Gradient jump penalty stabilisation of spectral/[formula omitted] element discretisation for under-resolved turbulence simulations

Abstract

One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG’s intrinsic (upwind) dissipation being biased towards high frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial diffusion such as spectral vanishing viscosity (SVV). However although SVV is recognised as very useful in CG-based high-fidelity turbulence simulations, this approach has been observed to be sub-optimal when compared to DG at intermediate polynomials orders (P≈3). In this paper we explore an alternative stabilisation approach through the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP’s P-dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier–Stokes equation, as we achieve higher quality flow solutions at P=3 using GJP, whereas SVV performs marginally worse at P=5 with twice as many degrees of freedom in total.