Theory of the 1-point PDF for incompressible Navier-Stokes fluids

TL;DR

This paper develops a statistical model for incompressible Navier-Stokes fluids using a 1-point velocity PDF within inverse kinetic theory, providing exact evolution equations and unique representations for multi-point PDFs, applicable to both regular and turbulent flows.

Contribution

It introduces a novel inverse kinetic theory framework that uniquely determines the initial and evolving 1-point PDF for incompressible fluids, including turbulent flows.

Findings

The initial PDF is generally non-Gaussian.

The PDF evolution follows a Liouville equation depending on finite velocity moments.

The model enables exact construction of stochastic-averaged and multi-point PDFs.

Abstract

Fundamental aspects of fluid dynamics are related to construction of statistical models for incompressible Navier-Stokes fluids. The latter can be considered either \textit{deterministic} or \textit{stochastic,} respectively for \textit{regular} or \textit{turbulent flows.} In this work we claim that a possible statistical formulation of this type can be achieved by means of the 1-point (local) velocity-space probability density function (PDF, $f_{1}$) to be determined in the framework of the so-called inverse kinetic theory (IKT). There are several important consequences of the theory. These include, in particular, the characterization of the initial PDF [for the statistical model ${f_{1},\Gamma} ]$ . This is found to be generally non-Gaussian PDF, even in the case of flows which are regular at the initial time. Moreover, both for regular and turbulent flows, its time evolution is provided by a Liouville equation, while the corresponding Liouville operator is found to depend only on a finite number of velocity moments of the same PDF. Hence, its time evolution depends (functionally) solely on the same PDF. In addition, the statistical model here developed determines uniquely both the initial condition and the time evolution of $f_{1}.$ As a basic implication, the theory allows the \textit{exact construction of the corresponding statistical equation for the stochastic-averaged PDF}and the \textit{unique representation of the multi-point PDF}'s in solely in terms of the 1-point PDF. As an example, the case of the reduced 2-point PDF's, usually adopted for the statistical description of NS turbulence, is considered.