Kinetic equation of turbulence from the Boltzmann equation

Abstract

We have shown how the kinetic equation for the velocity distribution function of an ensemble of turbulent velocities can be rigorously obtained from the Boltzmann kinetic equation with the classical collision integral. Compared to the Boltzmann equation on the left-hand side, the resulting kinetic equation of turbulence contains ten additional terms. Also, instead of the frequency of molecular collisions νcoll, the collision integral in the kinetic equation of turbulence includes the collision frequency νtp, which is significantly less than the frequency of molecular collisions. There are two key steps we have undertaken in obtaining the kinetic equation of turbulence. First, we used the invariance of the collision integral of the Boltzmann equation with respect to the Gaussian transformations. Second, we introduced the idea of fragmentation of turbulent flows into turbulent fluid quasiparticles. Each such quasiparticle is described by an equilibrium distribution of molecular velocities with fluctuating mean velocity. Also, each quasiparticle is characterized by its size, which is in the range of length scales larger than the mean free path of molecules λ and less than the typical length of spatial variation in the turbulence distribution function.