Numerical study of corner separation in a linear
compressor cascade using various turbulence models
Liu Yangwei, Yan Hao, Liu Yingjie
*, Lu Lipeng, Li Qiushi
Collaborative Innovation Center of Advanced Aero-Engine, Beihang University, Beijing 100083, China
National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100083, China
Received 14 May 2015; revised 30 June 2015; accepted 6 January 2016 Available online 10 May 2016
KEYWORDS Compressor cascade; Corner separation; Turbomachinery CFD; Turbulence anisotropy; Turbulence models
Abstract Three-dimensional corner separation is a common phenomenon that significantly affects compressor performance. Turbulence model is still a weakness for RANS method on predicting corner separation flow accurately. In the present study, numerical study of corner separation in a linear highly loaded prescribed velocity distribution (PVD) compressor cascade has been investigated using seven frequently used turbulence models. The seven turbulence models include
Spalart–Allmaras model, standardk–emodel, realizablek–e model, standardk–xmodel, shear
stress transportk–xmodel,v2–fmodel and Reynolds stress model. The results of these turbulence
models have been compared and analyzed in detail with available experimental data. It is found the
standard k–e model, realizable k–e model, v2–f model and Reynolds stress model can provide
reasonable results for predicting three dimensional corner separation in the compressor cascade.
The Spalart–Allmaras model, standardk–xmodel and shear stress transportk–xmodel
overesti-mate corner separation region at incidence of 0°. The turbulence characteristics are discussed and
turbulence anisotropy is observed to be stronger in the corner separating region.
Ó2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction
Three-dimensional (3D) corner separation in compressor cas-cade has been considered as an inherent flow phenomenon in compressor cascade.1 It has great impact on compressor performance, such as higher total pressure loss, static pressure rising limitation, compressor efficiency reduction, passage blockage, and stall and surge especially for highly loaded compressor.2 Many efforts have been made on studying the corner separation by experimental investigations3–5 and numerical simulations6,7 under various conditions over the past few years. Several flow controlling techniques have been * Corresponding author at: Collaborative Innovation Center of
Advanced Aero-Engine, Beihang University, Beijing 100083, China. Tel.: +86 10 82316455.
E-mail addresses: [email protected] (Y. Liu), yanhaovsdami@ 163.com(H. Yan),[email protected](Y. Liu),[email protected]. cn(L. Lu),[email protected](Q. Li).
Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier Chinese Journal of Aeronautics, (2016),29(3): 639–652
Chinese Society of Aeronautics and Astronautics
& Beihang University
Chinese Journal of Aeronautics
[email protected]www.sciencedirect.com
http://dx.doi.org/10.1016/j.cja.2016.04.013
1000-9361Ó2016 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
utilized to improve compressor performance recently.8–11The basic structures and characteristics of 3D corner separation are well-summarized12, thus their primary effects on compres-sor performance have been considered in designing process and have improved compressor performance greatly. In com-pressor routine design, the corner separation should be accu-rately predicted by employing computational fluid dynamics (CFD).
The CFD technique has been widely used and has played an increasingly important role in the aerodynamic design routine of compressor. The Reynolds-averaged Navier–Stokes equa-tions (RANS) method is still the most widely used approach in industrial CFD due to the computational cost and design schedule, though direct numerical simulation (DNS), large eddy simulation (LES) and hybrid LES/RANS (such as detached eddy simulation (DES)13, scale adaptive simulation (SAS)14) have been used to investigate flow mechanisms in rel-ative low Reynolds number with simple boundary.15–18 According to Spalart19,20, while it has been assumed that lim-itless increases in computing power will someday remove the need for turbulence modeling, the estimates for this milestone have been close to the year 2080, which is a long time away.
Turbulence model is one of the key elements and is cur-rently a weakness for RANS approach. Far less precision has been achieved in turbulence modeling since a mathematical model has been created to approximate the complicated phys-ical behavior of turbulent flows. It is generally acknowledged not any turbulence model is suitable for all kinds of complex flow problems.21 Hence, the performance of various widely used turbulence models should be assessed for a certain kind flow and then be improved.22–27 Then we could build up the scope of application for various turbulence models.
The next high-power-density generation aeroengines must be supported by more advanced compressors in the future and this requires that the RANS method applied in designing process can predict main flow structures in compressor cascade to decrease designing risk as much as possible. It is very practi-cal and critipracti-cal to simulate the corner separation accurately by using RANS approach. However, it is a big challenge for RANS method to precisely predict such complicated turbulent flow field, because the 3D corner separation contains complex vortices and large separation in cascade corner region.28
In this paper, seven turbulence models which are frequently applied in engineering have been used in the detailed numerical investigations using software FLUENT for a linear highly loaded compressor cascade experimentally studied by Gba-debo et al.29In order to minimize grid impacts on numerical results, great efforts for the grid generation have been made till grid independence was obtained. The numerical results of seven turbulence models have been compared with experimen-tal data carefully. The turbulence characteristics are discussed for analyzing the mechanisms for discrepancies. This study provides a worthy reference for using turbulence models prop-erly and helps designers understand numerical results further. It provides valuable reference on modifying turbulence models for 3D corner separation.
2. Numerical method
In this study, the same numerical platform and grid have been used to minimize other factors’ effects on numerical results
when assessing different turbulence models. The commercial flow solver package FLUENT30is applied to conduct numer-ical simulation in the cascade.
2.1. Governing equations and turbulence models
The governing equations of steady RANS method for incom-pressible fluid field are as follows:
@ui @xi ¼0 ð1Þ uj @ui @xj¼ 1 q @p @xiþ t@ 2 ui @x2 j þ1 q @ðqu0iu0jÞ @xj ð 2Þ whereqis density of the fluid,pis pressure of the fluid,uiis a
mean component of velocity in the directionxi.tis kinematic
viscosity. The additional fluctuation quantities u0iu0j are unknown Reynolds-stress tensors, while u0i represents the velocity fluctuation in i-direction, the bar above represents the time-averaged result of the invariant.
In the momentum equation, the Reynolds stress tensorsu0iu0j are unknown which makes this equation unclosed. The Rey-nolds stress tensors are modeled by turbulence model. Turbu-lence models utilized in the study except for the Reynolds stress model are based on the Boussinesq hypothesis, which is expressed as follows: qu0iu0j¼lt 2Sij 2 3dij @uk @xk 2 3dijqk ð3Þ
whereltis turbulent viscosity,dijis Kronecker delta function
(dij= 1 ifi=janddij= 0 ifi–j),kis the turbulent kinetic
energy. The repeated index implies summation from 1 to 3. Sijis the mean strain-rate tensor which is calculated as
Sij¼ 1 2 @ui @xj þ@uj @xi ð4Þ Turbulent viscosity lt are modeled by various turbulence
models. The transport equations are different from one turbu-lence model to another, so the turbuturbu-lence viscosity is quite dif-ferent which leads to discrepancies on the results of complicated turbulent flow field.
(1) Spalart–Allmaras model (SA)
As a simple one-equation model, the SA model solves a transport equation for the kinematic eddy viscosity. It has been proposed and developed by Spalart and All-maras31,32 since 1992. The transportation equation of this model in FLUENT is as follows:
@ @tðq~tÞ þ @ @xj ðq~tujÞ ¼cb1qS~~t þ 1 r~t @ @xj ðlþq~tÞ@~t @xj þcb2q @~t @xj 2 ( ) cw1qfw ~ t d 2 ð5Þ The turbulent viscosity is computed as
lt¼q~tft1 ð6Þ
~ S¼Sþ ~t j2d2ft2; S¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2XijXij p ; Xij¼ 1 2 @ui @xj @uj @xi ft1¼ v3 v3þc3 t1 ; ft2¼1 v 1þvft1 ; v¼~t t fw¼g 1þC6 w3 g6þC6 w3 " #1=6 ; g¼rþcw2ðr6rÞ; r¼ ~ t ~ Sj2d2 8 > > > > > > > > > < > > > > > > > > > : ð7Þ wheretdenotes time,~tis modified eddy viscosity, which is the working variable calculated in Eq. (5), l is the molecular viscosity,tis kinematic eddy viscosity andd denotes the distance from the field point to the nearest wall. cb1,cb2,r~t,ct1,cw1,cw2,cw3, andjare constant
coefficients. Their values are: cb1¼0:1355; cb2¼0:622 r~t¼2=3; ct1¼7:1 cw1¼cb1=j2þ ð1þcb2Þ=r~t cw2¼0:3; cw3¼2:0;j¼0:41 8 > > > < > > > : ð8Þ
(2) Standardk–emodel (SKE)
The SKE model was developed by assuming the flow is fully turbulent without consideration of molecular vis-cosity effects33,34. This model requires an additional model, which includes impacts of molecular viscosity for near wall regions. The transportation equations of this model in FLUENT are as follows:k-equation: @ @tðqkÞ þ @ @xj ðqkujÞ ¼ltS 2þ @ @xj lþlt rk @k @xj qe ð9Þ e-equation: @ @tðqeÞ þ @ @xj ðqeujÞ ¼ce1 e kltS 2 þ@@ xj lþlt re @e @xj ce2q e2 k ð10Þ
The turbulent viscosityltis computed as
lt¼qclfl k2
e ð11Þ
where e is dissipation rate. S¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij
p
. Sij refers to
Eq.(4).flis wall damping Function, which is defined as:fl= exp[3.4/(1 + 0.02Ret)
2
], whereRet=qk 2
/(le). The values of related parameters are
cl¼0:09; ce1¼1:44 ce2¼1:92; rk¼1:0 re¼1:3 8 > < > : ð12Þ
(3) Realizablek–emodel (RKE)
The RKE model has been developed from SKE and satis-fies mathematical constraints on Reynolds stress.35The RKE model can provide accurate predictions on flows that involve rotation, boundary layers under strong adverse pressure gradients, separation and recirculation. The transportation equations of this model in FLUENT are as follows:k-equation:
@ @tðqkÞ þ @ @xj ðqkujÞ ¼ltS 2þ @ @xj lþlt rk @k @xj qe ð13Þ e-equation: @ @tðqeÞ þ @ @xj ðqeujÞ ¼qc1Seþ @ @xj lþlt re @ e @xj c2q e2 kþpffiffiffiffite ð14Þ where c1¼max 0:43;gþg5 h i , g¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi2SijSij p k e, Sij refers to
Eq.(4).c2= 1.9The turbulent viscosity is computed as lt¼qclk
2
e ð15Þ
The constants (rk,re) have the same values as for stan-dardk–emodel. The parameterclcan be referred from User’s Guide in FLUENT.30
(4) Standardk–xmodel (SKW)
The SKW model contains two transport equations which correspond to turbulence kinetic energy and specific dissipation rate.36The transportation equations of this model in FLUENT are as follows:k-equation: @ @tðqkÞ þ @ @xj ðqujkÞ ¼ltS 2þ @ @xj lþ0:5lt ð Þ@k @xj qbxk ð16Þ x-equation: @ @tðqxÞ þ @ @xj ðqujxÞ ¼ ðax=kÞltS 2 þ@@ xj lþ0:5lt ð Þ@@x xj qbx2 ð17Þ
The turbulent viscosity is computed as
lt¼a
qk
x ð18Þ
wherexis specific dissipation rate. The related constant parameters are:
a¼1:0; a¼0:52
b¼9=100; b¼0:0072
ð19Þ (5) Shear stress transportk–xmodel (SST)
Menter37developed SST model, which calculates turbu-lent flow field with SKW model in the near wall region and switches to SKE model in the far field. Blending functions are used in this model. The transportation equations of this model in FLUENT are as follows:k -equation: @ @tðqkÞ þ @ @xjð qkujÞ ¼G~kþ @ @xj lþlt rk @ k @xj qbkx ð20Þ x-equation: @ @tðqxÞ þ @ @xjð qujxÞ ¼ ðax=kÞltS 2 þ@@ xj lþlt rx @ x @xj qbx2þ2ð1F 1Þqrx;21 x @k @xj @x @xj ð21Þ The turbulent viscosity is computed as
lt¼qk x 1 max 1 a; SF2 a1x ð22Þ
whereG~k¼minðltS 2;
10qbkxÞ. The constant parame-ter a1= 0.31, the other constants (a*, a, b*, b) have
the same values as for standardk–xmodel. The com-plex blending functions F1, F2 and related parameters rk, rx, rx,2 can be referred from User’s Guide in
FLUENT.30 (6) v2–fmodel (V2F)
Durbin38proposedv2–fmodel which requires solutions of three transport equations and one elliptic equation. This four-equation model based on transport equations for the turbulence kinetic energy (k), dissipation rate (e), a velocity variance scale (v2), and an elliptic relaxation
function (f). Thev2–f model is a low-Reynolds-number turbulence model which uses velocity scale (v2) instead of turbulent kinetic energy to evaluate turbulent viscos-ity.39,40The equations for turbulent kinetic energy and the dissipation rate are the same as those of the standard k–emodel, while the equations forv2andfcan be written as follows:Thev2-equation: @ @xj ðqujv2Þ ¼kfqqv2 k eþ @ @xj lþlt rk @ v2 @xj ð23Þ Thef-equation: L2@ 2 f @x2 i f¼1 T ð4:6Þ v2 k 0:8 3 0:3ltS 2 k ð24Þ where: T¼max k e; 6 ffiffiffi v e r ð25Þ L¼0:23 max k 3=2 e ; 70 v3 e 1=4 " # ð26Þ The turbulent viscosity is computed as:
lt¼qclv2T ð27Þ
Hererk= 1.0cl= 0.22 (7) Reynolds stress model (RSM)
The RSM model41,42 contains seven transportation equations which close RANS momentum equations by solving Reynolds stresses directly and combine with an equation for dissipation rate. The RSM is seldom selected in simulating complicated engineering applica-tions because the robustness and computational cost of the RSM are noticeably inferior to eddy-viscosity mod-els.43 The transport equations without the effects of buoyancy are shown as
@
@tðqu
0 iu0jÞ
|fflfflfflfflffl{zfflfflfflfflffl} Local time derivative
þ @ @xk ðquku0iu0jÞ |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} Cij¼Convection ¼ @@x kð qu0iu0ju0kþpðdkju0iþdkiu0jÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} DT;ij¼Turbulent diffusion þ@@x k l @ @xkð u0iu0jÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} DL;ij¼Molecular diffusion q u0iu0k @uj @xk þu0ju0k @ui @xk |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Pij¼Stress production þp @u 0 i @xjþ @u0j @xi |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} /ij¼Prssure strain 2l@u 0 i @xk @u0i @xk |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} DSij¼Dissipation 2qXijðu0jum0eikmþu0iu0mejkmÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Fij¼Production by system rotation ð28Þ
whereejkmis Levi–Civita symbol.The turbulent diffusion
DT,ijis defined by using a simplified version of the
gen-eralized gradient-diffusion model, which is DT;ij¼ @ @xk lt rk @ u0iu0j @xk ! ð29Þ Here,rkis a constant equal to 0.82.
The pressure-strain term/ijis defined as
/ij¼/ij;1þ/ij;2 ð30Þ
where the slow pressure-strain term/ij,1is defined as
/ij;1¼ 1:8qe k u 0 iu0j 2 3dijk ð31Þ and the rapid pressure-strain term/ij,2is defined as
/ij;2¼ 0:6 ðPijþFijCijÞ
2
3dijðPCÞ
ð32Þ where P= 0.5Pkk, C= 0.5Ckk, the repeated index implies
summation from 1 to 3.
The dissipation tensorDSijis defined as
DSij¼
2
3dijqe ð33Þ
The equation for dissipation rateeis the same as that of the standardk–emodel.
2.2. Linear compressor cascade
One prescribed velocity distribution (PVD) high-load linear compressor cascade, which was experimentally investigated at Cambridge University29,44–46, is used as a computational case in the current study. In our previous studies9,10, flow trol and flow mechanisms for corner separation were con-ducted in the same cascade. The parameters of this cascade are summarized in Table 1, where cis the chord length,s is the pitchwise distance,his the height of blade andtis the max-imum blade thickness.
In this study, one-blade passage is used to conduct the numerical simulation. Two sides of the flow passage are set as periodic boundary conditions. Because of no tip clearance in the cascade, the computational domain is cut to half span and the passage top surface is set as symmetric condition in order to reduce computation quantity.
The grid topology is O4H and the point distribution is shown inFig. 1(a). Hexahedral structural meshes are generated by software NUMECA, AutoGrid5TM, as shown inFig. 1(b). The y+of first grid adjacent to the wall is approximate 1.0.
Series cases with different grid numbers and grid distributions have been tested to examine the grid independence of the
Table 1 Geometric parameters of cascade.
Profile Value c(mm) 151.5 s/c 0.926 h/c 1.32 t/c 0.1 Camber angle (°) 42.0 Stagger angle (°) 15.0
solution. Finally, the case with 3 million grids is selected for the calculation and applied for different turbulence models.
Enhanced wall treatment30is applied to simulating the tur-bulent flow for the SKE, RKE and RSM in the near-wall region. In current studies, the enhanced wall treatment can possess the accuracy of the standard two-layer model in the near-wall region because of the fine near-wall meshes used in the simulations.
2.3. Numerical scheme and boundary conditions
The second-order upwind scheme is used for the convection terms and the viscous terms of each governing equation to minimize the numerical diffusion. Other numerical schemes can be described in detail in the FLUENT User’s Guide.30 According to the experiment, the velocity of the main flow (U1) is 23.0 m/s,Uis the local velocity at inlet. The profile of
boundary layer at inlet is shown in Fig. 2. The turbulent
intensity is 1.5% at inlet. Moreover, all the solid walls are set as non-slip conditions.
2.4. Definitions of flow field parameters
Several important performance parameters in compressor cas-cade are defined as follows. The static pressure coefficient is defined as Cp¼ ðpp1Þ 1 2qU 2 1 ð34Þ Another important parameter is total pressure loss which is defined as Yp¼ ðp01p0Þ 1 2qU 2 1 ð35Þ where the reference total pressure (p01) was measured in the
inlet free stream with a side wall static pressure (p1) at similar
location where the inlet velocity profile was measured. p0 is
total pressure at the local point.
The relative displacement thickness29 can represent the thickness of 3D separating layer, which is defined as
R¼ ðdðrÞ dmidspanÞ=c ð36Þ
wheredmidspanis the displacement thickness at midspan and the definition of displacement thickness is
dðrÞ ¼ Z d 0 1qUðr;sÞ q1U1 ds ð37Þ
wheredis boundary layer thickness,ris radial distance,q1is the density at inlet.U(r,s) is the local velocity.
3. Results and discussion
3.1. Validations of different turbulence models
The streamlines on the endwall shown inFig. 3can represent the corner separation scale in PVD cascade using different turbulence models. Compared with oil flow visualization in experiment, the corner separating scales from RSM, SKE, RKE and V2F models show good agreement with experiment and no reverse flow is observed on the endwall. The separating scales are obviously large in the results of SA, SKW and SST models and reverse flow appears on the endwall, which indi-cates that corner stall happens in the cascade. The low-speed area besides the rear part of blade suction surface can also reflect the corner separation scale. From the contour of veloc-ity magnitude shown inFig. 4, the low-speed areas in RSM, SKE, RKE and V2F models are obviously smaller than the results of SA, SKW and SST models at 0°incidence.
Fig. 5shows the static pressure coefficient (Cp) at mid-span
region (54% span) and hub region (89% span) using seven RANS turbulence models at incidences of7°and 0° respec-tively.Fig. 5(a) and (b) indicate that all turbulence models, except the SKW model, provide reasonable results at the inci-dence of7°at the mid-span region and hub region respec-tively.Cpin the SKW model exhibits a slight discrepancy at
the hub region (89% span). Given that the cascade works at the negative incidence, 3D corner separation is determined to be small. In this condition, these commonly-used turbulence Fig. 1 Grids of PVD cascade.
Fig. 3 Streamlines on endwall at 0°incidence.
models can provide reasonable results. Fig. 5(c) reveals that the RANS methods provide reasonable predictions at the blade mid-span region at the incidence of 0° because corner
separation is small. However, the discrepancy becomes signif-icant at the hub region. The lines of SA, SKW and SST model become flat behind 40% chord as shown inFig. 5(d), which
Fig. 5 Static pressure coefficient on blade surface.
indicates that these models predict earlier boundary separation and larger corner separation in the cascade than other models. Fig. 6 illustrates total pressure loss contour coefficient at 7° incidence. The total pressure loss in corner region is high-loss area in compressor cascade, which is called ‘‘loss core area”in this paper. The boundary layer thickness and the wake
from the results of the seven turbulence models are not as thick as those in the experiment. The loss core areas in the results of SA, V2F, SKE, RKE, SST, and RSM models are similar to those in the experiment, whereas the loss core area in SKW model is obviously larger than the other models. This finding indicates larger corner separation in the SKW model which
Fig. 7 Contour of total pressure loss at 0°incidence.
Fig. 8 Pitchwise mass-averaged total pressure loss.
matches to the results of the pressure coefficient inFig. 5(b). When the incidence reaches at 0°, the results of SKE, RKE, V2F and RSM agree better with measurement, while the results of SA, SKW and SST model predict unreasonably lar-ger loss core area in the cascade from Fig. 7. Relatively, the shape of corner region predicted by V2F is a little different from experiment. Fig. 8(b) shows that the pitchwise mass-averaged total pressure loss exhibits the same trend. The total pressure loss from SKW, SST and SA model is obviously higher than that from experiment near the endwall.
The pitchwise mass-averaged total pressure losses at 7° and 0°incidences are shown inFig. 8. When the cascade works at7°incidence, the corner separation region is not obvious and these turbulence models except for the SKW model can provide reasonable results. The total pressure loss from SKW model is a little higher than the experiment. When the incidence reaches at 0°, the corner separation appears and the discrepancies in the seven RANS results are obvious. Com-pared with the total pressure loss contour shown inFig. 7, the results of SA, SKW and SST model are much higher than that of the experiment. The trend is the same in the outflow angle shown in Fig. 9. The results of outflow angle show that SKE, RKE, V2F and RSM models are close to that of the experiment, whereas the SA, SKW and SST models display obviously larger outflow angle than that of the experiment at 0°incidence.
The relative displacement thickness predicted by SKE, RKE and RSM models show good consistency with the exper-imental results and the RSM model is the best. V2F model shows similar trend and reasonably good qualitative agree-ment with the measureagree-ment. Given that the SKW, SST and SA models predict a larger separating area at the corner region, the lines of relative displacement thickness rapidly increase and the trend of lines is different from the experiment shown inFig. 10.
Fig. 10 Relative displacement thickness at 0°incidence.
3.2. Turbulence characteristic analysis
The RANS approach requires that the Reynolds stresses are appropriately modeled in turbulence models. Except for RSM model, the other six turbulence models used in this study employ the Boussinesq hypothesis to relate the Reynolds stres-ses to the mean velocity gradients (referring to Eq.(3)), also called linear eddy viscosity turbulence models. If a turbulence model can predict Reynolds stress accurately, it will provide reasonable mean results on turbulence flow field. Moreover, turbulence characteristics, like turbulence viscosity and turbu-lent anisotropy, are considered differently in these turbulence models. Analyzing the results of turbulent characteristics is helpful to get more information about the performance of these turbulence models and can contribute to the modification of the turbulence models.
Fig. 11indicates that the Reynolds stress must be high at the corner region because the velocity fluctuation is obvious in this region. The streamwise normal stress w0w0 shows the
high-valued region at the corner in the seven turbulence mod-els. This indicates that the streamwise velocity fluctuation is
highly distinct at the corner separating region. However, the high-valued areas ofw0w0in these turbulence models are quite different. Thew0w0high-valued areas in RSM, SKE, RKE and
V2F models are small and locate near the blade suction sur-face. By contrast, the high-valued area in SKW, SST and SA models are large and gradually move away from the blade suc-tion surface. This phenomenon shows that these three models over predict the corner separation region and estimate stronger streamwise velocity fluctuation. The distributions of spanwise and pitchwise normal stress u0u0 and v0v0 (not shown here) are similar to those of w0w0. It can be found that w0w01:5u0u0¼1:5v0v0. Compared with the shear stress shown
inFig. 12, normal stress gives mainly contribution to the tur-bulence fluctuation because the scale of normal stress is signif-icantly higher than that of shear stress.
Fig. 12 demonstrates that the Reynolds shear stress u0v0
reveals different distributions in the seven turbulence models. The high-valued areas of shear stress in V2F, SKE, RKE and RSM models are small and close to the endwall. This observation provides a vivid contrast to the results of SA, SKW and SST models. Two high-valued shear stress regions
are observed in SKW model. This indicates that the shear stress is very active at the corner separating region. One shear stress core near the endwall is negative, whereas the other is positive. The high-valued shear stress regions in SST model and SA model are very distinct and the location of shear stress core is located away from suction surface of the blade. The distribution of the other two Reynolds shear stressv0w0 and u0w0(not shown here) are similar to that ofu0v0and it is found v0w04u0v02u0w0.
The preceding analysis verifies that SA, SKW and SST models overestimate the corner separation in the PVD cascade. The turbulent viscosity of these models is significantly larger than the results of V2F, SKE, RKE and RSM models shown inFig. 13. This finding is consistent with the analysis of Rey-nolds stress. When the ReyRey-nolds stress at corner region is higher, the velocity fluctuation is stronger, which in turn leads to high turbulence viscosity.
Another important turbulent characteristic is turbulent ani-sotropy, which is hardly to be considered correctly in the linear eddy viscosity turbulence models. It may be one reason why the linear eddy viscosity turbulence models could not predict corner separation correctly. As the turbulence models predict different Reynolds stress distributions in corner separation
region, the predicted turbulent anisotropy should be different. The analysis of turbulent anisotropy in the corner region can provide information on assessing the performance of these tur-bulence models.
The Lumley triangle47is proven to be a useful tool to study turbulent anisotropy. Lumley found the turbulent anisotropy can be derived from Reynolds stress tensor (u0iu0j) by
subtract-ing the isotropic part (s0ij) which is
s0ij¼u0iu0j
1 3u
0
ku0kdij ð38Þ
The non-dimensional form of the anisotropy tensorbij is
given by bij¼ u0iu0j u0ku0k 1 3dij ð39Þ
The tensorbijhas three scalar invariants. For
incompress-ible flows the first invariant equals 0, the second invariant (g) and third invariant (n) are expressed as
6g2¼b
ijbji ð40Þ
6n3¼bijbjkbki ð41Þ
The state of turbulence in a flow can be displayed with respect to its anisotropy by cross-plottinggandn. Thegand
ninvariants should settle in a triangle map for real physical turbulence flow. The left line of triangle describes axisymmet-ric turbulence in which one direction of velocity fluctuations is weaker than the other two. This is referred to as ‘‘disklike” type of turbulence. In contrast, the right line of triangle is characterized by one fluctuating component which dominates over the other two. It is called ‘‘rodlike”turbulence. The curve edge of the triangle represents 2D turbulence. Turbulence tends to be isotropic when the point is close to the original point.
In this study, the invariantsgandnare calculated from the line extracted in the corner region as shown in Fig. 14(a). The distribution of the points is shown below. The points in the RSM model gather near the right lines of the triangle which indicate that the turbulence type is ‘‘rodlike”. The point distribution in SKE and SKW models varies from each other. The points in SKE model are located along the left line of the triangle, whereas those in the SKW develop along the right line. The turbulence type is ‘‘disklike”in the SKE model and it is ‘‘rodlike”in SKW model. The result of RKE model is
sim-ilar to the results of SKE model. Points in V2F model move away from the origin point along the left line and then return close to the origin point. This indicates the turbulence tends to be anisotropic firstly and turns to be isotropic at the end. The results of RSM, SKE, RKE, SKW and V2F models are along the right or the left line of the triangle and turbulence types are axisymmetric. By contrast, the points in the SST model scatter within the triangle, thus the turbulence is considered anisotro-pic in the results of the SST model.
4. Conclusions
In this study, numerical study of corner separation in a linear highly loaded PVD compressor cascade has been investigated using seven frequently used turbulence models in FLUENT. The computational results of seven turbulence models, includ-ing Spalart–Allmaras model, standardk–emodel, realizablek–
e model, standard k–x model, shear stress transport k–x
model,v2–fmodel and the Reynolds stress model, are carefully discussed and analyzed with published experimental data. The conclusions are drawn as follows:
(1) In the steady simulations, all the RANS approaches pre-dict various scales and topologies for 3D corner separa-tion region. At7°incidence, the corner separation flow is weak in the cascade and these seven turbulence models except standard k–x model can provide satisfactory results on mean flow. At 0°incidence, 3D corner separa-tion gets strong, the discrepancies in these RANS mod-els become obvious. Among these modmod-els, the standard k–e model, realizable k–e model, v2–f model and the Reynolds stress model can still predict reasonable results on the mean flow, and the Reynolds stress model shows best performance; whereas the Spalart–Allmaras model, standard k–x model and shear stress transport k–x
model overestimate the corner separation.
(2) The distribution of Reynolds stress is quite different among the results of various turbulence models. Normal Reynolds stress significantly contributes to velocity fluc-tuation. Three components of the shear stresses are not in the same magnitude. The Spalart–Allmaras model, standard k–x model and shear stress transport k–x
model overestimate the Reynolds stress at the corner region. By contrast, the other turbulence models can provide relatively reasonable results.
(3) Turbulence anisotropy is explored in the study. In the corner region, the turbulence becomes more anisotropic with the development of corner separation. The stan-dard k–e model, realizable k–e model, standard k–x
model,v2–fmodel and Reynolds stress model are along the left line or the right line of the Lumley triangle and this indicates that turbulence type is axisymmetric ‘‘dis-like” or ‘‘rodlike”, whereas the points in shear stress transport k–xmodel are scattered in the triangle. The turbulence anisotropy for modifying turbulence models needs further study in corner separation.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 51376001, No. 51420105008, No. 51306013, No. 51136003), the National Basic Research Pro-gram of China (2012CB720205, 2014CB046405), the Beijing Higher Education Young Elite Teacher Project, and the Fun-damental Research Funds for the Central Universities. The work is also supported by the Innovation Foundation of BUAA for Ph.D. Graduates. The authors would like to thank Whittle Laboratory and Rolls-Royce Plc for providing their experimental results.
References
1. Lei VM, Spakovszky ZS, Greitzer EM. A criterion for axial compressor hub-corner stall.J Turbomach2008;130(3):031006–10. 2. Denton JD. Loss mechanisms in turbomachines. J Turbomach
1993;115:621–56.
3. Schulz HD, Gallus HE. Experimental investigation of the three-dimensional flow in an annular compressor cascade.J Turbomach 1988;110(4):467–78.
4. Guo S, Lu HW, Chen F, Wu CJ. Vortex control and aerodynamic performance improvement of a highly loaded compressor cascade via inlet boundary layer suction.Exp Fluids2013;54:1570. 5. Zhang HD, Wu Y, Li YH, Lu HW. Experimental investigation on
a high subsonic compressor cascade flow.Chin J Aeronaut2015;28
(4):1034–43.
6. Scillitoe AD, Tucker PG, Adami P. Evaluation of RANS and ZDES methods for the prediction of three-dimensional separation in axial flow compressors. Montre´al: ASME; 2015. Report No.: GT2015-43975.
7. Liu YW, Yan H, Lu LP. Numerical study of the effect of secondary vortex on three-dimensional corner separation in a compressor cascade.Int J Turbo Jet Eng2015. Available from: http://dx.doi.org/10.1515/tjj-2014-0039.
8. Guo S, Chen SW, Lu HW, Song YP, Chen F. Enhancing aerodynamic performances of a high-turning compressor cascade via boundary layer suction. Sci China Tech Sci 2010;53
(10):2748–55.
9. Gmelin C, Thiele F, Liesner K, Meyer R. Investigations of secondary flow suction in a high speed compressor cascade. Vancouver: ASME; 2011. Report No.: GT2011-46479.
10. Choi M. Effects of circumferential casing grooves on the perfor-mance of a transonic axial compressor. Int J Turbo Jet Eng 2015;32(4):361–71.
11. Liu YW, Sun JJ, Lu LP. Corner separation control by boundary layer suction applied to a highly loaded axial compressor cascade. Energies2014;7(12):7994–8007.
12. Lakshminarayan B.Fluid dynamics and heat transfer of turboma-chinery. New York: A Wiley-Interscience Publication; 1996. p. 322–7.
13. Spalart PR. Detached-eddy simulation. Annu Rev Fluid Mech 2009;41:181–202.
14. Menter F, Egorov Y. The scale-adaptive simulation method for unsteady turbulent flow predictions. Part 1: theory and model description.Flow Turbul Combust2010;85(1):113–38.
15. Castagna J, Yao YF, Yao J. Direct numerical simulation of a turbulent flow over an axisymmetric hill. Comput Fluids 2014;95:116–26.
16. Zhang HD, Yan C, Ye TH, Zhang JM, Chen YL. Large eddy simulation of unconfined turbulent swirling flow.Sci China Tech Sci2015;58(10):1731–44.
17. Liu J, Sun HS, Liu ZT, Xiao ZX. Numerical investigation of unsteady vortex breakdown past 80°/65°double-delta wing.Chin J Aeronaut2014;27(3):521–30.
18. Wang YF, Chen F, Liu HP, Chen HL. Large eddy simulation of unsteady transitional flow on the low-pressure turbuine blade.Sci China Tech Sci2014;57(9):1761–8.
19. Spalart PR. Philosophies and fallacies in turbulence modeling. Prog Aerosp Sci2015;74:1–15.
20. Spalart PR. Strategies for turbulence modelling and simulations. Int J Heat Fluid Flow2000;21:252–63.
21. Dunham J. CFD validation for propulsion system components. Neuilly-sur-Seine: AGARD; 1998. Report No.: AR-355. 22. Liu YW, Yu XJ, Liu BJ. Turbulence models assessment for
large-scale tip vortices in an axial compressor rotor.J Propul Power 2008;24(1):15–25.
23. Vijiapurapu S, Cui J. Performance of turbulence models for flows through rough pipes.Appl Math Model2010;34:1458–66. 24. Balabel A, Hegab AM, Nasr M, Samy El-Behery M. Assessment
of turbulence modeling for gas flow in two-dimensional conver-gent–divergent rocket nozzle.Appl Math Model2011;35:3408–22. 25. Liu YW, Lu LP, Fang L, Gao F. Modification of Spalart– Allmaras model with consideration of turbulence energy backscat-ter using velocity helicity.Phys Lett A2011;375(24):2377–81. 26. Heschl C, Inthavong K, Sanz W, Tu J. Evaluation and
improve-ments of RANS turbulence models for linear diffuser flows. Comput Fluids2013;71:272–82.
27. Ma L, Lu LP, Fang J, Wang QH. A study on turbulence transportation and modification of Spalart–Allmaras model for shock-wave/turbulent boundary layer interaction flow. Chin J Aeronaut2014;27(2):200–9.
28. Liu YW, Yan H, Lu LP.Investigation of corner separation in a linear compressor cascade using DDES. Montre´al: ASME; 2015. Report No.: GT2015-42902.
29. Gbadebo SA, Cumpsty NA, Hynes TP. Three-dimensional sepa-rations in axial compressors.J Turbomach2005;127(2):331–9. 30. Fluent 6. 3 User’s Guide. Fluent Documentation, Fluent Inc,
2006.
31. Spalart PR, Allmaras SR. A one-equation turbulence model for aerodynamic flows. Reno: AIAA; 1992. Report No.: AIAA-92-0439.
32. Spalart PR, Shur M. On the sensitization of turbulence models to rotation and curvature.Aerosp Sci Technol1997;5:297–302. 33. Launder BE, Spalding DP. The numerical computation of
turbulent flows.Comput Method Appl Mech1974;3:269–89. 34. Launder BE, Sharma BI. Application of the energy-dissipation
model of turbulence to the calculation of flow near a spinning disc. Lett Heat Mass Transfer1974;1:131–8.
35. Shih TH, Liou WW, Shabbir A. A newk–eeddy-viscosity model for high Reynolds number turbulent flows-model development and validation.Comput Fluids1995;24(3):227–38.
36. Wilcox DC. Turbulence modeling for CFD. San Diego: DCW Industries; 2006. p. 124–36.
37. Menter FR. Two-equation eddy-visocity turbulence models for engineering applications.AIAA J1994;32(8):1598–605.
38. Durbin PA. Near-wall turbulence closure modeling without damping functions.Theor Comput Fluid Dyn1991;3:1–13. 39. Durbin PA. Separated flow computations with thek–e–v2model.
AIAA J1995;33(4):659–64.
40. Behnia M, Parneix S, Shabany Y. Numerical study of turbulent. Int J Heat Fluid Flow1999;20:1–9.
41. Launder BE, Reece GJ, Rodi W. Progress in the development of a Reynolds-stress turbulence closure. J Fluid Mech 1975;3
(68):537–66.
42. Gibson MM, Launder BE. Ground effects on pressure fluctu-ations in the atmospheric boundary layer. J Fluid Mech 1978;86:491–511.
43. Smirnov PE, Menter FR. Sensitization of the SST turbulence model to rotation and curvature by applying the Spalart–Shur correction term.J Turbomach2009;131:0410104.
44. Gbadebo SA, Cumpsty NA, Hynes TP. Interaction of tip clearance flow and three-dimensional separations in axial com-pressors.J Turbomach2007;129(4):679–85.
45. Gbadebo SA, Cumpsty NA, Hynes TP. Control of three-dimen-sional separations in axial compressors by tailored boundary layer suction.J Turbomach2008;130(1):011004–8.
46. Gbadebo SA, Hynes TP, Cumpsty NA. Influence of surface roughness on three-dimensional separation in axial compressors.J Turbomach2004;126(4):455–63.
47. Lumely J. Computational modeling of turbulent flows.Adv Appl Math1978;18:123–76.
Liu Yangweiis currently an associate professor in the School of Energy and Power Engineering, Beihang University. His major research interests include CFD, complex flow mechanism and flow control in turbomachinery.
Yan Hao is a Ph.D. candidate in Beihang University, China. His research focuses on the complex flow in compressor cascade. Liu Yingjieis currently a senior engineer in AVIC Academy of Aero-nautic propulsion technology. Her major research interests include CFD, LES and numerical simulation of combustion.
Lu Lipengis currently a professor in the School of Energy and Power Engineering, Beihang University. His major research interests include turbulence, CFD and turbomachinery.
Li Qiushiis currently a professor in the School of Energy and Power Engineering, Beihang University. His major research interests include turbomachinery, vortex aerodynamic and flow instability.
