Probability densities in strong turbulence

In this work we, using Mellin’s transform combined with the Gaussian large-scale boundary condition, calculate probability densities (PDFs) of velocity increments P(δru,r)P(δru,r), velocity derivatives P(u′,r)P(u′,r) and the PDF of the fluctuating 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)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 the deviation of P(δru,r)P(δru,r) from PL(δru,r)PL(δru,r). An expression for the function D(h)D(h) of the multifractal theory, free from spurious logarithms recently discussed in [U. Frisch, M. Martins Afonso, A. Mazzino, V. Yakhot, J. Fluid Mech. 542 (2005) 97] is also obtained.