Inverse kinetic theory approach to turbulence theory

TL;DR

This paper introduces a deterministic inverse kinetic theory for 3D incompressible Navier-Stokes equations, demonstrating the existence of a Markovian statistical description of turbulence based on local velocity probability densities.

Contribution

It develops a novel inverse kinetic theory approach that establishes a Markovian statistical framework for turbulence using local velocity PDFs derived from first principles.

Findings

Existence of a Markovian statistical description of turbulence.

Exact equivalence between Lagrangian and Eulerian formulations.

Local velocity PDFs can be non-Gaussian and non-stationary.

Abstract

A fundamental aspect of turbulence theory is related to the identification of realizable phase-space statistical descriptions able to reproduce in some suitable sense the stochastic fluid equations of a turbulent fluid. In particular, a major open issue is whether a purely Markovian statistical description of hydrodynamic turbulence actually can be achieved. Based on the formulation of a \textit{deterministic inverse kinetic theory} (IKT) for the 3D incompressible Navier-Stokes equations, here we claim that such a \textit{Markovian statistical description actually exists}. The approach, which involves the introduction of the \textit{local velocity probability density} for the fluid (local pdf) - rather than the velocity-difference pdf adopted in customary approaches to homogeneous turbulence - relies exclusively on first principles. These include - in particular - the exact validity of the stochastic Navier-Stokes equations, the principle of entropy maximization and a constant H-theorem for the Shannon statistical entropy. As a result, the new approach affords an exact equivalence between Lagrangian and Eulerian formulations which permit local pdf's which are generally non-Maxwellian (i.e., non-Gaussian). The theory developed is quite general and applies in principle even to turbulence regimes which are non-stationary and non-uniform in a statistical sense.