Statistical Approach to Steady Homogeneous Turbulence, with Special Reference to Kolmogoroff’s Spectrum

Abstract

This paper studies steady homogeneous turbulence by means of a statistical approach. The Liouville equation for the probability distribution function is solved as a perturbation expansion from the Fokker-Planck equation based on turbulent viscosity, turbulent diffusion and three kinds of renormalized vertices. Five nonlinear simultaneous integral equations are derived for these quantities, under the assumption that in the perturbative solution only the first two terms contribute to the second-order velocity correlations (turbulent energy) and the third-order velocity correlations (energy transfer functions). It is shown that the introduction of three kinds of renormalized vertices gives the possibility of removing the effect of the energy-containing range upon the inertial range and of leading to Kolmogoroff’s spectrum consistently.