## The Kolmogorov-Obukov Theory of Turbulence

CIRF seminar on Wednesday, May 23, at 4pm in room ESB 2001

Bjorn Birnir

Center for Complex and Nonlinear Science and

Department of Mathematics,

Univ. of California, Santa Barbara

Abstract

The Kolmogorov-Obukov statistical theory of turbulence, with intermittency

corrections, is derived from a stochastic Navier-Stokes equation with generic

noise. In this talk we discuss how the laminar solution of the Navier-Stokes

equation becomes unstable for large Reynolds number and the unstable solution

is the solution of the stochastic Navier-Stokes equation. This is the unique

solution that describes fully-developed turbulence. In order to compare with

experiments and simulations, we find the solution of the stochastic Hopf’s

equation for the invariant measure. Gaussian noise and dissipation

intermittency produce the Kolmogorov-Obukov scaling of the structure

functions of turbulence. The Feynmann-Kac formula produces log-Poisson

processes from the stochastic Navier-Stokes equation. These processes,

first found by She, Leveque, Waymire and Dubrulle in 1995, give the

intermittency corrections to the structure functions of turbulence,

stemming from velocity fluctuations. The probability density function

(PDF) of the two-point statistics that can be compared to experiments

and simulations turn out to be similar to the generalized hyperbolic

distributions first suggested by Barndorff-Nilsen in 1977. We compare

the theoretical PDF with PDFs obtained from DNS simulations and

wind-tunnel experiments.