Measures of Anisotropy and the Universal Properties of Turbulence

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. The literature is replete with studies purporting to examine its validity and limitations. While, until the midseventies 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. This throws doubt on the existence of universal aspects of turbulence. Part of the problem in refining this loose statement is the absence until now of serious efforts to separate the isotropic component of any statistical object from its anisotropic components. These notes examine in some detail the isotropic and anisotropic contributions to structure functions by considering their SO(3) decomposition. After an initial discussion of the status of local isotropy (section 1) and the theoretical background for the SO(3) decomposition (section 2), we provide an account of the experimental data (section 3) and their analysis (sections 4, 5 and 6). 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.