Self-Tuning Dynamic Explicit Modal Filtering Based on Local Flow Characteristics for Large-Eddy Simulation

TL;DR

The paper addresses stabilization and energy dissipation in high-order DG-LES by introducing STDEMF, a self-tuning modal filter that leverages orthogonality to extract modal coefficients on general collocation grids. A hyperbolic tangent filter kernel and a dynamically computed cut-off mode $M$—based on the local Kolmogorov scale $\eta$, grid spacing $\Delta$, and strain/rotation invariants $Q_S$ and $Q_\Omega$—provide adaptive energy removal in unresolved regions while ensuring vanishing dissipation in laminar flow. The method is demonstrated on HID, Taylor-Green vortex, and periodic channel flow, consistently outperforming the DEMF and Smagorinsky models in accuracy and robustness; it achieves this with only modest computational overhead thanks to one-time transformation matrix setups. Overall, STDEMF offers a scalable, physics-informed, and grid-agnostic approach to LES stabilization on complex meshes, with potential for broader applicability and future machine-learning enhancements.

Abstract

This work improves upon our previously introduced explicit dynamic modal filter (DEMF) within the framework of the discontinuous Galerkin spectral element method (DGSEM) by introducing a mechanism for self-tuning of the model parameters. The new self-tuning dynamic explicit modal filter (STDEMF) also extends the methodology for obtaining modal values from nodal values beyond Chebyshev grids and polynomials to general collocation points and orthogonal polynomial bases by leveraging orthogonality. The generated modes are used to remove the built-up energy due to unresolved sub-grid scales (SGS) in large-eddy simulation (LES) of turbulent flows. The STDEMF improves the performance of DEMF in two ways. First, the filter kernel applied to the modes is adapted from a cut-off kernel to a hyperbolic tangent shape, which automatically adjusts the model for different polynomial orders. Second, the cut-off mode is computed dynamically for each element as a function of local flow characteristics, including the local Kolmogorov length scale and the second invariants of the strain and rotation rate tensors. The suggested formulation for the cut-off mode treats unresolved elements distinctly and improves performance by avoiding under- or over-dissipation. Moreover, the cut-off mode evolves over time within the same element as turbulent characteristics vary. The model is evaluated on three flows, homogeneous isotropic decaying, the Taylor-Green vortex, and periodic channel flow, each with distinct turbulent characteristics. Comparisons of the results show that the STDEMF model outperforms the DEMF model and the Smagorinsky eddy viscosity model.

Figures (18)

• Figure 1: Filter kernels for two elements with $\frac{\Delta}{\eta}=1.5$ and $\frac{\Delta}{\eta}=2.5$ ($N=19$, $X=0.6$, $c=0.4$ in Eq. (\ref{['eq:cutoff mode']})).
• Figure 2: Schematic of modal filtering for a 2D problem with $\mathcal{P}=2$ inside each element.
• Figure 3: Comparison of the turbulent kinetic energy in the HID flow: (a) DNS results between the HORSES3D and DSEM solvers; (b) DNS, coarse-DNS, Smagorinsky, DEMF, and STDEMF model results for $t=[0,8]$; and (c) a zoomed view of DNS, coarse-DNS, Smagorinsky, DEMF, and STDEMF model results for $t=[2,6]$.
• Figure 4: Temporal evolution of the STDEMF model parameters for two distinct elements in the $6\mathcal{P}7$ case of HID flow: (a) $\Delta/\eta$, (b) $X_\Omega$, (c) $X_S$, (d) $X$, and (e) the cut-off mode $M$.
• Figure 5: Probability density function of the cut-off mode for the $6\mathcal{P}7$ case of HID flow at different times.