Optimal fluxes and Reynolds stresses

TL;DR

This paper discusses the non-uniqueness of flux definitions in conservation laws and introduces a numerical method to compute fluxes that minimize their total magnitude, leading to more physically meaningful flux representations.

Contribution

It presents a novel numerical procedure to compute fluxes with minimized total magnitude, applicable to tensor fluxes like Reynolds stresses in turbulence.

Findings

Reynolds stresses differ significantly when computed with the new method.

The method reduces sterile flux circuits by deriving fluxes from a potential.

Application to turbulent channel flow demonstrates practical utility.

Abstract

It is remarked that fluxes in conservation laws, such as the Reynolds stresses in the momentum equation of turbulent shear flows, or the spectral energy flux in isotropic turbulence, are only defined up to an arbitrary solenoidal field. While this is not usually significant for long-time averages, it becomes important when fluxes are modelled locally in large-eddy simulations, or in the analysis of intermittency and cascades. As an example, a numerical procedure is introduced to compute fluxes in scalar conservation equations in such a way that their total integrated magnitude is minimised. The result is an irrotational vector field that derives from a potential, thus minimising sterile flux `circuits'. The algorithm is generalised to tensor fluxes and applied to the transfer of momentum in a turbulent channel. The resulting instantaneous Reynolds stresses are compared with their traditional expressions, and found to be substantially different.