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

摘要

A new approach to obtain the closure formulas for the von K\'arm\'an--Howarthand Corrsin equations is presented, which is based on the Lagrangianrepresentation of the fluid motion, and on the Liouville theorem associated tothe kinematics of a pair of fluid particles. This kinematics, characterized bythe finite--scale separation vector, is assumed to be statistically independentfrom the velocity field. Such assumption is justified by the hypothesis offully developed turbulence and by the property that this vector varies muchmore rapidly than the velocity field. This formulation leads to the closureformulas of von K\'arm\'an--Howarth and Corrsin equations in terms oflongitudinal velocity and temperature correlations following a demonstrationcompletely different with respect to the previous works. Some of the propertiesand the limitations of the closed equations are discussed. In particular, thetimes of evolution of the developed kinetic energy and temperature spectra areshown to be finite quantities which depend on the initial conditions.