Probability density functions of enstrophy and energy dissipation rate and their 1D surrogates in incompressible isotropic turbulence

Relation between the probability density functions (PDFs) of the three dimensional (3D) enstrophy and the one dimensional (1D) enstrophy surrogate in the incompressible isotropic turbulence is theoretically derived and verified by the direct numerical simulations. The relation indicates that the PDF of the 1D surrogate enstrophy has the longer tail than that of the PDF of 3D enstrophy and that their long tails are the stretched exponential with the same stretching exponents. Similar results for the PDFs of the 3D dissipation rate and the 1D dissipation surrogate are obtained and numerically verified. It is shown that the ratio of the moments of 3D dissipation to that of the 1D surrogate grows rapidly with the order but is independent of the Reynolds number.