PDF of Velocity Fluctuation in Turbulence by a Statistics based on Generalized Entropy

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the R'{e}nyi entropy or the Tsallis-Havrda-Charvat (THC) entropy. The parameters appeared in the PDF, including the index $q$ which appears in the measures of the R'{e}nyi entropy or of the THC entropy are determined self-consistently with the help of the observed value $\mu$ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions.