From Global, à La Kolmogorov 1941, Scaling to Local Multifractal Scaling in Fully Developed Turbulence

Abstract

In the first part of the paper a modern presentation of scaling ideas is made. It includes a reformulation of Kolmogorov’s 1941 theory bypassing the universality problem pointed out by Landau and a presentation of the multifractal theory with emphasis on scaling rather than on cascades. In the second part, various historical aspects are discussed. The importance of Kolmogorov’s rigorous derivation of the —(4/5)εℓ law for the third order structure function in his last 1941 turbulence paper is stressed; this paper also contains evidence that he was aware of universality not being essential to the 1941 theory.

An inequality is established relating the exponents ζ_2p of the structure functions of order 2p and the maximum velocity excursion. It follows that models (such as the Obukhov-Kolmogorov 1962 lognormal model), in which ζ_2p does not increase mono-tonically, are inconsistent with the basic physics of incompressible flow. This result is independent of Novikov’s 1971 inequality; in particular, the proof presented here does not rely on the (questionable) relation, proposed by Obukhov and Kolmogorov, between instantaneous velocity increments and local averages of the dissipation.