-wing multipropeller vtol/stol aircraft operating over a level, gravel-covered surface . a summary is presented of the damage experienced by a tilt-wing vtol stol aircraft as a result of operating from a level surface covered with loose gravel . the damage was inadvertently incurred as the aircraft was performing a taxiing-turn maneuver over an area of level macadam surface containing loose and embedded crushed stone . observers from a chase aircraft commented that the wing was tilted at approximately 76 with respect to the ground when the damage occurred . deposits of stone in the open fuselage, punctures in the rotor blade skin, and damage to the compressor blades of the engine occurred due to the circulation of the crushed gravel .
hangling qualities experience with several vtol research aircraft . all of the vtol research aircraft discussed in this paper have successfully demonstrated conversion from hovering to airplane flight and vice versa . however, control about one or more axes of these aircraft has been inadequate in hovering flight . furthermore, ground interference effects have been severe in some cases and have accentuated the inadequacy of control in hovering and very low speed flight . stalling of wing surfaces has resulted in limitations in level-flight deceleration and in descent, particularly for the tilt-wing aircraft, which in this case is a very rudimentary type . minor modifications to the wing leading edge have, however, produced surprisingly large and encouraging reductions in adverse stall effects . height control in hovering and in low-speed flight has proved to be a problem for the aircraft not having direct control of the pitch of the rotors . the other systems have shown undesirable time lags in development of a thrust change .
structural loads surveys on two tilt-wing vtol configurations . the results of two structural-loads surveys are summarized . the first loads program discussed concerns the airframe vibratory loads encountered during flight tests of the vz-2 tilt-wing vtol aircraft throughout the operational range from hover to cruise flight . the primary sources of airframe vibration were wing-stall buffeting and tail buffeting in descents . the second loads program discussed concerns the initial results of a structural-loads survey conducted as part of the wind-tunnel test of a large-scale tilt-wing research model . this loads program deals with the steady wing loads measured throughout simulated transition from hover to cruise .
the stability under axial compression and lateral pressure of circular cylindrical shells with a soft elastic core . the stability under axial compression and lateral pressure of a finite circular-cylindrical shell with an elastic core is treated by means of donnell’s equations . the stability criterion is investigated in detail for the general cylinder under axial compression or lateral pressure and for a particular cylinder under combined loading . comparisons are made with available experimental data .
elastic stability of circular cylindrical shells stabilized by a soft elastic core . the effect of a soft elastic core upon the buckling strength of a thin, circular, cylindrical shell is investigated experimentally . two types of loading are considered .. (a) axial compression, and (b) uniform radial-band loading, where the width of the band is small compared to the length of the shell . for each type of loading it is shown that the strengthening effect of the elastic core becomes more significant with the increasing values of the radius-thickness ratio . for example, it is shown that for the geometric and elastic constants considered it is possible, with the presence of the core, to increase the axial buckling stress by as much as 65 percent over the values found for those without an elastic core . the elastic core is even more effective in stabilizing the shell against buckling due to band loading, the peak pressure required to buckle the filled specimen being 7.30 times that required to buckle the unfilled shell .
the buckling of cylindrical shells under longitudinally varying loads . two problems illustrating the effect of nonuniformity of loading on the buckling characteristics of circular cylinders are investigated . the first problem deals with the effect of linearly varying axial compressive stress, such as would be produced by the weight of the propellant in a solid-propellant engine case . the results indicate that the ratio of the maximum critical compressive stress induced by the shear load to the critical uniform compressive stress varies from 1.9 for the curvature parameter z equal to 1.6 as z becomes infinite . in particular, the increase in stress is less than 20 per sq. ft. for z greater than 100 . the stability of thin cylinders loaded by lateral external pressure varying linearly in the longitudinal direction is also investigated . the results indicate that for z greater than 100, the buckling coefficients are proportional to square root z .
general theory of buckling . various aspects of the theory of buckling are expounded in many treatises (1 to 15) . the books of bleich (2) and salmon (3) contain large bibliographies . goodier developments in buckling theory . numerous references are appended to hoff’s article . nearly all publications on buckling of shells, available in the u.s.a. (to 1956), are listed in the bibliographies on shells by nash (18) . the section ready guide to recent literature .
stresses from local loadings in cylindrical pressure vessels . a short discussion is given of the possible methods for computing the stresses caused in cylindrical shells by local loadings . it is concluded that the method of developing the loads and displacements into double fourier series leads to formulas which are best suited for numerical evaluation . with this method the pertinent expressions for the displacements caused by radial loads are found by reducing the three partial differential equations of the shell theory to an eighth-order differential equation in the radial displacements, which is similar to, but not identical with, those derived by donnell and yuan . insertion of the fourier series for the radial displacements and the external loading in this equation leads directly to a double series expression of the radial displacement w in terms of the load factors of the radial load . this results in the pertinent expressions for the other displacements and for the bending moments and membrane forces . the cases of radial loading considered here and those which can be reduced to it are (a) a load uniformly distributed within a rectangle, tion, uniformly distributed over a short distance in the circumferential direction, (d) a moment in the circumferential direction, uniformly distributed over a short distance in the longitudinal direction . for all these loadings the load factors, which have to be used in the pertinent formulas for the displacements, bending moments, and membrane forces, are computed . for the case of tangential loading an eighth-order differential equation is derived in terms of the radial displacement and the tangential load . using this equation, formulas for the displacements, bending moments, and membrane forces for tangential loading within a rectangle are found .
bending tests of ring-stiffened circular cylinders . twenty-five ring-stiffened circular cylinders were loaded to failure in bending . the results are presented in the form of design curves which are applicable to cylinders with heavy rings that fail as a result of local buckling .
effects of rapid heating on strength of airframe components . results of several experimental investigations are presented which indicate the effects of rapid heating on the bending strength of multiweb beams and ring-stiffened cylinders . it is shown that thermal stresses reduce the bending load carried at buckling by both beams and cylinders . the influence of thermal stress on maximum load is found to depend largely on the mode of buckling . for beams that buckle locally, no apparent effect of thermal stress on the maximum load has been found . a reduction in maximum load has been observed for beams that buckle in the wrinkling mode and for cylinders .
buckling of ring-stiffened cylinders under a pure bending moment and a nonuniform temperature distribution . thirteen stainless-steel ring-stiffened cylinders were subjected to a pure bending load and heated rapidly until buckling occurred . for most of the cylinders the heating was not uniform around the circumference so that appreciable axial thermal stresses were present . elementary thermal stress theory was found to be inadequate for the prediction of these thermal stresses, but a method was developed that would give satisfactory thermal stress results . by properly accounting for the thermal stress, the buckling load could be correlated with a theory for the buckling of an axially compressed ring-stiffened cylinder that is uniformly heated .
a theory of asymmetric hypersonic blunt-body flows . two-dimensional asymmetric and three-dimensional inviscid blunt-body flows are analyzed using a new method . the method is inverse, that is, the shock-wave shape and freestream conditions are taken as known, and the body shape and flow field are to be determined . results at zero angle of attack are obtained as a special case of the general problem . solutions at zero angle are calculated for a variety of body shapes at freestream mach numbers ranging from infinity to 1.85 . the ratio of specific heats, is taken as 1.4 . comparison with results obtained using van dyke’s and garabedian’s numerical solutions indicates that the method under consideration is more accurate than the van dyke method for determining stand-off distance . solutions are obtained for parabolic and paraboloidal shock waves at small angle of attack and infinite freestream mach number,. assumes the values 1.4, 1.2, 1.1, and 1.05 . for all cases, the streamline that wets the body passes through the shock wave slightly above the point where the shock is normal and thus does not possess maximum entropy . these results provide counter examples to the conjecture that any isolated convex body in a supersonic stream is wetted by the streamline of maximum entropy .
approximate analysis of the slot injection of a gas in laminar flow . the laminar diffusion and combustion of a gas injected into a high-speed uniform stream by means of a wall slot are considered . the dorodnitzin-howarth transformation is employed to reduce the boundary layer equations to incompressible form,. the nonsimilar flow field is treated by a modified oseen approximation in conjunction with the integral method . thermal boundary conditions corresponding to an adiabatic wall and to constant wall enthalpy are discussed . the injection of homogeneous, heterogeneous, nonreactive, and reactive gases is treated . for the latter case, the models usually employed for chemical behavior, namely, frozen and equilibrium flow, are considered . the analysis is applicable to a wide variety of laminar flows, e.g., those involving cooling, thermal protection, skin-friction reduction, and supersonic deflagration . a numerical example of practical interest in connection with the venting of gaseous hydrogen boiloff from a rocket booster is presented .
steady magnetohydrodynamic flow past a non-conducting wedge . this paper presents a study of the steady two-dimensional magnetohydrodynamic flow of an infinitely conducting fluid past a nonconducting wedge with nonaligned flow and magnetic field . the flows considered are in the /superfast/ or fully hyperbolic regime . the flows consist of several regions of uniformity connected by shocks and expansion waves . because of the boundary condition on the magnetic field, the magnetic field must be the same in the regions above and below the wedge,. thus the flows in these regions are coupled, unlike in the case of ordinary supersonic gasdynamics . only small wedge angles and weak waves (characteristics) are considered . the problem thus is linearized, and explicit solutions are obtained which are qualitatively similar to the nonlinear solutions . some interesting and unexpected features arise, and they are discussed in detail .
an approximate solution for laminar boundary layer flow . this paper presents an approximate solution for two-dimensional, incompressible, laminar boundary layer flow with arbitrary pressure gradient . von mises’ form of the boundary layer equation is linearized by making a change in the coefficient of one of the terms . the linearized equation yields a solution that is accurate for the outer portion of the flow but inaccurate near the surface . a separate inner solution then is developed which is accurate at the surface and which joins with the outer solution at some point within the boundary layer . the method may be considered a major modification of one developed earlier by von karman and millikan, with changes in both outer and inner solutions, and the point at which the two solutions are joined . the changes improve the accuracy of the method and in some respects simplify the calculations . as examples, results are presented for flow with a linear variation of velocity (including flat plate and stagnation point flow as special cases), flow with sinusoidal variation of velocity, flow past a circular cylinder (heimenz’ velocity distribution), and flow past an ellipse (schubauer’s data) . agreement with theoretically exact solutions is good, and better than results obtained using the pohlhausen method .
laminar hypersonic trail in the expansion-conduction region . the usual procedure in calculating the cooling process in a wake behind a blunt object is to assume a region of pure expansion up to a distance where the pressure has reached its ambient value, followed by a region where the mechanism of pure heat conduction is operative . in the present paper both mechanisms are assumed to be valid simultaneously, and the result is compared with previous calculations . the following criterion is established .. the minimum radius of a hemisphere-cylinder configuration, above which a simultaneous conduction-expansion calculation is not needed, is given by the approximation where is the nondimensional value of the enthalpy at the axis of the wake below which the two methods of computation give the same result, and m is the flight mach number .
three dimensional effects in viscous wakes . three-dimensionality in wakelike or jetlike free mixing may stem from initial geometric configurations, nonuniformities in flow variables over a cross section, or boundary conditions along the flow . these may be generated by bodies at angle of attack, nonaxisymmetric bodies, mixing of nonaxisymmetric jets with an outer flow, finite wings, or more artificial means . this paper is devoted to studies bearing on such configurations . the first section deals with the general mathematical model, in which the boundary layer approximations are used, and with methods of solution . laminar and turbulent flow, compressibility, unsteadiness, and streamwise pressure gradients are admitted initially . the flux forms of the equations are given . algebraic integrals of the energy equations and the diffusion ( frozen-flow) equations are obtained . a simplification of the convective terms, roughly corresponding to the oseen approximation, is used in the asymptotic downstream region . the second section contains explicit solutions for specific configurations, in particular for flows whose initial isovels are of elliptic shape . these flows may be wakelike or jetlike . compressibility is admitted,. however, the flows must have uniform pressure and must be steady . the final section deals with interpretation and evaluation of the results .
an integral method for calculating heat and mass transfer in laminar boundary layers . an integral method previously used successfully in several kinds of boundary layer problems has been extended to treat simultaneous heat and mass transfer in a binary-mixture laminar boundary layer when the pressure is uniform . the principal results are two pairs of dual integral relations arising from solutions to the integral concentration and energy equations . one pair connects the surface mass transfer rate and surface concentration of injected gas,. the other relates surface temperature and heat transfer rate in the presence of mass transfer . only the cases of helium and air injection into an undissociated air stream are discussed in detail, but the method can be applied to problems involving other gases . the approximate results agree quite well with some numerical solutions and with recent experimental results for which no numerical solutions are available .
lift of slender delta wings according to newtonian theory . an approximate system of equations is derived to describe the inviscid flow past a flat slender wing at angle of attack, in the limit and . the aspect ratio is required to approach zero at the same rate as the mach angle in the flow behind the shock wave . only a single parameter appears in the resulting equations, and a similarity law therefore can be written expressing a correction to the newtonian normal-force coefficient . for the delta wing, a correlation of experimental data according to the similarity law is shown, and the first terms of the solution are derived under the assumption that the similarity parameter is small ( vertex angle much smaller than mach angle) .
shock-induced boundary layer separation in overexpanded conical exhaust nozzles . the flow in overexpanded supersonic nozzles is reviewed . although five essentially different flow regimes can be discerned, depending on the nozzle pressure ratio, the regime of most interest to the engine designer is the one characterized by oblique shock patterns in the nozzle and flow separation from the nozzle wall . it is shown that the pressure rise associated with the separation correlates well with the mach number at the separation point . a simple analytical formulation for the pressure rise required to separate the flow provides excellent agreement with experimental data over a wide range of nozzle operating conditions and allows prediction of overexpanded nozzle performance .
factors affecting lift-drag ratios at mach numbers from 5 to 20 . yawed-cone working charts and an engineering method are presented and used to calculate lift-drag ratios of flat-top conical wing-body arrangements at mach numbers from 5 to 20 . viscous interaction effects are considered, but bluntness effects are neglected . correlations of wind-tunnel data in the range show that boundary layer displacement corrections to surface pressure and skin friction are required to calculate lift-drag ratios by this method whenever is greater than 0.2 . is the freestream mach number and is the freestream reynolds number based on body length . double- and single-type shock patterns, transition from one pattern to the other, and the variation of inner-shock position with angle of attack are described . lift-drag ratios are calculated at selected flight design points for flat-top, conical body arrangements with triangular and hyperbolic wing planforms . the hyperbolic wing arrangement offers a potential l d benefit at mach 5 but not at mach 10 or above .
nonequilibrium flow past a wedge . an exact numerical solution is obtained for the chemically reacting flow past a wedge . the freestream is either in equilibrium or out of equilibrium but nonreacting . the attached shock wave is shown to be either concave, convex, or straight, depending on the values of the amount of dissociation in the freestream and a parameter describing the amount of energy contained in the freestream relative to the gas dissociation energy . numerical examples are presented illustrating these regimes . the flow field is characterized by the presence of an entropy layer and a relaxation layer, both easily identifiable in the presentation of the numerical results .
flow of a gas near a solid surface . the state of a gas near a solid surface is investigated . it is assumed that at a sufficiently large distance from the surface the particle distribution function is of the chapman-enskog form . the half-range analysis previously employed for parallel plate geometrics (symmetric problems) can be adapted to the single-plate problem . although the mathematical analysis differs, the slip coefficients are essentially identical with those obtained from the parallel plate problem (e.g., couette flow) . detailed calculations are presented for both hard sphere and maxwellian molecules . the recent work of bakanov and deryagin for hard sphere molecules, which is based on incorrect approximations, is discussed .
heat transfer to a hemisphere-cylinder at low reynolds numbers . measurements of the local heat flux to hemisphere-cylinder models in a supersonic rarefied air stream are presented . two different steady-state methods were developed, and five individual models were used . data were obtained throughout the mach number range of 2 to 6, with reynolds numbers (based on conditions behind the bow shock and model diameter) varying from 38 to 1730 . the stagnation point data indicated a gradual increase from continuum boundary layer theory at the higher reynolds numbers to about 10 above at the lower end of the range investigated . pressure distribution measurements on cooled and uncooled models were found to agree well with modified newtonian theory . local recovery factor measurements showed a small rarefaction effect at the lowest reynolds numbers .
an integral method for calculation of supersonic laminar boundary layer with heat transfer on yawed cone . an integral method for calculating the three-dimensional boundary layer over the surface of a cone at angle of attack is investigated . the numerical procedure of integration for that method on the basis of a simplifying assumption concerning the boundary layer development along the cone generator is developed and illustrated by applying the method to find the solutions of integral equations for a specific example . the results obtained for the example for the range of circumferential angle of 40 investigated are summarized and given as heat transfer coefficients, coefficients of friction, and other friction parameters . the distribution of heat transfer coefficients checked with available experimental data fairly well .
some exact solutions for cavitating curvilinear bodies . a special case of cavitating flow solutions is postulated and transformed to a semi-infinite plane . the complete, exact solution then is synthesized by superposition of singularities . the solution is relevant to a general, two-parameter family of curvilinear bodies . the parameters are the flow angles at the two points of flow separation . the body reduces, in the special case, to the rayleigh solution for a flat plate . the equations of the cavity boundaries are given in explicit form . the body form and the stagnation streamline are given as the locus of the roots of a cubic equation . local static pressures and, hence, lift and drag, also may be calculated . the generated solutions constitute a technique involving simple computation for exact solutions of a special family of cavitating curvilinear bodies at finite angles of attack .
magnetohydrodynamic flow past a thin airfoil . the steady flow of a perfectly conducting magnetohydrodynamic fluid past a thin nonconducting airfoil is studied with the usual model in which the fluid variables obey the lundquist equations linearized about a constant unperturbed flow . /hyperliptic/ flows, in which hyperbolic and elliptic fields are superimposed, are considered . results of grad, mccune and resler, and sears and resler are extended and considered in detail for the case of an arbitrarily inclined unperturbed field . the general solution contains four line singularities along the characteristics through the ends of the body and has two arbitrary constants . by a / generalized kutta-joukowski condition,/ these constants are fixed so that two of the line singularities disappear . specifically, it is required that the solution be locally square integrable . behavior of the exponents of the singularities is investigated by numerical computation and, in limiting cases, analytically . the singular parts of some flows are investigated numerically .
experiments with two-dimensional, transversely impinging jets . experiments on the interaction of transversely impinging two-dimensional jet flows were performed in which a low pressure control jet flow interacted with a relatively high pressure power jet flow . the ratio of the control jet to the power jet supply chamber gauge stagnation pressure was adjusted at 0, 10, and 15 . shadowgraphs of the power jet alone, as well as the corresponding interacting jet flows, were recorded to establish the nature of and changes in the shock structure . the jet flows were traversed by a pitot tube to record the pitot pressure distributions at various locations downstream of the power jet exit . it was discovered that with the addition of only a small percent control jet flow, the normal shock front of the highly underexpanded power jet flow changed to an oblique shock structure and, downstream of the previous location of the normal shock which appeared in the power jet flow alone, the maximum recovery stagnation pressures were proportionally much higher . the mechanism for this behavior of the normal shock is proposed . possible practical importance of this behavior of interacting jet flows with reference to aerodynamic noise, supersonic diffuser losses, etc., is also pointed out . for the power jet flow alone it was found that by considering the actual jet boundaries as simply an extension of the actual nozzle, the average axial flow quantities, computed from the area-mach number relation using the observed cross-sectional area of the jet flow, agreed quite favorably with the experimental results .
growth of the turbulent wake behind a supersonic sphere . experimental data are presented on the growth of turbulent wakes up to 8000 calibers behind and spheres traveling at supersonic velocities . experimental determination of the exponential coefficient in the growth law is very difficult, if not impossible . data are presented in the form of both . in the representation, two regions of different wake growths are observed . by means of a quasi-steady state assumption, the effect of drag deceleration is eliminated and growth of the far wake compared with theoretical predictions . the agreement with the lees-hromas theory in this region was found to be quite good .
unsteady aerodynamic forces on slender supersonic aircraft with flexible wings and bodies . the present paper derives generalized aerodynamic forces for slender supersonic aircraft on the basis of slender-body theory . particular consideration is given to configurations which are spanwise flexible . to treat configurations with flexible wings and bodies, the slender wing-body problem is first reduced to a simple body problem whose solution is well known, and a solution of the latter is obtained, utilizing the circle theorem or method of images and a known solution of the airfoil equation for a double interval . with this approach, it is not necessary to apply conformal mapping techniques, and the solution so obtained is valid for arbitrary spanwise downwash distributions . on the basis of slender-body theory, the velocity potential and, subsequently, the generalized aerodynamic forces are derived for a general class of spanwise flexible wing-body configurations .
the blunt-leading-edge problem in hypersonic flow . the present paper is mainly concerned with the hypersonic flow over a flat plate with a blunt nose . the analysis is based on the flow model in which the flow field behind the shock wave may be divided into two regions .. the inviscid-hypersonic-flow region and the entropy layer, across which the pressure has no appreciable change . the equations for the entropy layer can be reduced to those of the usual boundary-layer problem with the exception that the outer edge of the entropy layer, as well as the pressure remain unknown . these unknowns are determined so as to approximately match the entropy-layer solution with the inviscid hypersonic solution in which the shock wave has the shape of the power law of the distance from the leading edge . the assumed flow model is shown to be valid over a restricted range depending on the wall-to-stagnation temperature ratio and (where is the reynolds number based on half the thickness of nose t, m the freestream mach number, and c the chapman-rubesin constant . actual calculations have been carried out for the case with typical values of and the wall-to-stagnation temperature ratio . the calculated values for both the surface pressure and heat-transfer rate are compared with the experimental data . as regards surface pressure in particular, a satisfactory agreement with the data is obtained . the validity of the assumptions upon which the present analysis is based has been examined from the numerical results, and the region of the validity has been found to extend over a certain large range of the nondimensional distance from the leading edge .
theoretical investigations of a supersonic laminar boundary layer with foreign-gas injection . the phenomena arising from the uniform injection of helium, air, argon, and iodine into the laminar boundary layer of a supersonic stream of air in a tube were investigated theoretically . the partial differential equations describing the energy, mass, and momentum transfers through the boundary layer were obtained, and a series solution was found for the case of uniform injection through the tube wall . the results of the analysis are in the form of axial distributions of wall temperature and recovery factor and of radial distribution of concentration, velocity, static, and stagnation temperatures . the gas mixture was assumed to be a perfect gas . properties of the mixture were calculated in accordance with the gibbs-dalton rule and the mixing rules based on the kinetic theory of dilute gases . transport properties for pure air were taken from the n.b.s. tabulations . transport properties for the other gases were calculated by kinetic-theory methods, employing a lennard-jones 6-12 model for the interaction potential . the theoretical predictions for the recovery factor along the tube with air or argon injection agree with experimental data to within one percent . the theoretical predictions for helium injection indicate an 8-percent rise in the recovery factor along the tube, while experiments have shown only a 1-percent rise . these differences between theory and experiment are attributed to inaccuracies in the approximations to the transport properties of the binary mixtures .
hypersonic viscous flow over a sweat-cooled flat plate . this paper presents a theoretical analysis of the hypersonic viscous flow over a sweat-cooled flat plate . the physical system under consideration is the hypersonic laminar boundary layer over a porous flat plate with homogeneous, normal injection of a coolant into the external stream . a heat balance at the porous surface is made between the heat transferred to the surface and the heat absorbed by the coolant . the existence of similar solutions requires a nonuniform distribution of coolant injection . the method of solution consists of the integration of three simultaneous first-order equations, the momentum and the energy integral equations in the boundary layer, and the tangent-wedge approximation in the inviscid layer . first-order asymptotic formulas are given in both the strong and the weak pressure interaction regions for the induced surface pressure, the skin-friction coefficient, and the nusselt number . numerical results for three specific cases are presented and discussed .
a study of slender shapes of minimum drag using the newton-busemann pressure coefficient law . the problem of minimizing the drag of a slender, two- dimensional or axisymmetric body in hypersonic flow at zero angle of attack is considered under the assumption that the pressure coefficient law is newton’s impact law as modified by busemann in order to include centripetal acceleration effects . after the condition that the pressure coefficient be nonnegative is accounted for and after arbitrary conditions are imposed on, in addition to the thickness and the length, the enclosed area and the moment of inertia of the contour in the two-dimensional case and the wetted area and the volume in the axisymmetric case, the minimal problem is formulated as a problem of the mayer type and solved by the combined use of the euler-lagrange equations, the transversality condition, the erdmann-weierstrass corner condition, and the properties of the switching function . particular attention is devoted to the class of problems such that, among the four quantities being considered, two are prescribed while the remaining are free . for these problems, the extremal arc is composed of two subarcs .. one is characterized by a positive pressure coefficient and is called the regular shape,. the other is characterized by a zero pressure coefficient and is called the free layer . in this connection, the analysis shows the existence of two different types of solutions depending on whether the thickness is given or free . if the thickness is given, the expression for the regular shape is a power law, and the transition from the regular shape to the free layer occurs in the second half of the body . in the two- dimensional case, the exponent of the power law is 1 if the length is given if the enclosed area is given, and 3 if the moment of inertia of the contour is given,. the transition point from the power body to the free layer is located at 50 percent of the length if the length is given, at 66 percent if the enclosed area is given, and at the axisymmetric case, the exponent of the power law is if the length is given, 1 if the wetted area is given, and if the volume is given,. the transition point from the power body to the free layer is located at 60 percent of the length if the length is given, at 70 percent if the wetted area is given, and at 80 percent if the volume is given . on the other hand, for problems where the thickness is free, the equation governing the regular shape is not that of a power body, and the point of transition to the free layer is located in the first half of the body . in the two-dimensional case, the transition point is at 28 percent of the length if the length and the enclosed area are given, at 32 percent if the length and the moment of inertia of the contour are given, and at 45 percent if the enclosed area and the moment of inertia of the contour are given . in the axisymmetric case, the transition point is located at 35 percent of the length if the length and the wetted area are given, at 39 percent if the length and the volume are given, and at 46 percent if the wetted area and the volume are given . for all of the cases considered, analytical expressions are obtained for the optimum shapes, the thickness ratios, and the drag coefficients .
uniformly valid second-order solution for supersonic flow over cruciform surfaces . considered is the second-order supersonic flow over a cruciform configuration consisting of two intersecting rectangular wings of high aspect ratio . the practical interest is in application to supersonic inlets, wing-body junctions and vehicle fins . the fundamental interest centers about identification and adjustment of the severe local failures of the ordinary second-order theory . for wings with discontinuous slopes, discontinuous potentials occur across the planar shock and square-root singularities in the velocities occur at the intersection of these shocks with the cruciform surfaces . the problem is simple enough so that these interesting features stand out clearly . a second-order solution uniformly valid to first order is constructed by adjustment of the ordinary second-order solution obtained first . the uniformly valid solution has two different series representations in the thickness parameter . one is the ordinary second-order series in ascending integral powers of the thickness parameter which is valid in the interior of the vertex-centered undisturbed mach cone, and the other is a series containing fractional powers which is valid adjacent to and upstream of this mach cone . the uniformly valid solution gives the detailed wave structure and shows a flow regime upstream of the vertex-centered undisturbed mach cone not predicted by the ordinary theory . the two solutions are otherwise identical . the wave structure consists of a pyramidal arrangement of planar shocks adjacent to and upstream of the above cone, followed by weaker oblique expansion fans and finally by two extremely weak shocks coincident with the vertex-centered undisturbed mach cone . as an example of the above, detailed results are presented for the case of two intersecting wedges . application of the techniques to other quasi-cylindrical problems is discussed .
the propagation of a nonuniform magnetohydrodynamic shock wave into a moving monatomic fluid . an initially uniform magnetohydrodynamic shock wave of arbitrary strength propagates through a channel which consists of two portions of which one has uniform cross-sectional area while the other is of varying cross-sectional area . it is assumed that the flow in the nonuniform section in front of the shock is initially a uniform state and no perturbations (due to the area variations) of this flow reach the shock until the area variation is encountered . when the shock enters the nonuniform section, it is perturbed, the shock strength altered and the subsequent flow is nonisentropic . in addition to the perturbation due to the effect of the area variations on the initially uniform upstream flow, there are two further contributions–viz., a permanent perturbation caused directly by the area changes and a transient disturbance–which propagates with true sonic speed with respect to the flow behind the shock, due to reflections of the permanent perturbation at the shock . expressions for these various contributions are obtained . the results presented include as special cases propagation of a nonuniform conventional gas dynamic shock into a moving nonconduction fluid and propagation of a nonuniform hydromagnetic shock wave into a stationary fluid .
experimental effect of bluntness and gas rarefaction on drag coefficients and stagnation heat transfer on axisymmetric shapes in hypersonic flow . inverted hemispheres, circular discs (normal to stream), spheres, 26 total angle 0.368 blunt hemisphere cones, 18 total-angle sharp cones, and other axisymmetric shapes were run in a hypervelocity wind tunnel . hypersonic drag coefficients at zero angle of attack were measured in the air velocity range, 7,000- efficient is defined as drag force . knudsen number is defined as mean free path behind shock sphere shock detachment distance . in the case of nonsphere shapes, the knudsen number is defined as the knudsen number of a sphere with the same base diameter . these drag coefficients cover the range of gasdynamics to free molecule flow and are given in graphical form . the drag coefficients were measured by means of a ballistic balance in millisecond intervals, and referenced to the drag coefficient of a sphere in the gasdynamics region, for a gamma of 1.4, of 0.92 . tunnel stagnation conditions of pressure, temperature, density, and pressure drop with time were measured directly . in the tunnel test section, velocity, q density, total pressure, and static pressure were measured directly . these experimental curves have been found useful in the analysis of complex shapes if the complex shapes can be easily broken down into simple components with small interactions between components . heat-transfer distributions have also been obtained on these and other complex shapes in the hypervelocity wind tunnel, by means of a special paint which changes through several visible spectral orders within a heat transfer range of x10 for a single application . heat transfer rates, so obtained, have been performed in the hypersonic gasdynamic and slip flow regions and are presented for spheres . these data, in the vorticity interaction region, agree with the data of ferri and zakkay .
effects of cooling on boundary layer transition on a hemi- sphere in simulated hypersonic flow . an experimental investigation of the effects of cooling on boundary-layer transition on a 9-in. diameter hemisphere in simulated hypersonic flow is reported . the newtonian pressure distribution was obtained by use of a shroud and boundary layer cooling was achieved by internally cooling the model . transition was detected with hot wires and with a pitot tube at the surface . attained . transition was observed in the subsonic and near-sonic flow region at and upstream of n = 45 . in this region the stagnation reynolds number at which transition occurred when the surface was highly polished was only slightly affected by cooling within the temperature range . thus, transition reversal does not occur on a polished spherical surface within the range of these tests, and we therefore conclude that the cooling did not cause the linear stability of boundary layer to decrease significantly . an essential feature of transition studies with boundary-layer cooling is the close control of surface roughness . in the present experiments this control required, in addition to a highly polished surface, the necessity for low water vapor dewpoint, the avoidance of carbon dioxide condensation and the utilzation of every available means for removing the dust from the airstream .
magnetohydrodynamic mach cones . features of the surfaces of main disturbance created by a small object in steady motion through a conducting fluid are examined . these surfaces are found by drawing tangent cones from the object to the relevant wave-front diagrams . the outer wave cone (when present) is smooth, but the two inner cones have cross sections similar to the cusped figures of the inner wave-front diagram . it is conjectured that the disturbance may be concentrated along such line cusps . this has particular relevance in the application of known two-dimensional results to three-dimensional problems, say in the well-known techniques of aerodynamics . in mhd the omission of the large disturbance characteristics implicit in a two-dimensional solution may invalidate its use in any practical three-dimensional problem .
supersonic airfoil performance with small heat addition . an analytical method is presented which permits a very rapid evaluation of the acrodynamic effects arising from the addition of small amounts of heat near supersonic two-dimensional airfoils . this method applies to shockless inviscid flow without heat conduction . also, the mechanism by which the sesired heat addition is achieved is not considered . it is shown that even small amounts of heat generate a substantial pressure rise and thus cause appreciable changes in the acrodynamic coefficients . the results of this analysis compare favorably with those obtained by a more accurate, but also more tedious, graphical method of characteristics . two possible modes of application to an airplane design are considered from the energy requirements standpoint . in this connection, it is shown that the decrease of the required wing area resulting from heat addition may, in some cases, lead to savings in the rate of the fuel consumption . in general, however, one should not expect any substantial reduction in energy requirements resulting from the application of the wing heat addition .
a linearized analysis of the forces exerted on a rigid wing by a shock wave . solutions are obtained in closed form for the pressures exerted on a rigid half plane by an incident, plane acoustic shock wave . the angle of incidence of the wave front is arbitrary and the half plane is considered to be traveling at constant velocity, subsonic of supersonic with respect to the acoustic medium . a closed-form solution is obtained also for a rigid wedge which is motionless with respect to the acoustic medium . the analysis is carried out by transforming the wave equation to laplace’s equation by the busemann conical transformation and then applying conformal mapping .
aerodynamic processes in the downwash-impingement problem . theoretical and experimental data relating to the downwash impingement problem are examined in order to arrive at a coherent understanding of the process of entrainment of ground particles in the flow . it is demonstrated that a key mechanism in the process is the interaction of nonuniform flow in the ground boundary layer with bluff ground particles . this interaction produces a lift force which, under typical conditions, equals or exceeds the particle weight . in the interest of quantitative prediction of the conditions necessary for particle entrainment, four subsidiary problem areas in the impinging jet are examined . these are the viscous decay, the inviscid flow field, the ground boundary layer, and the forces on a bluff body in nonuniform flow . applicable theories are used in conjunction with experimental data to assess the accuracy and range of validity of the theories, and to define the stream conditions which will cause particle entrainment . available data are applied to the establishment of criteria for particle entrainment in the vicinity of the impinging-jet stagnation point . these criteria show that entrainment occurs in a finite annular region on the ground plane, and that the particles most readily entrained are those with a diameter equal to about two-thirds the thickness of the ground boundary layer . the configuration size is shown to influence the process in that the onset of entrainment is fixed by the jet diameter and velocity, and the size of the ground particles . the criteria established provide a quantitative estimate of the conditions causing entrainment and provide a basis for scaling experimental results to a variety of full-scale situations .
on slender airfoil theory for nonequilibrium flow . an exact linear theory for nonequilibrium flow past a thin airfoil is given . green’s function technique is used to solve the boundary value problem for the governing third order equation . upon satisfying the boundary condition on the airfoil surface an integral equation is obtained which has an exact solution . the final expression for the velocity potential, given as an integral over the source strength times the green’s function, shows that the solution is dependent not only on the slope variation of the airfoil but also on its curvature variation . this turns out to be the case for all free-stream mach numbers . as an example, the supersonic flow past a wedge is considered .
boundary layer transition at supersonic speeds-three-dimensional roughness effects (spheres). further experiments carried out in the 12-in. supersonic wind tunnel of the jet propulsion laboratory of the california institute of technology to investigate the effect of three-dimensional roughness elements (spheres) on boundary-layer transition on a the local mach number for these tests was 2.71 . the data show clearly that the minimum (effective) size of trip required to bring transition to its lowest reynolds number varies as the one-fourth power of the distance from the apex of the cone to the trip . use of available data for other mach numbers indicates that the mach-number influence for effective tripping is taken into account by the simple expression . some remarks concerning the roughness variation for transition on a blunt body are made .
effect of uniformly distributed roughness on turbulent skin-friction drag at supersonic speeds . an experimental program was carried out in the 18-in. by 20-in. supersonic wind tunnel of the jet propulsion laboratory to determine the effect of uniformly distributed sand-grain roughness on the skin-friction drag of a body of revolution for the case of a turbulent boundary layer . the mach number range covered was 1.98 to 4.54, and the reynolds number varied from about 3 x 10 to 8 x 10 . some data were also obtained at a mach number of 0.70 . at speeds up to a mach number of 5 and for roughness sizes such that the quadratic resistance law holds, the compressibility effect is indirect, and the skin-friction drag is a function of only the roughness reynolds number, exactly as in the incompressible case . it is shown that the entire compressibility effect is a reduction of the fluid density at the surface as the mach number increases . the critical roughness, below which the surface is hydraulically smooth, is,. this is equal to the thickness of the laminar sublayer for a smooth surface for both compressible and incompressible flow . over the range of roughness sizes considered here, there appears to be no wave drag associated with the drag due to roughness . the shift in the turbulent veocity profile for a rough surface at supersonic speeds is a function of only the roughness reynolds number and quantitatively follows exactly the same law as that for the incompressible case .
heat transfer to slender cones in hypersonic flow, including effects of yaw and nose bluntness . as part of a general study of the aerothermodynamic characteristics of flight of hypersonic vehicles, an investigation of laminar heat transfer to slender yawed cones has been conducted . experiments have been made in the cal 11- by 15-in. shock tunnel at mach numbers from 11 to 13 and at yaw angles up to were tested . the heat-transfer rates are compared with theoretical predictions . the effects on the local heat-transfer rates of the boundary-layer displacement thickness, transverse curvature, yaw, nose bluntness, and the entropy sublayer are discussed . it is shown that, at zero yaw, the experimental data for the sharp cone are in good agreement with theory when boundary-layer displacement and transverse-curvature effects are included . for the yawed sharp cone, the heat-transfer rates along the most windward streamline are in good agreement with reshotko’s theory for yaw angles up to 3 . at larger yaw angles, the experimental heat transfer was found to be greater than that predicted theoretically . however, at these yaw angles the heat-transfer distribution on the windward side was in good agreement with laminar-boundary-layer calculations based on an assumption of local similarity . the zero-yaw tests of the blunted cones showed qualitative agreement with cheng’s shock-layer theory for slender blunt-nose bodies .
the drag of elongated bodies over a wide reynolds number range . the resistance of bodies in motion through an incompressible viscous fluid is predictable from stokes- or oseen-type solutions in the creeping-motion range, while some test information is available in the boundary-layer range . with the exception of experimental results for spheres or circular cylinders and analytical and experimental results for flat plates, almost no information is available on other bodies, particularly in the intermediate range of reynolds numbers extending from unity to a million . experimental results as obtained from hydroballistic studies in water and glycerin-water solutions are presented for finned ellipsoids of fineness-ratio .4 over a 20,000-fold range and are correlated with available information on other bodies . although results do not extend down to the creeping-motion region where analytical predictions are available, comparison with the drag coefficient trends for spheres and flat plates indicates that an appropriate curve for the ellipsoid could be extended so as to cover the entire laminar-viscous range . less extensive results are presented on the drag of fineness-ratio 8 ellipsoids and on laminar-turbulent transition occurrences .
the effect of slip particularly for highly cooled walls . it is found that for boundary conditions on the velocity slip and on the temperature jump which are not oversimplified in an unrealistic way, the effect of these phenomena on the heat transfer and the shear at a stagnation point is of the order of the ratio of the mean free path outside the boundary layer to the boundary-layer thickness, even for highly cooled walls . a simplified theory of this effect is given, which puts the physical reasons for the results in evidence and agrees closely with the more accurate calculations . it is concluded that the effects of slip and jump are not negligible in comparison with other low-reynolds-number corrections, even for very cold walls .
pressure distribution in regions of step-induced turbulent separation . an analysis is made of the pressure distribution in the separated-flow region ahead of a step, using the concept of the turbulent mixing coefficient of crocco and lees and the /jet-flow/ model of chapman with some modification . on the basis of a variable mixing coefficient, a differential equation for the pressure distribution is derived, which gives the pressure rise as a function of the distance from the separation point . this equation contains the separation length as an unknown . a second equation is obtained by making a mass balance of the air entering and, leaving the /dead-air/ region ahead of the step . the pressure rise and the separation distance for a given mach number are determined by solving the two equations simultaneously . the analysis yields results which are in close agreement with the experimental data on steps, obtained at princeton, particularly for m = 3.85 . for lower mach numbers, a maximum variation of 5 percent is found between theory and experiment . use of the velocity profiles of jets, as required by the jet-flow model, necessarily restricts the applicability of the present study to flows with thin boundary layers at the separation point .
application of inequality constraints to variational problems of lifting re-entry . inequality constraints are introduced into the variational formulation of the optimum re-entry problem for a lifting vehicle to prevent human and or structural tolerances from being exceeded . these constraints consist of minimum and maximum angle of attack, maximum load factor, and maximum convective heat transfer (equilibrium temperature) . the equations have been programed for the ibm 704 computer, and sample trajectories are presented for which the total heat transferred to certain critical areas on the windward surface of the vehicle is minimized . these trajectories indicate the dominant effect of the constraints on the optimum flight path, which is shown to consist of both unconstrained and constrained arcs .
experimental lift and drag of a series of glide configurations at mach numbers 12 .6 and 17 .5. a series of semiballistic-type bodies consisting of three half sphere cones of 0.3 bluntness ratio with half-cone angles of 8.6, laboratory hypersonic shock tunnel at m = 17.5 and 12.4 . in addition, a representative winged glide configuration consisting of a sharp-edged, 60 swept delta wing with cone-segment the range of angle of attack for the half sphere-cone tests was the technique for force coefficient determination consists of analyzing high-speed motion pictures of the motion of very light balsa and isofoam plastic models which are literally free-flown for several milliseconds in the test section of the shock tunnel . because of viscous effects the newtonian prediction of half sphere cone drag is consistently less than, but generally parallel these bodies is generally well predicted by the newtonian theory except at small and moderate positive angles of attack where it is generally less than newtonian . this lift deficiency appears to increase with cone half angle . maximum lift-drag ratios fall considerably short of the newtonian predictions . several exploratory tests at mach 11.7 and low reynolds number ( approximately reduction in) on the 13 model produced an approximate doubling of minimum drag and a 35 percent decrease in (l d) max,. this demonstrates the importance of viscous effects for blunt bodies in the reynolds number range of these tests . the sharp leading-edge, 60 sweep delta wing-body configuration exhibited the same (l d) max, as the wing alone, about 2.80 at both positive and negative angles of attack .
determination of lift or drag programs to minimize re-entry heating . a study of single-pass re-entry from escape speed and from circular satellite speed is made to determine the lift program for a hypersonic glider and the drag-modulation program for a non-lifting vehicle that minimize the heating of the vehicles within acceleration or range constraints . a new method of numerical solution is used, similar to kelley’s /method of gradients,/ that permits rapid convergence to the optimum lift program starting with an original good estimate . this method avoids the two-point boundary-value problem of the calculus-of-variations formulation, and is applicable to any optimum-programing problem . an acceleration-tolerance limit is introduced which describes the human pilot’s capability to withstand acceleration more accurately than a simple acceleration limit .
boundary layer transition in the presence of streamwise vortices . results of an experimental investigation of instability leading to transition in the subsonic boundary layer flow along a flat plate are presented . a series of wings was placed outside the boundary layer to produce streamwise vortices, which in turn made the boundary layer three-dimensional–i.e., periodic in thickness in the spanwise direction . hot-wire measurements were made to trace the downstream development of the disturbance or wave created by the vibrating ribbon . as the wave travels downstream, it is deformed into a three-dimensional configuration by the three-dimensionality of the boundary-layer flow, but it is eventually damped out so long as it remains small in intensity . it is only after the wave intensity exceeds a certain amount (which depends on the degree of boundary-layer three-dimensionality) that the nonlinear effect manifests itself by the rapid amplification of wave intensity, the rapid increase in wave three-dimensionality, and the distortion in mean velocity profile . the appearance of nonlinear development inevitably leads to the breakdown of laminar flow, and hence the onset of turbulence . there is present a mechanism by which the energy is transferred from one spanwise position to another so that the breakdown of laminar flow occurs as a consequence of three-dimensional development of the wave as a whole .
steady flow of conducting fluids in channels under transverse magnetic fields, with consideration of hall effect . an approximate method of solution based on a minimum principle is presented for the steady laminar incompressible flow of an electrically conducting fluid through a straight channel of arbitrary cross section with conducting or nonconducting walls in the presence of a uniform transverse magnetic field . the hall effect is taken into account by making simplifying assumptions that the gas is fully ionized and that both reynolds number and magnetic reynolds number are small . numerical calculations are carried out for the case of a rectangular channel .
axisymmetric magnetohydrodynamic channel flow . the axisymmetric subsonic and supersonic flow fields, and the skin friction and heat transfer of an electrically conducting compressible fluid flowing in a channel of constant circular area through a magnetic field are investigated when the magnetic reynolds number is small . the inviscid-flow field for flow through a dipole field is solved by the method of characteristics in the supersonic case . for the subsonic case, linearized equations are derived for small values of the magnetic interaction parameter . numerical results are obtained by the relaxation method . the inviscid-flow-field solutions are used as boundary conditions for the laminar boundary layer along the wall, in which axial pressure gradients form an important feature . the exact continuum-flow equations are reduced by an order-of-magnitude analysis to the boundary-layer equations, which are solved numerically by an integral method using a fourth-degree velocity profile and a fifth-degree stagnation-enthalpy profile . pressure, temperature, and heat-transfer measurements are made with a shock tube under supersonic-flow conditions closely approaching those used in the numerical computations . general agreement is found between the theoretical and the experimental results .
inviscid-incompressible-flow theory of static two-dimensional solid jets, in proximity to the ground . the inviscid-incompressible-flow theory of static two- dimensional solid jets impinging orthogonally on the ground is presented using conformal mapping methods . it is shown that the thrust of a solid jet at constant power initially decreases as the ground is approached . the magnitude of the thrust out of ground effect is regained only at a very low height-to-jet width ratio (approximately 0.55) . the maximuin decrease is about 6 percent . the ground effect on solid jets is thus largely unfavorable .
on the plk method and the supersonic blunt-body problem . detailed analysis of the subsonic and transonic portious of the flow field about either very blunt or asymmetric configurations requires successive approximations,. these can be carried out in a systematic fashion only when an appropriate convergent perturbation procedure is available . the problem of producing successively refined sets of initial conditions for either /direct/ or /inverse/ analysis of the flow is formulated in the following terms .. given reasonable estimates for shock shape and pressure distribution on the body, can one determine the flow field of interest to any desired degree of approximation by a perturbation approach .qm a procedure to this effect is developed which involves stretching of coordinates in the spirit of the poincare-lighthill-kuo are transformed along body, shock, and intermediate lines so as to annul perturbations of the local resultant velocity,. (b) for the integral method the coordinate along the boundary of each strip is shifted so as to control perturbations of the velocity component that determines the critical point . the approach is justified by a study of the equations governing the direct method, and by consideration of model transonic flow problems for which closed form solutions are available . the range of validity of the proposed procedure is assessed by practical application and comparison with experiment,. results are presented for a disk set normal to a low-termperature air stream at m = 4.76, and for a highly asymmetric two-dimensional configuration at m = 8 .
the effect of adverse pressure gradients on the characteristics of turbulent boundary layers in supersonic streams . tests were conducted at mach numbers from 2.0 to 3.5 to determine the thickness and profile shape characteristics of turbulent boundary layers on two-dimensional and axisymmetric curved-surface models having adverse pressure gradients . the magnitude of the gradients relative to the boundary-layer thickness at the beginning of the gradient was varied by employing models having different radii of curvature and by changing the boundary-layer thickness at the beginning of the gradient . the overall pressure rise in most cases was greater than the value which would cause a turbulent boundary layer to separate if the pressure rise were created by an oblique shock wave . an analytical investigation was also conducted so that the results of the experimental investigation could be applied to the prediction of cases outside the range of the experiments . it is shown that boundary-layer momentum thickness can be predicted from the von karman boundary-layer momentum equation, but that measured values of boundary-layer profile shape are in poor agreement with values computed from procedures derived by extending conventional methods for predicting profile shape in subsonic flow . a new procedure for calculating boundary-layer profile shapes, developed in this paper, is shown to provide a good correlation between experimental and calculated values of boundary-layer profile shapes in adverse pressure gradients created by curved surfaces . this procedure is based on the experimental observation that the station at which high-energy free-stream flow actually mixes into a turbulent boundary layer in an adverse pressure gradient is well downstream of the station at which flow would have to mix in order to maintain a flat-plate profile .
heat transfer in the laminar boundary layer with ablation of vapor of arbitrary molecular weight . the reduction of heat transfer in the laminar boundary layer under the condition of vaporizing ablation is analyzed for arbitrary molecular weight of the vapor . primary assumptions are that the pressure gradient is zero, the individual components of the binary system are perfect gases, the prandtl number is constant, and the viscosity is proportional to temperature . variations through the boundary layer of the schmidt number for binary diffusion and the density-viscosity product, are included in the analysis . the wall temperature is held constant . numerical results are obtained for prandtl numbers of 0.75 and varying from 0.25 to 4.00, wall concentration of the foreign gas as high as 0.9 (corresponding to the high heat rates encountered during re-entry), and ratio of specific heats of foreign gas equal to that of air . kinetic theory is used to obtain schmidt number as a function of molecular weight and concentration . the departure of schmidt number and prandtl number from unity and the variation of reynolds analogy factor with prandtl number, blowing parameter, wall concentration, and molecular weight ratio are found to have relatively minor influence on the heat block ratio at high rates of ablation . the primary factor governing the influence of molecular weight ratio on the heat block ratio is the variation of across the boundary layer . little loss of accuracy is incurred, in the range of molecular weight ratios considered here, by assuming schmidt and prandtl numbers of unity as long as the variation is properly taken into account .
pressure-gradient effects on the preston tube in supersonic flow . this paper is concerned with an experimental investigation of the effects of a longitudinal pressure gradient in a supersonic stream of air over a bounding surface on the performance of a preston or impact-pressure tube at the surface . evidence is presented which indicates that for the mach number considered and for the range of pressure gradients covered, the preston tube functions in a completely satisfactory manner for the determination of local shear stress .
leading-edge separation of laminar boundary layers in supersonic flow . a brief description of the flow field is given for the interaction of shock wave and laminar boundary layer on a compression corner in supersonic flow . a special sub-case–that of leading-edge laminar separation–is analyzed by extension of chapman’s laminar mixing-layer theory . results are tabulated for ranges of mach number, and compression-corner angle . a limited region of possible leading-edge laminar separation with an attached leading-edge shock (or in certain cases an expansion) followed by a second shock due to the reattachment flow is found to exist . comparison with existing experimental data is found to be satisfactory in several cases .
the effect of sweep angle on hypersonic flow over blunt wings . a series of tests were carried out in the princeton university helium hypersoule wind tunnel on blunt two-dimensional wings at zero angle of attack with sweep angles up to 70 at mach numbers from 7 to 15 . the leading edge reynolds number varied from 3,000 to 25,000 . the measured pressure distributions were compared with the simple summation of the theoretical inviscid effect (based on blast wave theory using the normal mach number) added to the viscous effect (calculated as if no sweep were present) . for the unswept wing, the slope of the pressure decay was reasonably well predicted by the theoretical calculations . the viscous theory reasonably predicted the variation in the pressure distribution due to changes in leading-edge reynolds number . by subtracting the theoretical viscous effects, an inviscid mach number dependence of the 2.2 power was found as compared to the value of 2.0 predicted by the inviscid theory . the same approach for the swept wing did not give consistently satisfactory results . deviations avove and below the calculated value by as much as 40-50 percent were measured and there seemed to be no systematic variation with either mach number or reynolds number . at a constant high reynolds number, it was found that the pressure distribution varied with the distance along the wing with an exponent between about–0.53 and–0.58 except for a rather sharp decrease which occurred for the 70 sweep case . the pressure at a given station for a fixed mach number and given leading edge thickness varied as the cosine of the sweep angle to the 1.1 power as compared to the 1.3 power predicted from general geometrical considerations .
hypersonic nozzle expansion of air with atom recombination present . an experimental investigation on the expansion of high- temperature, high-pressure air to hypersonic flow mach numbers in a conical nozzle of a hypersonic shock tunnel has been carried out . the equilibrium temperature and pressure ranges after the reflected shock wave were 1400 to 6000 k and 100 to 1000 psia . static-pressure measurements, which are sensitive to the state of the gas, were made along the axis of the nozzle for different reservoir conditions . these results are compared with the calculated equilibrium and /frozen/ data for the same geometry and initial reservoir conditions . for reservoir pressures greater than 500 psia, the expansion of the air in the nozzle is essentially in equilibrium up to reservoir temperatures of about 4,500 k . for temperatures greater than almost frozen . at a given area ratio for the nozzle and reservoir pressure, the expansion process remains in equilibrium up to a certain reservoir temperature, and beyond this temperature the flow expansion deviates rapidly from the equilibrium process and approaches the frozen case .
hypersonic flow over an elliptic cone: theory and experiment . by applying hypersonic approximations to ferri’s linearized characteristics method, simple results were obtained for the shock shape and surface pressure distribution for an unyawed conical body of arbitrary cross section . calculations were carried out for an elliptic cone having a ratio of major to minor axes of, and a semivertex angle of about 12 in the meridian plane containing the major axis . an experimental investigation of the flow over this body conducted at a mach number of 5.8 in the galcit hypersonic wind tunnel showed that the surface pressure distribution at zero angle of attack agreed quite closely with the theoretical prediction . on the other hand, the simple newtonian approximation predicts pressures that are too low . surface pressure distributions and schlieren photographs of the shock shape were obtained at angles of attack up to 14 at zero yaw, and at angles of yaw up to 10 at zero pitch . at the higher angles of attack the newtonian approximation for the surface pressures is quite accurate .
the curtain jet . a detailed analytic study is made of the curtain jet, the two-dimensional fluid wall used to contain support pressure on the underside of ground effect machines . two variations of the jet are studied in detail–the bifurcated jet, in which a portion of the flow streams into the support pressure region, and the deflected jet, in which none of the flow penetrates into the support pressure region . kirchhoff-helmholtz free steamline analysis is used to construct the flow field, and quantitative results are presented for the effect of nozzle inclination and detailed geometry on flow requirements and support pressure differential at varying altitudes .
supersonic shear flow past an airfoil between two parallel walls . the supersonic flow with assigned mach number gradient in the span direction past a straight wing between two parallel walls is studied using the small-disturbance theory . the governing equation for the disturbance pressure on the airfoil, together with the boundary conditions on the airfoil and at the walls, is solved by the method of separation of variables . upon separation the problem is reduced to a sturm-liouville eigenvalue problem and to the solution of the telegraph equation . as an application, a certain mach number profile is selected and the resulting pressure distribution on a parabolic arc airfoil is computed .
direct calculation of pressure distribution on blunt hypersonic nose shapes with sharp corners . the method of belotserkovskii for calculating hypersonic flow fields past a circular cylinder is extended to deal with axially-symmetric flow past sharp-cornered nose shapes, in particular, spherical segments and flat-headed cylinders . results on spheres are also included . in the present paper belotserkovskii’s first approximation is considered, and comparison of calculated pressure distribution and shock shape with experimental results shows very good agreement .
a theory of the two dimensional laminar bounary layer over a curved surface . the purpose of this paper is to present a theory to account for surface curvature effects on the two-dimensional boundary-layer flow which approaches a potential flow at free stream . the problem of two-dimensional viscous flow is first formulated by using the streamlines and their orthogonal trajectories as the generalized coordinates . a boundary-layer approximation is applied to the navier-stokes equations and the gauss equation in the generalized coordinates to yield the boundary-layer equations . the conditions under which similar solutions of the boundary-layer equations exist are determined . by a simple transformation, the governing differential equation can be expressed in a form which reduces to the falkner-skan equation for zero surface curvature . numerical results for a similar solution which corresponds to a flow over a curved surface with zero surface pressure gradient have been obtained . the velocity profiles in the boundary layer and the wall skin-friction distribution for concave and convex surfaces are presented . the wall skin friction for a convex wall is found to be higher than the blasius value for a flat plate . on the other hand, for a concave wall, the skin friction will drop below the blasius value as the curvature increases, but it appears to reach a minimum, and beyond this minimum point it will increase again . the same flow problem was treated by murphy by a different method of analysis . comparison of murphy’s results with those obtained by the present method reveals some basic differences in the boundary-layer characteristics . in particular, murphy’s results indicate that the wall skin friction for a convex surface is smaller than the blasius value, while for a concave wall it is higher .
the stagnation point boundary layer in the presence of an applied magnetic field . similarity equations for axisymmetric compressible flow are obtained, assuming that the magnetic field is uniform, normal to the surface, and unaffected by the flow, and that the conductivity varies as the nth power of the enthalpy . numerical solutions are given for a number of values of n and of the field strength, and are used to modify the estimates of heat-transfer made by the author using inviscid theory (title source 26, 536-537, 1959) .
foreign-gas injection into a compressible turbulent boundary layer on a flat plate . the distributed injection of a foreign gas into a compressible, turbulent boundary layer in the absence of a pressure gradient is considered . the analysis is performed within the framework of the binary-mixture concept, that is, the primary fluid flowing over the surface represents one component while the injected species represents the second . calculations have been performed for the injection of helium into undissociated air . the results indicate an effect of mach number on surface shear and energy transfer when distributed light-gas injection normal to the surface exists . a comparison with experimental data indicates reasonable agreement over a range of mach numbers .
the newtonian approximation in magnetic hypersonic stagnation-point flow . the hypersonic flow of an electrically conducting fluid around the stagnation region of a sphere carrying a radial magnetic field is examined . by assuming a newtonian pressure distribution and constant density, the differential equation of the inviscid flow is integrated and a simple closed-form solution is obtained . it is found that the ratio of the stand-off distances of the shock wave for the magnetic and nonmagnetic cases does not depend explicitly on the magnetic parameter s (ratio of the ponderomotive force to the free-stream inertia force) nor on the density ratio (the value at the free stream divided by the value behind the shock wave) but on the product s at least for values of between and . the velocity gradient on the body is also calculated and the ratio of the magnetic to the nonmagnetic case is shown to depend on the parameter . the case of cylindrical shocks is also examined,. the same general conclusions are drawn .
body under lifting wing . an investigation is made of supersonic-aircraft configurations composed of a cambered body positioned a certain distance beneath an arbitrary lifting wing . the geometry of the wing is regarded as given and the geometry of the body may be given or optimum . expressions for the drag and lift are obtained from reverse-flow considerations,. these greatly implement such a study when interference cross flows must be cancelled . the drag advantage to be gained when a given body and wing assume a given orientation is studied . treated more extensively is the variational problem of determining the optimum wing incidence and optimum body shape, for the given volume and length, to yield the minimum drag for prescribed lift . numerical results are provided to indicate the significance of the large number of parameters appearing in the problem . of these, the gap between the wing and the body is found to be particularly important . it is found that at low gap moderate body distortions have a significant influence on the drag . drag reductions of up to 44 relative to the case of no interference have been found at a mach number of 2.24 in a configuration having a gap approximately equal to the maximum diameter of the body, and a wing chord of about three eighths of the length of the body . comparison is made with the conventional wing-body combination including the effects of skin friction, and it is concluded that the advantage suggested by the preceding considerations is not appreciably diminished . finally, it is shown that the configurations studied lead to bodies of fineness ratios much lower than are appropriate to conventional wing-body combinations . tests were made on an arrangement consisting of a scars-haack body located under a lifting rectangular diamond-profile wing . the mach number was 1.6 and the reynolds number was 9.17 x 10 based on the body length . it was found that the measured lift developed on the wing due to the flow field of the body agrees very well with the theoretical value . downstream of the impinging shock from the wing, flow separation was observed on the exterior of the body but not in the interior . the separation is attributed not to the pressure rise across the shock but to the pressure field arising from the reflection from the body of the shock-induced cross flow . further observations suggest that the separation can be avoided by pitching the body or by kinking the body at the shock wave to accommodate the shock-induced cross flow .
nonsimilar solutions of the compressible laminar boundary layer equations with applications to the upstream-transpiration cooling problem . a new method is presented for predicting the boundary-layer characteristics downstream of the porous region of an injection-cooled surface . the method consists of a general scheme for obtaining nonsimilar solutions of the compressible- laminar-boundary-layer equations and is formulated along the following lines . the viscous domain is divided into n curvilinear strips . the governing equations are then integrated along the coordinate normal to the body from the surface to the boundary of each strip . as a result, one obtains a set of independent integro-differential relations . the integration is carried out by expressing the integrands as polynomials, the coefficients of which are functions of the unknown values of the velocity and temperature on the strip boundaries as well as of the imposed boundary condition at the wall and at the outer edge . after the integration is performed, a set of ordinary first-order differential equations is obtained . the set of equations may be solved for given initial conditions by a numerical integration scheme such as the runge-kutta method . several numerical examples of interest are presented .
the turbulent boundary layer on chemically active ablating surfaces . incompressible turbulent-boundary-layer analysis is extrapolated analytically to the case of a compressible turbulent boundary layer with ablation or mass injection at the surface . the effects of chemical reactions such as dissociation and recombination as well as combustion are included . the analysis applies to blunt as well as sharp bodies which are either axisymmetric or two-dimensional . when the turbulent lewis and prandtl numbers are unity, it is found that, as in the laminar case, little detailed knowledge of the chemistry inside the boundary layer is required in most instances . the conditions at the surface and the outer edge of the boundary layer are often sufficient for prediction of heat and mass transfer . comparison is made with experiments on the combustion of graphite under turbulent flow conditions . prediction of ablation rates within about 30 percent accuracy is obtained when empirical constants obtained from incompressible velocity profiles with no mass injection are used .
some considerations on the laminar stability of time-dependent basic flows . as a stability criterion for infinitesimal disturbances in an incompressible, parallel but time-dependent basic flow, it is proposed to introduce the concept of /momentary stability,/ which is said to prevail at the instant if the kinetic energy of the disturbances, as a fraction of the kinetic energy of the basic flow, tends to decrease . the significance of such a criterion is briefly discussed . for special time-dependent basic flows which are described by similar velocity profiles at all times (except for changes in amplitude), in the inviscid limit only a change of the time scale is needed to reduce the solution essentially to that for the steady case . the disturbances may be of either the transverse-wave or the longitudinal-vortices type . the result indicates a very strong destabilizing influence of deceleration, which is likely to overshadow that of the velocity profile under normal circumstances . the observations of fales rotating cylinders) are believed to be largely due to the deceleration . at finite reynolds numbers, the usual procedure of calculating the stability solution on the basis of the instantaneous profile is further shown to be valid only for extremely slow acceleration or deceleration . even when the solution is acceptable, the condition for neutral stability may not be used without reservation . to calculate momentary stability properly, a procedure for a slowly varying but more general profile is also described .
supersonic boom of wing-body configurations . the supersonic boom in steady, level flight of a wing-body configuration is due to the effects of body volume, wing volume, wing incidence or lift, and wing-body interference . the contribution in the far field of each of these factors can, in any given azimuthal plane, be represented as that of an equivalent body of revolution . this concept is developed to investigate the possibilities of using interference among the components of a wing-body configuration to reduce or suppress the boom due to lift . results of wind tunnel experiments are also presented and discussed in light of the theoretical indications .
on the aerodynamic noise of a turbulent jet . a new model is advanced for analyzing the broad-spectrum noise of a turbulent jet . the shear layer bounding the turbulent jet is assumed to play an important role in modifying the / quadrupole sound radiation/ from the interior . to the sound- emitting small-scale turbulent eddies (with frequencies much higher than those of large-scale eddies), the laminar shear layer has an irregular contour, as if the large-scale turbulent motions were frozen . the linearized analysis is then applied to the laminar shear layer to relate the acoustic oscillations across it . the concept of geometrical acoustics is generalized to represent the passage of an acoustic ray through a laminar shear layer . acoustic rays may be traced across the shear layer as transmission and refraction, but they may also be apparently /absorbed/ or /generated/ by the laminar layer . this /generation/ is visualized as the schematic representation, within the framework of geometrical acoustics, of the action of the reynolds stress in transferring energy from the shearing mean flow to the acoustic waves . such action of the reynolds stress can be neglected in ordinary acoustics when the acoustic medium is not moving at speeds comparable to the speed of sound in the medium . however, this action is of crucial importance in the aerodynamic noise of high-speed turbulent jets where the reynolds stress is the fundamental element of the radiating quadrupoles, according to lighthill . those acoustic waves that become /stationary/ with respect to the local mean flow somewhere in the interior of the shear layer are significantly modified by the viscous action through the critical layer . the shear layer therefore serves as a selective amplifier of the acoustic waves passing through it . kinematically, the shear layer brings about the preferred downstream emission .. dynamically, the shear-layer augmentation significantly increases the polar peak noise level . the acoustic power output per unit solid angle for such downstream emissions augmented by the shear layer (including the polar peak) varies as, predicted by lighthill, but without lighthill’s convective corrections . on the other hand, the acoustic power output per unit solid angle nearly normal to the jet, due to the transmitted downstream-propagating waves, varies roughly as . heating the jet gas increases the shear-layer augmentation and may increase the polar peak noise level by several db . the silencing action of the edge notches and edge teeth may also be interpreted as due apparently to the result of possible distortion of the shear-layer profiles .
some aspects of nonequilibrium flows . in this paper are discussed some of the general features of nonequilibrium flow . in particular, vibrational relaxation is discussed in detail . this case is somewhat simpler than dissociation and ionization but it illustrates some of the main new features of nonequilibrium flow . those aspects of two-dimensional and axisymmetric flow behind shock waves are examined analytically which yield significant information without requiring numerical solution of the governing equations . the thermodynamics of a vibrational relaxing gas are discussed . the conditions for simulating flows are noted . crocco’s theorem and the characteristic equations are derived . then a simple method of obtaining the initial gradients of the flow variables behind a shock is shown . these gradients are used in discussing two particular flows . an exact solution for flow over a cusped body is obtained . flow over a wedge near the tip and far from the tip is considered . it is found that far from the tip a boundary-layer type phenomenon occurs .
solution of subsonic nonplanar lifting surface problems by means of high-speed digital computers . the method proposed in this paper is based on an approximate solution of the integral equation which represents the potential flow about a finite wing, with no restrictions beyond those necessary for linearization . after assuming the usual series representation of the wing surface vorticity distribution, the solution is achieved by approximating portions of the kernels of the transformed integral equation by single and double fourier series and performing termwise integrations analytically . this is followed by the routine inversion of the aerodynamic influence coefficient matrix, after satisfying appropriate boundary conditions at selected control points . in this procedure the number of control point used is limited only by the storage capacity of the computer . control points may be located so as to cover the entire wing surface, with due regard to certain physical requirements, permitting the accurate representation of complicated mean surface shapes . an evaluation of the proposed method is included . comparisons with other theoretical methods and electrical analogy tank results are used to substantiate the accuracy of the proposed method when applied to plane wings . a final evaluation involves a comparison of calculated surface pressure distribution with wind-tunnel measurements on a swept, tapered wing with a cambered and twisted mean surface . the agreement evidenced in the latter comparison has the same order of overall accuracy as similar comparisons on plane wing planforms . in either case, the results given by the proposed method are within the accuracy requirements for most aircraft design studies .
the supersonic boom of a projectile related to drag and volume . the whitham theory predicting the far-flow field around a projectile is used to derive body shapes which produce extreme bow shock-wave pressure jump or /boom,/ subject to constraining conditions regarding the drag due to the bow shock and fineness ratio of the bodies . it is found that the minimum drag body is also the minimum boom body . the body-volume effect and the effect of discontinuities in slope of the body meridian section on the boom intensity is investigated . as a general result of the investigation, it can be said that the boom of a projectile for given mach number and flight altitude is primarily determined by its length and fineness ratio . the maximum variation in the boom intensity for pointed bodies with given length and fineness ratio is of the order of 10 per cent . the geometry of the bodies is thus found to play a minor role .
an analytic extension of the shock-expansion method . the problem is considered of calculating approximately the inviscid rotational flow field and pressure distribution about a smooth two-dimensional airfoil with sharp leading and trailing edges in a uniform supersonic or hypersonic stream . the assumption of a perfect gas is made, and the basic flow pattern for the analysis is taken to be given by the simple isentropic shock-expansion method with straight characteristics . an elementary characteristics treatment is discussed to show when the simple shock-expansion method should be satisfactory for computing the surface pressure distribution, and under what circumstances it may be expected to break down . by utilizing characteristic variables the isentropic shock-expansion method is then formulated analytically, and an analytic result is obtained for the shock shape corresponding to this zero-order approximation . in the special case where hypersonic similitude is applicable, that is, for slender bodies and high mach numbers, the shock-shape expression for large distances is found to reduce to the result previously given by mahony, which for weak shocks and slender bodies in turn reduces to the simple-wave result first given by friedrichs . employing the analytic form of the isentropic shock-expansion method as a zero-order approximation, an analytically consistent perturbation method is developed by expanding the dependent flow variables in the exact partial differential equations in powers of the reflection coefficient for simple waves interacting with an oblique shock . the scheme by its nature helps to define those regions in which shock expansion can be used, in addition to taking into account in a perturbation sense the factors neglected in simple shock-expansion theory, namely, the curvature and reflection of the mach waves and the correct boundary conditions at the shock wave . analytic solutions are obtained for the first-order corrections, including the surface pressure distribution . the necessary numerical computation of the integrals involved is considerably simpler than a direct application of the method of characteristics . to illustrate the method and its accuracy, the zero-order shock shape and first-order pressure distribution are calculated for a family of parabolic arc airfoils at an infinite free-stream mach number . these results are compared with rotational characteristic solutions where available, and the present method is found to be in excellent agreement .
plasma flow over a thin charged conductor . the flow of a dense plasma over a wavy conducting wall of small amplitude is investigated where magnetic effects are negligible . these results are then used to analyze the flow over a thin conductor with cusped edges . it is found that the coulomb drag vanishes identically, while the fluid-pressure drag corresponds to the ackeret value for a neutral particle gas at the reduced-plasma mach number .
high-speed viscous corner flow . a boundary-layer integral method analysis is set up for compressible laminar flow in a symmetric corner with varying angle and streamwise pressure gradient . it represents an extension and modification of the constant density analysis of loitsianskii and bolshakov . the analysis is applied to the case of constant pressure, constant corner angle, and isothermal surfaces, for which the crocco velocity-enthalpy relation holds . although simplifying assumptions limit the quantitative accuracy outside the 60 to 120 angle range, some qualitative trends are probably correct outside this range . the limiting cases near 0 and 180 are not considered . favorable agreement between some results obtained by the integral method and by other methods is demonstrated for the isothermal, constant-density case . results show an increasingly sharp merger of the outermost isovels of streamwise velocity as the mach number increases . this sharp merging of the outer isovels is increased by increasing corner angle and by insulation of heating of the surfaces . within the interior of the viscous layer the spreading or contraction of the disturbed region of merging is influenced by surface heat-transfer conditions . surface shear and heat flux are decreased in the disturbed region, and are zero at the apex . for cases corresponding roughly to the higher mach numbers of wider corner angles, the /specific momentum-area/ exhibits the same decrease with mach number as its two-dimensional counterpart, whereas the /specific displacement-area,/ a measure of stream-tube dilation, increases more rapidly with mach number than the comparable two-dimensional parameter .
viscous flow past a quarter infinite plate . a study is made of the motion of an incompressible viscous fluid past a quarter-infinite plate, whose leading edge is perpendicular to and whose side edge is parallel to the undisturbed direction of the stream . it is assumed that the kinematic viscosity is small . the first approximation is taken to be the undisturbed motion, and successive approximations are obtained by iferation . the second approximation is the blasius shear layer necessary to satisfy the boundary conditions on the plate . in turn, this layer leads to a velocity component normal to the plate which needs a potential solution, in which the velocities are 0, to match with the conditions at infinity . further, the match at the edge of the blasius shear layer must be completed to 0 by introducing a secondary shear layer . the regions near the leading and side edges are considered separately,. in particular, the neighborhood of the side edge needs special care, because the determination of the chief terms is complicated by the presence of powers of log . in particular it is shown that the effect of the edge is to change the skin-friction coefficient by a factor
on the approach to chemical and vibrational equilibrium behind a strong normal shock wave . the concurrent approach to chemical and vibrational equilibrium of a pure diatomic gas passing through a strong normal shock wave is investigated . it is demonstrated that the equilibrium degree of dissociation behind the shock front, and hence the density, for the case where the vibrational degrees of freedom are frozen out can exceed the degree of dissociation, and hence the density, for the case where all degrees of freedom are in equilibrium . thus the necessary condition for a maximum of the density between the shock front and the position of full equilibrium flow downstream of the shock front is established . the sufficient condition that such a maximum be observable is shown to be that the approach to equilibrium of the vibrational degrees of freedom (or any other internal degrees of freedom) must lag the approach to dissociation equilibrium by a significant amount,. that is, there must be at least an order of magnitude difference in the respective relaxation times before such a maximum might be observed . an example calculation for a mach 13 strong shock wave in oxygen illustrates the appearance of such a maximum of the density and its dependency upon the relative values of the vibration and dissociation relaxation times .
hypersonic viscous flow near the stagnation point in the presence of magnetic field . the present study investigates the hypersonic viscous flow past blunt-nosed bodies with hydromagnetic interaction . local-similarity solutions of flow field and temperature distribution are near the stagnation-point region . the discussions may be grouped into two parts .. the two-dimensional problem (circular cylinder) and axisymmetric problem (sphere) . numerical computations have been carried out for the sphere problem for the /viscous-layer regime,/ with various magnetic field strengths and electrical conductivities .
combustion in the boundary layer on a porous surface . the position of the diffusion flame in a boundary layer with uniform mixture injection from a porous wall parallel to a uniform air stream is determined under the conditions of laminar, steady flow with zero streamwise pressure gradient . under the assumption of fast forward reaction rate, solutions of the boundary layer forms of the conservation laws of acrothermochemistry are obtained leading to a formula for the downstream velocity at the flame in terms of composition and flow variables . the rates of change of conditions at the wall in the streamwise direction are assumed to be relatively small . methods of treating complex reaction systems are described, and generalized form of the reynolds analogy is developed .
the flow about a charged body moving in the lower atmosphere . the flow about an electrically charged body traveling at high speeds through the lower ionosphere is analyzed . a simple gas model composed of electrons, ions, and neutral particles is used and the hydrodynamic description given is based on maxwell’s transfer equations for a mixture . the conditions under which local statistical equilibrium can be assumed are discussed, and different approaches to determine the gasdynamic force in the subsonic, supersonic, and hypersonic cases are indicated . the reciprocal action of the electric field of the flow on the body is also analyzed and a formula for the resultant electric force is given . the total force on the body is equal to the sum of the gasdynamic force and the electric force . the negative potential acquired by a plane body is also calculated . finally, the lack of validity of debye’s linearization in this case and the solution of the exterior nonlinear problem which characterize the electric potential and the electron distribution are discussed .
fluctuating lift and drag acting on a cylinder in a flow at supercritical reynolds numbers . the fluctuating lift and drag acting on a circular cylinder in a flow of an incompressible fluid at large reynolds numbers were measured . data on the root-mean-square values of the lift and drag coefficients, the extreme values of these coefficients, and their power spectra at various reynolds numbers are presented .
an optical boundary-layer probe . the applicability of the schlieren-photomultiplier technique to obtain quantitative density measurements in the laminar boundary layer induced by a traveling shock wave in a shock tube is investigated . tests were conducted at a mach number of 1.58 so that the data could be compared with the exact theoretical solution tabulated by mirels . the data obtained are in good agreement with the theory if the distance of the light beam above the floor of the shock tube is adjusted to fit the theoretical curve,. this would not be necessary if a larger shock tube were used . values of the transition reynolds number were also determined which are slightly less than those found by martin using an interferometer . it is shown that this technique is sensitive enough to detect changes in density that are only 0.000 per cent of atmospheric density .
heat transfer, recovery factor and pressure distributions around a circular cylinder normal to a supersonic rarefied air stream . measurements of the heat transfer, recovery factor, and pressure distributions around a circular cylinder normal to a supersonic rarefied-air stream (total temperature 300 k.) are described for the mach number range of 1.3 to 5.7, the reynolds number range of 37 to 4,100 and at cylinder wall average temperature levels of 90 k. and 210 k . study of the results yielded .. (1) a correlation equation for the stagnation-point nusselt number as a function of the reynolds number just after the normal part of the detached bow shock wave,. and (2) fourier series expressions for the heat-transfer coefficient and pressure coefficient distributions in terms of the stagnation point values . in comparing these measurements with predictions based on recent analytical studies, exceptionally good agreement for the heat-transfer coefficient distribution was obtained with lees’ theory . in the mach number range of 3.55 to 5.73 the pressure decreased less rapidly with distance from the stagnation point than predicted by the modified newtonian theory .
second-order theory for unsteady supersonic flow past slender pointed bodies of revolution . an analysis is made of the second-order effects of thickness on the unsteady aerodynamic forces on a slender pointed body of revolution in supersonic flow . the theory is restricted to harmonic oscillations for small angles of attack . the solution is obtained by approximating the nonlinear terms in the second-order potential equation by their first-order values and solving the resulting inhomogeneous partial differential equation, subject to more refined boundary conditions . the pressure equation is likewise refined and integrated to give the second-order corrections to lift and pitching moment coefficients . the analysis can be considered as an extension of the second-order, slender body theory of lighthill to the case of unsteady flow . the results indicate appreciable reductions in unsteady lift and damping moment coefficients when applied to slender cones . the present theory is estimated to be reliable provided that is less than 0.7 .
on the response of the laminar boundary layer to small fluctuations of the free-stream velocity . the linearized treatment of small time-dependent disturbances of a laminar boundary layer, initiated by lighthill, is extended in several ways . in particular, the high-frequency expansion is continued beyond the leading (stokes) term . several interesting questions of /joining/ occur, which are discussed but left unresolved . in addition, a practical method for obtaining the response to the laminar boundary layer to an impulsive change in velocity is presented . the methods are applied to the case in which the basic steady flow belongs to the falkner and skan family of similarity solutions .
a method of calculating velocity distribution for turbulent boundary layers in adverse pressure distributions . a new method of calculating the behavior of turbulent boundary layers in adverse pressure distributions is developed which permits direct determination of the velocity profile rather than the gross integral parameters normally used to infer the general character of the boundary layer . the method offers the simplicity of algebraic equations coupled with the use of charts rather than the laborious simultaneous solution of coupled differential equations required by existing methods . the method also affords, for the first time, a means of determining the total boundary-layer thickness, thus allowing calculation of the absolute as well as the nondimensional velocity distribution . the velocity profile is considered to be composed of two regions–an inner region which is described by the law of the wall and an outer region which is described by a function depicting the deviation from that law . the deviation function involves two parameters which are uniquely dependent upon the skin-friction coefficient and a third parameter which, for practical purposes, can be considered a constant . since the entire velocity distribution was found to be almost uniquely dependent upon the local skin friction, serious doubt is cast upon the generally accepted /history concept/ which considers the outer region of the boundary layer to be dependent on integrated upstream conditions . agreement between experimental velocity distributions and those calculated by the method presented here is generally very good . the analysis and calculation procedures which are presented are applicable to two-dimensional, pseudo-two- dimensional, and axisymmetric conical flows .
an extension of the linearized characteristics method for calculating the supersonic flow around elliptic cones . the method of linearized characteristics as applied by ferri to the flow about elliptic cones can be used to determine the surface pressure distribution, even when only linear terms are kept in the boundary conditions, provided an area rule requirement is satisfied . in addition, the method can be applied for angles of attack provided the elliptic body geometry is specified in a manner that does not distort the cross section . the surface pressure distribution obtained by this modified method is in reasonable agreement with experiment over the range of mach numbers and semidiameter ratios considered . experimental results for several conical bodies are presented .
turbulent heat transfer through a highly cooled, partially dissociated boundary layer . the problem of heat transfer from high-temperature air through a turbulent boundary layer to a cold surface is considered both analytically and experimentally . heat-transfer data obtained in shock tubes are presented and correlated by a semiempirical theory which includes the effect of atomic diffusion . the distinguishing characteristics of turbulent boundary layers with dissociation and large cooling are considered . it is shown that the equations governing such flow, after certain approximations, can be represented in a form similar to the classical equations for a turbulent boundary layer . an approximate theory is proposed for turbulent heat transfer for a highly cooled boundary layer on portions of the body where the pressure gradient is negligible in the case of blunted bodies of revolution in high-speed flight . experimental results obtained on the cylindrical portion of a hemisphere-cylinder model are presented for conditions simulating flight speeds to 21,350 ft. sec., where up to 30 per cent of the molecules are dissociated . reynolds numbers of 2.5 x 10, based on local fluid properties external to the boundary layer, were achieved . the larger values of reynolds number and flight speed were not obtained simultaneously, due to structural limitations of the shock tubes,. however, the experiments were conducted in such a way that the important effects of each could be determined . in the experiments the mach number external to the boundary layer varied between 1.7 and 2.2 . the corresponding mach number for blunted nonslender bodies in flight would have a maximum value between 2.5 and 4,. however, it is shown that these differences in mach number are not important for such bodies .
boundary layer transition and heat transfer in shock tubes . an experimental study is made of the wall boundary layer in a shock tube operated over a wide range of shock mach numbers and pressure levels in air, including those for which real-gas effects exist . transition distances are determined and correlated in terms of the transition reynolds number based on a characteristic length for this boundary layer . data from independent shock-tube studies are also included in this correlation . the results indicate a weak dependence of transition reynolds number on shock strength up to moderate values of shock mach number, followed by a larger stabilizing tendency . comparison of these data with transition data obtained in the same manner in argon indicate that the increased cooling rates are largely responsible for the stabilization . a dependence of transition reynolds number on the unit reynolds number is found at the lower shock strengths . specifically, higher transition reynolds numbers are achieved at larger unit reynolds numbers . the phenomenon of transition reversal does not appear within the range of the experiments reported . laminar- and turbulent-flow heat-transfer rates to the walls of the shock tube are determined experimentally . the results of the heat-transfer measurements substantiate existing theories in both the laminar- and turbulent-flow regimes .
on the thrust hypothesis for the jet flap including jet-mixing effects . this paper is concerned with the thrust generated by a jet flap . it is shown that a /linear/ thrust hypothesis can be obtained, provided linearized potential flow is assumed . in fact, the linearized problem of a jet-flap system is found to be the linear combination of a lift problem and a thrust problem . the lift problem gives all the lift generated, but it is of interest to note that the thrust problem would yield all the thrust developed by the jet flap within the limitation of the linearized theory . the mixing of the jet flap with the surrounding fluid is analyzed by the momentum-integral method . the analysis substantiates stratford’s suggestion for obtaining an increase of thrust by causing the jet to mix with the main stream in a region of high suction . finally, some approximate formulas, relating the thrust and the jet angle, are derived . the drag of the airfoil section and other viscous effects are, however, not considered .
minimum wing wave drag with volume constraint . a numerical method is developed for calculating the minimum thickness drag for a given wing planform and volume using linearized supersonic flow theory . the corresponding optimum volume distribution is also determined . the results show that considerable drag reduction is possible by improved volume distribution .
on supersonic flow past thick airfoils . the inviscid rotational supersonic flow behind the shock wave attached to the sharp leading edge of an airfoil is studied by a transformation of coordinates which introduces the crocco stream function as an independent variable . using expansions in the power series of, an iterative process is developed for the determination of pressure distribution along the airfoil surface .
stable combustion of a high-velocity gas in a heated boundary layer . it is generally recognized that stable combustion processes in heated boundary layers may be achieved by either of two conceptual mechanisms . in one mechanism it is pictured that the heat transfer to the wall quenches the propagating flame at a certain distance from the surface . the equality between the flow velocity and the normal burning velocity at this quenching distance determines the position of the propagating flame . in the second mechanism it is conceived that the hot surface provides a continuous source of ignition in much the same manner that the hot recirculation zone of a bluff body flame holder provides continuous ignition to the gas flowing around it . in this case it is the characteristic time during which the gas must be heated that determines the position of the flame . all experimental work reported to date has been concerned with conditions where the first picture has apparently been applicable . in the present paper, experiment and analysis are given that show under what conditions the continuous ignition mechanism provides the appropriate model and also how the two models are related . to differentiate the two mechanisms an experiment was set up to study flame stabilization in high-velocity boundary layers over a wall heated in the form of a step function . with a turbulent boundary layer and a wall temperature above 1,700f., the characteristic time was found to be a systematic and reproducible variable . these observations led to the conclusion that a continuous ignition mechanism governs stabilization in heated turbulent boundary layers . a rational explanation is made for the transition from the low-speed mechanism known to be applicable in unheated turbulent boundary layers and heated laminar boundary layers to the ignition mechanism applicable in heated turbulent boundary layers . as a further verification of the continuous ignition mechanism an apparent ignition energy was found . the logarithm of the heat added at the lower stability limit was found to be a linear function of the reciprocal of the limiting wall temperature . the activation energy derived from this arrhenius type of relation agreed reasonably well with the estimated value for the fuel used .
a study of supersonic combustion . steady, stable, plain, and oblique detonation waves were created in a high-temperature, steady flow supersonic tunnel . ignition conditions and properties across the wave were measured . the local-wave fluid-dynamic properties agree well with detonation theory . experimental data are presented in detail and compared with other studies and theory . experimental behavior of these detonations and their possible utility are discussed .
supersonic inlet dynamics . an approximation of the differential equation for compressible duct flow is presented . the equation is linear and of the second order . the duct transfer function and response characteristics are obtained by applying small-perturbation theory to the differential equation . the resulting equations describe duct natural frequency as a function of duct areas and volumes, and damping ratio as a function of the slope of the steady-state mass flow, pressure-recovery curve . the calcualted response agrees, to a first approximation, with measured response as obtained from tests of a fixed-geometry, sugar-scoop inlet model with hypass for matching airflows . testing was done in the 10 x 10 and 8 x 6 ft. supersonic tunnels at nasa lewis flight propulsion laboratory . further agreement was obtained during flight tests of the f8u-3 airplane .
theory of supersonic propeller aerodynamics . a supersonic propeller with blades attached to an infinite cylinder as a hub is studied . the forward speed may be subsonic, but the relative speed at each section is supersonic . the lightly loaded blades are represented by a surface distribution of appropriate /modified/ sources in a fashion similar to ordinary supersonic thin-wing theory . these sources are found by approximating the exact potential for a constant-strength compressible source traveling along a helical path . the usual relationship between the source strength and boundary condition is found,. and subsequently the source distribution is given, to the appropriate order, in terms of the blade geometry . tip effects are considered by extending the theory of evvard and krasilshchikova . the present investigation, however, is restricted to those planforms for which no vortex sheet appears off the tip . for points in the tip region, the potential is obtained through the appropriate distribution of /modified/ sources in the upwash region off the tip . by transforming to a curvilinear, nonorthogonal coordinate system coincident with the modified mach lines described by the infinities of the potential, an integral equation for the required source distribution in the upwash region is derived . without having to solve this equation, it is shown that the potential for a point in the tip region can be obtained in terms of an integration of known source distributions over the blade surface only . the case of a twisted flat plate of particular planform is treated, and a sample calculation is made of the pressure distribution at selected radial positions within the noncommunicating portion of the blade, as well as over the entire tip region . though this analysis is carried out explicitly for the supersonic propeller, it could also be extended to calculate various rotary derivatives for highspeed flight vehicles .
oscillatory aerodynamic coefficients for a unified supersonic hypersonic strip theory . this investigation presents a derivation of the oscillatory aerodynamic coefficients for wings with supersonic leading edges from the second-order, nonlinear, unsteady, supersonic flow theory of van dyke . the theory is considered applicable throughout the supersonic-hypersonic regime at mach numbers normal to the leading edge and reduced frequencies for which . the coefficients are modified for sweep, and a finite-span correction is suggested to increase the accuracy of strip-theory flutter analyses . the limiting values of the coefficients in steady flow are also discussed .
magnetohydrodynamic effects on the formation of couette, flow . this paper is concerned with the problem of the formation of couette flow–i.e., the problem of how the velocity profile varies with the time tending asymptotically to that of the steady flow of an electrically conducting viscous fluid in the presence of a magnetic field . the governing equations and boundary conditions are established and discussed . the cases of both vanishing and nonvanishing mean induced electric field strengths are solved in terms of complimentary error functions as well as some elementary functions . it is shown that the solutions are reducible to that of the steady case as the time approaches infinity, and to that of the nonmagnetic field as the hartmann number becomes zero . some numerical calculations are given . the results indicate that in the presence of a magnetic field the flow rate is reduced depending on the magnitude of the hartmann number, and that the magnetic field /assists/ the flow to reach its steady condition .
real gas effects in flow over blunt bodies at hypersonic speeds . a hypersonic shock tunnel has been developed to investigate the aerodynamic characteristics of flow over bodies at conditions comparable to those encountered by ballistic missiles and satellites re-entering the atmosphere . some results for a shock velocity of over 50,000 ft. per sec. in the shock tube portion of the facility are presented . static pressure investigations were made in the nozzle to determine the flow condition and the expansion process . the results of the investigation of representative blunt bodies at hypersonic mach numbers and nozzle stagnation temperatures up to approximately 6,000degreek. are presented . these include body pressure distributions, shock-wave shapes, detachment distances, and photographs of the luminous gas region in the shock layer . it is seen that the shock detachment distance is smaller at higher stagnation temperatures due to the real gas effects . for the hemisphere the pressure distribution was less than that predicted by the modified newtonian theory for all stagnation temperatures . for a 50degree cone-hemisphere the pressure distribution and the shock detachment distance were appreciably affected by the real gas effects . the observed shock-wave shape and the approximate boundary layer on a flat plate are compared with the analytical prediction . some preliminary results for the detached shock wave produced by a blunt two- dimensional body in a low density flow at a mach number of 19.6 are presented
flow about an unsteadily rotating disc . an analysis is made of the unsteady laminar flow about a rotating disc whose angular velocity may vary with time . the deviation of the actual instantaneous state of the flow from the quasi-steady state (instantaneous steady state) is determined . from this, a simplified criterion, is derived to define the conditions under which the flow can be considered as quasi-steady for the purposes of shear stress and torque computations . since a turbulent flow responds more rapidly than a laminar flow, the quasi-steady criterion found here should also serve for the turbulent situation .
a three-dimensional linearized analysis of the forces exerted on a rigid wing by a shock wave . the pressure distribution on a moving flat plate induced by an acoustic shock front striking the edge of the plate obliquely has been found in terms of the two-dimensional solution of the authors
a study of vortex cancellation . the cancellation of a vortex by means of another concentric vortex of equal strength but opposite spin is investigated . when such a cancellation occurs, there is a recovery of static pressure . the vortices are generated by means of two three-dimensional airfoils cantilevered from the duct wall, one being situated in the wake of the other . the airfoils have opposite effective angles of attack and therefore have trailing vortices of opposite spin, as required . it is demonstrated experimentally that there exists an optimum angle of attack for the second airfoil which cancels the vortex from the first airfoil and restores uniform flow downstream of the two airfoils . a theoretical solution of this optimum angle of attack is presented, and it is found to depend upon the angle of attack of the first airfoil and upon the geometrical properties of the wings . the pressure recovery accompanying the vortex cancellation is also studied . theoretical considerations, based on the model of a vortex filament in the center of a circular tube show that a maximum of 62 per cent of the static pressure drop across the first airfoil can be recovered . this maximum is imposed, irrespective of skin friction and separation losses, by the irreversibility associated with establishing a vortex field . experimental pressure recoveries of 50 per cent are realized . perhaps the primary value of the present study is the opportunity it provides to verify certain of the fundamental concepts of fluid mechanics which are brought into play when the trailing vortex system of a lifting wing is cancelled by a second wing .
transition in a separated laminar boundary layer . transition to turbulence was studied in a separated laminar boundary layer on a flat plate 24 in. long and thick . steps with a height of to were provided at a distance of 4 to transition was observed through a hot-wire anemometer . the author concludes that transition was always initiated by tollmien-schlichting waves . two types of transition were observed . in one type, bursts suddenly appeared in the wavy flow . the other type consists of amplification, distortion, and breaking up of the waves . which type of transition occurs depends on the value of the following parameter .. boundary-layer displacement thickness times step height times free-stream velocity squared divided by kinematic velocity squared . the burst type has been observed for values of this parameter larger than 4.2 x 10 . the separated laminar boundary layer becomes unstable and develops waves when the critical reynolds number based on boundary-layer displacement thickness at the step location exceeds a value of 350 . some conclusions on the development of separation bubbles on air foils are drawn from the present studies .
sublimation in a hypersonic environment . a priori knowledge of the response of materials subjected to a severe aerothermal environment is essential in the space age . the successful design of space and re-entry vehicles demands that the fundamental problem of the interaction between a material and dissociated air be properly formulated and solved . in this paper, the problem of sublimation in a hypersonic environment is considered . in this study of hypersonic ablation, the pertinent conservation equations are derived and the simultaneous processes of diffusion, convection, and thermal exchange are analyzed for the vaporization of a refractory material which is subjected to the environmental conditions encountered during hypersonic reentry . for simplicity, only the forward stagnation point of an axially symmetric body is treated . it is shown that the quantity, called the effective heat of vaporization, which includes all heat absorbing or heat blocking effects, is an increasing function of flight speed, independent of body size, except where nonequilibrium vaporization effects or radiative effects appear .
wings with minimum drag due to lift in supersonic flow . it has been shown by r. t. jones that, in order to produce minimum drag, the given lift must be distributed over the wing surface in such a way that the sum of the downwash induced by this distribution and the downwash induced in reversed flow is constant over the wing surface . this combined downwash can be expressed by an integral which contains the load as a function of the spanwise and chordwise coordinate . the problem of finding the appropriate load distribution is thus reduced to the problem of finding the solution of a rather cumbersome integral equation . the severe spanwise singularity of the kernel function is handled most easily, as in corresponding subsonic problems, by an approximate integration over interpolation polynomials . the chordwise load distribution is represented by a limited series development in legendre polynomials . the sigularity of the kernel function along the mach lines through any pivotal point can be avoided by a similar legendre development of the combined induced downwash which is constant . the integral equation is thus converted into a system of linear equations for the unknown coefficients of the legendre functions of the load distribution at a limited number of spanwise stations . practical calculations are carried out on an electronic computer . the solutions yield the optimum load distribution and the local incidence (twist, camber, etc.) necessary to realize this distribution . for many wing plan forms, considerable gains over a plane wing appear possible .
turbulent heat transfer on blunt-nosed bodies in two-dimensional and general three-dimensional hypersonic flow . recent results obtained for three-dimensional laminar boundary layers are extended to the turbulent case . it is shown that in the presence of highly cooled surfaces and of moderate mach numbers of the outer stream, the crossflow and the pertaining reynolds stresses in a general three-dimensional turbulent boundary layer are negligible even for large transverse pressure gradients . a correlation due to mager between two- dimensional compressible and incompressible turbulent boundary layers is extended to the problem in question . from a study of the transformation and of its implications, a rapid method for the analysis of the boundary-layer flow under the subject conditions is established . in the absence of general three- dimensional data, a comparison with experiments and with the predictions of other known analyses is carried out for several axisymmetric configurations,. the results of the method presented here exhibit good agreement with the data . the range of validity of the cold wall approximation for general three-dimensional problems is estimated qualitatively on the basis of recent measurements in laminar flow, the argument being that, for either zero or favorable streamwise pressure gradients, smaller three-dimensional effects are to be expected in a turbulent boundary layer, as compared to a laminar layer .
compressible flat-plate boundary-layer flow with an applied magnetic field . the laminar boundary-layer equations are formulated and solved for a flat plate in high-speed compressible air flow where equilibrium dissociation and ionization are assumed and where there is an applied magnetic field having its component normal to the plate proportional to . the skin-friction and heat-transfer characteristics are determined for free-stream velocities of up to 17,500 meters sec. and magnetic fields of up to about the results show that the skin friction and heat transfer at a given free-stream velocity decrease with increasing magnetic field strength, and the percentage reduction is constant along the length of the plate . they also exhibit the same hysteresis behavior as was first found in the case of magnetoaerodynamic couette flow,. however, for the flat plate the hysteresis effect disappears at a higher mach number . furthermore, it was found that the reduction in heat transfer with increasing field strength is opposite in behavior from that for couette flow .
on shearing flow between porous coaxial cylinders . the flow between concentric porous cylinders in relative axial motion with a pressure gradient is considered . the analysis is restricted by the assumption that the velocity distribution is a function of the radial coordinate only so that there is no net injection or withdrawal of fluid at any station . this assumption reduces the problem to a soluble system of ordinary differential equations . an associated heat-transfer problem is also discussed briefly .
the transition to tubulence in a boundary layer on a blunt cone in supersonic flow . experiments were made with a series of cones, each having an included angle of 15 degrees, and having different tip radii (sharp to blunt nose) . the cones were tested in streams, undisturbed mach numbers of 3.12 and 3.81, and the position of transition to turbulence was observed with a shadowgraph technique . for each mach number the distance to transition (distance downstream from the tip of the cone) increased with increase in tip radius, reached a maximum at a certain tip radius, and then decreased with increase in tip radius . a study indicates that a reynolds number based on the momentum thickness, instead of the length from tip to transition, is a more appropriate parameter for correlating the data . the research scientist active in this field probably would derive the most benefit from the original paper .
experiments at hypersonic speeds on circular cones at incidence . pressure distribution measurements on five circular cones with total apex-angles ranging from 25 to 45 degrees are described . the tests covered a range of angles of incidence from 0 to and 8.60 . the extent to which various analytical and empirical theories predict the measured pressures is assessed .
equilibrium real-gas performance charts for a shypersonic shock-tube wind-tunnel employing nitrogen . charts are presented covering a wide range of reflected-shock wind-tunnel operating conditions, using nitrogen as the working gas . a statistical-mechanical model of the gas is assumed which takes account of molecular vibration, electronic excitation and dissociation . the gas is assumed to be constantly in equilibrium that is, the reaction rates are taken to be infinitely fast . the equations of motion are solved with the aid of a digital computer, previously reported results for the state of the shock-processed gas in the shock-tube being used .
progress report on an experiment on the effect of surface flexibility on the stability of laminar flow . this paper describes the flexible surfaces whose properties have been examined and which have been tested on an aerofoil in a wind tunnel . the experiment has been rather inconclusive as no drag reductions have been found in turbulent flow, whilst the only rearward movements of transition occurred in conditions where the alteration has been inhibited by the onset of laminar separation . the limitations of the experiment are discussed carefully in order to clarify the next steps which are to be taken with more flexible surfaces with less damping .
analysis of the fluid mechanics of secondary injection for thrust vector control . an analysis is made of the interaction of an injected gas or liquid with a supersonic stream, and the force induced on an adjacent wall is predicted . the study deals only with the freestream-injectant interaction,. the modifications to the flow introduced by the boundary layer are not considered . in the case of liquids, it is shown that the momentum deficit of the injectant relative to the freestream may play a larger part in producing the side force than the volume generation by vaporization and reaction . the analytical results are compared with those obtained from experiments in a wind tunnel and in nozzles .
numerical technique to lifting surface theory for calculation of unsteady aerodynamic forces due to continuous sinusoidal gusts on several wing planforms at sobsonic speeds . a numerical lifting-surface method has been used to calculate direct gust forces and moments on wings of several planforms . the gust velocities are continuous and vary sinusoidally in the stream direction and are also uniform across the wing span . the procedure has the advantage of rapid machine calculation and includes the effects of wing planform, nonsteady subsonic flow, and induced flow effects . the method provides for calculation of gust forces on a basis consistent with that for the calculation of forces due to motion and deformation . the results include the in-phase and quadrature components of the following quantities .. (a) spanwise distribution of section lift coefficient, (b) total lift coefficient, and (c) total pitching-moment coefficient . in addition, generalized gust forces on approximate fundamental cantilever bending modes ( parabolic) are also included . results have been obtained for 60 and 75 delta wings, ratio 11.60, and an unswept wing of aspect ratio 6.00 . conditions for which calculations were made include two mach numbers reduced-frequency range of 0 to 1.0 . the direct gust forces and moments are in forms suitable to be inserted in equations of motion used in the calculation of the dynamic responses of flexible lifting vehicles to random turbulence and to be compared with results from other methods .
measured and calculated subsonic and transonic flutter characteristics of a 45 sweptback wing planform in air and in freon-12 in the langley transonic dynamics tunnel . in order to investigate the reliability of flutter data measured in the langley transonic dynamics tunnel, an experimental and theoretical subsonic and transonic flutter study has been conducted in air and in freon-12 in this facility . the wing planform employed had an aspect ratio of 4.0, a taper ratio of 0.6, and 45 of quarter-chord sweepback . a sting-mounted full-span model was tested in addition to three sizes of wall-mounted semispan models . a wide range of mass ratio was covered by the tests in air and by flutter calculations made by the modified strip-analysis method of naca research memorandum l57l10 . a limited amount of data was obtained in freon-12 . results of the tests in air and in freon-12 are in good agreement with the flutter calculations at all mach numbers . the test data compare favorably with previously published transonic flutter data for the same wing planform . the results indicate that flutter characteristics obtained in freon-12 may be interpreted directly as equivalent flutter data in air at the same mass ratio and mach number .
atmosphere entries with spacecraft lift-drag ratios modulated to limit decelerations . an analysis has been made of atmosphere entries for which the spacecraft lift-drag ratios were modulated to limit the maximum deceleration . the parts of the drag polars used during modulation were from maximum lift coefficient to minimum drag coefficient . five drag polars of different shapes were assumed for the spacecraft . the entries covered wide ranges of initial velocity, initial flight-path angle, initial and maximum lift-drag ratio . two-dimensional trajectory calculations were made for a nonrotating, spherical earth with an exponential atmosphere . the results of the analysis indicate for four of the five drag polars that, relative to the maximum deceleration of an unmodulated entry at maximum lift-drag ratio, the greatest reduction in maximum deceleration obtainable by modulation depends upon a single parameter . this parameter is the ratio of the value of the aerodynamic resultant-force coefficient at minimum drag coefficient to the value at maximum lift coefficient . thus, the reduction in maximum deceleration is independent of initial velocity, initial flight-path angle, initial maximum lift-drag ratio, and the shape of the drag polar . for the fifth drag polar, the reduction in maximum deceleration was found to depend upon the maximum lift-drag ratio . also, relative to the depth of a given deceleration-limited corridor, the greatest increase in corridor depth obtainable by modulation (for four of the five drag polars) depends upon the same ratio of aerodynamic resultant-force coefficients . the fractional increase in corridor depth can be expressed as an empirically determined analytical function of this ratio .
effect of jet pluming on the static stability of cone-cylinder-flare configurations at a mach number of 9. 65 . the effects of jet pluming on normal force and pitching moment of have been measured at a free-stream mach number of 9.65 with reynolds numbers based on model length of 500,000 to 600,000 . geometric variables included nose bluntness, flare half-angle, and nozzle geometry and exit displacement . two test nozzles with design mach numbers of 3.74 and 4.60 were operated with compressed air to simulate the initial jet-boundary shape of a particular solid-propellant rocket motor operating between altitudes of 165,000 and 215,000 feet . the ratio of the jet pressure to free-stream static pressure varied from a jet-off condition to approximately 1,300 for the nozzle with design mach number of 3.74, and from a jet-off condition to approximately 280 for the nozzle with design mach number of 4.60 . the angle-of-attack range was from 0 to approximately 6 . the results indicate that as the jet-pressure ratio was increased the size of the jet plume increased, and as a result the model static stability was decreased . increasing the angle of attack resulted in a reduction in static instability during the jet-on condition . increasing nose bluntness resulted in a more forward movement of the center of pressure when jet-plume interference was not present and a rearward movement in the center of pressure when jet interference was present . increasing the nozzle-area expansion ratio and displacing the nozzle exit downstream of the flare base resulted in a more rearward location of the center of pressure .
design of stiffened cylinders in axial compression . the problem of optimum design of axially compressed cylinders stiffened by rings and stringers is discussed . particular attention is devoted to configurations suitable for large launch vehicles . consideration is given to the analytical techniques for determining strength as well as the procedures for optimization .
non-linear shallow shell analysis by the matrix force method . the matrix force method of redundant structure analysis is currently being extended by various users to cover a number of non-linear problems . one of these is the non-linear analysis of heated cambered wings, such as might be used in advanced flight vehicles . in this case the approach used by the present authors is equally applicable to shallow shells, the formulation of the strain-displacement and equilibrium relations being a finite element equivalent to that used by marguerre . the solution is obtained by a combined iteration and step by step procedure utilizing a tangent flexibility matrix . divergence in the calculations indicates that the range of stable configurations has been exceeded . cambered plates subjected to several loadings are given as examples,. for one, an exact solution is available for comparison . it is believed that the basic concepts involved in this shallow shell analysis can be extended to apply to other, more general shell instability problems, and that useful solutions to the latter are probably within the capability of present day digital computers .
recent advances in nonequilibrium dissociating gasdynamics . the purpose of this paper is to review some recent advances in the study of gasdynamic problems including effects of chemical reactions . to provide a background for the study the general concepts shall be outlined briefly . the discussions of the recent developments are restricted to inviscid flow problems only, neglecting viscosity, heat conduction and diffusion . particular attention is directed to recent advances in analyses of nonequilibrium dissociating gas flows . in the hypersonic flight regime, high stagnation enthalpies sufficient to cause dissociation are realized . when the time to reach equilibrium is comparable with the time it takes for a fluid particle to pass through the flow, then there exist regions of the flow field where nonequilibrium states are encountered . a brief survey of both the linear and the nonlinear methods of treatment of these nonequilibrium flows, including some new developments that have not appeared elsewhere, will be presented .
non-equilibrium expansions of air with coupled chemical reactions . analysis and solutions of the streamtube gas dynamics involving coupled chemical rate equations are carried out . results are presented for airflows along the surface of blunt bodies and through hypersonic nozzles . speeds and altitudes corresponding to re-entry were selected to obtain initial conditions for the external flow calculations . conditions appropriate to hypersonic tunnel testing were chosen for the nozzle flow calculations . composition histories are shown for a kinetic mechanism including 6 species and 14 reactions . gas-dynamic effects of nonequilibrium processes qualitatively resemble those reported earlier . however, the freezing process is complicated by the coupling of the nitric oxide shuffle reactions with the dissociation-recombination reactions . in many cases of hypersonic nozzle flows where the energy in nitrogen dissociation is significant, the fast shuffle reactions prevent nitrogen-atom freezing which would otherwise occur if three-body recombination were the only process operating . nitric oxide concentrations undershoot the equilibrium values if the ratio of nitric oxide to oxygen molecule concentrations exceeds unity in the freezing region . this depletion of nitric oxide leads to nitrogen-atom freezing .
ionization nonequilibrium in expanding flows . approximate methods are developed for solving the electron-ion kinetic equations in expanding quasi one-dimensional airflows . results are obtained for inviscid nozzle flows at conditions appropriate to shock tunnel testing and are compared with exact numerical solutions . effects on rf transmission and d-c conductivity are examined . since two-body deionization never fully freezes in the flows considered here, the assumption of sudden freezing gives an upper bound on the residual ionization at large area ratios . the use of an asymptotic form of solution with the freezing criterion provides an improved estimate for such cases . ionization nonequilibrium is also considered for the plasma sheath associated with blunt hypersonic bodies flying at high altitudes . the influence of atomic ions is examined for typical re-entry conditions .
theory of radiation from luminous shock waves in nitrogen . the physical properties behind a normal shock in nitrogen are calculated as a function of time . these include the variation of temperature, composition, ionization, and the intensity of radiation from the n first negative band system . this calculation incorporates a rate equation for the dissociation of nitrogen, the conservation laws, an equation describing vibrational relaxation, and a method of coupling the vibrational relaxation with the dissociation rate . the n radiation is computed assuming excitation of the radiating state by collision with vibrationally excited nitrogen molecules . a particular case is considered for which experimental data are available, and regions sensitive to particular rates are indicated .
hypersonic viscous shock layer . a decade ago tsien (1) (as well as others) and, more recently, adams and probstein (2) have attempted to define the different regimes of gaseous interactions during high altitude flight . in this note some results are presented which are pertinent to the flight of hypersonic lifting vehicles composed of axially symmetric and two-dimensional elements,. sec. fig. 1 .
some effects of bluntness on boundary layer transition and heat transfer at supersonic speeds . large downstream movements of transition observed when the leading edge of a hollow cylinder or a flat plate is slightly blunted are explained in terms of the reduction in reynolds number at the outer edge of the boundary layer due to the detached shock wave . the magnitude of this reduction is computed for cones and wedges for mach numbers to 20 . concurrent changes in outer-edge mach number and temperature occur in the direction that would increase the stability of the laminar boundary layer . the hypothesis is made that transition reynolds number is substantially unchanged when a sharp leading edge or tip is blunted . this hypothesis leads to the conclusion that the downstream movement of transition is inversely proportional to the ratio of surface reynolds number with blunted tip or leading edge to surface reynolds number with sharp tip or leading edge . this conclusion is in good agreement with the hollow-cylinder result at mach 3.1 . application of this hypothesis to other mach numbers yields the result that blunting the tip of a slender cone or the leading edge of a thin wedge should produce downstream movements of transition by factors ranging from 2 at mach 3.0 to 30 at mach the possible reduction in over-all heat-transfer rate and friction drag for aircraft flying at high supersonic speeds . mach number profiles near the surfaces of blunted cones and wedges are computed for an assumed shape of the detached shock wave at flight mach numbers to 20 . the dissipation and stability of these profiles are discussed, and a method is described for estimating the amount of blunting required to produce the maximum possible downstream movement of transition .
compressible boundary layers on bodies of revolution . in a former paper (1) it has been shown that the behaviour of the laminar boundary layer on a body of revolution can be described mathematically by the same equations which are also applied to the processes in the laminar boundary layer in the two-dimensional flow along a body contour, the form of which is determined by the shape of the body of revolution . a simple relation exists between the two-dimensional boundary layers and the axially symmetrical ones . the flow had been assumed to be incompressible . in this report it shall be shown that this relation is still valid when the compressibility is taken into consideration . the distribution of velocity as well as that of temperature in the laminar boundary layer of a body of revolution can be calculated by solving the corresponding problem for the two-dimensional flow around a suitable contour . the method is made clear by the example of the supersonic flow towards a cone tip,. this example has already been treated by another method by hantzsche and wendt (2) .
the development of the boundary layer in supersonic shear flow . the development of the boundary layer in a velocity shear layer is discussed for two-dimensional flow and for axisymmetric flow of both compressible and incompressible fluids . it is shown that the solutions obtained by li and glauert for the two-dimensional flow of an incompressible fluid are applicable in the more general case after suitable transformations of coordinates have been made . new definitions are shown to be necessary, and are given, for the displacement and momentum thicknesses of such a boundary layer . reynolds numbers based on these thicknesses are given, and it is shown that any phenomenon which occurs at a constant value of such a reynolds number will occur at a point which, as the length scale of the flow increases, first moves down-stream and then moves slightly upstream . this is shown to be in qualitative agreement with experimental results on a blunt cone in a supersonic flow . a quantitative comparison of the theoretical and experimental values of displacement and momentum thicknesses is attempted, and no disagreement is obvious,. unfortunately the accuracy of the experiments so far available is insufficient to give positive confirmation of the theory of this note .
air pressure on a cone moving at high speeds . the cone is considered to be moving at a velocity higher than that of sound, so that there is in front of it a shock wave, moving with the same speed as the cone itself . in the first part of the paper, the case is investigated mathematically where the flow is irrotational, and the pressure, velocity and density of the air stream are each constant over the surfaces of cones coaxial with the moving solid cone . the complete solution is obtained in numerical form, for cones of semi-vertical angle of the paper, the results are compared with experiment, both in respect of pressure distribution as measured in a wind tunnel, and also (for the 30 cone) by comparison with photographs of bullets in flight . in the latter case the theory should only be applicable if the speed is 1.46 or more times the velocity of sound, and it is in fact found in the photographs, that the nature of the wave alters at about this velocity . the exact solution found, is compared with an approximation given recently by v. karman and moore . this should be valid for thin spindle-shaped bodies, and does in fact agree well in the case of the cone of 10 semi-vertical angle, but diverges increasingly from the truth as the angle is increased .
newtonian flow over a surface . a general method is presented for the study of a three-dimensional hypersonic flow about a body of arbitrary shape when . the manner of constructing a double asymptotic development in and is shown . formulae are given which enable the first three terms of this development to be obtained while neglecting . the theory is then applied to the case of a body of circular-cone shape . the pressure is given as a triple development in accordance with the preceding parameters and the angle of attack,. this development neglects . a. ferri’s vortical layer is brought into evidence . a second application is devoted to calculation of the total forces acting upon bodies of revolution at angles of incidence, while neglecting . general formulae are established for the coefficients of axial force, normal force and moments . the formulae are developed according to the powers of incidence, the first terms of each formula being of very simple form .
a proposed programme of wind tunnel tests at hypersonic speeds to investigate the lifting properties of geometrically slender shapes . a programme of tests at hypersonic speeds on slender bodies is described, which has the aim of investigating how lift is generated, and the compromises that may be enforced by aerodynamic heating . the programme is based on models of simple geometric shape, from which lifting configurations will later be built up .
experiments on circular cones at yaw in supersonic flow . pressure measurements made in the fort halstead supersonic tunnel on two circular cones, of semiapex angles 15 and coefficients are compared with corresponding values calculated by theoretical methods, and the relative merits of these methods are then discussed .
laminar heat-transfer and pressure measurements at a mach number of 6 on sharp and blunt 15 half-angle cones at angles of attack up to 90 . two circular conical configurations having 15 half-angles were tested in laminar boundary layer at a mach number of 6 and angles of attack up to 90 . one cone had a sharp nose and a fineness ratio of blunted nose with a bluntness ratio of 0.1428 and a fineness ratio of 1.66 . pressure measurements and schlieren pictures of the flow showed that near-conical flow existed up to an angle of attack of approximately near the base and the bow shock wave was considerably curved . comparison of the results with simply applied theories showed that on the stagnation line pressures may be predicted by newtonian theory, and heat transfer by local yawed-cylinder theory based on the yaw angle of the windward generator and the local radius of the cone . base effects increased the heat transfer in a region extending forward approximately circumferential pressure distributions were higher than the corresponding newtonian distribution and a better prediction was obtained by modifying the theory to match the pressure at 90 from the windward generator to that on the surface of the cone at an angle of attack of 0 . circumferential heat-transfer distributions were predicted satisfactorily up to about 60 from the stagnation line by using lees’ heat-flux distribution based on the newtonian pressure . the effects of nose bluntness at large angles of attack were very small in the region beyond two nose radii from the point of tangency .
a guide to the use of the m. i. t. cone tables . the second and third volumes of the m.i.t. cone tables have been found to be unsatisfactory in two respects . they have been criticized because of their inconvenient tabulation and because the theory on which they are based is inadequate near the cone surface . the former is climinated by means of a coordinate transformation . empirical evidence is presented to show that the latter may be ignored in practice . the exact nature of certain numerical errors in the table is also pointed out .
hypersonic flows past a yawed circular cone and other pointed bodies . a detailed treatment of inviscid hypersonic flow past a circular cone is given, for small and moderate yaw angles, within the framework of shock-layer theory . the basic problem of non-uniform validity associated with the singularity of the entropy field is examined and a valid first-order solution is obtained which provides an explicit description of a thin vortical layer at the inner edge of the shock layer . analytic formulas for pressure and circumferential velocity are given consistent to the second-order approximation including the non-linear yaw effect . the study of the entropy field (which is not restricted to the hypersonic case) also provides corrections to previous work on the yawed cone and confirms the validity of the linear yaw effect on pressure field in the stone theory . a related investigation of three-dimensional flow fields is presented with special reference to the flow structure near the surface of a pointed, but otherwise arbitrary body . the inviscid streamline pattern on the surface is given by the geodesics originting from the pointed nose as a leading approximation of shock-layer theory . associated with this streamline pattern is a vortical sublayer which exists generally at small as well as at large angle of attack . at the base of the sublayer, enthalpy and flow speed remain essentially uniform .
survey of inviscid hypersonic flow theory for geometrically slender shapes . a survey is made of existing theories for the calculation of pressure distributions on slender bodies at hypersonic speeds . no account is taken of boundary layer displacement effects which are expected to become important above a mach number of about 10 for a slender body . first the breakdown of linearised supersonic theory is demonstrated as mach number increases above about 5, and this is followed by a derivation of the hypersonic similarity rule . this section includes a description of the piston-analogy . next a physical interpretation of hypersonic flow is outlined and a simple derivation of the modified newtonian pressure formula is given . the equations of flow through an oblique shock wave are simplified by assuming a strong shock, and various results are thereby derived . these include the tangent-wedge and tangent-cone formulae . this is followed by a description of the newtonian approximation for slender bodies, including the effect of centrifugal forces, and the connection with newtonian flow theory is emphasized for . the shock-expansion method is described in some detail for both two-and three-dimensional bodies, and finally some remarks are made about the available data sheets and tables for estimating pressures on cones and ogive-cylinders in yaw . the note does not claim to be original, even in presentation . the aim has been to prepare a reasonably complete survey of available theory for hypersonic flow over slender bodies, excluding viscous and explicit real gas effects . this will provide the background for further work in which experimental data will be analysed and in conjunction with which it is hoped to produce accurate design methods for estimating pressures and forces on shapes intended for sustained flight at hypersonic speeds .
some simple solutions to the problem of predicting boundary layer self-induced pressures . simplified theoretical approaches are shown, based on hypersonic similarity boundary-layer theory, which allow reasonably accurate estimates to be made of the surface pressures on plates on which viscous effects are important . the consideration of viscous effects includes the cases where curved surfaces, stream pressure gradients, and leading-edge bluntness are important factors .
tabulated solutions of the equilibrium gas properties behind the incidents and reflected normal shock-wave in a shock-tube . tabulated solutions are presented for the equilibrium gas properties behind the incident and reflected normal shock-waves in the shock-tube, for nitrogen and oxygen . they cover the range of shock-wave mach numbers up to 12 at intervals of undisturbed gas pressure between 1 and the thermodynamic model of the gas used in the calculations is described in some detail, as is the method of solving the equations . the limitations of the assumption of thermodynamic equilibrium are discussed with regard to shock-tube applications, and the estimated accuracy of the tables is indicated .
on the flow in a reflected shock tunnel . the performance of a shock tunnel operated by the reflected-shock technique is examined theoretically neglecting viscous effects and high-temperature real-gas effects . particular attention is given to disturbances to the flow at the nozzle entry caused by waves reflected from the contact surface when the operating conditions depart from those for that the first disturbance reflected from the contact surface is weak enough to be tolerated only within a small range of primary-shock mach number, m /e.g., 5 7 m 6 3 if the pressure at entry to the nozzle is to remain constant to 10 per cent/ . within this range, running times much longer than those obtained in ‘straight-through’ shock tunnels are predicted, the limitation usually being imposed by the arrival of the expansion wave originating at the diaphragm . outside this range of mach number, the uniform-flow duration between the arrival at the nozzle entry of the primary shock and the first disturbance reflected from the contact surface is shown to be approximately equal to the time between the arrival of the primary shock and the contact surface in a ‘straight-through’ shock tunnel . at first sight it appears, therefore, that the advantages of reflected-shock operation are confined to a very narrow range of shock mach number, unless a heated driver gas is used in order to vary the mach number for further analysis suggests, however, that subsequent disturbances in the multiple wave reflection process between the contact surface and the end of the tube are relatively weak over a useful range of shock mach number . thus, if the flow after the arrival of the early reflected disturbances is used for test purposes, long running times seem possible in theory without severe restrictions to the shock mach number . experiments have been made in a shock tube and a shock tunnel to provide data for comparisons with the results of the simple theory . if allowance is made for viscous effects on the motion of the contact surface, fair agreement is found for the disturbances reflected and transmitted by the contact surface, and for the arrival of the expansion wave reflection process increases when the shock mach number is raised substantially above the ‘tailored’ value, and a limit to the usable flow duration may result . a striking feature of the results is a fall of pressure at the end of the tube immediately after reflection of the primary shock . this is attributed to attenuation of the reflected shock resulting from its interaction with the boundary layer on the wall of the tube . further research is required to check this explanation, and to investigate the effects of reynolds number and of the cross-sectional shape and size of the tube . the effects of the tail and reflected head of the expansion wave originating at the main diaphragm are discussed . it is shown that the arrival of the reflected head at the nozzle entry may impose a severe limitation to the duration of uniform conditions at low shock mach number, and that the arrival of the tail may limit the flow duration at high shock mach number . unless means can be devised to suppress the expansion wave, it is demonstrated that it is desirable to have alternative diaphragm positions in a tube required to operate over a range of shock mach number . it is concluded that running times of order 10 milliseconds at a shock mach number of 4, falling to, perhaps, 1 millisecond at a shock mach number of 8 seem possible in a shock tunnel of reasonable size by using reflected-shock operation with unheated hydrogen driving air . because of the simplifying assumptions of the theoretical investigations, and the deficiencies of the apparatus used for the experiments, the present investigation must, however, be regarded as preliminary in character . further research is required to check and extend the findings, and topics particularly requiring investigation are listed in the paper .
production of high temperature gases in shock tubes . this paper is intended to set forth aerodynamic and thermodynamic calculations which are useful in the production of strong shock waves . the experimental production of strong shock waves is discussed . comparison of the experimental shock strengths with the theoretical calcualtions is made, and finally, some preliminary results of shock tube studies in high temperature gases (up to 18,000k) are briefly surveyed .
performance estimates for the rae 6in . high-pressure shock tube . estimates are made of the performance of the rae 6” high pressure shock tube, with various driver gases, over a range of pressure ratios giving shock mach numbers from 6 to 22 . the calculations are based on a simplified model of shock tube flow, in which the working fluid (argon-free air) is assumed to be always in chemical equilibrium, and the driver gas (either hydrogen, or the products of combustion of a hydrogen-oxygen mixture) is assumed to behave as an ideal gas with constant specifiic heats . the results are presented in graphical form and comprise charts normal shock waves in argon-free air shock wave mach number and diaphragm pressure ratio under various initial conditions, and of the shock-induced flows, both in the uniform-sectioned shock tube, and when expanded in a divergent nozzle
temperature measurements of shock-waves by spectrum-line reversal, ii a double beam method . the sodium-line reversal method, as previously described, using a photomultiplier and oscillograph, has been modified . two light beams are now employed, and interference filters are used in front of the photomultipliers instead of a spectrograph . in one beam the background source is viewed directly, through the shock tube, and in the other beam the background source is viewed through the shock tube by a mirror system with a neutral filter interposed to reduce its effective brightness temperature . with a suitably chosen temperature for the background, one oscillograph trace indicates absorption and the other indicates emission of the sodium lines . it is thus possible, from the records of a single shock, to determine the temperature history behind the shock wave to about 20degreec . nitrogen and oxygen again show relaxation effects near the front . temperatures in argon tend to come low, owing to radiative disequilibrium,. excitation processes in argon are discussed . with this system it is possible to determine temperatures rather higher than that of the background source . some work has also been done, with a single-beam method, using a carbon arc as background and following reversal of the indium blue line . temperatures up to 3600degreek have been measured in shocks through nitrogen, but the time resolution is not so good .
shock-tube testing time . in a theoretical investigation of attenuation effects of the shock wave the conservation of mass equation led to an explanation of the difference between the ideal theoretical test time and the experimentally obtained time . a numerical example is given .
stagnation temperature measurements in a hypersonic gun tunnel using the sodium-line reversal method . the sodium line reversal /s.l.r./ method has recently been used to measure transient temperatures in the 1400degreek to 3000degreek range, for example ref.6 reports measurements of gun flash temperatures . in the gun tunnel, stagnation temperatures in the above range can be generated flow between a blunt body and its bow shock wave . the gas temperature in this zone is close to the stagnation value .
real gas effects in flow over blunt bodies at hypersonic speeds . a hypersonic shock tunnel has been developed to investigate the aerodynamic characteristics of flow over bodies at conditions comparable to those encountered by ballistic missiles and satellites re-entering the atmosphere . some results for a shock velocity of over 50,000 ft/sec in the shock tube portion of the facility are presented . static pressure investigations were made in the nozzle for different stagnation conditions in order to determine the flow condition and the expansion process . the results of the investigation on representative blunt bodies at hypersonic mach numbers and nozzle stagnation temperatures up to approximately 6000degreek are presented . these include body pressure distributions, shock wave shapes, detachment distances, and photographs of the luminous gas region in the shock layer . it is seen that the shock detachment distance is smaller at higher stagnation temperatures owing to the real gas effects . for the hemisphere the pressure distribution was less than that predicted by the modified newtonian theory for all stagnation temperatures . for a 50degree cone-hemisphere the pressure distribution and the shock wave detachment distance were appreciably affected by the real gas effects . the experimentally obtained shock wave shape and the approximate boundary layer on a flat plate are correlated with the analytical prediction . some preliminary results for the detached shock wave produced by a blunt two-dimensional body in a low density flow at a mach number of 19.6 are presented .
divergence of plate airfoils of low aspect ratio at supersonic speeds . in part (1), as a first approach to a theoretical investigation of low aspect ratio rectangular plate wings of constant thickness, the two assumptions are made that .. (a) the spanwise form of the structural distortion is known, leaving the chordwise distortion arbitrary,. and (b) the aerodynamic forces are approximations of the supersonic linearized theory . the form of the chordwise distortion is then deduced from the differential equation representing the state of neutral equilibrium for small displacements at the critical divergence speed . secondly, this problem is investigated using measured structural flexibility coefficients together with theoretical aerodynamic coefficients . thirdly, the usual series solution based on the rayleigh-ritz approach is discussed, using the same assumptions as in the first method . all the results of these methods are consistent and indicate that the transonic regime at m = 1 is the most critical for divergence . in part (2), it is established that sweeping the leading edge of a plate airfoil of constant thickness increases its stability . for angles of sweep less than 30, the critical conditions occur when the leading edge is sonic, but for angles greater than 30 the critical conditions occur when m = 1 .
effects of a flexible boundary on hydrodynamic stability . purpose of paper is to examine theoretically the use of coatings of elastic materials to prevent transition from laminar to turbulent flow . theory is extension to flexible boundary of the small-disturbance tollmien-schlichting stability theory and makes use of /tietjens function/ and other functions that occur in solution of orr-sommerfeld equation . it is shown how solutions for flexible wall can be obtained from solutions for rigid boundary . outline and discussion is given first for tollmien-schlichting stability theory for rigid wall, then for theory for flexible boundary . theory is given both for a nondissipative and a dissipative flexible boundary . behavior of flexible medium itself is also examined . practical requirements are discussed . for example, a conclusion is that to avoid tollmien-schlichting instability, the wave velocity of surface waves in absence of flow should coincide with tollmien-schlichting wave velocity at wavelength of /most dangerous/ tollmien-schlichting waves . moreover, damping should be large enough to prevent surface waves from developing but not so large that tollmien-schlichting waves are permissable . author states that a boundary that is both soft and light, one whose elastic constants are of same order as the dynamic pressure of the flow, may be practical for use at high speeds . this surface should have a small damping to avoid tollmien-schlichting type of instability and a large enough wave speed without flow to avoid surface wave instability . although paper is somewhat sketchy in places, it gives comprehensive coverage of stability of laminar flow over a flexible wall .
qualitiative solutions of the stability equation for a boundary layer in contact with various forms of flexible surface . an appropriate form of the boundary layer stability equation is developed for the condition where the fluid is in contact with an isotropic and homogeneous elastic medium, and various approximate analytical solutions obtained for certain types of surface, so as to reveal at least qualitatively the origin and characteristics of neutral oscillations . in the worked solutions the elastic medium is treated as nondissipative, and the interior boundary is supposed either fixed, or free of stress, or exposed to fluid .. the boundary layer, also, is treated as that over a flat-plate in an incompressible fluid . the results obtained show that the presence of such a resiliant surface introduces the possibility of a number of other modes of oscillation schlichting waves . most of these modes have speeds of propagation determined largely by the properties of the elastic material, and their presence may well be effectively a matter of ‘non-viscous’ flow stability -dash a subject not treated here . the tollmien-schlichting mode has its minimum reynolds number increased by the presence of the surface, but if the interior boundary is free there may be an upper limit as well . indeed, a sufficiently thin free surface, or one of low rigidity, apparently eliminates neutral oscillations of this mode altogether, only at the expense, however, of the introduction of a mode of flexural waves .
an investigation of the use of an auxiliary slot to re-establish laminar flow on low drag aerofoils . the use of an auxiliary slot on a laminar-flow aerofoil has been investigated to check whether laminar flow can be re-established by suction at the rear of the region of deposited dirt, flies, etc . results indicate that in the absence of unfavourable pressure gradients, it is possible to re-establish a laminar boundary layer by removing a little more than the whole turbulent layer reaching the slot, and preliminary estimates suggest that with efficient ducting it should be possible to achieve a reduction in overall effective drag coefficient by this means .
the effect on transition of isolated surface excrescences in the boundary layer . the effect of isolated surface excrescences in a laminar boundary layer in producing disturbances which may lead to turbulent flow has been examined experimentally by several methods . photographs of some of the flow patterns visualised by smoke and china-clay techniques are given . the critical heights of pimple which just give rise to spreading wedges of turbulent flow have been measured on a flat plate and on two aerofoils at several angles of incidence . the results are analysed and are presented in a form which enables approximate estimates to be made of the protuberances permissible on laminar-flow surfaces at full-scale flight reynolds numbers . the estimates suggest that at an altitude of 30,000 ft the critical pimple height is 0.004 in. for a speed of 350 m.p.h., whilst 0.002 in. may be permissible at all subsonic speeds . at sea-level, however, the tolerances are approximately halved .
experiments on the use of suction through perforated strips for maintaining laminar flow . transition and drag measurements . wind-tunnel tests are described in which suction is applied at perforated strips, as an alternative to porous strips or slots, in order to maintain a laminar boundary layer . a test was first carried out on a single row of perforations on a cambered plate, as a preliminary to the main tests which were performed on strips of multiple rows of perforations drilled through the surface of a low-drag-type aerofoil 13 per cent thick and of 5-ft chord . up to a wind speed of 180 ft sec it has been ascertained that suction may be safely applied to extend laminar flow provided the ratio of hole diameter to boundary-layer displacement thickness is less than 2, the ratio of hole pitch to diameter is less than 3 and there are at least three rows of holes in the strip . with less than three rows, the criteria are much more restrictive . it is possible to extend laminar flow by suction through perforations whose diameters and pitches exceed these values slightly, but only with the risk that excessive suction quantities will produce wedges of turbulent boundary layer originating at the holes . a uniform distribution of suction through the holes was necessary . this was successfully obtained by two methods, the use of cells and throttle holes, and with tapered holes . in particular, tests were carried out on some panels supplied by handley page, ltd., in which the cells and tapered holes had been constructed by commercial methods, and the suction distribution proved satisfactory . the resistance of some of the cellular arrangements was measured . it was found that when the suction quantities were the minimum required to maintain laminar flow, the additional losses in total head of the sucked air due to the resistance of the throttle holes could be made small compared with the loss in total head of the sucked boundary layer .
interaction of secondary injectants and rocket exhaust for thrust vector control . tests were conducted with 1300- to 1500-lb thrust solid rocket motors in order to investigate the side-force generation mechanisms associated with the injection of a secondary fluid into the expansion cone of a solid propellant rocket nozzle for thrust-vector control . the nozzles were 15 conicals with a nominal expansion ratio of . all firings were conducted in zero-flow ejectors . freon-12, water, and gascous nitrogen were used as the injectant . nozzle-wall pressure profiles, side thrust, and the nozzle-wall shock interface were recorded . the general character of the pressure disturbance was defined . the major portion of the side force was generated by the pressure disturbance downstream of the injector . the axial-thrust augmentation generated by the injectant was calculated . the effects of nozzle-expansion ratio and injector location on the side force were clearly illustrated .
on the propagation and structure of the blast wave . concerning blast waves with front surfaces of plane, cylindrical and spherical shape, the propagation velocity u and the distribution of hydrodynamical quantities are discussed . the solutions are constructed in the form of power series in (c u), where c is the sound velocity of undisturbed fluid . especially r, the distance of shock front from the charge, is represented as, where r is the characteristic length related to the energy of explosion, j and are constants, and a=0,1,2 correspond to plane, cylindrical and spherical case, respectively . in this paper the first approximations for a=0,1 are discussed (the case a=2 has been discussed by g. i. taylor) . the solution is obtained numerically for the case of the adiabatic index . the approximate solution is also considered . using these solutions, is found to be . the second approximation will appear in part 2 to be published subsequently .
the production of aerodynamic forces by heat addition on external surfaces of aircraft . within the framework of linearized flow theory an equivalence between a fluid mass source, a heat source, and streamwise body forces is developed . the equivalence between the fluid mass source and heat source was first noticed by hicks(2) and later by chu . (3) using the equivalence the flow field produced by heat addition and by magnetohydrodynamical body forces can be computed . examples for a two-dimensional flat plate, a delta wing, an axially symmetric slender body, and a wedge-shaped afterbody are computed at subsonic and supersonic speeds . the efficiency of lift or thrust production by surface heat addition is very low at subsonic speeds . at supersonic speeds the efficiency is compared with the efficiency of a conventional turbojet-powered aircraft configuration . it is found that the efficiency of lift or thrust production by heat addition on two-dimensional bodies is approximately the same as that for a turbojet-powered two-dimensional body . the efficiency is somewhat higher at low supersonic mach numbers and behaves as, decreasing to a constant value as increases . on the other hand, the efficiency of thrust production by heat addition increases linearly with mach number when heat is added on the rear surface of an axially symmetric afterbody of parabolic shape .
some aspects of non-stationary airfoil theory and its practical application . this paper consists of three notes on the theory of two- dimensional thin airfoils in non-uniform motion .. oscillating airfoil are collected from an earlier paper and are presented in convenient forms for practical application . rigid airfoil passing through a vertical-gust pattern having a sinusoidal distribution of intensity . the lift is determined as a function of the reduced frequency (which in this case is proportional to the ratio of the airfoil chord and the wave length of the gust pattern) and is presented in the form of a vector diagram . it is shown that the lift acts at the quarter-chord point of the airfoil at all times . calculation of the amplitude of torsional oscillation of a fan blade operating in the wake of a set of pre-rotation vanes . in a numerical example the amplitude is found to be small even when the vanes are spaced so that the exciting frequency coincides with the natural frequency of the fan blade .
on some fourier transforms in the theory of non-stationary flows . the growth of lift on a airfoil starting impulsively from rest to a uniform velocity has been given by wagner (1925) . the steady-state lift due to circulation on an airfoil oscillating sinusoidally and moving with uniform velocity has been given by theodorsen the present paper based essentially on the material of n. a. c. a. report no. 629 by the author, discusses some reciprocal relations of the nature of fourier transforms existing between the functions of wagner and theodorsen . kussner (1936) has already shown that wagner’s function may be derived from theodorsen’s function . by means of a superposition principle it is possible to utilize these fundamental functions to treat general problems in transient expression which is accurate to within 2 percent is given for wagner’s function . this expression leads to a good approximate expression for theodorsen’s function in terms of the exponential integral, instead of hankel functions . an analogy is drawn between transient hydrodynamic flows and transient electrical flows . kussner (1936) has introduced a function describing the growth of lift on an airfoil entering a sharp edged vertical gust region . this function bears a certain relation to wagner’s function which is briefly discussed .
calculated responses of a large sweptwing airplane to continuous turbulence with flight-test comparisons . calculated responses of symmetrical airplane motions, wing deformations, and wing loads due to gusts are shown to compare favorably with available flight-test results . these calculated responses are based on random-process theory, five degrees of freedom, lifting-surface aerodynamics, and one-dimensional vertical turbulence . the extent to which various degrees of freedom contribute to the responses is examined and in this connection the relative effects of static and dynamic aeroelasticity are determined .
calculated spanwise lift distributions, influence functions and influence coefficients for unswept wings in subsonic flow . spanwise lift distributions have been calculated for nineteen unswept wings with various aspect ratios and taper ratios and with a variety of angle-of-attack or twist distributions, including flap and aileron deflections, by means of the weissinger method with eight control points on the semispan . also calculated were aerodynamic influence coefficients which pertain to a certain definite set of stations along the span, and several methods are presented for calculating aerodynamic influence functions and coefficients for stations other than those stipulated . the information presented herein can be used in the analysis of untwisted wings or wings with known twist distributions, as well as in aeroelastic calculations involving initially unknown twist distributions .
aerodynamic forces on wings in non-uniform motion . the problem of determining the aerodynamic forces acting on wings of finite span in non-uniform motion in an incompressible, inviscid fluid is investigated . the underlying theory is outlined in 2, and some known results for the case of an aerofoil of infinite span are included in 3 . it is shown in 4, by the use of operational methods, that the growth of lift function k (s) corresponding to a sudden unit change of incidence can be derived from the lift function corresponding to simple harmonic translational motion . from results given by the writer for rectangular wings (1943) and tapered wings (1945) in simple harmonic motion the corresponding values of k (s) are determined . the growth of lift function k (s) for a wing penetrating a uniform vertical gust can then be estimated as shown in 4 and 5 . by the use of approximate formulae for the growth of lift curves given in fig. 2, the aerodynamic forces corresponding to damped and growing translational oscillations are derived . certain integrals involved in the theory are evaluated in appendix 1, and in appendix 2 the method of determining k (s), when k (s) is known, is discussed in detail . it is suggested that the aerodynamic forces acting on wings of finite span for any type of motion can best be derived from a knowledge of the forces corresponding to purely divergent motion, which can be calculated by the methods outlined in this report .
calculated spanwise lift distributions and aerodynamic influence coefficients for swept wings in subsonic flow . spanwise lift distributions have been calculated for 61 swept wings with various aspect ratios and taper ratios and with a variety of angle-of-attack or twist distributions, including flap and aileron deflections, by means of the weissinger method with eight control points on the semispan . also calculated for these plan forms were aerodynamic influence coefficients which pertain to a certain definite set of stations along the span . the information presented herein can thus be used both in the analysis of untwisted wings or wings with known twist distributions and in aeroelastic calculations involving initially unknown twist distributions . this paper supplements and is intended to be used in conjunction with naca tn 3014, where the same type of information, calculated in the same way, is presented for 19 unswept wings .
use of freon-12 as a fluid for aerodynamic testing . the thermodynamic properties of freon-12 have been investigated to determine the possibilities of the use of this gas as a fluid for aerodynamic testing . the values of velocity of sound in freon-12, which are less than one-half those in air, are presented as functions of temperatures and pressure, including measurements at room temperature . the density of freon-12 is about four times that of air . changes in state of freon-12 may be predicted by means of the ideal gas law with an accuracy of better than 1 percent at pressures below freon-12 is shown not to condense during an adiabatic expansion from normal conditions up to a mach number of 3 . the values of the ratio of specific heats for freon-12 are lower than that for air, and therefore an additional parameter is introduced, which must be considered when comparisons are made of aerodynamic tests using freon-12 with those using air . the time lag of the vibrational heat capacity of freon-12 to a change in temperature has been measured and found to be of the order of 2 x 10 second at atmospheric temperature and pressure . this time is so short that no important energy dissipations should result in most engineering applications .
studies of the use of freon-12 as a wind tunnel testing medium . a number of studies relating to the use of freon-12 as a substitute medium for air in aerodynamic testing have been made . the use of freon-12 instead of air makes possible large savings in wind-tunnel drive power . because of the fact that the ratio of specific heats is approximately 1.13 for freon-12 as compared with 1.4 for air, some differences exist between data obtained in freon-12 and in air . methods for predicting aerodynamic characteristics of bodies in air from data obtained in freon-12, however, have been developed from the concept of similarity of the streamline pattern . these methods, derived from consideration of two-dimensional flows, provide substantial agreement in all cases for which comparative data are available . these data consist of measurements throughout a range of mach number from approximately 0.4 to 1.2 of pressure distributions and hinge moments on swept and unswept wings having aspect ratios ranging from 4.0 to 9.0, including cases where a substantial part of the wing was stalled . the freon charging and recovery system used for the langley low-turbulence pressure tunnel is described .
study of effects of sweep on the flutter of cantilever wings . an experimental and analytical investigation of the flutter of sweptback cantilever wings is reported . the experiments employed groups of wings swept back by rotating and by shearing . the angle of sweep ranged from 0 to 60 and mach numbers extended to approximately 0.85 . a theoretical analysis of the air forces on an oscillating swept wing of high length-chord ratio is developed, and the approximations inherent in the assumptions are discussed . comparison with experiment indicates that the analysis developed in the present report is satisfactory for giving the main effects of sweep, at least for nearly uniform cantilever wings of high and moderate length-chord ratios . a separation of the effects of finite span and compressibility in their relation to sweep has not been made experimentally but some combined effects are given . a discussion of some of the experimental and theoretical trends is given with the aid of several tables and figures .
investigation to determine effects of center of gravity location on the transonic flutter characteristics of a 45degree sweptback wing . an experimental investigation has been conducted in the 26-inch langley transonic blowdown tunnel to determine effects of center-of-gravity location on the transonic flutter characteristics of a 45degree swept-back-wing plan form of aspect ratio 4.0 and taper ratio 0.6 . solid-construction models of the plan form with streamwise naca 65a004 airfoil sections and center-of-gravity locations at approximately 34 percent chord, 46 percent chord, and 58 percent chord, respectively, were fluttered at several mach numbers between 0.8 and 1.35 . it was found that, for streamwise mach numbers from 0.8 to 1.0, the variation with mach number of the ratio of experimental flutter speed to a calculated incompressible flutter speed was not affected by center-of-gravity location . however, for mach numbers from 1.0 to 1.35, there was an increase in flutter-speed ratio with mach number which was different for each center-of-gravity position . data from wings with successively more forward center-of-gravity locations showed successively larger values of flutter-speed ratio at mach numbers from 1.0 to
calculation of flutter characteristics for finite-span swept or unswept wings at subsonic and supersonic speeds by a modified strip analysis . a method has been developed for calculating flutter characteristics of finite-span swept or unswept wings at subsonic and supersonic speeds . the method is basically a rayleigh type analysis and is illustrated with uncoupled vibration modes although coupled modes can be used . the aerodynamic loadings are based on distributions of section lift-curve slope and local aerodynamic center calculated from three-dimensional steady-flow theory . these distributions are used in conjunction with the /effective/ angle-of-attack distribution resulting from each of the assumed vibration modes in order to obtain values of section lift and pitching moment . circulation functions modified on the basis of loadings for two-dimensional airfoils oscillating in a compressible flow are employed to account for the effects of oscillatory motion on the magnitudes and phase angles of the lift and moment vectors . flutter characteristics have been calculated by this method for 12 wings of varying sweep angle, aspect ratio, taper ratio, and center- of-gravity position at mach numbers from 0 to as high as 1.75 . comparisons of the results with experimental flutter data indicate that this method gives generally good flutter results for a broad range of wings .
method of controlling stiffness properties of a solid-construction model wing . a simple method is presented for controlling the bending and torsional stiffnesses of a solid-construction model wing . the method consists of weakening the wing by drilling holes through the wing normal to the chord plane . aerodynamic continuity is maintained by filling the holes with a relatively soft material . the important parameters controlling the stiffnesses are the amount of material removed by drilling, the ratio of hole diameter to wing thickness, and the plan-form pattern of the holes . data are given which may be used for predicting the stiffness of a model wing weakened in this manner .
investigation of wing flutter at transonic speeds for six systematically varied wing plan forms . an investigation of the effects of systematic variations in wing plan form on the flutter speed at mach numbers between 0.73 and 1.43 has been conducted in the 26-inch langley transonic blowdown tunnel . the angle of sweepback was varied from 0degree to 60degree on wings of aspect ratio 4, and the aspect ratio was varied from 2 to 6 on wings with experimental flutter speed and the reference flutter speed calculated on the basis of incompressible two-dimensional flow . this ratio, designated as the flutter-speed ratio, is plotted as a function of mach number for the various wings . it is found that the flutter-speed ratio increased rapidly past sonic speed for sweep angles of 45degree and less, indicating a favorable effect of mach number . for sweepback of mach number range of the tests . reducing the aspect ratio had a favorable effect on the flutter-speed ratio which was of the order of 100 percent higher for the aspect-ratio-2 wing than for the aspect-ratio-6 wing . this percentage difference was nearly constant throughout the mach number range, indicating that the effect of mach number was about the same for all aspect ratios tested .
the calculation of aerodynamic loading on surfaces of any shape . the object of the report is to establish a routine method for the calculation of aerodynamic loads on wings of arbitrary shape . the method developed is based on potential theory and uses a general mathematical formula for continuous loading on a wing which is equivalent to a double fourier series with unknown coefficients . in order to evaluate the unknown coefficients the continuous loading is split up into a regular pattern of horseshoe vortices, the strengths of which are proportional to the unknown coefficients and to standard factors which are given in a table . the total downwash at chosen pivotal points is obtained by summing the downwashes due to the individual vortices, a process which is simplified by the use of specially prepared tables of the properties of the horseshoe vortex . by equating the downwash to the slope of the wing at each pivotal point, simultaneous equations are obtained, the solution of which defines the unknown coefficients . the first layout involves a total of 76 vortices over the wing, and a second layout, involving a total of 84, is shown to be of superior accuracy . the effect on the solution of the number of pivotal points is investigated and it is concluded that by a suitable choice, it is unnecessary to use a large number . results for a rectangular wing at with those obtained by other workers and it appears that there may be errors in published results in at least one of these cases . immediate development includes the application to the calculation of the characteristics of actual sweptback wings, including rotary derivatives, and future development includes also applications in wind tunnel design and technique .
formulas for the supersonic loading, lift and drag of flat swept back wings with leading edges behind the mach lines . the method of superposition of linearized conical flows has been applied to the calculation of the aerodynamic properties, in supersonic flight, of thin flat, swept-back wings at an angle of attack . the wings are assumed to have rectilinear plan forms, with tips parallel to the stream, and to taper in the conventional sense . the investigation covers the moderately supersonic speed range where the mach lines from the leading-edge apex lie ahead of the wing . the trailing edge may lie ahead of or behind the mach lines from its apex . the case in which the mach cone from one tip intersects the other tip is not treated . formulas are obtained for the load distribution, the total lift, and the drag due to lift . for the cases in which the trailing edge is outside the mach cone from its apex (supersonic trailing edge), the formulas are complete . for the wing with both leading and trailing edges behind their respective mach lines, a degree of approximation is necessary . it has been found possible to give practical formulas which permit the total lift and drag to be calculated to within 2 or 3 percent of the accurate linearized-theory value . the local lift can be determined accurately over most of the wing, but the trailing-edge-tip region is treated only approximately . charts of some of the functions derived are included to facilitate computing, and several examples are worked out in outline .
atmospheric entries with vehicle lift-drag ratio modulated to limit deceleration and rate of deceleration vehicles with maximum lift-drag ratio of 0. 5. an analysis has been made of atmosphere entries for which the vehicle lift-drag ratio was modulated to maintain specified maximum decelerations and or maximum deceleration rates . the part of the vehicle drag polar used during modulation was from maximum lift coefficient to minimum drag coefficient . the entries were at parabolic velocity and the vehicle maximum lift-drag ratio was 0.5 . two-dimensional trajectory calculations were made for a nonrotating, spherical earth with an exponential atmosphere . the results of the analysis indicate that for a given initial flight-path angle, modulation generally resulted in a reduction of the maximum deceleration to 60 percent of the unmodulated value or a reduction of maximum deceleration rate to less than 50 percent of the unmodulated rate . these results were equivalent, for a maximum deceleration of 10g, to lowering the undershoot boundary 24 miles with a resulting decrease in total convective heating to the stagnation point of 22 percent . however, the maximum convective heating rate was increased 18 percent,. the maximum radiative heating rate and total radiative heating were each increased about 10 percent .
the use of aerodynamic lift during entry into the earth’s atmosphere . by employing aerodynamic lift during entry into the earth’s atmosphere at either orbital or /escape/ velocity, the range of allowable entry angles for a prescribed peak deceleration is greatly increased, while the total heat energy transferred to the vehicle can be held to about the same value as for a nonlifting vehicle . only modest lift-drag ratios are required beyond peak g to prevent the deceleration from exceeding the peak value, or to prevent the vehicle from skipping out of the earth’s atmosphere . thus, the difficult guidance and control problem is greatly alleviated,. in particular, for return from the moon or other planets the necessity for multiple-pass drag braking is eliminated .
modulated entry . the technique of modulation, or variable coefficients, is discussed and the analytical formulation is reviewed . representative numerical results of the use of modulation are shown for the lifting and nonlifting cases . these results include the effects of modulation on peak acceleration, entry corridor, and heat absorption . results are given for entry at satellite speed and escape speed . the indications are that coefficient modulation on a vehicle with good lifting capability offers the possibility of sizable loading reductions or, alternatively, wider corridors,. thus, steep entries become practical from the loading standpoint . the amount of steepness depends on the acceptable heating penalty . the price of sizable fractions of the possible gains does not appear to be excessive .
approximate analysis of atmospheric entry corridors and angles . a simple closed-form solution for the achievable corridor depths and entry angles as a function of g-load limit, entry velocity, and vehicle aerodynamics and thermodynamics is developed for two modes of vehicle operation, constant angle of attack and modulated angle of attack . for constant angle of attack, operation at maximum negative lift coefficient on the overshoot bound, and at an angle of attack between zero and that for maximum lift-drag ratio on the undershoot bound, gives the deepest corridor . for modulated angle of attack, operating at maximum negative lift coefficient on the overshoot bound and modulating the angle of attack from maximum positive lift coefficient to zero on the undershoot bound give the deepest corridor . the modulated angle of attack gives corridor depths two to four times larger than the fixed angle of attack . for both cases the corridor depth is increased by increasing maximum lift-drag ratio, increasing g limit, and decreasing entry velocity . consideration of hot-gas radiation places a limit on the maximum angle of attack for either mode of operation . if a maximum free-stream reynolds number limit must be placed on the vehicle to ensure a laminar boundary layer, the deep atmospheric penetrations associated with configurations with high lift-drag ratio may be ruled out . both of these thermodynamic considerations reduce the acceptable corridor depth below the value calculated from aerodynamic considerations alone .
radiative heat transfer during atmosphere entry at parabolic velocity . stagnation point radiative heating rates for manned vehicles entering the earth’s atmosphere at parabolic velocity are presented and compared with corresponding laminar convective heating rates . the calculations were made for both nonlifting and lifting entry trajectories for vehicles of varying nose radius, weight-to-area ratio, and drag . it is concluded from the results presented that radiative heating will be important for the entry conditions considered .
effects of simulated rocket jet exhaust on stability and control of a research type airplane configuration at a mach number of 6. 86 . an investigation has been undertaken in the langley 11-inch hypersonic tunnel at a free-stream mach number of 6.86 to determine the jet-interference effects at high jet-static-pressure ratios on the stability and control of a research-type airplane configuration . compressed-air tests with a jet exhausting from the base of the fuselage were conducted over a reynolds number range of 0.57 x 10 to and over a jet-static- pressure-ratio range of 0 to 1460 . the results of these tests indicated that the operation of the jet induced a sizable separated-flow region over the vertical- and horizontal-tail surfaces which could be approximately duplicated at low angles of attack by use of metal jet-boundary simulators . the results of force tests, during which these metal jet-boundary simulators were used, indicated that this separated-flow region caused a large reduction in the longitudinal stability and control and a smaller reduction in the lateral and directional stability and control . by extending the divergent section of the nozzle and thus reducing the jet-static-pressure ratio, these losses were diminished .
effects of jet billowing on stability of missile-type bodies at mach 3. 85 . the interference effects of a billowing jet on the forces and moments of two missile-type bodies were investigated in the nasa lewis 2-by 2-foot mach 3.85 wind tunnel . to simulate a rocket jet, pressurized nitrogen was exhausted from an annular sonic nozzle . the results indicate that for both models the stability parameter moment coefficient with angle of attack) in the region of zero angle of attack was favorably influenced by the interference resulting from separation due to jet billowing . schlieren photographs are presented that show the separation due to the jet billowing at various pressure ratios and angles of attack .
exploratory tests of the effects of jet plumes on the flow over cone- cylinder flare bodies . schlieren photographs have been taken of the flow over cone-cylinder-flare bodies to study the extent of boundary-layer separation due to the presence of rocket jet plumes . tests were made of three cone-cylinder-flare configurations in the langley 11-inch hypersonic tunnel at a mach number of 9.65 and in the langley unitary plan wind tunnel at a mach number of 4.65 with two additional configurations . the stream reynolds number varied from approximately 317,000 to 582,000 based on model length . the conical flares had half-angles of 7 or 13 and contained one of two test nozzles with a design mach number of 3.72 or 4.53 . the test nozzles were operated with compressed air and were designed to simulate a solid-propellant rocket motor operating at altitudes between to free-stream static-pressure ratio varied from jet off to 1,150 for the test nozzle with a design mach number of 3.72 and from jet off to mach number of 4.53 . for most of the tests the angle-of-attack range was 0 to -4,. some additional tests were made at 2 and 4 . measurements taken from flow pictures indicated that at zero angle of attack on all configurations tested with jet on the boundary layer separates ahead of the flare-cylinder juncture and the separation point moves toward the cone-cylinder juncture with an increase in pressure ratio . increasing angle of attack reduced the extent of boundary-layer separation on the windward side as did increasing the stream mach number from 4.65 to 9.65 . other parameters which tended to reduce the extent of boundary-layer separation were .. number, (b) decreasing stream reynolds number, and (c) displacing nozzle exit rearward .
aerodynamic investigation of a parabolic body of revolution at mach number of 1. 92 and some effects of an annular supersonic jet exhausting from the base . an aerodynamic investigation of a parabolic body of revolution was conducted at a mach number of 1.92 with and without an annular supersonic jet exhausting from the base . measurements with the jet inoperative were made of lift, drag, pitching moment, radial and longitudinal pressure distributions, and base pressures . with the jet in operation, measurements were made of the pressures over the rear of the body with the primary variables being angle of attack, ratio of jet velocity to freestream velocity, and ratio of jet pressure to stream pressure . the results with the jet inoperative showed that the radial pressures over the body varied appreciably from the distribution generally employed in most approximate theories . the linearized solutions for lift, pitching moment, and center of pressure gave relatively poor predictions of the experimental results . an analysis of several theoretical methods for calculating pressure distribution and wave drag showed that some methods gave results in considerable disagreement with experimental values . maximum effects of the jet were obtained at the lower ratio of jet velocity to stream velocity and the highest ratio of jet pressure to stream pressure . these effects amounted to a slight decrease in fore-drag, a reduction in lift, and a shift of center of pressure in a destabilizing direction .
investigation of a two-step nozzle in the langley 11in . hypersonic tunnel . flow surveys have been made in the first of several nozzles to be investigated in the langley 11-inch hypersonic tunnel . the nozzle was designed by the method of characteristics for a mach number of 6.98 . two step expanded the air in the horizontal plane to a mach number of 4.36 and the second in the vertical plane to a mach number of 6.98 . the test results showed that, although a maximum mach number of about 6.5 was obtained, the flow in the test section was not sufficiently uniform for quantitative wind-tunnel test purposes . deviations from the design flow were traced to the presence of a thick boundary layer which developed in the first step along the parallel walls .
investigation of the flow through a single stage two dimensional nozzle in the langley 11in . hypersonic tunnel . flow surveys have been made in the second of several nozzles to be investigated in the langley 11-inch hypersonic tunnel . the single-stage, two-dimensional nozzle was designed by the method of characteristics for a mach number of 7.08 without boundary-layer corrections . the test results show that reasonably uniform flow at an average mach number of about 6.86 was obtained in a central region of the stream at the test section . this region had a cross section nearly 5 inches square and had a deviation from uniform flow of less than 1 percent in mach number and 0.3 in flow angle . an increase in mach number of about 3 percent occurred during test runs of about 60 seconds duration because of distortions of the boundaries at the first minimum due to nonuniform heating of the nozzle blocks during the tests .
boundary layer displacement effects in air at mach numbers of 6. 8 and 9. 6. measurements are presented for pressure gradients induced by a laminar boundary layer on a flat plate in air at a mach number of 9.6 and for the drag of thin wings at a mach number of about 6.8 and zero angle of attack . the pressure measurements at a mach number of 9.6 were made in the presence of substantial heat transfer from the boundary layer to the plate surface . the measured pressure distribution on the surface of the plate was predicted with good accuracy by a modification to insulated-plate displacement theory which allows for the effect of the heat transfer and temperature gradient along the surface on the boundary-layer displacement thickness . the total drag of thin wings with square and delta plan forms was measured at a nominal mach number of 6.8 over a reasonably wide range of reynolds numbers . the total drag was found to be greater than can be explained by adding a classical value of laminar skin friction to the estimated pressure drag . the difference is, in general, explained by the increase in skin friction (20 to 40 percent) caused by the boundary-layer-induced pressures .
secondary flow fields embedded in hypersonic shock layers . when a ramp or other compression surface is located in a locally supersonic region behind a hypersonic bow shock wave, it generates a secondary shock wave . the ramp flow disturbance may be viewed as an embedded newtonian impact flow if the embedded shock layer is thin . examination of the applicability of newtonian flow theory to cones and wedges in uniform streams suggests that this theory can be expected to give a useful approximation to the surface pressures . a pressure equation based on this concept predicts a number of interesting things .. first, pressures can differ from simple newtonian theory by factors of 1 5 to 3,. for example, on flare stabilizers on blunt-nosed bodies of revolution, pressures are lower than newtonian and diminish with increasing flight speed in the hypersonic speed range . the calculated pressures vary over the flare surface as a result of the nonuniformity of its incident stream, and depend on the axial location of the flare . in the case of a flap mounted on a large-angled blunt-nosed cone, the pressure coefficients vary from 1 to 5 through the variable entropy layer . a pressure coefficient of 5 greater than the maximum possible in newtonian flow can occur because the compression process is more efficient than a single shock wave process . on areas of the flap that protrude through the main bow wave, the pressure coefficient should revert to the simple newtonian value . equations are developed for the initial slopes of the normal-force and pitching-moment curves of a flare stabilizer . in the simplest case these differ from conventional newtonian theory by the ratio of local dynamic pressure to free-stream dynamic pressure . this ratio takes values as low as 0.1 in some of the examples considered .
compressive buckling of simply supplorted plates with longitudinal stiffeners . charts are presented for the analysis of the stability under compression of simply supported rectangular plates with one, two, three, and an infinite number of identical equally spaced longitudinal stiffeners that have zero torsional stiffness .
compressive buckling of simply supported plates with transverse stiffeners . charts are presented for the analysis of the stability under longitudinal compression of simply supported rectangular plates with several equally spaced transverse stiffeners that have both torsional and flexural rigidity .
compression tests on circular cylinders stiffened longitudinally by closely spaced z-section stringers . six circular cylinders stiffened longitudinally by closely spaced z-section stringers were loaded to failure in compression . the results obtained are presented and compared with available theoretical results for the buckling of orthotropic cylinders . the results indicate that the large disparity that exists between theory and experiment for unstiffened compression cylinders may be significantly smaller for stiffened cylinders .
simplified analysis of general instability of stiffened shells in pure bending . although much work has been done to develop a theory for the failure of shells by general instability, there is at present no simple method by which the size of the frames may be determined for any given diameter, bending moment,.and frame spacing . such a method is needed in determining the optimum design for stiffened shells, to be used as a basis for weight analysis of fuselages, and other shell structures . in an extension of the work done for the rand corporation, a simple coefficient has been determined for this purpose . since it appears that this method may also be useful in design calculations, a brief description is presented below .
large deflections of structures subjected to heating and external loads . the method of direct formulation of the stiffness matrix is extended to include the effects of nonuniform heating and large deflections . the purpose is to develop an analytical tool for the treatment of actual structures . in the solution of aeroelastic problems the relations between forces and deflections must be determined . the usual stiffness matrix formulation of this relationship is limited to small temperature changes and small deflections . for large temperature changes additional terms are required . also the problem becomes geometrically nonlinear when large deflections are involved . to overcome the inherent difficulties of the nonlinear problem for practical structures either an iterative or a step- by-step procedure must be used . the force-deformation relations necessary for this step-by-step or iterative approach are derived for an axially loaded member and for a plate element including the effects of thermal strains .
non-linear analysis of heated, cambered wings by the matrix force method . various extensions of the matrix force method for complex structure analysis are presented and illustrated with the objective of expanding its range to handle the problems likely to be encountered in advanced vehicle wing design . methods are covered in detail for (1) determining the change in flexibility that occurs when thermal stresses are present, and also how large these stresses must be to cause buckling, (2) including the non-linear effect of large deflections by an iterative procedure, and (3) analyzing a wing that is initially slightly cambered and warped with either or both of the aforementioned effects present . formulas are given for calculating the input matrix terms as are the matrix equations and supporting theoretical discussion . an example illustrates the nature and magnitude of the effects being examined .
a characteristic type of instability in the large deflections of elastic plates . part 1. from a general equation governing the bending of thin elastic plates into certain types of surfaces of revolution are derived expressions for the behaviour of rectangular plates with initial curvatures, subjected to pure bending about one axis . it is found that such plates exhibit the type of instability characteristic of thin-walled structures which depend for their stiffness on curvature . curves are drawn showing the deformation suffered by such plates, and an expression for the critical bending moment at which instability occurs is obtained . experimental results show satisfactory agreement . part 2. the analysis of part 1 is extended to deal with the case of flat square or rectangular plates loaded by distributed bending moments applied to all four edges . curves are drawn to describe their behaviour, and they are found to exhibit the characteristic instability displayed by thin-walled curved structures . experimental verification is satisfactory .
an experimental investigation of the interaction between shock waves and boundary layers . an account is given of an investigation into the interaction between the boundary layer on a flat plate and a shock wave produced either externally, by a wedge in the supersonic mainstream, or from within the boundary layer, by a wedge held in contact with the plate . a wide range of free-stream mach numbers, boundary-layer reynolds numbers, and shock strengths has been covered, shock strength being defined as the ratio of the static pressure downstream of the shock to the static pressure upstream of it . variations in these parameters can have large effects on the interaction, and there are also large differences between cases with externally generated shocks and cases where the shock is generated from within the boundary layer . the investigation has thrown light on the physical mechanisms involved . it is found that many of the major features of the interaction arise because the boundary layer separates from the surface ahead of the shock wave . the conditions under which separation occurs and the behaviour of the separated boundary layer thus have important effects, in terms of which, for example, the differences between the interactions observed with laminar and with turbulent boundary layers may be explained .
approximate calculation of the laminar boundary layer . after analyzing a large class of boundary-layer velocity-profiles, the author discovered that the functions l(m) and h(m) for all such cases differ only slightly from each other over the whole range of positive and negative pressure gradients . here l, m and h are defined by being the velocity-component in the x direction and u the value of u at the edge of the boundary-layer and and the displacement and momentum thickness, respectively . based on this discovery, an approximate method is proposed by constructing two universal curves l(m) and h(m) for all conceivable boundary-layer flows found in practice . once these are chosen, karman’s momentum-integral can be written in the form, v being the kinematic viscosity coefficient, and can be integrated numerically . as examples, both howarth’s and hartree’s tained is considered good for practical purposes .
the compressible laminar boundary layer with heat transfer and arbitrary pressure gradient . an approximate method for the calculation of the compressible laminar boundary layer with heat transfer and arbitrary pressure gradient, based on thwaites’ correlation concept, is presented . the method results from the application of stewartson’s transformation to prandtl’s equations, which yeilds a nonlinear set of two first-order differential equations . these equations are then expressed in terms of dimensionless parameters related to the wall shear, the surface heat transfer, and the transformed free-stream velocity . thwaites’ concept of the unique interdependence of these parameters is assumed . the evaluation of these quantities is then carried out by utilizing exact solutions recently obtained . with the resulting relations, methods are derived for the calculation of the two-dimensional and axially symmetric laminar boundary layer with arbitrary free-stream velocity distribution . mach number, and surface temperature level . the combined effect of heat transfer and pressure gradient is demonstrated by applying the method to calculate the characteristics of the boundary layer on thin supersonic surfaces and in a highly cooled, convergent-divergent, axially symmetric rocket nozzle .
a theoretical investigation of the effects of mach number, reynolds number, wall temperature and surface curvature on laminar separation in supersonic flow . laminar separation in supersonic flow is investigated by an extension of stratford’s method . it is assumed that separation is of the usual practical type, taking place upstream of the shock wave or other agency provoking it . the results of the analysis agree well in most respects with experiment .
three dimensional viscous wakes . the velocity fields of three-dimensional viscous wakes are examined with the use of the boundary-layer approximations, osoen’s linearization of the convective terms, and the assumption of constant fluid properties . transform methods yield solutions for general types of initial conditions . as an illustration, the axial velocity distribution of a wake whose initial isovels (lines of constant velocity) are of elliptic shape and their decay to axial symmetry are demonstrated . both laminar and turbulent flows are considered .
steady motion of a sphere., oseens’s criticism and solution . the formula of stokes for the resistance experienced slowly moving sphere has been employed in physical researches of fundamental importance, as a means of estimating the size of minute globules of water, and thence the number of globules contained in a cloud of given mass . consequently the conditions of its validity has been much discussed both from the experimental and from the theoretical side .
some remarks on the flat plate boundary layer . the authors discuss the solutions for the flow of a viscous incompressible fluid near the leading edge of a semi-infinite flat plate without pressure gradient . the oseen linearization is employed which approximates the equations of motion and continuity by where are the coordinate directions, the corresponding velocity components and the uniform free stream velocity which is parallel to the plate . defining a perturbation stream function by the differential equation to be solved is with boundary conditions far from the plate and when y=0 and . the authors discuss the problem by applying the two-dimensional fourier transform and obtain an explicit solution for the velocity gradient at the plate which is in disagreement with the result of the blasius solution . from this the authors conclude that it would be more appropriate to use a velocity other than in the linearization of the equations of motion and suggest replacing by where . this choice does not affect the solution far from the plate but gives on the plate and in comparison with blasius solution indicates that c=0.35 . the solution of the modified oseen equation with this value of c then seems acceptable as the approximate solution in the region intermediate between the stokes flow and the free stream . on the basis of these considerations, the authors suggest an iteration procedure for obtaining the exact solution for the above problem as well as a solution for the plate of finite length .
axisymmetric free mixing with swirl . viscous laminar axially-symmetric free-mixing with small, moderate, and large swirl is investigated by a boundary layer type of analysis with integral methods . moderate and small swirls are formally the same, differing only in the order of their associated radial pressure gradients . neither induces significant axial pressure gradients,. consequently their effect on the axial flow is negligible . for moderate and small swirl an interesting feature is the swirl decay . in both compressible and incompressible flow, it is shown that jet swirl decays more rapidly than wake swirl whereas both swirls decay more rapidly than the non-uniformity in axial velocity . large swirl generates axial pressure gradients as well as large radial pressure gradients, and therefore alters the streamwise flow . examples calculated for incompressible flow, show that the wake is lengthened by large swirl . it is expected that this effect will be diminished in the presence of higher free-stream mach numbers which lead to decreased densities, due to decreased centrifugal effects, decreased radial pressure gradients, and decreased axial pressure gradients .
on axially symmetric, turbulent, compressible mixing in the presence of initial boundary layer . recent experimental results have shown that the mixing of heterogeneous gases having an initial velocity ratio close to unity occurs faster than is predicted by classical eddy-viscosity theory . the theoretical analysis of two uniform streams of different gases but of nearly equal velocity, performed with the usual assumptions for eddy viscosity and prandtl number equal to a constant, shows that mixing will take place very slowly, i.e., at the rate corresponding to laminar diffusion . it has been suggested that the difference between analysis and experiment could be attributed to the presence of a boundary layer in the experiments . it is the purpose of this note to show that the use of the classical eddy-viscosity law, admitting the existence of a boundary layer, is not sufficient to explain the rapid mixing that is observed physically . instead, it is shown that rapid mixing can be explained on the basis of a different eddy-viscosity law, as was suggested in ref. 1 . these conclusions are obtained through application of the analysis presented briefly below .
nose drag in free-molecule flow and its minimization . the superaerodynamic nose drag of a body in a free-molecule flow involves two parameters .. the speed ratio s between ordered and random molecular motions (modified mach number), and the temperature ratio between the solid surface and undisturbed gas . simplifications of the drag formula are obtained at hypersonic as well as low-subsonic extremes . to minimize the drag on a nose of specified length and base radius, the ordinary method of calculus of variation was found inadequate . a generalized approach has, accordingly, been developed, and the specification of end conditions is discussed at length . results of the present investigation indicate that in all cases an optimum nose requires a flat tip . the optimum nose curve for the hypersonic extreme does not depend on the temperature ratio, but that for the low-subsonic extreme varies in the following manner .. for a hot body the curve is convex,. for a cold body, concave . an optimum solution exists in a restricted range of specification only . with prescribed tip and base radii the admissible nose length is bounded below for the cases of hypersonic and low-subsonic hot body and bounded above for the case of low-subsonic cold body . a vanishing tip radius leads to an infinitely long nose in the former and a vanishing nose in the latter case . optimum nose curves for several temperature ratios at the low-subsonic extreme, as well as the one for hypersonic extreme, are presented . it is observed that at the low-subsonic extreme, with, the hot-body solution asymptotically approaches the hypersonic solution–i.e., a slender conventional warhead with a flat tip,. whereas with, the cold-body solution asymptotically approaches the minimal-surface solution–i.e., tip radius, a flat disc .
theoretical analysis of turbulent mixing of reactive gases with application to supersonic combustion of hydrogen . the turbulent mixing of an axisymmetric jet of a reactive gas is considered . by assuming a convenient model for the compressible eddy viscosity, the momentum equation is reduced to a form amenable to approximate solution . the resulting velocity distribution in both incompressible and compressible flows is in reasonable agreement with experiment . the usual assumptions with respect to chemical behavior, namely either frozen or equilibrium flow, and to unity lewis numbers and prandtl number are employed . the theoretical results for chemical equilibrium are shown to be in reasonable agreement with experimental data from low speed hydrogen flames . a numerical example of interest in connection with a hypersonic, air breathing vehicle is carried out in detail .
an approximate solution for the axisymmetric jet of a laminar compressible fluid . an extension of the modified-oseen method of carrier, based on the linearization of the viscous term of the von mises transformation, is presented . the method is employed to determine the velocity field associated with the laminar axisymmetric jet flow of a compressible gas with an arbitrary but constant external flow . the approximate solution is shown to be in good agreement with the exact numerical calculation of pai . in many boundary layer problems it is not possible to make the assumption of flow similarity . the solution in these cases can be obtained either by laborious finite difference techniques or by resort to approximate solutions . carrier and lewis (1), and more recently carrier (2), have suggested a method of obtaining approximate solutions to problems involving convection and diffusion . this method, termed by carrier /the modified-oseen method/, overcomes an essential difficulty of integral methods, namely, the generation of reasonable profiles . it is well known that the integral method gives accurate results only if the analytical profiles represent closely the true profiles . according to the modified-oseen method the convective operator in the original partial differential equation is replaced by a linear one . the resulting equation for the boundary layer problem is the heat conduction equation which can be treated by well-known techniques . it is the purpose of this paper to indicate a modification of this procedure and to demonstrate its simplicity and accuracy by treating the axisymmetric laminar flow of a compressible gas with arbitrary but constant external flow . the modification is based on the use of the von mises transformation with a subsequent linearization of the viscous term, rather than the linearization of the convective term . pai’s problem (3), originally treated by a finite difference technique, is considered to illustrate the effectiveness of this method .
some applications in physics of the p-function . the mathematical background and typical applications in physics are presented for a recently tabulated function . because of its properties, the p function should prove to be a useful aid in the solution of certain problems in applied mathematics involving surface integrations in cylindrical coordinates . a tabulation of the function in its normalized form is appended . particular attention is paid to the application of the p function to multiple scattering problems involving circular symmetry .
theoretical investigation of the flow field about blunt nosed bodies in supersonic flight . a numerical method ofr obtaining the solution to the inverse problem of the flow behind a given detached shock to any desired accuracy is presented . the cases of zero and small incidence are considered . the combination of sets of such solutions satisfying prescribed boundary conditions (body shapes) is described . particular attention is devoted to the analysis of the sonic and subsonic region of the flow field . convergence and stability of the stepwise integration from the shock in the elliptic region are discussed . numerical examples are also included .
blunt-cone pressure distributions at hypersonic mach numbers . the static pressure distributions on the surface of a blunted 7.5 degree-half-angle cone have recently been experimentally determined in the cal 48 inch hypersonic shock tunnel . this facility and the associated instrumentation are described in detail in ref. 1 . these tests covered a mach number range of 8 to 18 at a reynolds number per foot of approximately 1 times 10 to the 5th power., the models included one flat-faced cone and two hemispherically blunted cones .
hypersonic flight and the re-entry problem . paper reviews the possibilities and some of the main problems of three types of long-range vehicle, the ballistic, the glide, and the skip rocket . performance assessments are made on the basis of an ingenious, if debatable, conversion of the vehicle characteristics to the breguet formula . the problems of aerodynamic heating, aerodynamic loads and stability are briefly discussed, and other aspects of hypersonic flight free molecule flow-dash are touched upon . the results on the whole favor the glide vehicle for manned flight . the subsequent discussion on the paper includes references to multistaging and the use of high-energy fuels .
the problem of obtaining high lift-drag ratios at supersonic speeds . the importance of the lift to drag ratio is well known to all aircraft designers since it gives, to a great extent, the aerodynamic efficiency of the airplane . aerodynamic efficiency, however, is only one component of the grand compromise that a completed airplane represents . at subsonic speeds, lift-drag ratios of well over 200 have been measured in wind tunnels on airfoil sections., but few powered aircraft have attained (lift to drag ratio) value of 20 . it is invariably true that the requirements of stability and control, structure, and flight operation all contribute to reducing the design (lift to drag ratio) considerably below those exotic values which can be predicted from unrestricted aerodynamic theory . if, however, a certain range or operating efficiency is required, there is most certainly a minimum if we examine the range equation we see that range is proportional to the lift-drag ratio, the thermopropulsive efficiency, and the logarithm of the initial to final weight ratio . the appearance of the lift-drag ratio as a linear factor in the range equation indicates that every attempt should be made to increase (lift to drag ratio)., however, the search for higher (lift to drag ratio) may lead to strange and unorthodox configurations . most frequently, such configurations are ruled out by the adverse effects of their geometry on the weight ratios . in the present paper, we will deal with the maximum lift-drag ratio problem for conventional configurations having a wing and a body in close proximity to each other . no attempt will be made to select a particular configuration as being the best . however, the promising direction to go from the aerodynamic view will be stressed with the understanding that the other factors may outweight the aerodynamics .
effect of mach number on boundary layer transition in the presence of pressure rise and surface roughness on an ogive-cylinder body with cold wall conditions . the effect of mach number variation from 1.8 to 7.4 on boundary-layer transition was investigated on a slender fin-stabilized ogive-cylinder body in free flight at a constant length reynolds number of 13.8 million . the wall to free-stream temperature ratio was constant at a value of 1.0 below mach number 4.5 and at a value of of the test showed that increasing mach number had a very favorable effect of increasing the extent of the laminar boundary layer for a given surface roughness . the transition data, when plotted as a function of a factor indicative of heat transfer, showed that heat transfer was possibly responsible for a good deal of the increase in transition reynolds number with mach number . transition was found to occur farther forward on the sheltered side of the body than on the windward side for angles of attack as low as 0.4 and for all mach numbers . the pressure rise along sheltered-side stream-lines was examined and it was found that the pressure-rise coefficient at the transition point, showed no variation with mach number . data from other sources for different test conditions, when reduced to values of pressure-rise coefficient, were also found to correlate well with that of the present investigation with the exception of data at low subsonic mach numbers . these present results also show that mach number, surface roughness, pressure rise, and length reynolds number all affected boundary-layer transition in the region of theoretical infinite laminar stability to small two-dimensional disturbances as calculated for a flat plate with zero pressure gradient .
the solution of the equations of the laminar boundary layer for schubauer’s observed pressure distribution for an elliptic cylinder . the solution of the equations of the laminar boundary layer has been carried out for the pressure distribution for an elliptic cylinder of axial ratio 2.96 .. 1 with its major axis in the direction of the incident stream . the solution has been obtained by the method of hartree and womersley . in applying this method the derivatives parallel to the boundary are replaced by finite differences, so that the partial differential equation of the boundary layer is replaced by an ordinary equation relating the velocity distribution through the boundary layer at one section to that at another, at an interval upstream . by two independent integrations covering the same range by finite intervals of different sizes, it is possible to estimate the errors involved in replacing the derivatives by finite differences, and so to correct for these errors . the process of solution requires the values of the pressure gradient along the solid boundary, and there is a certain tolerance in the derivation of the pressure gradient distribution from a limited number of observed values of pressure . an analysis of schubauer’s pressure distribution is outlined, and the results were used for the main solution calculated . it is found that the solution, for the distribution of pressure gradient so derived, does not give separation of the boundary layer from the solid boundary, whereas the actual flow does separate . it is found that the calculated solution is very sensitive to the pressure distribution, and a comparatively small modification of the pressure distribution gives a solution which does indicate separation close to the point at which separation is observed to occur . the solution with this pressure distribution also gives very good agreement with the observed velocity distribution through the boundary layer at points upstream from separation .
on the theory of laminar boundary layer involving separation . the paper presents a mathematical discussion of the laminar boundary layer, which was developed with a view of facilitating the investigation of those boundary layers in particular for which the phenomenon of separation occurs . the treatment starts with a slight modification of the form of the boundary layer equation first published by von mises . two approximate solutions of this equation are found, one of which is exact at the outer edge of the boundary layer while the other is exact at the wall . the final solution is obtained by joining these two solutions at the inflection points of the velocity profiles . the final solution is given in terms of a series of universal functions for a fairly broad class of potential velocity distributions outside of the boundary layer . detailed calculations of the boundary layer characteristics are worked out for the case in which the potential velocity is a linear function of the distance from the upstream stagnation point . finally the complete separation point characteristics are determined for the boundary layer associated with a potential velocity distribution made up of two linear functions of the distance from the stagnation point . it appears that extensions of the detailed calculations to more complex potential flows can be fairly easily carried out by using the explicit formulae given in the paper .
a theoretical calculation of the laminar boundary layer around an elliptic cylinder and its comparison with experiment . the author, in conjunction with th. von karman, has recently given a new method of approximate integration of the prandtl boundary layer equations, which was developed in order to treat cases in which separation of a laminar boundary layer might be expected . the method was developed because some doubt was felt as to the accuracy with which the well-known pohlhausen analysis would describe conditions in the neighborhood of such a separation point . numerical calculations were carried out for certain cases involving theoretical simplifications, and very considerable discrepancies were found between the results of the new and pohlhausen methods . the method was also used in developing a theory for the maximum lift coefficient of certain classes of airfoils . this theory gave satisfactory agreement with experiment but no direct experimental check on the boundary layer analysis itself has been given up to the present .
air flow in a separating laminar boundary layer . the speed distribution in a laminar boundary layer on the surface of an elliptic cylinder, of major and minor axes 11.78 and 3.98 inches, respectively, has been determined by means of a hot-wire anemometer . the direction of the impinging air stream was parallel to the major axis . special attention was given to the speed distribution in the region of separation and to the exact location of the point of separation . an approximate method, developed by k. pohlhausen for computing the speed distribution, the thickness of the layer, and the point of separation, is described in detail,. and speed-distribution curves calculated by this method are presented for comparison with experiment . good agreement is obtained along the forward part of the cylinder, but pohlhausen’s method fails shortly before the separation point is reached and consequently cannot be used to locate this point . the work was carried out at the national bureau of standards with the cooperation and financial assistance of the national advisory committee for aeronautics .
analysis and calculation by integral methods of laminar compressible boundary layer with heat transfer and with and without pressure gradient . a survey of integral methods in laminar-boundary-layer analysis is first given . a simple and sufficiently accurate method for practical purposes of calculating the properties layer in an axial pressure gradient with heat transfer at the wall is then presented . for flow over a flat plate, the method is applicable for an arbitrarily prescribed distribution of temperature along the surface and for any given constant prandtl number close to unity . for flow in a pressure gradient, the method is based on a prandtl number of unity and a uniform wall temperature . a simple and accurate method of determining the separation point in a compressible flow with an adverse pressure gradient over a surface at a given uniform wall temperature is developed . the analysis is based on an extension of the karman-pohlhausen method to the momentum and thermal energy equations in conjunction with fourth- and especially higher degree velocity and stagnation-enthalpy profiles . from the equations derived here, conclusions regarding the effect of pressure gradient, mach number, and wall temperature on the boundary-layer characteristics are derived and illustrated . in particular the effects on skin-friction, heat-transfer coefficient, separation point in an adverse pressure gradient, and stability of the laminar boundary layer are analyzed .
the buckling of a square panel under shear, when one pair of opposite edges is clamped and the other pair is simply supported . reasons for investigation.–for an efficient design of spar with thin sheet web it is important to know the load which will just cause the web to buckle . as stiffeners divide the web into panels, it is required to find the buckling stress of rectangular panels bounded on two sides by spar flanges and on the other two sides by stiffeners . boundary conditions which represent closely this type of edge fixing are clamping (along the flanges) and simple support critical shear stress for a square panel held in this way . conclusions and further development.–it is found that the value of the critical shear stress is almost midway between its values when all four edges are clamped and all four edges are simply supported . the method of solution developed in this report is of very general application, and can be used to investigate the stability of rectangular panels when the loading is any combination of shear and compression or tension, and the edges are clamped or simply supported, and not necessarily all clamped or all simply supported . by an easy extension the method of solution can also be used to find the periods of transverse vibration of rectangular panels for the same types of loading and edge fixing .
a process for the step-by-step integration of differential equations in an automatic digital computing machine . it is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known . from a general theory given by kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers . it is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions . the errors are studied and a simple example is given .
numerical construction of detached shock waves . this article proposes a new method for solving the problem of the detached shock wave . if the shock wave is assumed known, a cauchy problem for a system of partial differential equations arises . this has been solved by several authors in the region where the system is elliptic (near the peak of the shock wave) . considering the plane stationary case, the author seeks an analytic continuation of the propagation function (x,y) in the complex plane y = y1 + y2, x real . in the plane (x, y2) the propagation function satisfies a hyperbolic equation (near the peak of the shock wave) . a new cauchy problem is solved and the solution of the original problem obtained by analytic continuation . a numerical example is treated with many details .
on the numerical calculation of detached bow shock waves in hypersonic flow . a method is described for calculating examples of hypersonic flow with a detached bow shock wave past a bluff axially symmetric body . the form of the shock wave is assumed, and the analysis is based on a cauchy problem for the stream function in the subsonic region, where the motion is governed by a partial differential equation of elliptic type . through analytic continuation into the complex domain, the cauchy problem is reformulated in such a manner that it becomes properly set in the subsonic region . this leads to a stable scheme for computation of the flow by finite differences . numerical examples at freestream mach number 5.8 are presented in which the flow is determined throughout the subsonic region, and, in particular, the detachment distance, the location of the sonic line, and the pressure distribution along the body are calculated . these results are in excellent agreement with experimental data obtained at the california institute of technology .
shock wave and flow field development in hypersonic re-entry . a study is made of when and how a shock wave and continuum-type flow field develop in the nose region of a highly cooled blunt body re- entering the atmosphere at hypersonic speed and in a free molecular flow regime . the various types of flow regimes encountered down to low altitude conditions are delineated, and the nature of the flow field and behavior of some of the aerodynamic characteristics are discussed . it is shown that for a highly cooled body, free molecule flow conditions occur at a higher altitude than previously indicated . based on available evidence, it is suggested that kinetic theory solutions, which are essentially modified free molecule results, along with the navier-stokes equations with no surface slip, serve to define all of the flow regimes except for a narrow transitional layer regime which has a height of less than one factor of 10 in free stream density change . it is also suggested that the appearance of a definable shock wave occurs very rapidly in terms of density change near the beginning of the transitional layer regime, and that its location, as in continuum flow, is governed principally by the body geometry, whereas its thickness is determined by a local mean free path .
the solution of small displacement, stability or vibration problems concerning a flat rectangular panel when the edges are either clamped or simply supported . this report describes an energy method for the exact solution of problems concerning the small displacements, stability or vibration of a flat rectangular panel when the edges are either clamped or simply supported . the influence of stiffeners which are parallel to one pair of edges, and situated in pairs on opposite sides of the panel so that the neutral axis of each stiffener pair lies in the middle surface of the panel, is taken into account . the method is not only applicable to isotropic panels but also to aeolotropic panels when the material of the panel has two directions of elastic symmetry parallel to the edges . the final solution of the problems depends on an infinite set of linear equations for small displacement problems or on an infinite determinantal equation for stability and vibration problems . the important feature of the analysis given is that it enables a direct approach to be made to these equations in any particular problem . it is not in general possible to obtain a direct solution of the final equations and it is necessary to approximate and consider a finite set of linear equations or a finite determinantal equation derived from the more important terms in the analytical expression for the transverse displacement of the panel . here, physical intuition and, if available, experimental data serve as a guide and the accuracy of the final results so obtained is gauged by the rate of convergence with the increase in the number of terms considered . the general method of solution is applied first to the free vibration of a square panel when all the edges are clamped, and second to the buckling of a square panel under shear when three edges are clamped and one edge is simply supported .
heat transfer near the forward stagnation point of a body of revolution . in order to determine the temperature distribution over a body moving through the atmosphere, a knowledge of the local heat-transfer coefficients is required . for slender sharp-nosed bodies, the heat-transfer coefficients are frequently approximated by using the comparable flat-plate values . however, for blunt-nosed bodies, flat-plate solutions are not applicable near the forward stagnation point . since the greatest rate of heat transfer may occur at the forward stagnation point, its value should be investigated . in this note a theoretical solution is given for the heat transfer near the forward stagnation point of a body of revolution assuming laminar, incompressible, low-speed flow . the comparable solution for two-dimensional flow has been given by squire . in the case of a blunt-nosed body moving with supersonic velocity, the flow behind the central portion of the bow wave is subsonic, and it is possible that a low-speed solution, using as /free-stream/ conditions those behind the center of the bow wave, will apply near the stagnation point .
stagnation point heat transfer measurements in hypersonic low density flow . in hypersonic, low reynolds number flow around a blunt body, the boundary-layer thickness approaches the shock-layer thickness (shock standoff distance) within the region of continuum flow . in this instance, the customary boundary-layer approximations no longer apply . hoshizaki and probstein have obtained solutions to the incompressible navier-stokes equations in the stagnation region of a blunt body in this hypersonic low reynolds number flow . the results indicate that heat-transfer rates are substantially higher than those predicted by incompressible boundary-layer theory . probstein indicated that the actual heat-transfer rates would be correspondingly higher than the predictions of fay and riddell . these findings are of particular importance in the atmospheric entry phase of recoverable satellites .
low density stagnation point heat transfer measurements in the hypersonic shock tunnel . presents absolute heat-transfer measurements, using resistance thermometer, for hemisphere-cylinder at mach numbers 9.2 to 11.2 . results show vorticity-interaction and viscous-layer effects increase heat transfer above values predicted by boundary-layer theory . data are correlated using cheng’s reynolds-number- dependent parameter . investigation covers vorticity-interaction to incipient-merged-layer regimes (free-stream unit reynolds numbers
shear buckling of clamped and simply-supported infinitely long plates reinforced by transverse stiffeners . the paper presents a solution to the buckling of infinitely long plates clamped along the edges, together with an extension of the solution obtained by stein and fralich for the case when the edges are simply-supported . it is shown that, as a consequence of increasing the edge support from that of a simple support to one in which both deflection and rotation are prevented, the rigidity which an intermediate transverse stiffener must possess in order to support the plate effectively is much reduced . agreement between the theoretical relationships and existing experimental data is good .
critical shear stress of an infinitely long simply supported plate with transverse stiffeners . a theoretical solution is given for the critical shear stress of an infinitely long, simply supported, flat plate with identical, equally spaced, transverse stiffeners of zero torsional stiffness . results are obtained by means of the lagrangian multiplier method and are presented in the form of design charts . experimental results are included and are found to be in good agreement with the theoretical results .
stability of rectangular plates under shear and bending forces . the author first discusses the problem of a plane, simply supported rectangular plate loaded by shearing forces in the plane of the plate on all four edges . there are two stiffeners attached one third and two thirds of the way along the plate . the critical load is calculated for various stiffener rigidities . also, the rigidity necessary to keep the stiffeners straight when the plate buckles is found . this stiffener rigidity is found to be slightly larger than that necessary for a plate with one stiffener and the same panel dimensions as the plate with two stiffeners . the second problem discussed by the author is that of a plane, simply supported rectangular plate loaded by uniformly distributed edge shearing forces in the plane of the plate and linearly distributed tension and compression in the plane of the plate at the ends . the end forces vary from tension, at one corner to, at the other corner, so that their resultant is a bending moment . the presence of the edge shearing forces is found to diminish the critical bending stress in this case . calculations are made for various magnitudes of bending and shearing forces for plates of various proportions .
buckling of transverse stiffened plates under shear . this paper presents an analysis of buckling of simply supported rectangular plates reinforced by any number of transverse stiffeners and subjected to shearing forces uniformly distributed along the edges . two cases are considered .. (a) the case of a plate with a finite length,. ing stresses in both cases are expressed in similar forms, that is, in equation (13), and k in equation (24), respectively . design curves are drawn as shown in figs. 2,3, and 5 .
the buckling shear stress of simply-supported infinitely long plates with transverse stiffeners . this report is an extension of previous theoretical investigations of the elastic buckling in shear of flat plates reinforced by transverse stiffeners . the plates are treated as infinitely long and simply-supported along the long sides . stiffeners are spaced at regular intervals, dividing the plate into a number of panels of uniform size . the effect ob bending and torsional stiffnesses of the stiffener upon the buckling shear stress is calculated for the complete range of stiffnesses, for panels with ratios of width to stiffener spacing of graphical forms .