From consideration of turbulence anisotropy dynamics due to spatial or temporal variations in the mean strain rate, a new Reynolds stress closure for nonequilibrium effects in turbulent flows has been developed. This closure, formally derived from the Reynolds stress anisotropy transport equation, results in an effective strain rate tensor that accounts for the strain rate history to which the turbulence has been subjected. In contrast to prior nonequilibrium models that have sought to address nonequilibrium effects via changes in the eddy viscosity, the present approach accounts for nonequilibrium effects in the fundamental relation between the anisotropy tensor and the strain rate tensor. The time-local form of the nonequilibrium closure can be readily implemented in place of the classical equilibrium Boussinesq closure on which most existing computational frameworks are currently based. This new closure is applied here to four substantially different classes of nonequilibrium test problems. Results show dramatically improved agreement with experimental and computational data, without the need to vary any model parameters, when compared with the standard equilibrium closure and with various prior nonequilibrium closures.
REFERENCES
1.
C. G.
Speziale
, “Analytical methods for the development of Reynolds-stress closures in turbulence
,” Annu. Rev. Fluid Mech.
23
, 107
(1991
).2.
T. B.
Gatski
, “Constitutive equations for turbulent flows
,” Theor. Comput. Fluid Dyn.
18
, 345
(2004
).3.
A. V.
Johansson
, “Engineering turbulence models and their development, with emphasis on explicit algebraic Reynolds stress models
,” in Theories of Turbulence: CISM Courses and Lectures No. 442
(Springer
, New York
, 2002
), pp. 253
–300
.4.
J. L.
Lumley
, “Toward a turbulent constitutive relation
,” J. Fluid Mech.
41
, 413
(1970
).5.
T. B.
Gatski
and C. G.
Speziale
, “On explicit algebraic stress models for complex turbulent flows
,” J. Fluid Mech.
254
, 59
(1993
).6.
S. S.
Girimaji
, “Fully explicit and self-consistent algebraic Reynolds stress model
,” Theor. Comput. Fluid Dyn.
8
, 387
(1996
).7.
S.
Wallin
and A. V.
Johansson
, “An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows
,” J. Fluid Mech.
403
, 89
(2000
).8.
D.
Yu
and S. S.
Girimaji
, “Direct numerical simulations of homogeneous turbulence subject to periodic shear
,” J. Fluid Mech.
566
, 117
(2006
).9.
J.
Bardina
, J. H.
Ferziger
, and W. C.
Reynolds
, “Improved turbulence models based on large-eddy simulation of homogeneous, incompressible turbulent flows
,” Stanford University
Report No. TF-19, 1983
;J.
Bardina
, “Improved turbulence models based on large eddy simulation of homogeneous, incompressible, turbulent flows
,” Ph.D. thesis, Stanford University
, 1983
.10.
I.
Hadzic
, K.
Hanjalic
, and D.
Laurence
, “Modeling the response of turbulence subjected to cyclic irrotational strain
,” Phys. Fluids
13
, 1739
(2001
).11.
J.
Chen
, C.
Meneveau
, and J.
Katz
, “Scale interactions of turbulence subjected to a straining-relaxation-destraining cycle
,” J. Fluid Mech.
562
, 123
(2006
).12.
B. E.
Launder
and D. B.
Spalding
, “The numerical computation of turbulent flows
,” Comput. Methods Appl. Mech. Eng.
3
, 269
(1974
).13.
14.
F. R.
Menter
, “Two-equation eddy-viscosity turbulence models for engineering applications
,” AIAA J.
32
, 1598
(1994
).15.
J.
Kim
, P.
Moin
, and R.
Moser
, “Turbulence statistics in fully developed channel flow at low Reynolds number
,” J. Fluid Mech.
177
, 133
(1987
).16.
Z. S.
She
, E.
Jackson
, and S. A.
Orszag
, “Structure and dynamics of homogeneous turbulence: Models and simulations
,” Proc. R. Soc. London, Ser. A
434
, 101
(1991
).17.
K. K.
Nomura
and G. K.
Post
, “The structure and dynamics of vorticity and rate of strain in incompressible homogeneous turbulence
,” J. Fluid Mech.
377
, 65
(1998
).18.
J. G.
Brasseur
and W.
Lin
, “Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence
,” Fluid Dyn. Res.
36
, 357
(2005
).19.
P. E.
Hamlington
, J.
Schumacher
, and W. J. A.
Dahm
, “Local and nonlocal strain rate fields and vorticity alignment in turbulent flows
,” Phys. Rev. E
77
, 026303
(2008
).20.
K. A.
Buch
and W. J. A.
Dahm
, “Experimental study of the fine-scale structure of conserved scalar mixing in turbulent flows. Part 1.
,” J. Fluid Mech.
317
, 21
(1996
).21.
S. B.
Pope
, Turbulent Flows
(Cambridge University Press
, Cambridge
, 2000
).22.
C. G.
Speziale
, S.
Sarkar
, and T. B.
Gatski
, “Modeling the pressure strain correlation of turbulence: An invariant dynamical systems approach
,” J. Fluid Mech.
227
, 245
(1991
).23.
S. B.
Pope
, “A more general effective-viscosity hypothesis
,” J. Fluid Mech.
72
, 331
(1975
).24.
W.
Rodi
, “A new algebraic relation for calculating the Reynolds stresses
,” Z. Angew. Math. Mech.
56
, T219
(1976
).25.
A.
Yoshizawa
, “Statistical analyis of the deviation of the Reynolds stress from its eddy-viscosity representation
,” Phys. Fluids
27
, 1377
(1984
).26.
C. G.
Speziale
, “On nonlinear
and models of turbulence,” J. Fluid Mech.
178
, 459
(1987
).27.
R.
Rubinstein
and J. M.
Barton
, “Nonlinear Reynolds stress models and the renormalization group
,” Phys. Fluids A
2
, 1472
(1990
).28.
D. B.
Taulbee
, “An improved algebraic Reynolds stress model and corresponding nonlinear stress model
,” Phys. Fluids A
4
, 2555
(1992
).29.
V.
Yakhot
, S. A.
Orszag
, S.
Thangam
, T. B.
Gatski
, and C. G.
Speziale
, “Development of turbulence models for shear flows by a double expansion technique
,” Phys. Fluids A
4
, 1510
(1992
).30.
S.
Sarkar
and C. G.
Speziale
, “A simple nonlinear model for the return to isotropy in turbulence
,” Phys. Fluids A
2
, 84
(1990
).31.
C. G.
Speziale
and R. M. C.
So
, The Handbook of Fluid Dynamics
(Springer
, New York
, 1998
), Chap. 14, pp. 14
–1
14
–111
.32.
R. B.
Bird
, R. C.
Armstrong
, and O.
Hassager
, Dynamics of Polymeric Liquids
, 2nd ed. (Wiley
, New York
, 1987
).33.
J. C.
Rotta
, “Recent attempts to develop a generally applicable calculation method for turbulent shear flow layers
,” North Atlantic Treaty Organization
Report No. AGARD-CP-93, 1972
.34.
A.
Yoshizawa
and S.
Nisizima
, “A nonequilibrium representation of the turbulent viscosity based on a two-scale turbulence theory
,” Phys. Fluids A
5
, 3302
(1993
).35.
A.
Yoshizawa
, S.
Nisizima
, Y.
Shimomura
, H.
Kobayashi
, Y.
Matsuo
, H.
Abe
, and H.
Fujiwara
, “A new methodology for Reynolds-averaged modeling based on the amalgamation of heuristic-modeling and turbulence-theory methods
,” Phys. Fluids
18
, 035109
(2006
).36.
M. E.
Olsen
and T. J.
Coakley
, “The lag model, a turbulence model for nonequilibrium flows
,” AIAA
Paper No. 2001-2564, 2001
.37.
T.
Jongen
and T. B.
Gatski
, “A unified analysis of planar homogeneous turbulence using single-point closure equations
,” J. Fluid Mech.
399
, 117
(1999
).38.
C. G.
Speziale
and X. -H.
Xu
, “Towards the development of second-order closure models for non-equilibrium turbulent flows
,” Int. J. Heat Fluid Flow
17
, 238
(1996
).39.
S. C.
Crow
, “Viscoelastic properties of fine-grained incompressible turbulence
,” J. Fluid Mech.
33
, 1
(1968
).40.
B. A.
Pettersson-Reif
, T. B.
Gatski
, and C. L.
Rumsey
, “On the behavior of two-equation models in nonequilibrium homogeneous turbulence
,” Phys. Fluids
18
, 065109
(2006
).41.
B. E.
Launder
, G.
Reece
, and W.
Rodi
, “Progress in the development of a Reynolds stress turbulence closure
,” J. Fluid Mech.
68
, 537
(1975
).42.
M. M.
Gibson
and B. E.
Launder
, “Ground effects on pressure fluctuations in the atmospheric boundary layer
,” J. Fluid Mech.
86
, 491
(1978
).43.
A. G.
Fredrickson
, Principles and Applications of Rheology
(Prentice-Hall
, Englewood Cliffs, NJ
, 1964
).44.
J. D.
Goddard
and C.
Miller
, “An inverse for the Jaumann derivative and some applications to the rheology of viscoelastic fluids
,” Rheol. Acta
5
, 177
(1966
).45.
S.
Lee
, S. K.
Lele
, and P.
Moin
, “Interaction of isotropic turbulence with shock waves: Effect of shock strength
,” J. Fluid Mech.
340
, 225
(1997
).46.
K.
Mahesh
, S. K.
Lele
, and P.
Moin
, “The influence of entropy fluctuations on the interaction of turbulence with a shock wave
,” J. Fluid Mech.
334
, 353
(1997
).47.
K.
Sinha
, K.
Mahesh
, and G. V.
Candler
, “Modeling shock unsteadiness in shock/turbulence interaction
,” Phys. Fluids
15
, 2290
(2003
).48.
M. E.
Olsen
, R. P.
Lillard
, and T. J.
Coakley
, “The lag model applied to high speed flows
,” AIAA
Paper No. 2005-101, 2005
.49.
A. J.
Revell
, S.
Benhamadouche
, T.
Craft
, and D.
Laurence
, “A stress-strain lag eddy viscosity model for unsteady mean flow
,” Int. J. Heat Fluid Flow
27
, 821
(2006
).50.
Y. -N.
Huang
and H. -Y.
Ma
, “Reynolds stress model involving the mean spin tensor
,” Phys. Rev. E
70
, 036302
(2004
).© 2008 American Institute of Physics.
2008
American Institute of Physics
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