Statistical mechanics of the Burgers model of turbulence

Abstract

The velocity field of the Burgers one-dimensional model of turbulence at extremely large Reynolds numbers is expressed as a train of random triangular shock waves. For describing this field statistically the distributions of the intensity and the interval of the shock fronts are defined. The equations governing the distributions are derived taking into account the laws of motion of the shock fronts, and the self-preserving solutions are obtained. The number of shock fronts is found to decrease with time t as t−α, where α (0 [les ] α < 1) is the rate of collision, and consequently the mean interval increases as tα. The distribution of the intensity is shown to be the exponential distribution. The distribution of the interval varies with α, but it is proved that the maximum entropy is attained by the exponential distribution which corresponds to α = ½. For α = ½, the turbulent energy is shown to decay with time as t−1, in good agreement with the numerical result of Crow & Canavan (1970).