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TECHNICAL PAPERS

Robust Implicit Multigrid Reynolds-Stress Model Computation of 3D Turbomachinery Flows

[+] Author and Article Information
G. A. Gerolymos

Institut d’Alembert, Université Pierre-et-Marie-Curie, 75005 Paris, Francegeg@ccr.jussieu.fr

I. Vallet

Institut d’Alembert, Université Pierre-et-Marie-Curie, 75005 Paris, Francevallet@ccr.jussieu.fr

J. Fluids Eng 129(9), 1212-1227 (Mar 31, 2007) (16 pages) doi:10.1115/1.2754320 History: Received June 16, 2006; Revised March 31, 2007

The purpose of this paper is to present a numerical methodology for the computation of complex 3D turbomachinery flows using advanced multiequation turbulence closures, including full seven-equation Reynolds-stress transport models. The flow equations are discretized on structured multiblock grids, using an upwind biased (O[ΔxH3]MUSCL reconstruction) finite-volume scheme. Time integration uses a local dual-time-stepping implicit procedure, with internal subiterations. Computational efficiency is achieved by a specific approximate factorization of the implicit subiterations, designed to minimize the computational cost of the turbulence transport equations. Convergence is still accelerated using a mean-flow-multigrid full-approximation-scheme method, where multigrid is applied only on the mean-flow variables. Speed-ups of a factor 3 are obtained using three levels of multigrid (fine plus two coarser grids). Computational examples are presented using two Reynolds-stress models, and also a baseline kε model, for various turbomachinery configurations, and compared to available experimental measurements.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Grid-convergence study of computations with the SV–RSM (34) of the NASA_37 rotor (3,48-49), using progressively finer grids (1.15×106 points grid_B, 1.96×106 points grid_C, 3.19×106 points grid_D3, and 9.24×106 points grid_E3 (Table 1)): Operating map of total-pressure ratio πT-T between stations 1 and 4 versus mass flow ṁ, and distributions of pitchwise-averaged total pressure ptM, total temperature TtM, and absolute-flow angle αxθM, at station 4, and isentropic efficiency between stations 1 and 4 ηisM versus span ς (ṁ=20.12kgs−1; Tui=3%; ℓTi=5mm; δTC=0.356mm)

Grahic Jump Location
Figure 2

Convergence (a) of the error pseudonorms for the mean flow (eMF, Eq. 15) and for the turbulence variables (eRSM; Eq. 16), as a function of both the number of iterations nit and of CPU time (2Gflops sustained performance), (b) of mass flow at rotor inflow ṁi and outflow ṁo versus CPU, and (c) of the number of subiterations Mit(nit,ℓGRD) required for an error reduction of rMF≤rOBJ=−2 on each grid, for multigrid (LGRD=3; [CFL,CFL*,rOBJ]=[150,15,−2]) computations of the NASA_37 rotor (3,48-49) using the SV–RSM (34) (ΥOBJ=6250Paskg−1; ṁ=20.12kgs−1; Tui=3%; ℓTi=5mm; δTC=0.356mm; 9.24×106 points grid_E3; Table 1)

Grahic Jump Location
Figure 3

Convergence as a function of CPU-time (2Gflops sustained performance) of mass flow at rotor inlet ṁi, mass-averaged total pressure at rotor outlet p̑to, and entropy at rotor outlet s̑o−sISA, for monogrid (LGRD=1) and multigrid (LGRD=3) computations of the NASA_37 rotor (3,48-49), using two wall-normal-free RSMs (SV–RSM (34) and WNF–LSS–RSM (11)) and the Launder-Sharma k−ε two-equation model (33), using appropriate ΥOBJ throttle boundary conditions (ṁ=20.12kgs−1; Tui=3%; ℓTi=5mm; δTC=0.356mm; 3.2×106 points grid_D3; QN[CFL*,rOBJ]=[10,−2])

Grahic Jump Location
Figure 4

Comparison of computed and measured operating map of total-pressure ratio πT-T between stations 1 and 4 versus mass flow ṁ, and spanwise (ς) distributions of pitchwise-averaged absolute-flow angle αxθM, total pressure ptM and total temperature TtM at station 4, and isentropic efficiency between stations 1 and 4 ηisM, of the NASA_37 rotor (3,48-49), using two wall-normal-free RSMs (GV–RSM (25) and SV–RSM (34)) and the Launder–Sharma k–ε two-equation model (33) (ṁ=20.12kgs−1; Tui=3%; ℓTi=5mm; δTC=0.356mm; 3.2×106 points grid_D3)

Grahic Jump Location
Figure 5

Meridional view of the CREATE_1312 stage compressor (50-52), showing measurement planes and interfaces between grid blocks (ℓBR=1,…,10)

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Figure 6

View of the computational grid at midspan of the CREATE_1312 stage compressor (50-52) (11×106 points grid_C3)

Grahic Jump Location
Figure 7

Experimental and computed, using the SV–RSM (34) (both with basic and mxout MPs; cf. Sec. 32) and the Launder-Sharma k−ε two-equation model (33), performance map (total-pressure-ratio πT-T between planes 300 and 250 versus massflow ṁ) of the CREATE_1312 stage compressor (50-52) (Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); QN[CFL*,rOBJ]=[20,−2]; LGRD=3; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 8

Convergence as a function of CPU time (2Gflops sustained performance) of mass flow at compressor inlet ṁi, mass-averaged total pressure at compressor outlet p̑to, and entropy at compressor outlet s̑o−sISA, for monogrid (LGRD=1) and multigrid (LGRD=3) computations, using the SV–RSM (34) with basic MPs (cf. Sec. 32), of the CREATE_1312 stage compressor (50-52) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 9

Comparison of computed and measured spanwise (ς) distributions of pitchwise-averaged total pressure ptM for the CREATE_1312 stage compressor (50-52), using the SV–RSM (34) (both with BASIC and MXOUT MPS ; cf. Sec. 32) and the Launder–Sharma k−ε two-equation model (33) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); LGRD=3; QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 10

Comparison of computed and measured spanwise (ς) distributions of pitchwise-averaged absolute-flow angle αM for the CREATE_1312 stage compressor (50,52), using the SV–RSM (34) (both with BASIC and MXOUT MPs ; cf. Sec. 32) and the Launder-Sharma k−ε two-equation model (33) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm ( IGV, R1, R2, R3); LGRD=3; QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 11

Comparison of computed and measured spanwise (ς) distributions of pitchwise-averaged total temperature TtM for the CREATE_1312 stage compressor (50,52), using the SV–RSM (34) (both with BASIC and MXOUT MPs ; cf. Sec. 32) and the Launder–Sharma k−ε two-equation model (33) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); LGRD=3; QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 12

Comparison of computed and measured spanwise (ς) distributions of pitchwise-averaged relative-flow-angle βM for the CREATE_1312 stage compressor (50,52), using the SV–RSM (34) (both with BASIC and MXOUT MPs ; cf. Sec. 32) and the Launder–Sharma k−ε two-equation model (33) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); LGRD=3; QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

Grahic Jump Location
Figure 13

Comparison of computed and measured spanwise (ς) distributions of pitchwise-averaged axial-velocity VxM for the CREATE_1312 stage compressor (50-52), using the SV–RSM (34) (both with BASIC and MXOUT MPs ; cf. Sec. 32) and the Launder–Sharma k−ε two-equation model (33) (ṁ=0.96ṁCH, ΥOBJ=23,150Paskg−1; Tui=2%; ℓTi=10mm; δTC=0.3mm, 0.53mm, 0.57mm, 0.52mm (IGV, R1, R2, R3); LGRD=3; QN[CFL*,rOBJ]=[20,−2]; 11×106 points grid_C3; Table 2)

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