On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods

Abstract

We analyse numerical errors (dissipation and dispersion) introduced by the discretisation of inviscid and viscous terms in energy stable discontinuous Galerkin methods. First, we analyse these methods using a linear von Neumann analysis (for a linear advection-diffusion equation) to characterise their properties in wave-number space. Second, we validate these observations using the 3D Taylor-Green Vortex Navier-Stokes problem to assess transitional/turbulent flows. We show that the dissipation introduced by upwind Riemann solvers affects primarily high wave-numbers. This dissipation may be increased, through a penalty parameter, until a critical value. However, further augmentation of this parameter leads to a decrease of dissipation, reaching zero for very large values. Regarding the dissipation introduced by second order derivatives, we show that this dissipa- tion acts at low and medium wave-numbers (lower wave-numbers compared to upwind Riemann solvers). In addition, we analyse the Spectral Vanishing Viscosity (SVV) technique, previously used in continuous discretisations (e.g. Fourier), to find that with an appropriate kernel (which damps selected modes) it is possible to control the amount of dissipation introduced in the low and medium wave-number range. Combining these ideas, we finally propose a DG-SVV approach that uses a Smagorinsky model to compute the numerical viscosity. This DG-SVV approach is tested in an isotropic laminar/turbulent under-resolved scenario. Combining the SVV technique with a low dissipation Riemann solver, we obtain a scheme capable of maintaining low dissipation levels for laminar flows, whilst providing the correct dissipation for all wave-number ranges in turbulent regimes.