Probability density functions of dissipation rate and enstrophy in turbulence

Variation of the probability density functions (PDFs) for the dissipation rate and enstophy from very law to high Reynolds numbers in the isotropic steady turbulence is studied numerically and theoretically. It is shown that the asymptotic PDFs at very high Reynolds numbers are both the stretched gamma distribution. The left tails of the PDFs are of the power law with the exponents 3/2 for the dissipation rate and 1/2 for the enstrophy, on the other hand, the right tails are stretched exponential with the same exponents which are smaller than unity but with different decay rates. It is theretically shown that the right PDF tail of the enstrophy is longer than that of the dissipation at all Reynolds number. Implications of these asymptotic PDFs are also discussed.