||Turbulent models provide closure equations that relate the Reynolds stress with kinematic
tensors. In this study, we present a methodology to quantify the dependence of the
Reynolds stress tensor on mean kinematic tensor basis. The methodology is based upon
tensor decomposition theorems which allow to extract from the anisotropic Reynolds
stress tensor the part that is linear or nonlinear in the strain rate tensor D, and the parts
that are in-phase (sharing the same eigenvectors) and out-of-phase with the strain rate.
The study was conducted using direct numerical simulation (DNS) data for turbulent
plane channel (from Reτ = 180 to Reτ = 1000) and square duct flows (Reτ = 160).
The results have shown that the tensorial form of the linear Boussinesq hypothesis
is not a good assumption even in the region where production and dissipation are in
equilibrium. It is then shown that the set of tensor basis composed by D, D2 and the
persistence-of-straining tensor D · (W−ΩD) − (W−ΩD) · D, whereWis the vorticity
tensor and ΩD is the rate of rotation of the eigenvectors of D, is able to totally reproduce
the anisotropic Reynolds stress.
With the proposed methodology, the scalar coefficients of nonlinear algebraic turbulent
models can be determined, and the adequacy of the tensorial dependence of the
Reynolds stress can be quantified with the aid of scaled correlation coefficients.