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 ( δ r u , r ) , velocity derivatives 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 P L ( δ r u , 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 ( δ r u , r ) from P L ( δ r u , r ) . An expression for the function 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.