Using LES in a Discontinuous Galerkin method with constant and dynamic SGS models

Abstract

The Discontinuous Galerkin (DG) method provides numerical solutions of the NavierStokes equations with high order of accuracy in complex geometries and allows for highly efficient parallelization algorithms. These attributes make the DG method highly attractive for large eddy simulation (LES). The main goal of this work is to investigate the feasibility of adopting an explicit filter to the numerical solution of the Navier-Stokes equations to increase the numerical stability of underresolved simulations such as LES and to use the explicit filter in dynamic subgrid scale (SGS) models for LES. The explicit filter takes advantage of DG’s framework where the solution is approximated using a polynomial basis. The higher modes of the solution correspond to a higher order polynomial basis, therefore by removing high order modes the filtered solution contains only lower frequency content. The explicit filter is successfully used here to remove the effects of aliasing in underresolved simulations of the Taylor-Green vortex case at a Reynolds number Re = 1600. The de-aliasing is achieved by evaluating a solution at a higher order polynomial (effectively increasing the number of quadrature points used for integration) and then projecting the solution down to a lower order polynomial. The SGS models investigated include the constant coefficient Smagorinsky Model (CCSM), Dynamic Smagorinsky Model (DSM), and Dynamic Heinz Model (DHM). The Taylor-Green Vortex case exhibits a laminar-turbulent transition and it is shown that the dynamic SGS models capture this transition more accurately than the CCSM when a sufficiently high polynomial order is used. The explicit test-filter operation for the dynamic models introduces a commutation error. A brief comparison of the effects of the commutative error that exists with this filter implementation is shown although further investigation is needed to determine the more appropriate order of operations.