Abstract
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 byShe-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.
Highlights
It has become clear, since Kolmogorov and Obukhov [16, 15, 21] proposed a statistical theory of turbulence based on dimensional arguments, that noise plays a big role in turbulent flow
It is easy to check that the moments of the invariant measure for the two-point statistics give the estimates for the structure functions above
We present a derivation to show how the probability distribution functions (PDFs) for the different moments depend on the intermittency corrections τp for the pth moment
Summary
Since Kolmogorov and Obukhov [16, 15, 21] proposed a statistical theory of turbulence based on dimensional arguments, that noise plays a big role in turbulent flow. The additive noise describes the mean of the dissipation process in turbulent flow It is a Fourier series with coefficients that are independent Brownian motions in the first term of the additive noise above. The noise in fully developed turbulence is nondegenerate, and this is what experiments and simulations indicate This is not a complete description of the mean, there frequently is a bias in the flow and associated are large deviation of the dissipation process. These large deviations are described by the second term in the additive noise that is derived from the large deviation principle or Cramer’s theorem, see [11]. The equation (8) defines the fluid velocity u as a stochastic process, taking its values in L2(T3) for each t
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