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.