parameter study on numerical simulation of corner separation in lmfa naca65 linear compressor cascade

CJA 769 24 December 2016 Chinese Journal of Aeronautics, (2016), xxx(xx): xxx–xxx No of Pages 17 Chinese Society of Aeronautics and Astronautics & Beihang University Chinese Journal of Aeronautics cja@buaa.edu.cn www.sciencedirect.com Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade Gao Feng a, Ma Wei b, Sun Jinjing c, Jerome Boudet d, Xavier Ottavy d, Liu Yangwei c,*, Lu Lipeng c, Shao Liang d a 11 Department of Mechanical Engineering Sciences, University of Surrey, Guildford GU2 7XH, UK School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China c School of Energy and Power Engineering, Beihang University, Beijing 100191, China d Laboratoire de Me´canique des Fluides et d’Acoustique, E´cole Centrale de Lyon, E´cully 69134, France 12 Received 12 December 2015; revised September 2016; accepted September 2016 10 b 13 15 16 KEYWORDS 17 Corner separation; Influencing parameter; LES; Linear compressor cascade; RANS 18 19 20 21 22 Abstract Large-eddy simulation (LES) is compared with experiment and Reynolds-averaged Navier-Stokes (RANS), and LES is shown to be superior to RANS in reproducing corner separation in the LMFA-NACA65 linear compressor cascade, in terms of surface limiting streamlines, blade pressure coefficient, total pressure losses and blade suction side boundary layer profiles However, LES is too expensive to conduct an influencing parameter study of the corner separation RANS approach, despite over-predicting the corner separation, gives reasonable descriptions of the corner separated flow, and is thus selected to conduct a parametric study in this paper Two kinds of influencing parameters on corner separation, numerical and physical parameters, are analyzed and discussed: second order spatial scheme is necessary for a RANS simulation; incidence angle and inflow boundary layer thickness are found to show the most significant influences on the corner separation among the parameters studied; unsteady RANS with the imposed inflow unsteadiness does not show any non-linear effect on the corner separation Ó 2016 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/) * Corresponding author E-mail address: liuyangwei@126.com (Y Liu) Peer review under responsibility of Editorial Committee of CJA Production and hosting by Elsevier Introduction 23 For the economic and ecological purpose, researchers work at reducing the weight of turbomachines in aircrafts This leads to an increase of compression ratio per compressor stage, and thus of the blade loading However, the rise of the blade loading results in the strengthening of three-dimensional phenomena, e.g., corner separations, clearance flows, shock waves 24 http://dx.doi.org/10.1016/j.cja.2016.09.015 1000-9361 Ó 2016 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/) Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 25 26 27 28 29 CJA 769 24 December 2016 No of Pages 17 F Gao et al Nomenclature Roman symbols B characteristic point of blade pressure coefficient c blade chord length ca axial chord length CL blade lift coefficient CP static pressure coefficient C Pt total pressure loss coefficient CÃPt pitchwise-mass-averaged total pressure loss coefficient CPt;global global mass-averaged total pressure loss coefficient f frequency h blade span i incidence angle k turbulent kinetic energy P static pressure Pt absolute total pressure Rec chord-based Reynolds number s pitch s* arc length from leading edge Tu turbulence intensity U velocity magnitude us tangential velocity component to blade suction side un wall normal velocity component to blade suction side x, y, z coordinates 54 and other secondary flows, which highly restrict the efficiency and stability of compressor.1,2 The corner separation has great effect on compressor performance, such as passage blockage, limiting on static pressure rise, total pressure loss, and eventually stall and surge especially for highly loaded compressor Hence, recently the flow mechanism and flow control for corner separation have been investigated by many researchers using experiment3–5 and computational fluid dynamics.6–8 Associated with high pressure gradients and boundary layer separations, corner separation is quite difficult to reproduce with a conventional Reynolds-averaged Navier-Stokes (RANS) approach.9,10 Large-eddy simulation (LES) and hybrid RANS/LES have proven to be capable of simulating turbomachinery flows,9,11–13 and are found to be superior to RANS in simulating the corner separation Nevertheless, LES is still too expensive to investigate the influencing parameters of the corner separation In the present work, both LES and RANS are used to study the corner separation and compared with the experimental results RANS, despite over-estimating the extent and intensity of the corner separation, can give a reasonable prediction, and is an alternative to conduct the influencing parameter study In this study, two kinds of influencing parameters on corner separation, numerical and physical parameters, are analyzed and discussed based on RANS approach 55 Review of influencing parameters on corner separation 56 Corner separation has been investigated by many researchers, and so its influencing parameters Some known influencing 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 57 Greek symbols d1 displacement boundary layer thickness Dy+ wall distance of the 1st grid layer, in wall unit e4 fourth-order artificial viscosity coefficient x specific turbulent dissipation rate Subscript reference quantity Acronyms AUSM advection upstream splitting method BL boundary layer CFD computational fluid dynamics DRSM differential Reynolds stress model LES large-eddy simulation LMFA Laboratoire de Me´canique des Fluides et d’Acoustique MUSCL monotone upwind schemes for conservation law-s RANS Reynolds-averaged Navier-Stokes SA Spalart-Allmaras model SISM shear-improved Smagorinsky model SLAU simple low-dissipation AUSM TKE turbulent kinetic energy URANS unsteady RANS parameters are loading, inflow boundary layer, free-stream turbulence intensity, clearance flow, Reynolds number, Mach number, rotating effect, surface roughness and real blade geometry A literature review of these parameters is listed in Table 58 Experimental and numerical configuration 63 3.1 Experimental configuration 64 The experiments have been made in the LMFA-NACA65 linear compressor cascade wind tunnel.36,37 A schematic of the test section and the blade geometry is drawn in Fig Thirteen NACA65 blades are installed to ensure the periodicity in the middle passage The free stream velocity is set to 40 m/s, yielding a chord-based Reynolds number Rec = 3.82  10.5 In order to force the boundary layers as turbulent on the blade surface (as expected in real compressors), two pieces of sandpaper are pasted near the blade leading edge on both the pressure and suction sides In this paper, particular attention is paid to the incidence angle 4°, considered as a reference, where a three-dimensional corner separation has been clearly observed More information about the compressor cascade and experimental details could be found in Ref 36 65 3.2 Numerical setup 79 Two different solvers have been used to conduct the numerical studies: an in-house code Turb’Flow developed in the 80 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 59 60 61 62 66 67 68 69 70 71 72 73 74 75 76 77 78 81 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade Table Review of influencing parameters on corner separation Parameter References Description Loading 14, 15 Increasing the blade loading, a corner separation developed into a full-span separation on the rotor In the second-stage stator, increasing the blade loading resulted in a dramatic growth of the stator corner separation, and the blockage due to the corner separation reached nearly 40% with an extension of nearly 70% of the span The same trends were observed: increasing compressor loading generally increases the spread and the intensity of corner separation 16–19 Inflow boundary layer 20, 21 20 22 Free-stream turbulence intensity 16, 23 24 Clearance flow 25 18 26 Reynolds number 27 28 Mach number 29 30 The size of corner separation and the losses increase when the incoming boundary layer is thickened Through RANS simulations with mixing length model, Gbadebo presumed that increasing the turbulence level within the thickened inlet boundary layer brought high momentum fluid from the freestream into the boundary layer, thus suppressing the further growth of separation, and the extra losses were generated by the turbulent mixing within the boundary layer It is observed that the size of the corner separation decreases when the incoming boundary layer skewness increases The high turbulence intensity suppressed the laminar-turbulent transition bubble on the blade suction side The massive corner separation and the losses near the hub significantly decreased, mostly owing to the wakes-induced transition at the blade leading edge which suppressed the transition bubble When turbulence intensity increased, the laminar-turbulent transition bubble was removed, and the bypass transition became dominant At the same time, the transition location moved upstream The authors of the present paper observed in the figures of Ref 24 that the upstream movement of the laminar-turbulent transition can reduce the corner separation and the losses The stator corner separation was significantly reduced by a hub clearance (the hub is not rotating), because the high momentum leakage flow through the gap from the pressure side to the suction side reenergized the low-momentum flow on the suction side and thus decreased corner separation A stator hub clearance provides great impact on the corner separation, and the losses It helps increase the flow turning and decrease the diffusion factor near the hub, therefore leading to a reduction of the corner separation With a small clearance of about 0.2% of chord length, the losses were predicted to be the highest, which could be also associated to the increase of the critical points When the clearance is increased to about 0.58%, which is comparable to the displacement thickness of the inlet boundary layer, the losses are significantly reduced, and the critical points as well as the horseshoe vortex are found to disappear As the clearance is increased well beyond 0.58%, a strong tip-leakage vortex is formed, which prevents the end-wall low momentum fluid from interacting with the blade suction surface and thereby inhibits the corner separation Within a range of Reynolds number from 50,000 to 200,000, there is no significant effect of Reynolds number on the cascade performance for fully separated configurations Above a critical Reynolds number in the neighborhood of 200,000, the losses and the flow deflection (i.e., the cascade performance) are constant for a cascade that is not separated The losses are insensitive to the Reynolds number for the smoothing blades, while for the rough blades, the losses increase when the Reynolds number is augmented In a numerical study on a stator row of a high loading core compressor with a subsonic design inlet Mach number distribution around 0.72, a corner separation was formed close to the leading edge at high attack angle due to the shock that follows the leading edge local acceleration zone When the inlet Mach number was reduced, the exit losses were reduced, and the leading edge corner separation was eliminated as well A violent corner separation induced by a strong 3D shock system was identified experimentally and numerically in a compressor cascade at an inlet Mach number of 1.09 and a Reynolds number of 1.9  106 Rotating effect 14 Under low rotating speed condition, low total pressure fluid accumulates at blade-hub corner due to the passage vortex, which leads to a big corner separation However, under high rotating speed condition, a large spanwise redistribution of fluid occurs, and low energy fluid is centrifuged radially outward, which results in a smaller corner separation Surface roughness 31 The blade roughness induces an earlier laminar-turbulent transition, as well as a considerable frictional drag into the flow, which leads to the premature thickened boundary layer on the blade suction side This thickened boundary layer encounters the passage adverse pressure gradient, and finally leads to the increase of the corner separation and the losses The decrease of the compressor cascade performance depends mostly on the blade suction surface roughness For Reynolds number above 500,000, increasing the blade roughness will further increase losses and blockage 28 (continued on next page) Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 CJA 769 24 December 2016 No of Pages 17 F Gao et al Table (continued) Parameter References Description Real blade geometry 32–34 In the investigation of Friedrichs et al., a stator was designed with advanced conception rules, i.e., an aftswept leading edge with increasing sweep angle near hub and shroud It was observed that this modern design tends to reduce cross-passage pressure gradient, and therefore reduces the corner separation and losses by an induced new secondary flow Corner separation and its corresponding losses are very sensitive to the turbulent transition process between 5% and 30% span near the leading edge Blade geometric changes which cause suction surface transition to move toward the leading edge in this region will result in a large growth of the corner separation and its impact on losses 35 Fig 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Schematic of experimental test section and blade geometry LMFA,12,38 and a commercial solver ANSYS FLUENT The reason is that only two k-x turbulence models are currently available in Turb’Flow, and ANSYS FLUENT is thereby used as a complement to provide more results with other turbulence models A large-eddy simulation,39 consisting of  108 grid points (the grid size is Dx+ 60, Dy+ and Dz+ 30), has been carried out with Turb’Flow A flat plate simulation was running with the simulation of the compressor cascade domain, in order to feed the inflow condition of the latter The approach is depicted in Fig The blade surface sandpaper used in the experiment is reproduced by removing some grid points at the same position A 4-point Jameson centered spatial scheme with an artificial viscosity coefficient40,41 of 0.002 is implemented for the inviscid fluxes interpolation, while the viscous fluxes are discretized by a two-point centered scheme A three-step Runge-Kutta scheme is used for temporal discretization with a fixed time step of 2.5  10À8 s, corresponding to a Courant number (Courant-Friedrichs-Lewy Fig condition) of 0.95 Finally, the shear-improved Smagorinsky model (SISM)42 is utilized to represent the subgrid-scale motions Similar configurations are applied to the RANS simulations with Turb’Flow, where the grid-point number is reduced to about 2.8  106 The near wall grid size is Dy+ = Two available turbulence models are tested: the Wilcox k-x model43 and the Kok k-x model.44 To complement the RANS results with different turbulence models, and to assess the sensitivity to the numerical solver, ANSYS FLUENT is used to carry out two simulations with the Spalart-Allmaras (SA) model45 and the differential Reynolds stress model (DRSM).46 The mesh used in these two simulations consists of 1.6Â106 grid points, and the grid size Dy+ is set to A standard scheme is applied to the pressure term discretization, while second order upwind schemes are used for modified turbulent viscosity and energy term interpolation Another two FLUENT RANS simulations on the same mesh are conducted with two spatial interpolation schemes of two different scheme Sketch map of LES inflow feeding Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade 123 orders to check the influence of the spatial scheme order on corner separation More details about the computational settings involved in this paper are listed in Table 124 Influencing parameters of corner separation 120 121 122 125 126 127 128 129 130 131 132 133 134 135 136 The influencing parameters of the corner separation are classified into two categories: (1) numerical parameters, which concern the numerical resolution, such as turbulence model, numerical scheme and boundary condition type; (2) physical parameters, e.g., incidence angle, inflow turbulent kinetic energy, inflow perturbations and inflow boundary layer thickness Before the investigation on the influencing parameters of the corner separation, a subsection on the mean aerodynamics comparison will be firstly presented In this subsection, results are compared among the experiment, LES and RANS in order to make a sense on the capacity of the different numerical Table methods and turbulence models for predicting the corner separation in the reference configuration (incidence angle: 4°) Comparisons are made on the wall static pressure coefficient and the total pressure losses, which are good indicators of the separation Surface flow visualizations and blade suction side boundary layer profiles are also presented for the experiment, LES and reference RANS in order to emphasize the computational accuracy Apparently, the influence of turbulence model will also be included in this part It will be followed by a small synthesis Then, the other influencing parameters will be discussed 137 4.1 Mean aerodynamics comparison (influence of turbulence model) 148 Surface flow visualizations are usually used to qualitatively identify the corner separation that occurs in compressor cascades As illustrated in Fig 3, the LES and reference RANS surface limiting streamlines are compared with the experimen- 150 List of computations Code Incidence angle Trip Spatial scheme Artificial viscosity Turbulence model Inflow Outlet LES Turb’Flow 4° Yes 0.002 SISM d1(Exp.) RANS reference RANS SA RANS DRSM DRSM 1storder DRSM 3rdorder AUSM AUSM+-up Turb’Flow 4° Yes 0.020 Wilcox k-x d1(Exp.) Mixed P outlet P outlet FLUENT FLUENT 4° 4° No No 4-pt Jameson center 4-pt Jameson center 2nd-order upwind 2nd-order upwind SA DRSM d1(Exp.) d1(Exp.) P outlet P outlet FLUENT 4° No 1st-order upwind DRSM d1(Exp.) P outlet FLUENT 4° No DRSM d1(Exp.) P outlet Turb’Flow Turb’Flow 4° 4° Yes Yes Wilcox k-x Wilcox k-x d1(Exp.) d1(Exp.) P outlet P outlet Roe SLAU Viscosity 0.01 Turb’Flow Turb’Flow Turb’Flow 4° 4° 4° Yes Yes Yes 0.010 Wilcox k-x Wilcox k-x Wilcox k-x d1(Exp.) d1(Exp.) d1(Exp.) P outlet P outlet P outlet Mixed outlet Turb’Flow 4° Yes 0.020 Wilcox k-x d1(Exp.) Kok k-x Turb’Flow 4° Yes 0.020 Kok k-x d1(Exp.) Mixed P outlet P outlet i = À2° Turb’Flow À2° Yes 0.020 Wilcox k-x d1(Exp.) P outlet i = 0° Turb’Flow 0° Yes 0.020 Wilcox k-x d1(Exp.) P outlet i=2° Turb’Flow 2° Yes 0.020 Wilcox k-x d1(Exp.) P outlet i = 6° Turb’Flow 6° Yes 0.020 Wilcox k-x d1(Exp.) P outlet 0d1 Turb’Flow 4° Yes 0.020 Wilcox k-x d1(Exp.) P outlet 1.5d1 Turb’Flow 4° Yes 0.020 Wilcox k-x 1.5 d1(Exp.) P outlet 2.0TKE Turb’Flow 4° Yes 0.020 Wilcox k-x 2k(Exp.) P outlet Inflow fluctuation Turb’Flow 4° Yes 3rd-order MUSCL 3rd-order AUSM 3rd-order AUSM+-up 3rd-order Roe 3rd-order SLAU 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 4-pt Jameson center 0.020 Wilcox k-x d1(Exp.) with inflow P outlet fluctuations Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 138 139 140 141 142 143 144 145 146 147 149 151 152 153 CJA 769 24 December 2016 No of Pages 17 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 F Gao et al tal oil visualization, on both the endwall and blade suction surface On the endwall, a good qualitative agreement is achieved among the experiment, LES and RANS, in terms of the reverse flow structure and the singular points LES gives better prediction on the outset of the endwall separation line, which occurs around 50% axial chord position on the blade suction side A second pair of flow visualizations shows on the blade suction surface It should be noticed that an excellent symmetry has been achieved in the experiment Recirculation regions are observed among experiment, LES and RANS near the trailing edge of the blade suction sides beside the endwalls Again, RANS shows a larger reverse flow area, but qualitatively agrees with the experiment and LES The mean static pressure coefficient CP = (P À P1)/ (Pt,1 À P1) is a key parameter to determine the compressor cascade performance The area enclosed by the static pressure coefficient on the pressure and suction sides represents the blade loading A comparison among the experimental, LES and RANS results is shown in Fig The left figure shows the static pressure coefficient at midspan, while the distribution close to the endwall is plotted on the right figure At midspan (in Fig 4(a)), both LES and RANS match with the experimental data The numerical oscillations, which appear near the leading edge on both the pressure and suction sides, are due to the square meshing of the tripping bands By carefully inspecting the results, it can be observed that the DRSM model gives the best prediction of CP The slight over-estimate by LES may relate to the difficulty in simulating transition using tripping bands It is more interesting to investigate what happens close to the endwall in Fig 4(b) The results are quite different among the different turbulence models Encouragingly, LES provides a very good prediction near the endwall On the suction side, a flat region (indicating the separation) begins at about x/ca = 0.6 in both the experiment and the LES Among the four RANS simulations, the DRSM model gives the best prediction, though the onset of separation is located earlier at about x/ca = 0.4 The largest corner separation is predicted Fig by the SA model The results with the Kok k-x model are very close to those with the Wilcox k-x model, but Kok’s model predicts a slightly lower blade loading than Wilcox’s model It is observed that the earlier the corner separation occurs, the lower the blade loading is A second key indicator of the compressor cascade performance is the total pressure loss coefficient CPt = (Pt,1 À Pt)/ (Pt,1 À P1) Contour maps of the total pressure loss coefficient are compared in Fig The losses are associated with the blade wake (around y/s = 1) and the corner separation wake (below z/h = 0.2) LES is found quite powerful to predict both the strength and extent of the losses, while RANS models fail in reproducing the experimental total pressure losses Among the RANS results, the SA model is seen to predict the highest losses, and it is consistent with the early separation observed on the static pressure coefficient CP The total pressure losses are then weighted averaged by mass flow along the pitchwise direction (CÃPt ), and comparison is plotted in Fig A very good agreement is observed between the LES and the experiment In contrast, the RANS models over-estimate the losses downstream of the corner separation This is consistent with the over-prediction of the separation observed through CP A good prediction of the blade wake losses is obtained by the RANS models Among the RANS models, the DRSM model gives the best prediction on CÃPt Further, the boundary layer profiles along blade suction side are compared among the experiment, LES and reference RANS The measurement stations are depicted in Fig The velocity vectors V on those measurement stations are presented in tangential velocity components us and wall normal velocity components un, and the velocity decomposition method is drawn in Fig as well The velocity profiles at two different blade span positions, z/h = 48.6% and z/ h = 2.7%, are discussed here, and they are plotted in Fig Excellent agreements are observed in Fig 8(a) and (b) at z/ h = 48.6% for both us and un At z/h = 2.7%, LES results Surface flow pattern visualizations (top: endwall; bottom: blade suction surface) Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade Figure Fig 227 228 229 230 231 232 233 234 235 Mean static pressure coefficient Mean total pressure loss coefficient (x = 1.363ca) agree with the available PIV measurements RANS predicts an earlier separation outset: the first negative us values appear on the measurement station s* = 0.41 The separation outset predicted by LES is observed on the measurement station s* = 0.80 Although RANS predicts an earlier separation outset, it shows similar velocity profiles to LES on the last two measurement stations within the separation region This builds confidence in using RANS for further parametric investigations 4.2 Synthesis 236 The comparison among the experiment, LES and RANS is concluded as follows: (1) regarding the surface flow visualizations, both LES and RANS qualitatively match the experiment; (2) a good prediction of the static pressure coefficient and the total pressure losses is obtained by LES throughout the half span; (3) RANS predicts an earlier separation outset but shows similar velocity profiles on the measurement stations 237 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 238 239 240 241 242 243 CJA 769 24 December 2016 No of Pages 17 Fig F Gao et al Pitchwise-mass-averaged total pressure loss coefficient Fig Schematic of boundary layer profile measurement stations and velocity decomposition method Fig close to the blade trailing edge Among the four RANS models, the DRSM works better than the others Finally, the largest corner separation is given by the SA model Although LES gives the best prediction of the corner separation, it is still too expensive to conduct the influencing parameter studies RANS over-predicts the corner separation, but gives qualitatively reasonable trends Along with the analysis about incidence angle effect in Section 4.4.1, the results in this section heighten confidence in using RANS approach to continue the parametric studies in the following sections 244 4.3 Numerical parameters 254 4.3.1 Spatial interpolation scheme 255 It is interesting to study if the spatial scheme influences the prediction of the corner separation Four different upwind spatial schemes are studied in comparison with the four-point centered scheme and artificial viscosity (Jameson33) chosen as reference in this work These four upwind schemes are Roe scheme,47 AUSM scheme,48 AUSM+-up scheme49 and simple low-dissipation AUSM scheme (SLAU).50 Besides, in order to bring some insights into the influence of the spatial scheme order, a 1st-order upwind scheme and a 3rd-order monotone upwind scheme for conservation laws (MUSCL) scheme are also compared with a 2nd-order upwind scheme (as reference), using FLUENT with DRSM model The comparison of the static pressure coefficient on the blade is drawn in Fig The results of the first five different spatial schemes are strictly superimposed In Fig 10, the total pressure loss coefficient CPt contours are illustrated on a plane downstream of the compressor cascade Their pitchwise-massaveraged values CÃPt are plotted in Fig 11 Again, it shows no discrepancy for the first five different spatial schemes In the present RANS simulation, the corner separation is insensitive to these first five spatial interpolation schemes Moreover, it is believed that the spatial scheme is not the cause of the overpredicting of the corner separation 256 Blade suction side velocity profiles Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 245 246 247 248 249 250 251 252 253 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade 279 280 281 282 283 284 285 286 287 288 289 290 291 Fig Influence of spatial interpolation scheme on CP Fig 10 Influence of spatial interpolation scheme on CPt The last three spatial schemes are compared using FLUENT with DRSM model Differences between them and the first five spatial schemes may be due to the different solvers and different RANS models At midspan, as plotted in Fig 9, the CP lines of the DRSM results are overlapping, suggesting that all of the three orders of spatial scheme are able to capture the flow physics However, some discrepancies appear close to the endwall on the suction side (see Fig 9(b)) from x/ ca = 0.4 to the trailing edge The results of 2nd-order and 3rdorder schemes are superimposed, differing from the 1st-order scheme It means that in this case, the 1st-order scheme is insufficient, while the 3rd-order scheme is lavish as it uses more resources and provides the same results compared with the 2nd-order scheme The same conclusion can be drawn through the total pressure loss comparison The total pressure loss coefficient contours are shown in Fig 10 The 1st-order scheme’s results are observed different from the 2nd-order and 3rdorder ones The mixing process is slower for the 1st-order scheme, which shows a gradual gradient across the high loss region The plot of CÃPt of the last three spatial schemes is depicted in Fig 11, and the 1st-order scheme is found to differ from others throughout the spanwise direction 292 4.3.2 Artificial viscosity of centered spatial scheme 301 When a centered spatial scheme is used to simulate a flow for the convection terms of each governing equation, it is neces- 302 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 293 294 295 296 297 298 299 300 303 CJA 769 24 December 2016 No of Pages 17 10 304 305 306 307 308 310 311 312 313 314 315 316 317 318 319 320 321 322 323 F Gao et al sary to employ an artificial viscosity to stabilize the calculation The definition of the numerical dissipative flux Fdj , at the face indexed j À 0.5 for a conservative quantity q, can be found in Ref 41:  à Fdj ẳ Vu a ỵ cs kakị e4 qjỵ1 3qj ỵ 3qj1 qj2 ị=8 1ị where e4 is the 4th-order artificial viscosity coefficient, while V, u, a, cs and j are the cell volume, velocity vector, contravariant vector, speed of sound and index of grid, respectively Smati51 suggests to set e4 between 0.01 and 0.15 for a RANS simulation Nevertheless, it is desirable to know how the artificial viscosity influences the simulation of the corner separation Two simulations, with e4 = 0.02 (reference) and 0.01, are carried out to investigate the influence of the artificial viscosity on the description of the corner separation The comparison of CP, CPt and CÃPt is shown in Figs 12and 13 No discrepancy can be seen between the ‘‘RANS reference” case and the ‘‘viscosity 0.01” case Within the present range of values of e4, there is no sensitivity to the artificial viscosity Fig 11 Influence of spatial interpolation scheme on CÃPt Fig 12 4.3.3 Outlet boundary condition 324 Outlets need to be carefully treated in numerical simulations because the outlet boundary condition controls the confinement of the waves Moreover, if the outlet region is not long enough, it may impact the mixing process In the present study, the computational domain extends over 2c downstream of the blade trailing edge, and the mesh is stretched near the outlet plane Two outlet boundary conditions are tested here: one is a standard pressure outlet condition; the other is the pressure outlet condition mixed with a non-reflection outlet condition, which allows a partial evacuation of the waves out of the computational domain The comparison between these two outlet boundary conditions is available in Figs 12 and 13 No difference is observed between the results This implies that there is no spurious confinement effect in the simulations, since it would be influenced by the change of the outlet condition 325 4.4 Physical parameters 341 4.4.1 Incidence angle 342 Incidence angle is an important physical parameter of corner separation Numerical results for five incidence angles are investigated in comparison with the experimental results The static pressure coefficient around the blade, at midspan and near the endwall is plotted in Fig 14 The evolution of the static pressure coefficient at midspan is fairly predicted by RANS As proposed by Ma,52 a characteristic point is identified on the blade suction side near the leading edge, denoted by B in Fig 14(a) at x/ca = 0.2 CP at this point never varies whatever the incidence angle changes Upstream of the point B, CP decreases with the incidence angle, while CP increases with the incidence angle downstream of this point The location of B is fairly well identified by RANS These good results at midspan suggest that the incidence angle in the experiment, which is rather difficult to precisely evaluate, is indeed the same as that in the simulations The distribution of CP near the endwall is shown in Fig 14(b) When the incidence angle increases, the outset of the constant CP region on the blade suction side moves upstream, suggesting an earlier outset of the corner separation The extent of the separation region is thus increased by augmenting the incidence angle The characteristic point B is again identified on the blade suction side; 343 Influences of artificial viscosity and outlet boundary condition on CP Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade Fig 13 Influences of artificial viscosity and outlet boundary condition on CPt and CÃPt Fig 14 Fig 15 365 366 367 368 369 370 11 Influence of incidence angle on CP Evolution of lift coefficient against incidence angle however discrepancies appear between the experimental and RANS results The RANS characteristic point BRANS is located at about x/ca = 0.06, upstream of the experimental characteristic point BExp at x/ca = 0.14 A relatively good agreement is achieved in the endwall region between the RANS and the experiment at i = 2° When the incidence angle increases, RANS over-predicts the corner separation, which seems to push BRANS upstream The blade lift coefficient CL (expressed in Eq (2)) is presented against the incidence angle i in Fig 15 It is clearly observed that CL increases with the incidence angle from i = À2° until i = 4° From i = 4° to i = 6°, CL decreases due to the large corner separation The lift is globally underestimated by RANS, which is relevant to the over-prediction of the corner separation However, the evolution with the incidence angle is well described R h=2 R ca CP ðx; zÞn Á ydxdz ð2Þ CL ¼ R h=2 R ca dxdz 0 371 This study has been carried out from both the experimental and RANS results It appears that RANS over-estimates the size of the corner separation, but reproduces right trends of the lift coefficient This gives confidence to use RANS for further physical parameter studies, for which experimental results might not be available The total pressure loss coefficient contours are compared in Fig 16 for the five incidence angles A gradual increase of high loss region is observed with the rise of incidence angle The 384 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 372 373 374 375 376 377 378 379 380 381 383 385 386 387 388 389 390 391 392 CJA 769 24 December 2016 No of Pages 17 12 F Gao et al Fig 16 Fig 17 393 394 395 396 397 398 399 400 401 402 403 404 Influence of incidence angle on CPt Tangential velocity profiles us most significant change is found at the position downstream of the corner separation, while the blade wake losses close to the midspan only have minor variation It means that the increase of total pressure losses with incidence angle is mostly due to the formation and growth of the corner separation To further investigate the influence of incidence angle on corner separation, blade suction boundary layer profiles us are plotted in Fig 17 at midspan (z/h = 48.6%) and close to the endwall (z/h = 2.7%) The measurement stations and velocity decomposition method are illustrated in Fig At midspan, the velocity profiles have a slight change with incidence angle, which results in the slight variation of the blade wake losses observed in Fig 16 Significant change of the tangential velocity profiles is observed in Fig 16(b) close to the end-wall On the first measurement station s* = 0.21, none of the boundary layers separates At s* = 0.31, the boundary layer under the operating condition with an incidence angle 6° shows negative velocity values, implying that the boundary layer is separate With the rise of incidence angle, suction side boundary layer separates gradually: i = 4° at s* = 0.41; i = 2° at s* = 0.5; i = 0° at s* = 0.6; i = À2° at s* = 0.7, all of the cases studied are separate This also explains the reason why significant increases of total pressure losses are dominated by corner separation rather than the incidence angle itself Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 405 406 407 408 409 410 411 412 413 414 415 416 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade Fig 18 Fig 19 Influences of inflow TKE level and inflow fluctuations on CP Influences of inflow TKE level and inflow fluctuations on CPt and CÃPt Fig 20 Influence of inflow boundary layer thickness on CP 417 4.4.2 Inflow TKE level 418 In the present work, the inflow boundary layer profile is extracted from a flat plate simulation, and the inlet condition of this flat plate simulation is a uniform velocity profile with the same free-stream TKE as in the experiment Therefore, the extracted velocity profile is coherent with the TKE profile In this part, the TKE profile is set to be twice its initial value to investigate its influence The comparisons of CP, CPt and CÃPt 419 420 421 422 423 424 13 are shown in Figs 18 and 19 No visible discrepancy could be found in the figures, which means that the corner separation in this study is insensitive to a doubling of the inflow TKE 425 4.4.3 Inflow fluctuations 428 Inflow angle fluctuations are generally found in real compressors, since they can be generated by both rotor wakes and inflow instabilities In recent years, unsteady RANS (URANS) 429 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 426 427 430 431 CJA 769 24 December 2016 No of Pages 17 14 F Gao et al 452 method is found to improve the numerical solutions in simulating turbomachinery flows, compared with steady RANS approach.53–55 Thus, the idea is to investigate the sensitivity of the corner separation to realistic inflow perturbations in URANS solutions Perturbations are imposed on the inlet plane by varying the inflow angle The inflow angle in the x– y plane varies sinusoidally when a constant mass flow rate is assumed The fluctuating amplitude is prescribed as Di = arctan(2Tu) = 0.92°, and the frequency: f = U1/h = 108 Hz, where Tu is the free-stream turbulence intensity measured at inflow A constant global time step  10À8 s was used for the URANS simulation, corresponding to a Courant number of for the minimum grid cell The averaged static pressure coefficient CP, total pressure loss coefficient contours CPt and the pitchwise-mass-averaged total pressure loss coefficient CÃPt are plotted in Figs 18 and 19 In these figures, no discrepancy is shown between the steady RANS results and the averaged URANS results with inflow fluctuations This suggests that the mean static pressure and total pressure losses, simulated by URANS, are insensitive to the inflow perturbations imposed in this study 453 4.4.4 Inflow boundary layer thickness 454 Herein, the influence of the inflow boundary layer thickness is investigated with three different values: 0d1,1, 1.0d1,1 and 1.5d1,1, where d1,1 is the measured value The different displacement thicknesses of the inflow boundary layer and the corresponding velocity profiles used in this study are obtained by adjusting the axial length of the boundary layer simulation The free-stream velocities are different for the three cases because an identical mass flow rate is imposed on the inlet plane The comparisons of CP at midspan and close to the endwall are shown in Fig 20 At midspan, CP decreases a bit with increase of d1 Close to the endwall, thickening the inflow d1 again decreases the static pressure on the blade pressure side On the suction side, the thickened inflow d1 pushes the outset of the corner separation upstream The lowest blade loading is found in the case with the thickest inflow boundary layer This observation may help to understand the phenomenon that occurs at midspan The stronger corner separation pushes the flow toward the midspan, increases the velocity outside 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 Fig 21 the boundary layers, and therefore reduces the static pressure, according to Bernoulli’s principle The comparison of CPt is plotted in Fig 21 No difference is observed between the simulations from z/h = 0.25 to z/ h = 0.5, where the blade weak losses dominate CÃPt Close to the endwall, the results are different: the case with the uniform inflow (no boundary layer) creates little losses on the endwall and reduces as well the high loss region extent The largest losses are found in the case with 1.5d1,1 The corner separation is found sensitive to the inflow boundary layer thickness, which should be taken into account when a compressor is designed The pitchwise-mass-averaged total pressure loss coefficient CÃPt is plotted in Fig 21(d) For the case with uniform inflow, a vertical increase of CÃPt is observed from z/ h = 0.03 to z/h = 0.12 This part has smaller losses than the reference case, and the losses indicated by the area enclosed by the solid and dashed lines are due to the contribution of the inflow boundary layer A global mass-averaged total pressure loss coefficient is introduced in Eq (3) to evaluate the contribution of inflow boundary layer to the total pressure losses CPt;global of the three cases are listed in Table It is shown that Table Global mass-averaged total pressure loss coefficient Boundary layer thickness 0d1,1 1.0d1,1 1.5d1,1 CPt;global 0.054 0.080 0.096 Fig 22 Tangential velocity profiles us at z/h=8.1% (close to endwall) Influence of inflow boundary layer thickness on CPt and CÃPt Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 CJA 769 24 December 2016 No of Pages 17 LMFA-NACA65 linear compressor cascade 494 495 496 498 32.5% of the total pressure losses in the reference case come from the contribution of the inflow boundary layer R h=2 R s CPt ðy; zÞuðy; zÞdydz ð3Þ CPt;global ẳ R h=2 Rs uy; zịdydz 0 510 A comparison of the tangential velocity profiles us is plotted in Fig 22 in the corner separation region (z/h = 8.1%) Differences are clearly observed Compared to the reference case with 1.0d1,1 (separates at s* = 0.6), the case without inflow boundary layer separates later at s* = 0.7 The case with 1.5d1,1 may separate slightly earlier than the reference case Finally the boundary layer profiles become more similar on the last measurement station before leaving the blade trailing edge It implies that the thickened inflow boundary layer can push upstream the outset of suction side boundary layer separation, and the rear part of the corner separation is less sensitive to the inflow boundary layer thickness 511 Conclusions 499 500 501 502 503 504 505 506 507 508 509 513 512 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 (1) LES is shown superior to the RANS method (with SA, Wilcox k-x, Kok k-x and DRSM models) in reproducing the corner separation observed on the LMFANACA65 linear compressor cascade configuration, in terms of surface flow visualization, mean static pressure coefficient, mean total pressure losses and blade suction side boundary layer profiles (2) RANS over-estimates the corner separation, but gives reasonable trends in the affordable computational resource consumption (compared with LES), which allows the investigation on the parameters controlling the corner separation (3) Concerning the numerical parameters, the corner separation is found to be insensitive to some spatial interpolation schemes, and to the artificial viscosity (within a reasonable range) Besides, the 1st-order spatial scheme is shown insufficient to capture the corner separation, while the 2nd-order scheme is enough compared with the 3rd-order one The RANS turbulence modeling is considered as being mainly responsible for the misprediction of the corner separation Among the commonly used RANS models, DRSM model gives the best prediction of the corner separation (4) Regarding the physical parameters, the incidence angle is shown to increase the corner separation as expected The mean results of the corner separation appear to be insensitive to the increase of the inflow TKE (twice the original value) and the prescribed inflow perturbations by URANS More interestingly, the boundary layer thickness is also observed to increase the separation, which should be taken into account during the design of a compressor The mechanisms that how the parameters affect the corner separation are also discussed through blade suction side boundary layer evolution 547 548 Acknowledgements 549 The authors dedicate this paper to Prof Francis Leboeuf, the supervisor of Feng Gao, Wei Ma and Xavier Ottavy, an efficient and energetic colleague, a credible friend He devoted 550 551 15 all his energies and enthusiasm to developing the relationship in scientific research between the laboratories of the five E´coles Centrale in France and Beihang University in China Without his contributions, this work would not have been possible, and the authors would not have the opportunity to name together on this paper His elegance, his kindness, his enthusiasm, his contagious optimism .and of course his brown 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693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 CJA 769 24 December 2016 LMFA-NACA65 linear compressor cascade 735 736 737 738 739 740 741 742 743 744 745 746 51 Smati L Contribution au de´veloppement d’une me´thode nume´rique d’analyse des e´coulements instationnaires Applications aux turbomachines [dissertation] Lyon: E´cole Centrale de Lyon; 1997 52 Ma W Experimental investigation of corner stall in a linear compressor cascade [dissertation] Lyon: E´cole Centrale de Lyon; 2012 53 Boudet J, Autef VND, Chew JW, Hills NJ, Gentilhomme O Numerical simulation of rim seal flows in axial turbines Aeronaut J 2005;109(1098):373–83 54 Tre´binjac I, Kulisa P, Bulot N, Rochuon N Effect of unsteadiness on the performance of a transonic centrifugal compressor stage J Turbomach 2008;131(4):1835–45 No of Pages 17 17 55 Gougeon P, Ngo Boum G Aerodynamic interactions between a high-pressure turbine and the first low-pressure stator J Turbomach 2014;136(7):1–14 Gao Feng is a research fellow at Department of Mechanical Engineering Sciences, University of Surrey, Guildford, UK He received his Ph.D degree in E´cole Centrale de Lyon His main research interests are large-eddy simulation, compressor corner separation and turbine rim seal Liu Yangwei is currently an associate professor at School of Energy and Power Engineering, Beihang University, Beijing, China His major research interests include CFD, complex flow mechanism and flow control in turbomachinery Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA-NACA65 linear compressor cascade, Chin J Aeronaut (2016), http://dx.doi.org/10.1016/j.cja.2016.09.015 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 ... No of Pages 17 LMFA- NACA65 linear compressor cascade Table Review of in? ??uencing parameters on corner separation Parameter References Description Loading 14, 15 Increasing the blade loading, a corner. .. boundary condition on CP Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA- NACA65 linear compressor cascade, Chin J Aeronaut (2016),... profiles on the measurement stations 237 Please cite this article in press as: Gao F et al Parameter study on numerical simulation of corner separation in LMFA- NACA65 linear compressor cascade, Chin