On the anti-aliasing properties of entropy filtering for discontinuous spectral element approximations of under-resolved turbulent flows

TL;DR

The paper addresses aliasing in under-resolved turbulent flows simulated with high-order discontinuous spectral element methods and assesses entropy filtering as an adaptive, parameter-free anti-aliasing technique. It compares EF against over-integration and modal filtering in implicit large-eddy simulations of a NACA0021 deep-stall case, using the entropy criterion $\sigma = P\rho^{-\\gamma}$ to regulate filtering. The results show entropy filtering stabilizes the simulation with accuracy comparable to over-integration and often more robust than modal filtering, while avoiding tunable parameters; higher per-step cost is offset by the ability to take larger time steps, resulting in lower total computational cost. Overall, the work supports entropy filtering as a robust, potentially unified anti-aliasing and shock-capturing tool for high-Reynolds-number flows in aero-dynamics applications.

Abstract

For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In Dzanic and Witherden (J. Comput. Phys., 468, 2022), an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.

Figures (8)

• Figure 1: Cross-section of the mesh used for the $\mathbb P_3$ (left) and $\mathbb P_4$ (right) simulations.
• Figure 2: Average surface pressure coefficient distribution computed using a $\mathbb P_3$ approximation (left) and $\mathbb P_4$ approximation (right) with over-integration (OI), modal filtering (MF), and entropy filtering (EF). Experimental results of Swalwell2005 shown for reference.
• Figure 3: Contours of streamwise velocity computed using a $\mathbb P_3$ approximation with over-integration (left), modal filtering (middle), and entropy filtering (right). Zero velocity represented by red isocontour.
• Figure 4: Contours of streamwise velocity computed using a $\mathbb P_4$ approximation with over-integration (left), modal filtering (middle), and entropy filtering (right). Zero velocity represented by red isocontour.
• Figure 5: Profiles of streamwise velocity (top row) and normal velocity (bottom row) at $x/c = 1$ (left) and $x/c = 2$ (right) computed using a $\mathbb P_3$ approximation with over-integration (OI), modal filtering (MF), and entropy filtering (EF).