The purpose of the present note is to contribute in clarifying the relation between representation bases used in the closure for the redistribution (pressure-strain) tensor φij, and to construct representation bases whose elements have clear physical significance. The representation of different models in the same basis is essential for comparison purposes, and the definition of the basis by physically meaningful tensors adds insight to our understanding of closures. The rate-of-production tensor can be split into production by mean strain and production by mean rotation $Pij=PS¯ij+PΩ¯ij$. The classic representation basis $B[b,S¯,Ω¯]$ of homogeneous turbulence [e.g. Ristorcelli, J. R., Lumley, J. L., Abid, R., 1995, “A Rapid-Pressure Covariance Representation Consistent with the Taylor-Proudman Theorem Materially Frame Indifferent in the 2-D Limit,” J. Fluid Mech., 292, pp. 111–152], constructed from the anisotropy $b$, the mean strain-rate $S¯$, and the mean rotation-rate $Ω¯$ tensors, is interpreted, in the present work, in terms of the relative contributions of the deviatoric tensors $PS¯ij(dev):=PS¯ij-23Pkδij$ and $PΩ¯ij(dev):=PΩ¯ij$. Different alternative equivalent representation bases, explicitly using $PS¯ij(dev)$ and $PΩ¯ij$ are discussed, and the projection rules between bases are calculated using a matrix-based systematic procedure. An initial term-by-term a priori investigation of different second-moment closures is undertaken.

## References

1.
Rotta
,
J.
, 1951, “
Statistische Theorie nichthomogener Turbulenz - 1. Mitteilung
,”
Z. Phys.
,
129
, pp.
547
572
.
2.
Reynolds
,
W. C.
, 1974, “
Recent Advances in the Computation of Turbulent Flows
,”
,
9
, pp.
193
246
.
3.
Lumley
,
J. L.
, 1978, “
Computational Modeling of Turbulent Flows
,”
,
18
, pp.
123
176
.
4.
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.
,
227
, pp.
245
272
.
5.
Chou
,
P. Y.
, 1945, “
On Velocity Correlations and the Solutions of the Equations of Turbulent Fluctuations
,”
Am. Math. Soc.
,
3
, pp.
38
54
.
6.
Ristorcelli
,
J. R.
,
Lumley
,
J. L.
, and
Abid
,
R.
, 1995, “
A Rapid-Pressure Covariance Representation Consistent with the Taylor-Proudman Theorem Materially Frame Indifferent in the 2-D Limit
,”
J. Fluid Mech.
,
292
, pp.
111
152
.
7.
Fu
,
S.
, and
Wang
,
C.
, 1997, “
Second-Moment Closure Modelling of Turbulence in a Non-Inertial Frame
,”
Fluid Dyn. Res.
,
20
, pp.
43
65
.
8.
Naot
,
D.
,
Shavit
,
A.
, and
Wolfshtein
,
M.
, 1973, “
2-Point-Correlation Model and the Redistribution of Reynolds-Stresses
,”
Phys. Fluids
,
16
(
6
), pp.
738
743
.
9.
Launder
,
B. E.
,
Reece
,
G. J.
, and
Rodi
,
W.
, 1975, “
Progress in the Development of a Reynolds-Stress Turbulence Closure
,”
J. Fluid Mech.
,
68
, pp.
537
566
.
10.
Launder
,
B. E.
, 1989, “
Second-Moment Closure: Present And Future?
,”
Int. J. Heat Fluid Flow
,
10
, pp.
282
300
.
11.
Naot
,
D.
,
Shavit
,
A.
, and
Wolfshtein
,
M.
, 1970, “
Interactions between Components of the Turbulent Velocity Correlation Tensor due to Pressure Fluctuations
,”
Israel J. Technol.
,
8
(
3
), pp.
259
269
.
12.
Dafalias
,
Y. F.
, and
Younis
,
B. A.
, 2009, “
Objective Model for the Fluctuating Pressure-Strain-Rate Correlations
,”
ASCE J. Eng. Mech.
,
135
(
9
), pp.
1006
1014
.
13.
Fu
,
S.
, 1988, “
Computational Modelling of Turbulent Swirling Flows with Second-Moment Closures
,” Ph. D. thesis, University of Manchester Institute of Science and Technology, Manchester, Great Britain.
14.
Craft
,
T. J.
, and
Launder
,
B.
, 2001, “
Principles and Performance of TCL-Based Second-Moment Closures
,”
Flow Turb. Comb.
,
66
, pp.
355
372
.
15.
Lo
,
C.
, 2011, “
Fermeture de la Turbulence au Second-Ordre Proche Paroi Basée sur L’analyse des Données DNS
,” Ph. D. thesis, Université Pierre-et-Marie-Curie, Paris, France.
16.
Shima
,
N.
, 1998, “
Low-Reynolds-Number Second-Moment Closure without Wall-Reflection Redistribution Terms
,”
Int. J. Heat Fluid Flow
,
19
, pp.
549
555
.
17.
Rivlin
,
R. S.
, 1955, “
The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua
,”
Indiana Univ. Math. J.
,
4
, pp.
681
702
.
18.
Spencer
,
A. J. M.
, and
Rivlin
,
R. S.
, 1959, “
The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua
,”
Arch. Rat. Mech. Anal.
,
2
, pp.
309
336
.
19.
Smith
,
G. F.
, 1971, “
On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors
,”
Int. J. Eng. Sci.
,
9
, pp.
899
916
.
20.
Rivlin
,
R. S.
, and
Ericksen
,
J. L.
, 1955, “
Stress-Deformation Relations for isotropic Materials
,”
Indiana Univ. Math. J.
,
4
, pp.
323
425
.
21.
Gerolymos
,
G. A.
,
Sauret
,
E.
, and
Vallet
,
I.
, 2004, “
Contribution to the Single-Point-Closure Reynolds-Stress Modelling of Inhomogeneous Flow
,”
Theor. Comp. Fluid Dyn.
,
17
(
5-6
), pp.
407
431
.
22.
Gibson
,
M. M.
, and
Launder
,
B. E.
, 1978, “
Ground Effects on Pressure Fluctuations in the Atmospheric Boundary-Layer
,”
J. Fluid Mech.
,
86
, pp.
491
511
.
23.
So
,
R. M. C.
,
Aksoy
,
H.
,
Yuan
,
S. P.
, and
Sommer
,
T. P.
, 1996, “
Modeling Reynolds-Number Effects in Wall-Bounded Turbulent Flows
,”
ASME J. Fluids Eng.
,
118
, pp.
260
267
.
24.
Simonsen
,
A. J.
, and
,
P. Å.
, 2005, “
Turbulent Stress Invariant Analysis: Classification of Existing Terminology
,”
Phys. Fluids
,
17
, pp.
088103(1
4)
.
25.
MAXIMA. A Computer Algebra System. http://maxima.sourceforge.net
26.
Crow
,
S. C.
, 1968, “
A Suggestion for the Numerical Computation of the Steady Navier-Stokes Equations
,”
J. Fluid Mech.
,
33
, pp.
1
20
.
27.
Hoyas
,
S.
, and
Jiménez
,
J.
, 2008, “
Reynolds Number Effects on the Reynolds-Stress Budgets in Turbulent Channels
,”
Phys. Fluids
,
20
, pp.
101511(1
8)
.
28.
Gerolymos
,
G. A.
, and
Vallet
,
I.
, 2001, “
Wall-Normal-Free Near-Wall Reynolds-Stress Closure for 3-D Compressible Separated Flows
,”
AIAA J.
,
39
(
10
), pp.
1833
1842
.
You do not currently have access to this content.