The Kolmogorov–Obukhov Statistical Theory of Turbulence

Abstract

In 1941 Kolmogorov and Obukhov proposed that there exists a statistical theory of turbulence that should allow the computation of all the statistical quantities that can be computed and measured in turbulent systems. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier-Stokes equation. The additive noise in the stochastic Navier-Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections. Then we compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

The KOSL Scaling, Invariant Measure and PDF of Turbulence

“…The Fourier coefficients of the first series contain independent Brownian motions b k t and thus the noise is white in time in the infinitely many directions in function space. The noise cannot be white in space, hence the decaying coefficients c 1/2 k and d k , because if it was the small scale velocity u would be discontinuous in 3 dimension, see [5]. This is contrary to what is observed in nature.…”

“…The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen [1,2], and compare well with PDFs from simulations and experiments. The details of the proofs can be found in [5] and the background material can be found in [6].…”

“…Once the form of the noise in the stochastic Navier-Stokes equation for the small scales is determined, we can estimate the structure functions of turbulence and establish the Kolmogorov-Obukhov '62 scaling hypothesis with the She-Leveque intermittency corrections [5]. Then one can compute the invariant measure of turbulence writing the stochastic Navier-Stokes equation as an infinite-dimensional Ito process and solving the linear Kolmogorov-Hopf [8] functional differential equation for the invariant measure.…”

Strong solutions for the stochastic 3D LANS-α model driven by non-Gaussian Lévy noise

“…is, in fact, equivalent to the usual gradient norm. We denote by H the closure in (L 2 (D)) 3 of the set V = {v ∈ (D(D)) 3 : ∇v = 0 in D}, and by V the closure of V in (H 1 0 (D)) 3 . H is a Hilbert space equipped with the inner product of (L 2 (D)) 3 , and V is a Hilbert subspace of (H 1 0 (D)) 3 .…”

Incremental similarity and turbulence

“…This distribution was introduced in [1] and has since found a multitude of applications in a variety of fields. Recently the fact that N IG describes the velocity increments to high precision has found a theoretical counterpart in studies of Birnir concerning a stochastic version of the Navier-Stokes equations, see [7,8,9].…”