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

Technical Summary: Exact Expressions for the Unresolved Stress in a Finite-Volume Based Large-Eddy Simulation

Problem Statement
Large-eddy simulation (LES) aims to model turbulent flows by resolving large scales while modeling the effects of unresolved sub-grid scales. Traditionally, the closure problem in LES focuses on the commutator error arising from the non-commutation of spatial filtering and nonlinear advection terms, resulting in the classical residual stress tensor (RST), $\tau_{ij} = \overline{u_i u_j} - \bar{u}_i \bar{u}_j$. However, in practical simulations, LES is solved using numerical discretization schemes, most commonly the Finite Volume Method (FVM).

A critical issue arises because the filter width $\Delta$ in LES is often of the same order of magnitude as the grid spacing $h$ (i.e., $\Delta \approx h$). In such regimes, discretization errors—arising from numerical fluxes and discrete divergence operators—are of the same order as the physical sub-filter stresses. Classical approaches typically define the closure model based on continuous equations and then discretize, or treat discretization errors as separate convergence issues. This separation leads to a mismatch between the target data (the true residual) and the model, particularly when the discretization itself introduces significant non-local or non-symmetric effects that standard symmetric, local closure models (like Smagorinsky) cannot capture.

Methodology
The authors propose a unified mathematical framework termed LES-FVM, which integrates the LES filtering process directly with the FVM discretization. The core methodology involves:

1. Unified Formalism: Instead of applying LES filtering to continuous equations and then discretizing, the authors apply the FVM filter (a volume average over control volumes) to the LES-filtered equations. This creates a "double-filtered" system where the filter width $\Delta$ and grid spacing $h$ are treated as independent parameters.
2. Derivation of Exact Residuals: By utilizing a "discrete filter-swap" property (where the finite difference operator is equivalent to a filtered exact derivative), the authors derive exact expressions for the residual flux in the discretized equations.
3. Decomposition of Residuals: The total residual flux $\tau_{\Delta,h}$ is decomposed into three distinct components:
   • Classic Term: The standard commutator error associated with the double filter ($\Delta$ and $h$).
   • Flux Term: Errors arising from the difference between the exact flux and the numerical flux ($r$ vs. $r_h$).
   • Divergence Term: Errors arising from the difference between the exact divergence and the discrete divergence.
4. Extension to 3D Incompressible Flow: The framework is extended to the 3D incompressible Navier-Stokes equations. This requires handling the pressure term via projection operators to ensure the velocity field remains divergence-free in the discrete setting. The authors derive a projected RST that accounts for the interaction between the pressure projection and the discretization.

Key Contributions

• Exact Discretization-Informed RST: The paper derives an exact expression for the unresolved stress tensor in an FVM-based LES. Unlike the classical RST, this new expression explicitly includes contributions from numerical fluxes and discrete divergence operators.
• Non-Symmetry and Non-Locality: For incompressible flows, the authors demonstrate that the proposed LES-FVM RST is inherently non-symmetric and non-local (due to the pressure projection). This contrasts sharply with classical LES RSTs, which are typically assumed to be symmetric and local.
• Zero A-Posteriori Error: In "DNS-aided" experiments (where the exact residual is computed from Direct Numerical Simulation data and injected into the LES equation), the proposed RST expression yields zero a-posteriori error. In contrast, classical RST expressions result in errors that accumulate over time.
• Improved Closure Modeling: When the new RST is used as target data to fit the coefficient of a Smagorinsky model, the resulting coefficient is higher than when fitted to the classical RST. This leads to improved performance in terms of energy spectra and relative error, particularly for small filter-to-grid ratios ($\Delta/h$).

Results
The authors validate their theory using two test cases: the 1D viscous Burgers equation and 3D decaying isotropic turbulence.

• 1D Burgers Equation:

  • A-Priori Analysis: The study shows that for common filter ratios ($\Delta \le 4h$), discretization errors (flux and divergence terms) constitute a significant portion (up to 70-80%) of the total residual. Neglecting these terms leads to a poor representation of the true residual.
  • A-Posteriori Analysis: Using the exact discretization-informed residual as a closure term results in a "perfect" reconstruction of the filtered DNS solution.
  • Smagorinsky Model: Fitting the Smagorinsky coefficient to the new RST yields a higher coefficient value compared to fitting to the classical RST. Simulations using this informed coefficient show lower relative errors and energy spectra that better match the filtered DNS, particularly at low $\Delta/h$ ratios.

• 3D Navier-Stokes Equations:

  • The derivation confirms that the 3D RST is non-symmetric and non-local.
  • DNS-aided experiments on 3D decaying turbulence confirm that the proposed framework correctly captures the residual stress, validating the extension of the 1D theory to full incompressible flow.

Significance and Claims
The paper claims that the traditional separation between "modeling the physics" (LES closure) and "modeling the numerics" (discretization) is flawed when the filter width and grid spacing are comparable. By unifying these processes into the LES-FVM framework, the authors provide a mathematically rigorous definition of the unresolved stress that is consistent with the numerical scheme.

The primary significance lies in the demonstration that discretization-induced errors are not merely numerical noise but are integral parts of the residual stress that must be accounted for in the closure model. The authors argue that using the classical RST as a target for data-driven or tuned closure models introduces a fundamental inconsistency, leading to suboptimal model coefficients and accumulated errors. Their proposed "discretization-informed" RST provides the correct target data, enabling the development of more accurate and stable closure models, particularly for implicit LES or scenarios with coarse grids where $\Delta \approx h$.

The authors remain modest regarding the immediate application of these findings to complex industrial flows, noting that while the new RST improves the performance of the simple Smagorinsky model, the full potential of this framework will likely be realized with more expressive, non-local, and non-symmetric closure models (e.g., machine learning-based approaches) that can fully exploit the structure of the derived residual.