A local energy transport equation for isotropic turbulence

Abstract

In this paper it is shown that a Boltzmann-type equation for turbulent energy transport in wave number, may be approximated by a local differential equation, which is an analogue of the Fokker-Planck equation. The principal assumptions are that the dissipation integral is mainly determined by the inertial range form of the energy spectrum (i.e. the Kolmogoroff distribution) and that the response function, required for closure of the moment hierarchy, may be approximated by its inertial range form only. It is shown that the resulting equation yields the Kolmogoroff distribution for the case of the infinite Reynolds number limit, as it must, to be consistent. In addition, the general asymptotic form of the spectrum is obtained and found to be E ( K ) = 4 πqh 2/3 k -5/3 [( k / k d ) 1/2 Z 1/4 ( k 2 / k d 2 )], where q is the Kolmogoroff constant, h the dissipation rate and Z a modified Bessel function of the second kind. This form behaves like k -5/3 in the inertial range and as an exponential decay in the viscous range of wave numbers where k ~ k d .