An infinity of possible invariants for decaying homogeneous turbulence

TL;DR

This paper explores the infinite possible invariants in decaying homogeneous turbulence, revealing implications for the asymptotic behaviour of velocity correlations and the nature of turbulence decay.

Contribution

It identifies an infinite set of invariants related to the von Karman-Howarth equation and analyzes their impact on turbulence decay solutions.

Findings

Multiple invariants influence turbulence decay behavior

Presence of two invariants prevents self-similar decay

Asymptotic velocity correlation behaviors determine invariant finiteness

Abstract

The von Karman-Howarth equation implies an infinity of invariants corresponding to an infinity of different asymptotic behaviours of the double and triple velocity correlation functions at infinite separations. Given an asymptotic behaviour at infinity for which the Birkhoff-Saffman invariant is not infinite, there are either none, or only one or only two finite invariants. If there are two, one of them is the Loitsyansky invariant and the decay of large eddies cannot be self-similar. We examine the consequences of this infinity of invariants on a particular family of exact solutions of the von Karman-Howarth equation.