Fluid flow dynamics under location uncertainty

TL;DR

This paper develops a stochastic Navier-Stokes model incorporating uncertain velocity components, providing a new perspective on subgrid modeling and enabling stochastic versions of shallow water equations and reduced systems.

Contribution

It introduces a novel stochastic decomposition of velocity fields and derives a corresponding Navier-Stokes model that accounts for location uncertainty in fluid flow.

Findings

Provides a stochastic Navier-Stokes framework with uncertainty-driven subgrid formulas.

Explains the connection between uncertainty modeling and existing eddy diffusion models.

Derives stochastic shallow water equations and reduced order models from the formalism.

Abstract

We present a derivation of a stochastic model of Navier Stokes equations that relies on a decomposition of the velocity fields into a differentiable drift component and a time uncorrelated uncertainty random term. This type of decomposition is reminiscent in spirit to the classical Reynolds decomposition. However, the random velocity fluctuations considered here are not differentiable with respect to time, and they must be handled through stochastic calculus. The dynamics associated with the differentiable drift component is derived from a stochastic version of the Reynolds transport theorem. It includes in its general form an uncertainty dependent "subgrid" bulk formula that cannot be immediately related to the usual Boussinesq eddy viscosity assumption constructed from thermal molecular agitation analogy. This formulation, emerging from uncertainties on the fluid parcels location, explains with another viewpoint some subgrid eddy diffusion models currently used in computational fluid dynamics or in geophysical sciences and paves the way for new large-scales flow modelling. We finally describe an applications of our formalism to the derivation of stochastic versions of the Shallow water equations or to the definition of reduced order dynamical systems.