Design of a Smagorinsky spectral Vanishing Viscosity turbulence model for discontinuous Galerkin methods

Abstract

We present a new closure model for Large Eddy Simulation to introduce dissipation in under–resolved turbulent simulation using discontinuous Galerkin (DG) schemes applied to the compressible Navier–Stokes equations. The development of the method is based on a thorough analysis of the numerical dissipation mechanisms in DG schemes. In particular, we use upwind Riemann solvers for inter–element dissipation, and a Spectral Vanishing Viscosity (SVV) method for interior dissipation. First, these mechanisms are analysed using a linear von Neumann analysis (for a linear advection–diffusion equation) to characterise their properties in wave–number space. Second, their behaviour is tested using the three–dimensional Taylor–Green Vortex Navier–Stokes problem to assess transitional/turbulent flows. The results of the study are subsequently used to propose a DG–SVV approach that uses a mode-selection Smagorinsky LES model to compute the turbulent viscosity. When the SVV technique is combined with a low dissipation Riemann solver, the scheme is capable of maintaining low dissipation levels for laminar flows, while providing the correct dissipation for all wave–number ranges in turbulent regimes. The developed approach is designed for polynomial orders N ≥ 2 and is specially well suited for high order schemes. This new DG–SVV approach is calibrated with the Taylor–Green test case; to then show its accuracy in an under–resolved (y+>8) channel flow at Reynolds number Reτ=183.