Measures of Anisotropy and the Universal Properties of Turbulence

TL;DR

This paper investigates the validity of local isotropy in turbulence at small scales, using SO(3) decomposition to analyze anisotropic contributions and challenging the assumption that isotropy dominates at high Reynolds numbers.

Contribution

It provides a detailed analysis of anisotropic effects in turbulence structure functions, showing that isotropy dominates up to order 6 but anisotropic parts decay more slowly than previously assumed.

Findings

Isotropic part dominates small scales up to order 6

Anisotropic contributions decay with higher-order sectors

Anisotropic parts decrease less sharply than dimensional predictions

Abstract

Local isotropy, or the statistical isotropy of small scales, is one of the basic assumptions underlying Kolmogorov's theory of universality of small-scale turbulent motion. While, until the mid-seventies or so, local isotropy was accepted as a plausible approximation at high enough Reynolds numbers, various empirical observations that have accumulated since then suggest that local isotropy may not obtain at any Reynolds number. These notes examine in some detail the isotropic and anisotropic contributions to structure functions by considering their SO(3) decomposition. Viewed in terms of the relative importance of the isotropic part to the anisotropic parts of structure functions, the basic conclusion is that the isotropic part dominates the small scales at least up to order 6. This follows from the fact that, at least up to that order, there exists a hierarchy of increasingly larger power-law exponents, corresponding to increasingly higher-order anisotropic sectors of the SO(3) decomposition. The numerical values of the exponents deduced from experiment suggest that the anisotropic parts in each order roll off less sharply than previously thought by dimensional considerations, but they do so nevertheless.