The relation between Markov process theory and Kolmogoroff's theory of turbulence and the extension of Kolmogoroff's laws

Abstract

The whole paper consists of two parts (Part I and Part II). In part I, we shall analyze the relation between the two theories of turbulence involving large Reynolds number: the Markov process theory from the Lagrangian point of view and Kolmogoroff's theory from the Eulerian point of view. It will be pointed out that the Reynolds number needed for the Markovian description of turbulence should be as large as that needed for Kolmogoroff's second hypothesis, that the eddies of the period of order T₁ (the self-correlation time scale of the random velocity u) and the eddies of the period of order T₁ (the self-correlation time scale of the random force f) correspond to the energy-containing eddies and the eddies in the dissipation range respectively, and that T_*⩽t≪β⁻¹, the time interval for the applicability of Richardson's law during twoparticle's dispersion, corresponds to the inertial subrange in Kolmogoroff's theory. Thus, these two theories reflect the property of the turbulence involving very large Reynolds number arguing from different aspects.

In Part II, by using physical analysis in Part I, we shall establish in a certain way the quantitative relation between these two theories. In terms of this relation and the results of the study of two-particle's dispersion motion, we shall obtain the structure functions, the correlation functions and the energy spectrum, which are appliciable not only to the inertial subrange, but also to the whole range with the wave number less than that in the inertial subrange. Kolmogoroff's “2/3 law” and ”−5/3 law” are the asymptotic solutions with respect to the present result in the inertial subrange. Thus, the present result is an extension of Kolmogoroff's laws.