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We analyse numerical errors (dissipation and dispersion) introduced by thediscretisation of inviscid and viscous terms in energy stable discontinuousGalerkin methods. First, we analyse these methods using a linear von Neumannanalysis (for a linear advection-diffusion equation) to characterise theirproperties in wave-number space. Second, we validate these observations usingthe 3D Taylor-Green Vortex Navier-Stokes problem to assesstransitional/turbulent flows. We show that the dissipation introduced by upwindRiemann solvers affects primarily high wave-numbers. This dissipation may beincreased, 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 bysecond order derivatives, we show that this dissipa- tion acts at low andmedium wave-numbers (lower wave-numbers compared to upwind Riemann solvers). Inaddition, we analyse the Spectral Vanishing Viscosity (SVV) technique,previously used in continuous discretisations (e.g. Fourier), to find that withan appropriate kernel (which damps selected modes) it is possible to controlthe amount of dissipation introduced in the low and medium wave-number range.Combining these ideas, we finally propose a DG-SVV approach that uses aSmagorinsky model to compute the numerical viscosity. This DG-SVV approach istested in an isotropic laminar/turbulent under-resolved scenario. Combining theSVV technique with a low dissipation Riemann solver, we obtain a scheme capableof maintaining low dissipation levels for laminar flows, whilst providing thecorrect dissipation for all wave-number ranges in turbulent regimes.

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