Boundary conditions for probability density function transport equations in fluid mechanics

TL;DR

This paper investigates boundary conditions for the probability density function transport equation in fluid mechanics, highlighting the need for additional terms to maintain normalization and examining discontinuities at probability space limits.

Contribution

It introduces a new boundary term for the PDF transport equation to ensure normalization in nonstationary processes and analyzes boundary discontinuities.

Findings

A new boundary term is necessary for nonstationary processes.

Particle methods naturally incorporate the new boundary term.

Discontinuities at probability space limits are significant in PDF transport equations.

Abstract

The behavior of the probability density function (PDF) transport equation at the limits of the probability space is studied from the point of view of fluid mechanics. Different boundary conditions are considered depending on the nature of the variable considered (velocity, scalar, and position). A study of the implications of entrance and exit conditions is performed, showing that a new term should be added to the PDF transport equation to preserve normalization in some nonstationary processes. In practice, this term is taken into account naturally in particle methods. Finally, the existence of discontinuities at the limits is also investigated.