Spatial eigenanalysis of spectral/hp continuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows

Abstract

This study considers the spatial eigensolution analysis of spectral/hp continuous Galerkin (CG) schemes, complementing a recent work by Moura et al. (2016) [15] which addressed CG's temporal analysis. While the latter assumes periodic boundary conditions, the spatial approach presumes inflow/outflow type conditions and therefore provides insights for a different class of problems. The linear advection-diffusion problem is considered for a wide range of Péclet numbers, allowing for viscous effects at different intensities. The inviscid (linear advection) case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous Galerkin (DG) schemes in the limit of strong over-upwinding. These effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under-resolved simulations at high Reynolds numbers. The spectral vanishing viscosity (SVV) technique is subsequently considered as a natural stabilization strategy, in the context of linear advection. An optimization procedure is employed to match SVV diffusion levels to those of DG at appropriate polynomial orders. The resulting CG-SVV discretisations are tested against under-resolved computations of spatially developing vortex-dominated flows and display excellent robustness at high Reynolds numbers along with superior eddy-resolving characteristics at higher polynomial orders. This highlights the importance of appropriate stabilization techniques to improve the potential of spectral/hp CG methods for high-fidelity simulations of transitional and turbulent flows, including implicit LES / under-resolved DNS approaches.