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From global scaling, à la Kolmogorov, to local multifractal scaling in fully developed turbulence. (English) Zbl 0727.76064
Summary: 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 $$-(4/5)\epsilon l$$ 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 exponent $$\zeta_{2p}$$ 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 E. A. Novikov’s inequality [Prikl. Mat. Mekh. 35, 266–277 (1971; Zbl 0263.76043)]; 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.

##### MSC:
 76F99 Turbulence 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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