JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 2, pp. 635-647, Warsaw 2017 DOI: 10.15632/jtam-pl.55.2.635
A NUMERICAL APPROACH TO PREDICT THE ROTATING STALL IN THE VANELESS DIFFUSER OF A CENTRIFUGAL COMPRESSOR USING THE EIGENVALUE METHOD
Chenxing Hu, Pengyin Liu, Xiaocheng Zhu
Co-Innovation Center for Advanced Aero-Engine, Shanghai Jiao Tong University, Shanghai, China e-mail: ryanhu@sjtu.edu.cn; pengyinliu@163.com; zhxc@sjtu.edu.cn
Hua Chen
National Laboratory of Engine Turbocharging Technology, Tianjin, China e-mail: huachen204887@163.com
Zhaohui Du
Co-Innovation Center for Advanced Aero-Engine, Shanghai Jiao Tong University, Shanghai, China e-mail: zhdu@sjtu.edu.cn
A two-dimensional incompressible flow model is presented to study the occurrence of rotating stall in vaneless diffusers of centrifugal compressors. The diffuser considered has two parallel walls, and the undisturbed flow is assumed to be circumferentially uniform, isentropic, and to have no axial velocity. The linearized 2D Euler equations for an incompressible flow in a fixed frame of the coordinate system are considered. After discretization by a spectral collocation method based on Chebyshev-Gauss-Lobatto points, the generalized eigenvalue problem is solved through the QZ algorithm. The compressor stability is judged by the imaginary part of the eigenvalue obtained. Based on the 2D stability analysis, the influence of inflow angle, radius ratio and wave number are studied. The results from the present stability analysis are compared with some experimental measurement and Shen’s model. It is showed that diffuser instability increases rapidly and the stall rotational speed decreases quickly with an increase in the diffuser radius ratio. The largest critical inflow angle can be obtained when the wave number is around 3 ∼ 5 for the radius ratio between 1.5 to 2.2. It is also verified that the stability model proposed in this paper agrees well with experimental data and has the capability to predict the onset of rotating stall, especially for wide diffusers.
Keywords: instability, vaneless diffuser, eigenvalue problem, spectral method 1. Introduction
Rotating stall in radial vaneless diffuser is one of the most common flow instabilities in centrifugal compressors and it can significantly influence the performance of the compressors. The nature of flow instability, especially rotating stall associated with vaneless diffusers have been extensively investigated by numerous researchers (Day, 2016; Everitt and Spakovszky, 2013; Spakovszky and Roduner, 2009; Ubben and Niehuis, 2015).
For decades, efforts have been made by researchers to explain the mechanism and predict the occurrence of rotating stall within the vaneless diffuser. Quantities of theoretical methodologies based on different assumptions and simplifications on the base flow have been proposed du- ring that time. Generally speaking, one of the basic distinctions between the theoretical models on vaneless diffusers is whether the influence of the boundary layer is taken into considera- tion. The first type of approaches which was adopted by Jansen (1964), Senoo and Kinoshita (1977) and Frigne and Braembussche (1985) is that the three-dimensional wall boundary layer
636
C. Hu et al.
is supposed to be responsible for the occurrence of rotating stall in a narrow vaneless diffuser. According to the experimental and theoretical study of Jansen (1964), the flow was assumed to be symmetric with respect to the diffuser depth and the local inward radial velocity com- ponent was treated as the inception of rotating stall. While in Senoo’s vaneless diffuser model, the flow was no longer assumed to be symmetric and a non-uniform distribution of inlet velo- city along the axial direction was taken into account. In his calculations, the critical velocity angle was defined for which reverse flow started in the vaneless diffuser (Senoo and Kinoshita, 1978).
On the other hand, an other theory for the occurrence of rotating stall in a vaneless diffu- ser, which was employed by Abdelhamid (1980) and Moore (1989), indicates that the stall is associated with two-dimensional core flow instability in vaneless diffusers which usually have the width radius ratio above 0.1. The two-dimensional numerical model developed by Moore is based on calculation of 2D incompressible Euler’s equations. Neutrally stable rotating distur- bances with low speed were found and a dense set of resonant solutions in which large pressure perturbations were taken as the criterion of rotating stall in Moore’s work. Chen et al. (2011) extended the 2 dimensional model of Moore into a 3 dimensional model with consideration of the distribution of inlet velocity along the axial direction. Sun et al. (2013, 2016) proposed sta- bility models for axial and centrifugal compressors on the basis of the eigenvalue approach. The comparisons with the results from experiments validated the effectiveness and accuracy of their models.
In addition, the significant influence of diffuser geometry and flow parameters on the va- neless diffuser performance and structure of the stall pattern have been also numerically and experimentally investigated in the recent years. The experiments carried out by Abdelhamid (1983) and Bianchini et al. (2013) confirmed that the local reverse flow did not necessari- ly lead to stall. The dependency of flow instability on the diffuser width ratios and diame- ter were also verified. The critical flow coefficient becomes larger with a decrease in diffu- ser width according to experiments of Abidogun (2002). Besides those experimental resear- ches, the diffuser stability was also numerically investigated by Everitt (2010) through con- ducting isolated diffuser simulations, and the volute was found to have potential in delaying the onset of diffuser instability. Although the CFD method has an advantage over theoreti- cal methods by providing a direct and vivid flow field with a relatively high accuracy, there is no a certain way for numerical simulation to capture the complicated disturbance with dif- ferent frequency, amplitude and length scale due to unsteadiness and complexity of the flow field.
According to Jansen (1964), unsteady inviscid motion of the fluid is analyzed with the assumption that the disturbances can be expressed in terms of periodic waves. The equations are then reduced and solutions are sought for the resulting eigenvalue problem. However, it is mentioned in his paper that the prediction of the number of stall cells and the determination of the constant in the velocity equation requires a subsequentstudy.
In the present paper, firstly a 2D vaneless diffuser theoretical model based on stability ana- lysis of an incompressible base flow is established and the full eigenvalue spectrum is obtained. Then the influence of collocation points and geometric parameters of the vaneless diffuser on the stability is investigated. Finally, the comparison between the results obtained from the present stability model and those from several experiments and models reported previously is perfor- med. Compared to unsteady CFD simulations which are quite time and resources consuming and Senoo’s stability model, which is unable to provide the stall number, the present analy- sis is able to predict the occurrence and stall number cheaply and quickly. And compared to Shen’s model, the present model is easier to be extended to the stability problem concerning the sensitivity.
A numerical approach to predict the rotating stall… 637 2. Theoretical model
2.1. Numerical methodology
In the present analysis, the stability discussed here is assessed by linearizing 2D Euler’s equations for a base flow and determined by an operator that describes the evolution of small perturbations superposed on the base flow. The eigenvalues of this operator give the frequency and time growth rate of the perturbations. The corresponding eigenfunctions yield the mode shapes. The perturbation with the largest growth rate is the one that dominates the stability in the long term, and the positive growth rate means the occurrence of instability.
The diffuser considered has two parallel walls, and the undisturbed flow is assumed to be circumferentially uniform, isentropic, and to have no axial velocity. To further simplify the calculation, the perturbations and velocity distribution of the base flow along the axial direction are neglected. The flow sketch is shown in Fig. 1. The inflow circumferential angle α is defined at the diffuser inlet as illustrated in Fig. 1b. Then, a 2D flow model for study of the flow stability in the vaneless diffuser can be developed.
Fig. 1. Sketch of a vaneless diffuser: (a) base flow, (b) inflow angle
2.2. Implementation of the numerical model
2.2.1. Linearization of Euler’s equations
To begin with, viscosity and compressibility are neglected in our study. The flow field is described by non-dimensionalized 2-D unsteady incompressible Euler’s equations as shown
vr +∂vr +1∂vθ =0 r ∂r r∂θ
∂vr ∂vr vθ ∂vr vθ2 ∂p ∂t +vr ∂r + r ∂θ − r =−∂r
∂vθ +v ∂vθ +vθ∂vθ +vθvr =−∂p ∂t r∂r r∂θ r r∂θ
(2.1)
Here, the velocity is non-dimensionalized by the impeller tip velocity U. The length is non- dimensionalized by the corresponding inlet radius r1 and the time t is non-dimensionalized by r1/U. Thus, vr and vθ mean the non-dimensionalized radial and circumferential velocities. In the stability analysis, the flow field is assumed to consist of the base flow and a small disturbance. Then, the velocity and pressure can be rewritten as follows
p=p+p′ vr =vr +vr′ vθ =vθ +vθ′ (2.2)
638
C. Hu et al.
where p represents base flow pressure and p′ are small disturbances of pressure. For the base flow, Vr and Vθ represent the inlet radial and circumferential velocity of the base flow, respectively.
Before Eqs. (2.2) are substituted into Eqs. (2.1), emphasis on the small perturbation theory is made: 1) any disturbances of the 2nd and higher orders are neglected, 2) the base flow is treated as steady, so ∂•/∂t = 0, c) since the mean flow is circumferentially symmetric, ∂•/∂θ = 0. And Eqs. (2.1) are still appropriate for the base flow. Then the linearized non-dimensional Euler’s equations can be derived as follows
v r′ ∂ v r′ 1 v θ′
r + ∂r + r θ = 0
∂vr′∂vr∂vr′ ∂vr′2
v θ v θ′ ∂ p ′ ∂t+∂rvr′+vθr∂θ+vr∂r − r +∂r=0
(2.3)
∂v′ ∂vv∂v′ ∂v′vv′∂p′ θ+vr′ θ+θ+vθθ+vrθ+rθ+ =0
∂t ∂r r r∂θ ∂r r r∂θ
2.2.2. Establishment of the eigenvalue problem
In general, the separation between spatial and temporal coordinates allows making use of Fourier modes in time when the variables are homogeneous. Then the perturbations can be written in form of eiθ where the homogeneous variables (including time) are taken into account and the inhomogeneous variables are taken as the amplitude function. The specific form of the Fourier mode assumed is that the disturbances are normal modes in the circumferential direction, the disturbances can be written as follows
p′ = P (r)ei(−ωt+mθ) vr′ = A(r)ei(−ωt+mθ) vθ′ = W (r)ei(−ωt+mθ) (2.4) where m is the azimuthal wavenumber and ω = ωr + iωi. Taking p′ for instance as
p′ = P(r)eimθe(−iωr+ωi)t (2.5)
it can be seen that corresponding to eωit dominates the growth rate of disturbances with time. Namely, if the imaginary part ωi > 0, the disturbances will grow exponentially with time, and the mode will be unstable. If ωi < 0, the disturbances will decay with time and the mode will be stable. And is the angular frequency.
The rotational speed of the disturbances in the circumferential direction non-dimensionlized by the impeller rotational speed is
f = −ωr (2.6) m
Substituting equations (2.4) into (2.3), the ordinary differential equations for A, W, P which represent A(r), W(r) and P(r) with the subscript omitted, can be obtained as follows (A′r = dA/dr etc.)
A ′r + A + i m W = 0 rr
−iωA+ ∂vrA+vrA′r + imvθA − 2
v θ W + P r′ = 0 ∂r rr
( 2 . 7 )
− i ω W + A ∂ v θ + v θ + v r W r′ + i m v θ W + v r W + i m P = 0 ∂rr rrr
Since there is no inlet disturbances coming from the outside of the system, homogeneous ordinary differential equations (2.7) can also expressed as
Mφ = ωJφ (2.8)
A numerical approach to predict the rotating stall...
639
where φ = [A, W, P ]T , and M and J are the corresponding coefficient matrices
000 J = i 0 0
0i0
1+∂ im 0 r∂r r
vθ ∂ vr +v ∂ +imvθ −2
M = d
drr∂rr r ∂r
( 2 . 9 )
d
vθ + vθ v ∂ + imvθ + vr im
drr r∂rrrr
Solving equation (2.8) leads to a generalized eigenvalue problem, and the eigenvalue is used
to determine the instability of the vaneless diffuser.
2.2.3. Establishment of the base flow
According to conservation of angular momentum and mass, the circumferential and radial velocity can be expressed as follows
vr12πr1 = vr2πr vθ1r1 = vθr (2.10) Therefore, the expressions of circumferential and radial velocity can be obtained as
vr =Vr vθ =Vθ (2.11) rr
where Vr and Vθ are the radial and circumferential velocity distributions at the diffuser inlet. 2.3. Discretization and calculation
2.3.1. Spatial discretization
Among the widely used spatial discretization methods, spectral methods have played a pro- minent role in early instability analyses. At present they are particularly useful. In this study, the spectral method is accomplished with Chebyshev-Gauss-Lobatto points. The two-dimensional li- near eigenvalue problems posed above can be solved by constructing an operator that discretizes a n-th order linear ODE in the form of
Lij = an(ri)D(n) + . . . + a1(ri)D(1) + a0(ri) (2.12) ij ij
where D(m), m = 1,2,...,n, is the m-th derivative matrix corresponding to the collocation ij
points ri, and a0, a1, . . . , an are evaluated at ri.
When using spectral collocation methods, the value of the functions at collocation points is
expressed as follows
N
j=0
where φ(r) is the value of the function at the point r obtained by using interpolant polynomials constructed for the variables in terms of their values at the collocation points, and φ(r) is known as the basis function.
The collocation points in the calculation domain Ω are chosen as
rΩ = cos πj j = 0,1,...,N (2.14) N
φ(r) =
φj(r)φ(ri) (2.13)
640
C. Hu et al.
The extrema of the N-th order Chebyshev polynomial TN are defined in the interval
−1 < rΩ < 1. Then the interpolant φj(rΩ) for the Chebyshev scheme is given as φ j ( r Ω ) = 1 − r Ω2 j T N′ ( r Ω ) ( − 1 ) j + 1
r−rΩi N2cj
c0 =cN =2 cj =1 0
Rf
β [◦]
m
α [◦]
experiment
Shen
present model
1.4
24.6
3
5.2
7.925
6.657
1.67
24.6
3
5.5
12.82
11.631
Fig. 7. The experimental rig for a vaneless diffuser in (Abidogun, 2002)
It is illustrated in Table 3 that the inlet critical flow velocity at the onset of stall derived by Abidogun (2002), Shen’s model and the present stability analysis shows the same trend of increase with the diffuser radial ratio Rf. For a smaller radius ratio of the diffuser Rf, the prediction results of the present model and Shen’s model show a higher accuracy than those for the radius ratio 2.0. Compared with the results in (Kinoshita and Senoo, 1985), the predicted critical inlet velocity for different radius ratios obtained in this test are all much closer to the experimental results. One main reason for this lies in that the present analysis and Shen’s model are both based on the core flow assumption, which is more suitable for vaneless diffusers with large width.
646
C. Hu et al.
Table 3. Comparison of the inlet critical flow in (Abidogun, 2002), Shen’s and the present stability analysis
5. Conclusions
In this research, a two dimensional stability model of a vaneless diffuser in a centrifugal compres- sor based on stability analysis is proposed. The independence of the number collocation points and the influence of geometric parameters on the stability of the vaneless diffuser are verified. The capability of the present model to predict the onset of stall with different radius ratios is validated against several experimental data. The present model is effective in stall prediction for a wide vaneless diffuser. Compared to Moore’s and Shen’s stability models, the advantage of the present model lies in direct calculation and providing both the complex eigenspectrum and rotational speed instead of multiple trials and iterations. Another remarking superiority is that the present model investigates the instability induced by the inviscid main flow, and provides the fundamental research for further study of the unsteady interaction of the inviscid main flow and the boundary layers.
In our investigation, the significant effects of the diffuser radius ratio on diffuser stability are confirmed. The stability deteriorates rapidly with an increase in the radius ratio. The largest critical inflow angle is obtained when the wave number m is around 3-5 for the radius ratio between 1.5 to 2.2. Through model assessment, this model has the capability of predicting the onset of stall in vaneless diffusers and can be applied in the cases without considering the axial distribution of inlet velocity. However, in the cases where the axial distribution of inlet velocity plays significant role, the application of a 3D stability model is required.
References
1. Abdelhamid A.N., 1980, Analysis of rotating stall in vaneless diffusers of centrifugal compressors, Canadian Aeronautics and Space Journal, 26, 118-128
2. Abdelhamid A.N., 1983, Effects of vaneless diffuser geometry on flow instability in centrifugal compression systems, Canadian Aeronautics and Space Journal, 29, 259-266
3. Abidogun K.B., 2002, Effects of vaneless diffuser geometries on rotating stall, Journal of Propul- sion and Power, 22, 3, 542-549
4. Bianchini A., Biliotti D., Belardini E., Giachi M., Tapinassi L., Vannini G., Ferrari L., Ferrara G., 2013, A systematic approach to estimate the impact of the aerodynamic force induced by rotating stall in a vaneless diffuser on the rotordynamic behavior of centrifugal compressor, Journal of Engineering for Gas Turbines and Power, 135, 11
5. Canuto B.C., Hussaini M.Y., Quarteroni A., Zang T.A., 2010, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer
6. Chen H., Shen F., Zhu X.C., Du Z.H., 2011, A 3D compressible flow model for weak rotating waves in vaneless diffusers. Part I: The model and Mach number effects, Journal of Turbomachinery, 134, 4, 041010
7. Day I.J., 2016, Stall, surge, and 75 years of research, Journal of Turbomachinery, 138
Rf
m
Vrc
Abidogun
Shen
present model
1.5
3
0.16
0.1525
0.152
1.7
3
0.216
0.2150
0.215
2.0
3
0.255
0.2925
0.294
A numerical approach to predict the rotating stall… 647
8. Everitt J.N., 2010, Investigation of stall inception in centrifugal compressors using isolated dif-
fuser simulations, M.S. Thesis, MIT, Cambridge, MA
9. Everitt J., Spakovszky Z., 2013, An investigation of stall inception in centrifugal compressor
vaned diffuser, Journal of Turbomachinery, 135
10. Frigne P., Braembussche R.V.D., 1985, A theoretical model for rotating stall in the vaneless diffuser of a centrifugal compressor, Journal of Engineer for Gas Turbines and Power, 107, 2. 507-513
11. Jansen W., 1964, Rotating stall in a radial vaneless diffuser, ASME Journal of Basic Engineering, 86, 4, 750-758
12. Kinoshita Y., Senoo Y., 1985, Rotating stall induced in vaneless diffusers of very low specific speed centrifugal blowers, Journal of Engineering for Gas Turbines and Power, 107, 514-521
13. Meerbergen K., Roose D., 1997, The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues, SIAM Journal on Matrix Analysis and Applications, 18, 1, 1-20
14. Moler C.B., Stewart G.W., 1973, An algorithm for generalized matrix eigenvalue problems, SIAM Journal on Numerical Analysis, 10, 2, 241-256
15. Moore F.K., 1989, Weak Rotating flow disturbances in a centrifugal compressor with a vaneless diffuser, Journal of Turbomachinery, 111, 442-449
16. Peters G., Wilkinson J.H., 1979, Inverse iteration, Ill-conditioned equations and Newton’s method, SIAM Review, 21, 3, 339-360
17. Senoo Y., Kinoshita Y., 1977, Influence of inlet flow condition and geometries of a centrifugal vaneless diffuser on critical flow angle for reverse flow, Journal of Fluids Engineering, 99, 1, 98-103
18. Senoo Y., Kinoshita Y., 1978, Limits of rotating stall and stall in vaneless diffuser of centrifugal compressors, ASME Paper 78-GT-19
19. Sleijpen G.L.G., Van der Vorst H.A., 1994, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Review, 17, 2, 267-293
20. Spakovszky Z., Roduner C., 2009, Spike and modal stall inception in an advanced turbocharger centrifugal compressor, Journal of Turbomachinery, 131
21. Sun X.F., Liu X.H., Sun D.K., Sun X., 2013, A theoretical model for flow instability inception in transonic fan/compressors, ASME Turbo Expo 2013: Turbine Technical Conference and Exposition
22. Sun X.F., Ma Y.F., Liu X.H., Sun D.K., 2016, Flow stability model of centrifugal compressor based on eigenvalue approach, AIAA Journal
23. Ubben S., Niehuis R., 2015, Experimental investigation of the diffuser vane clearance effect in a centrifugal compressor stage with adjustable diffuser geometry. Part I: Compressor performance analysis, Journal of Turbomachinery, 137
Manuscript received July 5, 2016; accepted for print December 17, 2016