Tensorial representations of Reynolds-stress pressure-strain redistribution

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 $\phi_{ij}$, 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 meaningfull 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 $P_{ij}=P_{\bar S_{ij}}+P_{\bar\Omega_{ij}}$. The classic representation basis $\mathfrak{B}[\tsr{b}, \tsrbar{S}, \tsrbar{\Omega}]$ of homogeneous turbulence {\em [{\em eg} Ristorcelli J.R., Lumley J.L., Abid R.: {\it J. Fluid Mech.} {\bf 292} (1995) 111--152]}, constructed from the anisotropy $\tsr{b}$, the mean strain-rate $\tsrbar{S}$, and the mean rotation-rate $\tsrbar{\Omega}$ tensors, is interpreted, in the present work, in terms of the relative contributions of the deviatoric tensors $P^{(\mathrm{dev})}_{\bar S_{ij}}:=P_{\bar S_{ij}}-\tfrac{2}{3}P_\mathrm{k}\delta_{ij}$ and $P^{(\mathrm{dev})}_{\bar\Omega_{ij}}:=P_{\bar\Omega_{ij}}$. Different alternative equivalent representation bases, explicitly using $P^{(\mathrm{dev})}_{\bar S_{ij}}$ and $P_{\bar\Omega_{ij}}$ are discussed, and the projection rules between bases are caclulated, using a matrix-based systematic procedure. An initial term-by-term {\em a priori} investigation of different second-moment closures is undertaken.