American Institute of Mathematical Sciences

September  2014, 13(5): 1737-1757. doi: 10.3934/cpaa.2014.13.1737

The Kolmogorov-Obukhov-She-Leveque scaling in turbulence

 1 Department of Mathematics, University of California, Santa Barbara

Received  September 2013 Revised  December 2013 Published  June 2014

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
Citation: Björn Birnir. The Kolmogorov-Obukhov-She-Leveque scaling in turbulence. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1737-1757. doi: 10.3934/cpaa.2014.13.1737
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