Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling

TL;DR

This paper introduces a stochastic formulation of the Navier-Stokes equations based on a modified Reynolds transport theorem, leading to new large-scale flow models that outperform classical approaches in numerical simulations.

Contribution

It develops a novel stochastic large-scale modeling framework derived rigorously from flow randomness, generalizing traditional eddy viscosity models with new subgrid tensors.

Findings

Models outperform classical large-eddy simulations in Green-Taylor vortex flow.

The proposed models incorporate anisotropic subgrid tensors and drift corrections.

Numerical results demonstrate the effectiveness of the stochastic approach.

Abstract

We explore the potential of a formulation of the Navier-Stokes equations incorporating a random description of the small-scale velocity component. This model, established from a version of the Reynolds transport theorem adapted to a stochastic representation of the flow, gives rise to a large-scale description of the flow dynamics in which emerges an anisotropic subgrid tensor, reminiscent to the Reynolds stress tensor, together with a drift correction due to an inhomogeneous turbulence. The corresponding subgrid model, which depends on the small scales velocity variance, generalizes the Boussinesq eddy viscosity assumption. However, it is not anymore obtained from an analogy with molecular dissipation but ensues rigorously from the random modeling of the flow. This principle allows us to propose several subgrid models defined directly on the resolved flow component. We assess and compare numerically those models on a standard Green-Taylor vortex flow at Reynolds 1600. The numerical simulations, carried out with an accurate divergence-free scheme, outperform classical large-eddies formulations and provides a simple demonstration of the pertinence of the proposed large-scale modeling.