Transport and mixing in control volumes through the lens of probability
File(s)
Author(s)
Craske, John
Mannix, Paul
Type
Journal Article
Abstract
A partial differential equation governing the evolution of the joint probability distribution of multicomponent flow observations, drawn randomly from one or more control volumes, is derived and applied to examples involving irreversible mixing. Unlike local probability density methods, this work adopts an integral perspective by regarding a control volume as a sample space with an associated probability distribution. A natural and general definition for the boundary of such control volumes comes from the magnitude of the gradient of the sample space distribution, which can accommodate Eulerian or Lagrangian frames of reference as particular cases. The resulting equation exposes contributions made by uncertain or stochastic boundary fluxes and internal cross-gradient mixing in the equation governing the observables’ joint probability distribution. Advection and diffusion over a control volume’s boundary result in source and drift terms, respectively, whereas internal mixing, in general, corresponds to the sign-indefinite diffusion of probability density. Several typical circumstances for which the corresponding diffusion coefficient is negative semidefinite are identified and discussed in detail. The framework is a natural setting for examining available potential energy, the incorporation of uncertainty into bulk models and establishing a link with the Feynman-Kac formula and Kolmogorov equations that are used to analyse stochastic processes.
Date Issued
2025-10-10
Date Acceptance
2025-08-16
Citation
Journal of Fluid Mechanics, 2025, 1020
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
1020
Copyright Statement
© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Identifier
10.1017/jfm.2025.10631
Subjects
turbulent mixing
coupled diffusion and flow
general fluid mechanics
Publication Status
Published