Fully developed isotropic turbulence: symmetries and exact identities

TL;DR

This paper uncovers gauge symmetries in the Navier-Stokes field theory for isotropic turbulence, deriving exact identities and relations between correlation functions, including generalizations of known turbulence relations.

Contribution

It identifies gauge symmetries in the turbulence field theory and derives a set of exact, local relations between correlation functions, generalizing previous turbulence identities.

Findings

Identified gauge symmetries in Navier-Stokes turbulence theory

Derived Ward identities linking correlation functions

Generalized the Kármán-Howarth relation and pressure-velocity correlation

Abstract

We consider the regime of fully developed isotropic and homogeneous turbulence of the Navier-Stokes equation with a stochastic forcing. We present two gauge symmetries of the corresponding Navier-Stokes field theory, and derive the associated general Ward identities. Furthermore, by introducing a local source bilinear in the velocity field, we show that these symmetries entail an infinite set of exact and local relations between correlation functions. They include in particular the K\'arm\'an-Howarth relation and another exact relation for a pressure-velocity correlation function recently derived in Ref. [G. Falkovich, I. Fouxon, Y. Oz,J. Fluid Mech. 644, 465 (2010)], that we further generalize.