A rational length scale for large-eddy simulation of turbulence on anisotropic grids

TL;DR

The paper addresses the lack of a universally robust subgrid length δ for LES on anisotropic and unstructured grids. It derives a rational length scale δ_rls from the entanglement between numerical discretization and LES filtering, computed directly at cell faces, with an optional dissipation-equivalent variant δ̃_rls that leverages velocity-gradient invariants. Numerical experiments on decaying isotropic turbulence and a wall-bounded channel flow show that δ_rls markedly reduces sensitivity to mesh anisotropy and yields convergent, physically plausible results where conventional δ_vol-based closures fail or diverge. The approach is simple to implement, geometry-agnostic, and suitable for unstructured grids and DES, offering a practical path to more reliable LES in complex geometries with strong grid anisotropy.

Abstract

Due to the prohibitive cost of resolving all relevant scales, direct numerical simulations of turbulence remain unfeasible for most real-world applications. Consequently, dynamically simplified formulations are needed for coarse-grained simulations. In this regard, eddy-viscosity models for Large-Eddy Simulation (LES) are widely used both in academia and industry. These models require a subgrid characteristic length, typically linked to the local grid size. While this length scale corresponds to the mesh step for isotropic grids, its definition for unstructured or anisotropic Cartesian meshes, such as the pancake-like meshes commonly used to capture near-wall turbulence or shear layers, remains an open question. Despite its significant influence on LES model performance, no consensus has been reached on its proper formulation. In this work, we introduce a novel subgrid characteristic length. This length scale is derived from the analysis of the entanglement between the numerical discretization and the filtering in LES. Its mathematical properties and simplicity make it a robust choice for reducing the impact of mesh anisotropies on simulation accuracy. The effectiveness of the proposed subgrid length is demonstrated through simulations of decaying isotropic turbulence and a turbulent channel flow using different codes.

Figures (13)

• Figure 1: Example of a one-dimensional mesh.
• Figure 2: Left: face normal and neighbor labeling criterion. Right: definition of the volumes, $\mathsf{\Omega}_s$, associated with the the face-normal velocities, $\boldsymbol{u}_s$. Thick dashed rectangle is the volume associated with the staggered velocity $\mathrm{U}_{4} = [ \boldsymbol{u}_s ]_{4}$, i.e.,$[ \mathsf{\Omega}_s ]_{4,4} = A_{4} \delta_{4}$ where $A_{4}$ is the face area and $\delta_{4} = | \boldsymbol{n}_{4} \cdot \overrightarrow{{c1} {c2}} |$ is the projected distance between adjacent cell centers. Thin dash-dotted lines are placed to illustrate that the sum of volumes is exactly preserved $\mathrm{tr}{\mathsf{\Omega}_s}=\mathrm{tr}{\mathsf{\Omega}}= d \mathrm{tr}{\mathsf{\Omega}_{c}}$ ($d=2$ for 2D and $d=3$ for 3D) regardless of the location of the cell nodes.
• Figure 3: Flowcharts to compute the eddy-viscosity at the cell faces, $\boldsymbol{\nu}_{t,s}$.
• Figure 4: Comparison between $\tilde{\delta}_{\mathrm{rls}}$ and $\delta_{\mathrm{vol}}$ for the simple 2D flow defined in Eq.(\ref{['simpleflow']}) with different values of $\beta=\{1/5,1/2,2,5,10\}$.
• Figure 5: Scaling of different definitions of $\delta$ for a Cartesian mesh with $\Delta x=\Delta y=1$ and $\Delta z=\alpha$. Average results of $\delta_{\mathrm{lsq}}$ and $\tilde{\delta}_{\mathrm{rls}}$ have been obtained from a large enough sample of random traceless velocity gradient tensors.

Theorems & Definitions (2)

• Remark 1
• Remark 2