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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