Turbulent energy cascade through equivalence of Euler and Lagrange motion descriptions and bifurcation rates

Abstract

This work analyses the homogeneous isotropic turbulence by means of the equivalence between Euler and Lagrange representations of motion, adopting the bifurcation rates associated with Navier--Stokes and kinematic equations, and an appropriate hypothesis of fully developed chaos. The equivalence of these motion descriptions allows to show that kinetic and thermal energy cascade arise both from the convective term of Liouville equation. Accordingly, these phenomena, of nondiffusive nature, correspond to a transport in physical space linked to the trajectories divergence. Both the bifurcation rates are properly defined, where the kinematic bifurcation rate is shown to be much greater than Navier--Stokes bifurcation rate. This justifies the proposed hypothesis of fully developed chaos where velocity field and particles trajectories fluctuations are statistically uncorrelated. Thereafter, a specific ergodic property is presented, which relates the statistics of fluid displacement to that of velocity and temperature fields. A detailed analysis of separation rate is proposed which studies the statistics of radial velocity component along the material separation vector. Based on previous elements, the closure formulas of von K\'arm\'an--Howarth and Corrsin equations are finally achieved. These closures, of nondiffusive kind, represent a propagation phenomenon, and coincide with those just presented by the author in previous works, corroborating the results of these latter.