From global scaling, à la Kolmogorov, 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 -${\textstyle\frac{4}{5}}\epsilon $l 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 $\zeta _{2p}$ of the structure functions of order 2p and the maximum velocity excursion. It follows that models (such as the Obukhov-Kolmogorov 1962 log-normal model), in which $\zeta _{2p}$ does not increase monotonically, 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.