5 Probability Densities in Strong Turbulence

According to modern developments in turbulence theory, the ”dissipation” scales (u.v. cut-offs) η form a random field related to velocity increments δηu. In this work we, using Mellin’s transform combined with the Gaussain large -scale boundary condition, calculate probability densities (PDFs) of velocity increments P (δru, r) and the PDF of the dissipation scales Q(η,Re), where Re is the large-scale Reynolds number. The resulting expressions strongly deviate from the Log-normal PDF PL(δru, r) often quoted in the literature. It is shown that the probability density of the small-scale velocity fluctuations includes information about the large (integral) scale dynamics which is responsible for deviation of P (δru, r) from PL(δru, r). A framework for evaluation of the PDFs of various turbulence characteristics involving spatial derivatives is developed. The exact relation, free of spurious Logarithms recently discussed in Frisch et al (J. Fluid Mech. 542, 97 (2005)), for the multifractal probability density of velocity increments, not based on the steepest descent evaluation of the integrals is obtained and the calculated function D(h) is close to experimental data. A novel derivation (Polyakov, 2005), of a well-known result of the multi-fractal theory [Frisch, ”Turbulence. Legacy of A.N.Kolmogorov”, Cambridge University Press, 1995)) , based on the concepts described in this paper, is also presented.