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

TL;DR

The work develops a discretization-aware framework that unifies large-eddy simulation with finite-volume methods to derive an exact residual-stress tensor (RST) for LES-FVM. For incompressible flows, the resulting LES-FVM RST is non-local and non-symmetric due to discretization and pressure projection, and it provides zero a-posteriori error in DNS-aided tests when used as a closure target. The approach demonstrates that discretization effects are central to accurate closure modeling and shows that Smagorinsky coefficients fitted to the discretization-informed RST improve performance. The results, validated in Burgers’ equation and 3D Navier–Stokes turbulence, offer a principled path to discretization-consistent closures and potential data-driven enhancements. Overall, this work bridges implicit and explicit LES closures by making discretization a first-class component of the residual and its closure.

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.

Figures (15)

• Figure 1: Three different routes to a closed system of discrete equations that simulate the large-scales of a turbulent flow $u$. A change of color indicates an approximation step, while a preservation of color indicates an exact operation. The grid points are $(x^h_i)_{i \in \mathbb{Z}}$. Classical LES-FVM relies on two separate approximation steps, whereas the FVM and our proposed LES-FVM framework only rely on one approximation step.
• Figure 2: Spatial kernels of LES filter $f^\Delta$ (top-hat and Gaussian), FVM grid filter $f^h$ (always top-hat), and LES-FVM double filter $f^{\Delta, h}$ for $\Delta \in \{ 1 h, 2 h, 3 h\}$. In the top-left plot, we have $f^\Delta = f^h$.
• Figure 3: Spectral transfer functions of LES filter $f^\Delta$ (top-hat and Gaussian), FVM grid filter $f^h$ (always top-hat), and LES-FVM double filter $f^{\Delta, h}$ for $\Delta \in \{ 1 h, 2 h, 3 h\}$. In the top-left plot, we have $f^\Delta = f^h$.
• Figure 4: Initial and final solution to the Burgers equation.
• Figure 5: Relative contribution of the different flux parts in the decomposition \ref{['eq:tau-Delta-h-decomposition']}.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5