Algorithms and Models for Turbulence Not at Statistical Equilibrium

TL;DR

This paper introduces a new derivation and correction method for eddy viscosity models to better capture complex turbulence phenomena, including backscatter, especially when turbulence is not at statistical equilibrium.

Contribution

It presents a novel derivation of eddy viscosity models from variance evolution equations and proposes corrections to improve their accuracy in complex turbulent flows.

Findings

Corrected models preserve key Reynolds stress features.

Numerical tests show successful backscatter representation.

Algorithms enable easy integration into existing codes.

Abstract

Standard eddy viscosity models, while robust, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium. This report gives a new derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow. The new derivation also shows how to correct eddy viscosity models. The report proves the corrected models preserve important features of the true Reynolds stresses. It gives algorithms for their discretization including a minimally invasive modular step to adapt an eddy viscosity code to the extended models. A numerical test is given with the usual and over diffusive Smagorinsky model. The correction (scaled by $10^{-8}$ ) does successfully exhibit intermittent backscatter.