The purpose of this paper is to assess the importance of the explicit dependence of turbulent diffusion on the gradients of mean-velocity modeling in second moment closures on three-dimensional (3D) detached and secondary flows prediction. Following recent theoretical work of Younis, Gatski, and Speziale, 2000, [Proc. Royal Society Lon. A, 456, pp. 909–920], we propose a triple-velocity correlation model, including the effects of the spatial gradients of mean velocity. A model for both the slow and rapid parts of the pressure-diffusion term was also developed and added to a wall-normal-free Reynolds-stress model. The present model is validated against 3D detached and secondary flows. Further developments, especially on the echo terms (which should appear in the formulation of pressure-velocity correlation), are discussed.

1.
Daly
,
B. J.
, and
Harlow
,
F. H.
, 1970, “
Transport Equations in Turbulence
,”
Phys. Fluids
0031-9171,
13
, pp.
2634
2649
.
2.
Hanjalić
,
K.
, and
Launder
,
B. E.
, 1972, “
A Reynolds Stress Model of Turbulence and its Application to Thin Shear Flows
,”
J. Fluid Mech.
0022-1120,
52
, pp.
609
638
.
3.
Lumley
,
J. L.
, 1978, “
Computational Modeling of Turbulent Flows
,”
Adv. Appl. Mech.
0065-2156,
18
, pp.
123
176
.
4.
Hirt
,
C. W.
, 1969, “
Generalized Turbulence Transport Equations
,”
Int Sem. of Int. Center for Heat and Mass Transfer
, Herceg Novi, Yugoslavia.
5.
Cormack
,
D. E.
,
Leal
,
L. G.
, and
Seinfeld
,
J. H.
, 1978, “
An Evaluation of Mean Reynolds Stress Turbulence Models: The Triple Velocity Correlation
,”
ASME J. Fluids Eng.
0098-2202,
100
, pp.
47
54
.
6.
Amano
,
R. S.
, and
Goel
,
P.
, 1986, “
Triple-Velocity Products in a Channel With Backward Facing Step
,”
AIAA J.
0001-1452,
24
, pp.
1040
1043
.
7.
Schwarz
,
W. R.
, and
Bradshaw
,
P.
, 1994, “
Term-by-Term Tests of Stress-Transport Turbulence Models in a 3-D Boundary Layer
,”
Phys. Fluids
1070-6631,
6
, pp.
986
998
.
8.
Hanjalić
,
K.
, 1994, “
Advanced Turbulence Closure Models: A View of Current Status and Future Prospects
,”
Int. J. Heat Fluid Flow
0142-727X,
15
, pp.
178
203
.
9.
Kim
,
J.
,
Moin
,
P.
, and
Moser
,
R.
, 1987, “
Turbulence Statistics in Fully Developed Channel Flow at Low-Reynolds-Number
,”
J. Fluid Mech.
0022-1120,
177
, pp.
133
166
.
10.
Moser
,
R. D.
,
Kim
,
J.
, and
Mansour
,
N. N.
, 1999, “
Direct Numerical Simulation of Turbulent Channel Flow up to Reτ=590
,”
Phys. Fluids
1070-6631,
11
, pp.
943
945
.
11.
Gatski
,
T. B.
, 2004, “
Constitutive Equations for Turbulent Flows
,”
Theor. Comput. Fluid Dyn.
0935-4964,
18
, pp.
345
369
.
12.
Younis
,
B. A.
,
Gatski
,
T. B.
, and
Speziale
,
C. G.
, 2000, “
Towards a Rational Model for the Triple Velocity Correlations of Turbulence
,”
Proc. R. Soc. London, Ser. A
1364-5021,
456
, pp.
909
920
.
13.
Fu
,
S.
, 1993, “
Modelling of the Pressure-Velocity Correlation in Turbulence Diffusion
,”
Comput. Fluids
0045-7930,
22
, pp.
199
205
.
14.
Straatman
,
A. G.
, 1999, “
A Modified Model for Diffusion in Second-Moment Turbulence Closures
,”
ASME J. Fluids Eng.
0098-2202,
121
, pp.
747
756
.
15.
Demuren
,
A. O.
,
Rogers
,
M. M.
,
Durbin
,
P.
, and
Lele
,
S. K.
, 1996, “
On Modelling Pressure Diffusion in Non-homogeneous Shear Flows
,”
Studying Turbulence Using Numerical Simulation Databases-VI, Proceeding of the CTR Summer Program 1996
, Center for Turbulence Research,
NASA Ames/Stanford University
.
16.
Sauret
,
E.
, and
Vallet
,
I.
, 2007, “
Near-Wall Turbulent Pressure Diffusion Modeling and Influence in 3-D Secondary Flows
,”
ASME J. Fluids Eng.
0098-2202, to be published.
17.
Suga
,
K.
, 2004, “
Modeling the Rapid Part of the Pressure-Diffusion Process in the Reynolds Stress Transport Equation
,”
ASME J. Fluids Eng.
0098-2202,
126
, pp.
634
641
.
18.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2001, “
Wall-Normal-Free Near-Wall Reynolds-Stress Closure for 3-D Compressible Separated Flows
,”
AIAA J.
0001-1452,
39
, pp.
1833
1842
.
19.
Gerolymos
,
G. A.
,
Sauret
,
E.
, and
Vallet
,
I.
, 2004, “
Contribution to the Single-Point-Closure Reynolds-Stress Modelling of Inhomogeneous Flow
,”
Theor. Comput. Fluid Dyn.
0935-4964,
17
, pp.
407
431
.
20.
Launder
,
B. E.
, and
Sharma
,
B. I.
, 1974, “
Application of the Energy Dissipation Model of Turbulence to the Calculation of Flows Near a Spinning Disk
,”
Lett. Heat Mass Transfer
0094-4548,
1
, pp.
131
138
.
21.
Launder
,
B. E.
,
Tselepidakis
,
D. P.
, and
Younis
,
B. A.
, 1987, “
A Second-Moment Closure Study of Rotating Channel Flow
,”
J. Fluid Mech.
0022-1120,
183
, pp.
63
75
.
22.
Shir
,
C. C.
, 1973, “
A Preliminary Numerical Study of Atmospheric Turbulent Flows in the Idealized Planetary Boundary Layer
,”
J. Atmos. Sci.
0022-4928,
30
, pp.
1327
1339
.
23.
Gibson
,
M. M.
, and
Launder
,
B. E.
, 1978, “
Ground Effects on Pressure Fluctuations in the Atmospheric Boundary-Layer
,”
J. Fluid Mech.
0022-1120,
86
, pp.
491
511
.
24.
Craft
,
T. J.
, and
Launder
,
B.
, 1996, “
A Reynolds-Stress Model Designed for Complex Geometries
,”
Int. J. Heat Fluid Flow
0142-727X,
17
, pp.
245
254
.
25.
Rotta
,
J.
, 1951, “
Statistische Theorie Nichthomogener Turbulenz—1. Mitteilung
,”
Chem. Eng. Prog.
0360-7275,
129
, pp.
547
572
.
26.
Naot
,
D.
,
Shavit
,
A.
, and
Wolfshtein
,
M.
, 1970, “
Interactions Between Components of the Turbulent Velocity Correlation Tensor Due to Pressure Fluctuations
,”
Isr. J. Technol.
0021-2202,
8
, pp.
259
269
.
27.
Launder
,
B. E.
, and
Shima
,
N.
, 1989, “
2-Moment Closure for the Near-Wall Sublayer: Development and Application
,”
AIAA J.
0001-1452,
27
, pp.
1319
1325
.
28.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 1997, “
Near-Wall Reynolds-Stress 3-D Transonic Flows Computation
,”
AIAA J.
0001-1452,
35
, pp.
228
236
.
29.
Lumley
,
J. L.
, 1979, “
Second Order Modeling of Turbulent Flows
,”
Prediction Methods for Turbulent Flows
,
VKI Lecture Series
, von Kármán Institute for Fluid Dynamics, 72 Chaussée de Waterloo, 1640 Rhode-sur-Génèse.
30.
Launder
,
B. E.
,
Reynolds
,
W. C.
,
Rodi
,
W.
,
Mathieu
,
J.
, and
Jeandel
,
D.
, 1984,
Turbulence Models and Their Applications II
,
No. 56 in Collection de la Direction des Etudes et Recherches d’Electricité de France
,
Eyrolles
,
Paris
.
31.
Chou
,
P. Y.
, 1945, “
On Velocity Correlations and the Solutions of the Equations of Turbulent fluctuations
,”
Q. Appl. Math.
0033-569X,
3
, pp.
38
54
.
32.
Jovanović
,
J.
, 2004,
The Statistical Dynamics of Turbulence
,
Springer
,
New York
.
33.
Piquet
,
J.
, 1999,
Turbulent Flows—Models and Physics
,
Springer-Verlag
,
Berlin
.
34.
Pope
,
S. B.
, 2000,
Turbulent Flows
,
Cambridge University Press
,
Cambridge
.
35.
Le
,
H.
,
Moin
,
P.
, and
Kim
,
J.
, 1997, “
Direct Numerical Simulation of Turbulent Flow Over a Backward-Facing Step
,”
J. Fluid Mech.
0022-1120,
330
, pp.
349
374
.
36.
Yao
,
Y. F.
,
Thomas
,
T. G.
, and
Sandham
,
N. D.
, 2001, “
Direct Numerical Simulation of Turbulent Flow Over a Rectangular Trailing Edge
,”
Theor. Comput. Fluid Dyn.
0935-4964,
14
, pp.
337
358
.
37.
Mellor
,
G. L.
, and
Herring
,
H. J.
, 1973, “
A Survey of the Mean Turbulent Field Closure Models
,”
AIAA J.
0001-1452,
11
, pp.
590
599
.
38.
Rogers
,
M. M.
, and
Moser
,
R. D.
, 1994, “
Direct Simulation of a Self-Similar Turbulent Mixing Layer
,”
Phys. Fluids
1070-6631,
6
, pp.
903
923
.
39.
Gessner
,
F. B.
, and
Emery
,
A. F.
, 1981, “
The Numerical Prediction of Developing Turbulent Flow in Rectangular Ducts
,”
ASME J. Fluids Eng.
0098-2202,
103
, pp.
445
455
.
40.
Wellborn
,
S. R.
,
Reichert
,
B. A.
, and
Okiishi
,
T. H.
, 1994, “
Study of the Compressible Flow in a Diffusing S-Duct
,”
J. Propul. Power
0748-4658,
10
, pp.
668
675
.
41.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2005, “
Mean-Flow-Multigrid for Implicit Reynolds-Stress-Model Computations
,”
AIAA J.
0001-1452,
43
, pp.
1887
1898
.
42.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2007, “
Advances in the Numerical Computation of Complex Flows Using Reynolds-Stress Models
,”
AIAA Computational Fluid Dynamics Conference
, Jun 25–28, Miami.
43.
So
,
R. M. C.
, and
Yuan
,
S. P.
, 1999, “
A Geometry Independent Near-Wall Reynolds-Stress Closure
,”
Int. J. Eng. Sci.
0020-7225,
37
, pp.
33
57
.
44.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2007, “
Reynolds-stress Model Computations of Aircraft Engine Inlet s-Duct Flows
,”
AIAA Fluid Dynamics Conference
, Jun 25–28, Miami.
45.
Gerolymos
,
G. A.
,
Sauret
,
E.
, and
Vallet
,
I.
, 2004, “
Influence of Inflow-Turbulence in Shock-Wave/Turbulent-Boundary-Layer Interaction Computations
,”
AIAA J.
0001-1452,
42
, pp.
1101
1106
.
46.
Fujii
,
S.
, and
Okiishi
,
T. H.
, 1972, “
Curved Diffusing Annulus Turbulent Boundary-Layer Development
,”
J. Aircr.
0021-8669,
9
, pp.
97
98
.
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