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# Tensorial Representations of Reynolds-Stress Pressure-Strain Redistribution

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

Université Pierre-et-Marie-Curie (UPMC), 4 place Jussieu, 75005 Paris, Francegeorges.gerolymos@upmc.fr

C. Lo

Université Pierre-et-Marie-Curie (UPMC), 4 place Jussieu, 75005 Paris, Franceceline.lo@upmc.fr

I. Vallet

Université Pierre-et-Marie-Curie (UPMC), 4 place Jussieu, 75005 Paris, Franceisabelle.vallet@upmc.fr

Notice that Launder [10] has suggested to further include the advection tensor $Cij:=ρDtui'uj'¯$ in the representation, but this has not become standard practice. In the case of spatially evolving stationary quasi-homogeneous turbulence, $Cij=ρu¯ℓ∂xℓui'uj'¯≠0$, while in DNS studies of time-evolving spatially homogeneous turbulence, $Cij=ρ∂tui'uj'¯≠0$, implying that the suggestion of including $Cij$ in the representation merits further study.

Launder et al.  [9] attribute to Crow [Ref. 26, Eq. (3.6), p. 7] the behavior of the rapid part of redistribution at the limit of isotropic turbulence, and this has been since repeated by several authors [7,12]. Notice, however, [8] that the same constraint had also been given by Rotta [Ref. 1, p. 558].

J. Appl. Mech 79(4), 044506 (May 11, 2012) (10 pages) doi:10.1115/1.4005558 History: Received January 14, 2011; Revised June 21, 2011; Posted January 30, 2012; Published May 11, 2012; Online May 11, 2012

## Abstract

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.

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## Figures

Figure 1

Locus, in the (IIIb ,−IIb )-plane of the invariants of the anisotropy tensor [3], of the condition -23IIb=827-IIIb for which the representation T⋆(n)=∑m=18aTFnmF⋆(n) (Eq. 10) of the mean strain-rate tensor S⋆ in B[b,P⋆S¯,P⋆Ω¯] would be singular (the line -23IIb=827-IIIb lies outside of Lumley’s realizability triangle [3,24], but includes the 1-C point)

Figure 2

A priori term-by-term analysis of SMCs (LRR [9], LRR-IP [9], SSG [4], DY [12]) for the components of the redistribution tensor φij , eventually augmented by the anisotropy of dissipation (φij-2ρɛbɛij:=φij-ρ(ɛij(μ)-23ɛ)), with respect to DNS data [27] for the log and outer regions (y+  ≥ 30) of fully developed incompressible plane channel flow at friction-Reynolds-number Reτw=2003 [27]

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