von K\`arm\`an--Howarth and Corrsin equations closure based on Lagrangian description of the fluid motion

Abstract

A new approach to obtain the closure formulas for the von K\'arm\'an--Howarth and Corrsin equations is presented, which is based on the Lagrangian representation of the fluid motion, and on the Liouville theorem associated to the kinematics of a pair of fluid particles. This kinematics, characterized by the finite--scale separation vector, is assumed to be statistically independent from the velocity field. Such assumption is justified by the hypothesis of fully developed turbulence and by the property that this vector varies much more rapidly than the velocity field. This formulation leads to the closure formulas of von K\'arm\'an--Howarth and Corrsin equations in terms of longitudinal velocity and temperature correlations following a demonstration completely different with respect to the previous works. Some of the properties and the limitations of the closed equations are discussed. In particular, the times of evolution of the developed kinetic energy and temperature spectra are shown to be finite quantities which depend on the initial conditions.