Symmetry and Scaling of Turbulent Mixing

TL;DR

This paper explores the approximate $SL(n, R)$ symmetry in the stationary correlation functions of a passive scalar in turbulent flow, challenging classical theories and providing testable predictions for experiments.

Contribution

It introduces an $SL(n, R)$ symmetry framework for analyzing turbulent scalar mixing and discusses its implications for anisotropy and universality in turbulence.

Findings

Large scale anisotropy is relevant, contradicting Kolmogorov theory.

Quantitative predictions for correlation functions are formulated.

Exponents are non-universal, varying with flow conditions.

Abstract

The stationary condition (Hopf equation) for the ($n$+1) point correlation function of a passive scalar advected by turbulent flow is argued to have an approximate $SL(n, R)$ symmetry which provides a starting point for the perturbative treatment of less symmetric terms. The large scale anisotropy is found to be a relevant field, in contradiction with Kolmogorov phenomenology, but in agreement with the large scalar skewness observed in shear flows. Exponents are not universal, yet quantitative predictions for experiments to test the $SL(n, R)$ symmetry can be formulated in terms of the correlation functions.