A Statistical Formulation of Non-Stationary Homogeneous Turbulence, Based on Time-Dependent Turbulent Viscosity

Abstract

A statistical formulation is presented for non-stationary homogeneous turbulence whose initial statistical quantities are given. Nonlinear effects of turbulence are taken into account in the form of the time-dependent turbulent viscosity. An integral equation is derived for the time-dependent turbulent viscosity from the Liouville equation, under an assumption that in the perturbative solution, only the leading term contributes to the second-order velocity correlations or the turbulent energy. This equation leads to a closed integral equation for the energy spectrum. It is shown that the solution of this equation always gives the positive energy spectrum and that Malfliet’s equation for the energy spectrum (W. P. M. Malfliet: Physica 45 (1969) 257) based on the method of multiple-time scales can be recovered from the present equation.