The Cameron—Martin—Wiener method in turbulence and in Burgers’ model: general formulae, and application to late decay

Abstract

We apply the Cameron—Martin—Wiener (formerly ‘Wiener—Hermite’) expansion of a random velocity field to the analytical study of turbulence. The kernels of this expansion contain all statistical information about the ensemble. Complete expressions are derived for constructing statistical quantities in terms of the kernels, and for the equations of motion of the kernels. We rigorously prove the Gaussian trend of the velocity field of the Navier—Stokes equation in the very late stage when the non-linear term is neglected. The n-dependence (n is the order of derivative) of the flatness factor, minus three for derivatives of the velocity field, shows a rapid increase with n in this stage.

The late decay problem of the Burgers model of turbulence is studied analytically with a view to obtaining suggestive guidelines for fitting the non-linear aspects of the model turbulence. We can divide the energy spectrum density into two parts, the larger of which is a kind of steady solution, which we call the ‘equilibrium state’, which remains self-similar in time in terms of an appropriate variable. The deviation from this ‘equilibrium solution’ satisfies the Kármán—Howarth equation. As initial velocity field, we take two particular cases: (a) a pure Gaussian, and (b) a non-Gaussian velocity field. With these two cases a detailed spectral analysis has been obtained. The energy spectrum deviation from ‘equilibrium’ declines exponentially to zero for all wave-numbers. The Gaussian case shows that the flatness factor minus three increases rapidly with n, while the non-Gaussian case does not show any marked dependence on n.