Abstract

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.

1.
Hah
,
C.
, and
Krain
,
H.
, 1990, “
Secondary Flows and Vortex Motion in a High-Efficiency Backswept Impeller at Design and Off-Design Conditions
,”
ASME J. Turbomach.
0889-504X,
112
(
1
), pp.
7
13
.
2.
Sieverding
,
C. H.
, 1985. “
Recent Progress in the Understanding of Basic Aspects of Secondary Flows in Turbine Blade Passages
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
107
, pp.
248
257
.
3.
Reid
,
L.
, and
Moore
,
R. D.
, 1978. “
Design and Overall Performance of 4 Highly Loaded, High-Speed Inlet Stages for an Advanced High-Pressure-Ratio Core Compressor
,” Tech. Paper No. TP–1337, NASA, Lewis Research Center, Cleveland, Oct.
4.
Hah
,
C.
, and
Loellbach
,
J.
, 1999. “
Development of Hub Corner Stall and Its Influence on the Performance of Axial Compressor Blade Rows
,”
ASME J. Turbomach.
0889-504X,
121
(
1
), pp.
67
77
.
5.
Volino
,
R. J.
, 2002. “
Separated Flow Transition Under Simulated Low-Pressure Turbine Airfoil Conditions—Part 1: Mean-Flow and Turbulence Statistics
,”
ASME J. Turbomach.
0889-504X,
124
, pp.
645
655
.
6.
Volino
,
R. J.
, 2002. “
Separated Flow Transition Under Simulated Low-Pressure Turbine Airfoil Conditions—Part 2: Turbulence Spectra
,”
ASME J. Turbomach.
0889-504X,
124
, pp.
656
664
.
7.
Copenhaver
,
W. W.
,
Puterbaugh
,
S. L.
, and
Hah
,
C.
, 1997. “
Unsteady Flow and Shock Motion in a Transonic Compressor Rotor
,”
J. Propul. Power
0748-4658,
13
(
1
), pp.
17
23
.
8.
Volino
,
R. J.
, 1998. “
A New Model for Freestream Turbulence Effects on Boundary Layers
,”
ASME J. Turbomach.
0889-504X,
120
, pp.
613
620
.
9.
Speziale
,
C. G.
, 1998. “
Turbulence Modeling for Time-Dependent RANS and VLES: A review
,”
AIAA J.
0001-1452,
36
(
2
), pp.
173
184
.
10.
Walters
,
D. K.
, and
Leylek
,
J. H.
, 2004, “
A New Model for Boundary-Layer Transition Using a Single-Point RANS Approach
,”
ASME J. Turbomach.
0889-504X,
126
(
1
), pp.
193
202
.
11.
Gerolymos
,
G. A.
Sauret
,
E.
, and
Vallet
,
I.
, 2004, “
Contribution to the Single-Point-Closure Reynolds-Stress Modeling of Inhomogeneous Flow
,”
Theor. Comput. Fluid Dyn.
0935-4964,
17
(
5–6
), pp.
407
431
.
12.
Scholz
,
N.
, ed., 1977,
Aerodynamics of Cascades
, AGARD, Neuilly-sur-Seine, No. AGARD–AG–220 in AGARDograph.
(translated and revised by A. Klein from the original german Aerodynamik der Schaufelgitter).
13.
Chenault
,
C. F.
Beran
,
P. S.
, and
Bowersox
,
R. D. W.
, 1999, “
Numerical Investigation of Supersonic Injection Using a Reynolds-Stress Turbulence Model
,”
AIAA J.
0001-1452,
37
(
10
), pp.
1257
1269
.
14.
Batten
,
P.
Craft
,
T. J.
Leschziner
,
M. A.
, and
Loyau
,
H.
, 1999, “
Reynolds-Stress-Transport Modeling for Compressible Aerodynamics Applications
,”
AIAA J.
0001-1452,
37
(
7
), pp.
785
797
.
15.
Chassaing
,
J. C.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2003, “
Efficient and Robust Reynolds-Stress Model Computation of 3-D Compressible Flows
,”
AIAA J.
0001-1452,
41
(
5
), pp.
763
773
.
16.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2005, “
Mean-Flow-Multigrid for Implicit Reynolds-Stress-Model Computations
,”
AIAA J.
0001-1452,
43
(
9
), pp.
1887
1898
.
17.
Dawes
,
W. N.
, 1992, “
Toward Improved Throughflow Capability: The Use of 3-D Viscous Flow Solvers in a Multistage Environment
,”
ASME J. Turbomach.
0889-504X,
114
, pp.
8
17
.
18.
Turner
,
M. G.
, and
Jennions
,
I. K.
, 1993, “
An Investigation of Turbulence Modeling in Transonic Fans Including a Novel Implementation of an Implicit k−ε Turbulence Model
,”
ASME J. Turbomach.
0889-504X,
115
(
2
), pp.
249
260
.
19.
Jennions
,
I. K.
, and
Turner
,
M. G.
, 1993, “
3-D Navier-Stokes Computations of Transonic Fan Flow Using an Explicit Flow Solver and an Implicit k−ε Turbulence Model
,”
ASME J. Turbomach.
0889-504X,
115
(
2
), pp.
261
272
.
20.
Rhie
,
C. M.
Gleixner
,
A. J.
Spear
,
D. A.
Fischberg
,
C. J.
, and
Zacharias
,
R. M.
, 1998, “
Development and Application of a Multistage Navier-Stokes Solver—Part I: Multistage Modeling Using Bodyforces and Deterministic Stresses
,”
ASME J. Turbomach.
0889-504X,
120
, pp.
205
214
.
21.
LeJambre
,
C. R.
Zacharias
,
R. M.
Biederman
,
B. P.
Gleixner
,
A. J.
, and
Yetka
,
C. J.
, 1998, “
Development and Application of a Multistage Navier-Stokes Solver—Part II: Application to a High-Pressure Compressor Design
,”
ASME J. Turbomach.
0889-504X,
120
, pp.
215
223
.
22.
Adamczyk
,
J. J.
, 2000, “
Aerodynamic Analysis of Multistage Turbomachinery Flows in Support of Aerodynamic Design
,”
ASME J. Turbomach.
0889-504X,
122
, pp.
189
217
.
23.
Gier
,
J.
Stubert
,
B.
Brouillet
,
B.
, and
de Vito
,
L.
, 2005, “
Interaction of Shroud Leakage Flow and Main Flow in a 3-Stage LP Turbine
,”
ASME J. Turbomach.
0889-504X,
127
(
4
), pp.
649
658
.
24.
Gerolymos
,
G. A.
Neubauer
,
J.
Sharma
,
V. C.
, and
Vallet
,
I.
, 2002, “
Improved Prediction of Turbomachinery Flows Using Near-Wall Reynolds-Stress Model
,”
ASME J. Turbomach.
0889-504X,
124
(
1
), pp.
86
99
.
25.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2002, “
Wall-Normal-Free Reynolds-Stress Model for Rotating Flows Applied to Turbomachinery
,”
AIAA J.
0001-1452,
40
(
2
), pp.
199
208
.
26.
Rautaheimo
,
P. P.
Salminen
,
E. J.
, and
Sikonen
,
T. L.
, 2003, “
Numerical Simulation of the Flow in the NASA Low-Speed Centrifugal Compressor
,”
Int. J. Turbo Jet Engines
0334-0082,
20
, pp.
155
170
.
27.
Chriss
,
R. M.
Hathaway
,
M. D.
, and
Wood
,
J. R.
, 1996, “
Experimental and Computational Results From the NASA Lewis Low-Speed Centrifugal Impeller at Design and Part-Flow Conditions
,”
ASME J. Turbomach.
0889-504X,
118
(
1
), pp.
55
65
.
28.
Speziale
,
C. G.
Sarkar
,
S.
, and
Gatski
,
T. B.
, 1991, “
Modelling the Pressure-Strain Correlation of Turbulence: An Invariant Dynamical Systems Approach
,”
J. Fluid Mech.
0022-1120,
227
, pp.
245
272
.
29.
Shima
,
N.
, 1988, “
A Reynolds-Stress Model for Near-Wall and Low-Reynolds-Number Regions
,”
ASME J. Fluids Eng.
0098-2202,
110
, pp.
38
44
.
30.
Borello
,
D.
Hanjalić
,
K.
, and
Rispoli
,
F.
, 2005, “
Prediction of Cascade Flows With Innovative Second-Moment Closures
,”
ASME J. Fluids Eng.
0098-2202,
127
, pp.
1059
1070
.
31.
Manceau
,
R.
, and
Hanjalić
,
K.
, 2002, “
Elliptic Blending Model: A New Near-Wall Reynolds-Stress Turbulence Closure
,”
Phys. Fluids
1070-6631,
14
(
2
), pp.
744
754
.
32.
Borello
,
D.
Rispoli
,
F.
, and
Hanjalić
,
K.
, 2006, “
Prediction of Tip-Leakage Flow in Axial Flow Compressor With Second Moment Closure
,” June, ASME Paper No. GT–2006–90535.
33.
Patel
,
V. G.
Rodi
,
W.
, and
Scheuerer
,
G.
, 1985, “
Turbulence Models for Near-Wall and Low-Reynolds-Number Flows: A Review
,”
AIAA J.
0001-1452,
23
, pp.
1308
1319
.
34.
Sauret
,
E.
, and
Vallet
,
I.
, 2007, “
Near-Wall Turbulent Pressure Diffusion Modelling and Influence in 3-D Secondary Flows
,”
ASME J. Fluids Eng.
0098-2202,
129
(
5
), pp.
634
642
.
35.
Gerolymos
,
G. A.
Tsanga
,
G.
, and
Vallet
,
I.
, 1998, “
Near-Wall k−ε Computation of 3-D Transonic Turbomachinery Flows With Tip-Clearance
,”
AIAA J.
0001-1452,
36
(
10
), pp.
1769
1777
.
36.
Gerolymos
,
G. A.
, and
Hanisch
,
C.
, 1999, “
Multistage 3-D Navier-Stokes Computation of Off-Design Operation of a 4-Stage Turbine
,”
Proc. Inst. Mech. Eng., Part A
0957-6509,
213
, pp.
243
261
.
37.
Anderson
,
W. K.
Thomas
,
J. L.
, and
Van Leer
,
B.
, 1986, “
Comparison of Finite-Volume Flux-Vector-Splittings for the Euler Equations
,”
AIAA J.
0001-1452,
24
(
9
), pp.
1453
1460
.
38.
Schumann
,
U.
, 1977, “
Realizability of Reynolds-Stress Turbulence Models
,”
Phys. Fluids
0031-9171,
20
, pp.
721
725
.
39.
Jameson
,
A.
, 1983, “
Solution of the Euler Equations for 2-D Transonic Flow by a Multigrid Method
,”
Appl. Math. Comput.
0096-3003,
13
, pp.
327
355
.
40.
Mottura
,
L.
Vigevano
,
L.
, and
Zaccanti
,
M.
, 2000, “
Factorized Implicit Upwind Methods Applied to Inviscid Flows at High Mach Number
,”
AIAA J.
0001-1452,
38
(
10
), pp.
1846
1852
.
41.
Chakravarthy
,
S. R.
, 1983, “
Euler Equations—Implicit Schemes and Boundary Conditions
,”
AIAA J.
0001-1452,
21
(
5
), pp.
699
706
.
42.
Giles
,
M. B.
, 1990, “
Nonreflecting Boundary Conditions for Euler Equation Calculations
,”
AIAA J.
0001-1452,
28
(
12
), pp.
2050
2058
.
43.
Goyal
,
R. K.
, and
Dawes
,
W. N.
, 1993, “
A Comparison of the Measured and Predicted Flowfield in a Modern Fan-Bypass Configuration
,”
ASME J. Turbomach.
0889-504X,
115
, pp.
273
282
.
44.
Chima
,
R. V.
, 1998, “
Calculation of Tip Clearance Effects in a Transonic Compressor Rotor
,”
ASME J. Turbomach.
0889-504X,
120
(
1
), pp.
131
140
.
45.
Saxer
,
A. P.
, and
Giles
,
M. B.
, 1994, “
Predictions of 3-D Steady and Unsteady Inviscid Transonic Stator/Rotor Interaction With Inlet Radial Temperature Nonuniformity
,”
ASME J. Turbomach.
0889-504X,
116
, pp.
347
357
.
46.
Gerolymos
,
G. A.
,
Neubauer
,
J.
, and
Michon
,
G. J.
, 2002, “
Analysis and Application of Chorochronic Periodicity for Turbomachinery Rotor/Stator Interaction Computations
,”
J. Propul. Power
0748-4658,
18
(
2
), pp.
1139
1152
.
47.
Leclercq
,
M. P.
, and
Stoufflet
,
P. L.
, 1993, “
Characteristic Multigrid Method Application to Solve the Euler Equations With Unstructured and Unnested Grids
,”
J. Comput. Phys.
0021-9991,
104
, pp.
329
346
.
48.
Davis
,
R. L.
,
Delaney
,
R. A.
,
Denton
,
J. D.
,
Giles
,
M. B.
,
Strazisar
,
A. J.
, and
Wisler
,
D. C.
, 1993, “
CFD Code Assessment in Turbomachinery—Author’s Information Package
,” ASME Turbomachinery Committee, ASME–IGTI, Atlanta.
49.
Strazisar
,
A. J.
, 1994, “
Data Report and Data Diskette for NASA Transonic Compressor Rotor 37
,” NASA, Lewis Research Center, Cleveland, Nov.
50.
Arnaud
,
D.
,
Ottavy
,
X.
, and
Vouillarmet
,
A.
, 2004, “
Experimental Investigation of the Rotor/Stator Interactions Within a High-Speed Multistage Axial Compressor—Part 1: Experimental Facilities and Results
,” ASME Paper No. 2004–GT–53764.
51.
Arnaud
,
D.
,
Ottavy
,
X.
, and
Vouillarmet
,
A.
, 2004, “
Experimental Investigation of the Rotor/Stator Interactions Within a High-Speed Multistage Axial Compressor—Part 2: Modal Analysis of the Interactions
,” ASME Paper No. 2004–GT–53778.
52.
Touyeras
,
A.
, and
Villain
,
M.
, 2004, “
Aerodynamic Design and Test Result Analysis of a 3-Stage Research Compressor
,” June, ASME Paper No. GT2004–53940.
53.
Shabbir
,
A.
,
Celestina
,
M. L.
,
Adamczyk
,
J. J.
, and
Strazisar
,
A. J.
, 1997, “
The Effect of Hub Leakage on 2 High Speed Axial Flow Compressor Rotors
,” June, ASME Paper No. 97–GT–346.
54.
Toro
,
E. F.
, 1997,
Riemann Solvers and Numerical Methods for Fluid Dynamics
,
Springer-Verlag
, Berlin.
55.
Suder
,
K. L.
, 1998, “
Blockage Development in a Transonic Axial Compressor Rotor
,”
ASME J. Turbomach.
0889-504X,
120
, pp.
465
476
.
56.
van de Wall
,
A. G.
,
Kadambi
,
J. R.
, and
Adamczyk
,
J. J.
, 2000, “
A Transport Model for the Deterministic Stresses Associated With Turbomachinery Blade-Row Interactions
,”
ASME J. Turbomach.
0889-504X,
122
(
4
), pp.
593
603
.
57.
Hanjalić
,
K.
, 2005, “
Will RANS Survive LES? A View of Perspectives
,”
ASME J. Fluids Eng.
0098-2202,
127
, Sept., pp.
831
839
.
58.
Chaouat
,
B.
, and
Schiestel
,
R.
, 2005, “
A New Partially Integrated Transport Model for Subgrid-Scale Stresses and Dissipation Rates for Turbulent Developing Flows
,”
Phys. Fluids
1070-6631,
17
(
6
), pp.
065106
-1–
19
.
59.
Cornelius
,
C.
,
Volgmann
,
W.
, and
Stoff
,
H.
, 1999, “
Calculation of 3-D Turbulent Flow With a Finite Volume Multigrid Method
,”
Int. J. Numer. Methods Fluids
0271-2091,
31
, pp.
703
720
.
60.
Lien
,
F. S.
, and
Leschziner
,
M. A.
, 2003, “
Multigrid Convergence Acceleration for Complex Flow Including Turbulence
,”
Multigrid Methods III
,
Hackbusch
,
W.
, and
Trottenberg
,
U.
, eds.,
International Series on Numerical Mathematics
, No. 98,
Birkhäuser
, pp.
277
288
.
You do not currently have access to this content.