Assessment of the Gradient Jump Penalisation in Large-Eddy Simulations of Turbulence

TL;DR

This study evaluates gradient jump penalisation (GJP) as a stabilization for continuous Galerkin spectral element method–based large-eddy simulations (LES) when coupled with active subgrid-scale models. GJP introduces a penalty term in the weak form to damp inter-element gradient jumps, with a tunable parameter and hex-element formulation, and is tested on the Taylor–Green vortex, periodic hills, and channel flow at representative Reynolds numbers. The results show GJP reduces non-physical wiggles and smooths spectra at high wavenumbers but tends to oversmooth some fine-scale features and can modify low-wavenumber energy, indicating GJP provides targeted high-k dissipation while leaving the large scales under-resolved. Overall, GJP improves spectral smoothness without altering global energetics significantly, but additional dissipation mechanisms or filtering strategies are needed for more complete stabilization in SEM-LES with active SGS models.

Abstract

This research investigates the efficacy of the gradient jump penalisation (GJP) in large eddy simulations (LES) when coupled with active subgrid-scale (SGS) models. GJP is a stabilisation method tailored for the continuous Galerkin spectral element method, aiming at mitigating non-physical oscillations induced by discontinuous velocity gradients across element interfaces. We demonstrate that GJP effectively smoothens fields from LES without a salient impact on flow dynamics for the Taylor--Green vortex (TGV) at $Re=1600$, periodic hill flows at bulk Reynolds numbers $Re_b=10595$ and $37000$, as well as turbulent channel flow at $Re_τ \approx 550$. In the TGV case, the application of GJP results in decreased fluctuations at only high wavenumbers compared to simulations without GJP. The periodic hill flow simulations indicate the applicability of GJP in wall-resolved LES (WRLES) involving curved geometries, though it tends to dissipate some of the finer details in the solution. Finally, in the analysis of the canonical turbulent channel flow cases, GJP leads a higher resolved turbulent kinetic energy than simulations without GJP and direct numerical simulations. GJP's mechanism is identified as providing enhanced dissipation at high wavenumbers but accompanied with insufficient dissipation at low wavenumbers, leading to a pronounced spectral cut-off. Non-physical oscillations on element interfaces are reflected as spikes in the power spectral density. By evaluating the sharpness of the strongest spike, GJP is shown to smoothen the spectra, however without completely removing the gradient jumps at low computational resolution.

Figures (17)

• Figure 1: An illustration of the determination of $h_{\Omega_e}\vert_{P_1,P_2}$ for a hexahedral element with $P=2$
• Figure 2: Volume-averaged kinetic energy and enstrophy, obtained using the classical Smagorisnky model with $C_S=0.17$ and $\Delta=(V_{\Omega_e}/(P+1)^3)^{1/3}$.
• Figure 3: Turbulent kinetic energy $\sqrt{U^2+V^2+W^2}/V_0$ extracted in the plane at $y=-L\pi$ and $t/t_c=17$.
• Figure 4: Spatial energy spectrum at two time instants, with superimposed $-5/3$ power law.
• Figure 5: (a) Sketch of the geometry definitions taken from Rapp2009Phill and (b) cross-section (x-y) of the mesh showing the elements for the LES case at $Re_b=37000$.