Fokker-Planck Kinetic description of small-scale fluid turbulence for classical incompressible fluids

TL;DR

This paper develops a Fokker-Planck kinetic model for small-scale fluid turbulence, extending inverse kinetic theory to describe stochastic incompressible flows with non-Gaussian statistics.

Contribution

It introduces a stochastic inverse kinetic theory framework that uniquely determines the evolution of turbulent fluid fields using a local phase-space pdf.

Findings

The local pdf obeys a kinetic equation that is generally non-Markovian.

The approach allows approximation by a Fokker-Planck equation in velocity space.

The local pdf typically exhibits non-Gaussian behavior.

Abstract

Extending the statistical approach proposed in a parallel paper \cite% {Tessarotto2008-aa}, purpose of this work is to propose a stochastic inverse kinetic theory for small-scale hydrodynamic turbulence based on the introduction of a suitable \textit{local phase-space probability density function} (pdf). In particular, we pose the problem of the construction of Fokker-Planck kinetic models of hydrodynamic turbulence. The approach here adopted is based on the so-called IKT approach (inverse kinetic theory), developed by Ellero et al. (2004-2008) which permits an exact phase-space description of incompressible fluids based on the adoption of a local pdf. We intend to show that for prescribed models of stochasticity the present approach permits to determine uniquely the time evolution of the stochastic fluid fields. The stochastic-averaged local pdf is shown to obey a kinetic equation which, although generally non-Markovian, locally in velocity-space can be approximated by means of a suitable Fokker-planck kinetic equation. As a side result, the same pdf is proven to have generally a non-Gaussian behavior.