New insight into the functional dependence rules in turbulence modelling

This paper is intended to revisit the Euclidean invariance of turbulence models, in particular of the algebraic Reynolds stress models (ARSM). The approach is motivated by the existing different interpretations of the Euclidean invariance of an equation. We show that these interpretations are not suitable to rule out functional dependencies in turbulence. We use the concept of Euclidean invariance with respect to a constitutive or flow rule class of a closure relation to analyze the Euclidean invariance properties of the ARSM. It turns out that the existing ARSM are Euclidean invariant with respect to two different classes, that are basically determined by the equilibrium assumption made in an inertial and a non-inertial reference frame, respectively. The first one is more restrictive and does not include the relative environment dependence of the Reynolds stress tensor in the implicit ARSM. The second class endeavors to involve this environment dependence. After this first step, the issue how to formulate a unified rationale for a consistent reduction procedure based on extended thermodynamics to rule out functional dependencies of the ARSM relations for Reynolds-averaged Navier–Stokes equations (RANS) or large eddy simulations (LES) is left for future work.