Exact expressions for the unresolved stress in a finite-volume based large-eddy simulation

TL;DR

This paper introduces new discretization-informed expressions for the residual stress tensor in finite-volume large-eddy simulations, accounting for numerical flux, divergence, and pressure contributions, leading to improved LES accuracy.

Contribution

The paper presents a novel, non-symmetric, non-local residual stress tensor formulation that incorporates discretization effects, enhancing LES closure modeling and accuracy.

Findings

Including discretization effects reduces LES residual stress errors to zero.

The new RST leads to higher Smagorinsky coefficients and better LES results.

Classical RST expressions show increasing errors over time.

Abstract

In this article we propose new discretization-informed expressions for the residual stress tensor (RST) in a finite-volume based large-eddy simulation (LES-FVM). In addition to the classical RST $\overline{u u} - \bar{u} \bar{u}$ resulting from the non-commutation between filtering and the nonlinear stress, our RST also contains contributions from the numerical flux, discrete divergence, and pressure terms. Unlike the classical RST, our proposed RST is non-symmetric and non-local. The proposed form of the RST is important for generating appropriate reference data for LES closure modeling. Based on DNS results of the 1D Burgers and 3D incompressible Navier-Stokes equations, we show that the discretization-induced parts of the RST play an important role in the LES-FVM equation for common LES filter widths. When the discrete contribution is included, our RST expression gives zero a-posteriori error in LES, while existing RST expressions give errors that increase over time. For a Smagorinsky model, we show that the Smagorinsky coefficient is higher when fitted to our new RST than when fitted to the classical RST and gives improved results.