Modelling the pressure–strain correlation of turbulence: an invariant dynamical systems approach

TLDR

In standard turbulence closure models, the pressure–strain correlation is assumed to be a linear function of mean velocity gradients with coefficients algebraically dependent on the anisotropy tensor. The study aims to develop improved second‑order closure models for the pressure–strain correlation by examining it from a basic theoretical standpoint and briefly discussing alternative approaches. The authors apply invariance considerations and elementary dynamical systems theory to analyze the standard hierarchy of closure models. They prove that for plane homogeneous turbulence the equilibrium structure reduces to a simple quadratic model (SSG) that outperforms the Launder–Reece–Rodi and other complex nonlinear models in five homogeneous flows, but still fails for rotating shear flows, suggesting that adding more complex nonlinearities may be of questionable value.

Abstract

The modelling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - appears to yield improved results over the Launder, Reece & Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in five independent homogeneous turbulent flows. However, some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modelling of the pressure–strain correlation. Possible alternative approaches are discussed briefly.