A High-Order p-adaptive Algorithm for Large-Eddy Simulation Based on a Discontinuous Galerkin Method

The aim of this paper is to present and analyse a static p-adaptation algorithm based on a modal high-order discontinuous Galerkin method. To this end, we first define appropriate error estimation strategies that provide information on the necessary time-independent local resolution quality. Two error indicators are employed, the Small-Scale Energy Density (SSED) indicator proposed in Naddei (2019) and the novel Small-Scale Lifted (SSL) indicator developed in this work. Based on the SSED estimator, three different strategies are considered to extend these indicators to perform static p-adaptive simulations of unsteady flow problems. The first approach consists in applying the SSED estimator to the time-averaged solution (SSED-A). The second and third approaches consist in evaluating the temporal $ L<^>2 $ L2 and $ L<^>\infty $ L infinity-norms of the indicator computed from the instantaneous solution, called respectively $ L<^>2 $ L2-SSED and $ L<^>\infty $ L infinity-SSED. These strategies are compared by performing simulations of the periodic laminar flow past a cylinder at $ {\rm Re}=100 $ Re=100 and Mach = 0.1, and of the turbulent flow over periodic hills at $ {\rm Re}=2\,800 $ Re=2800. The outcome from these computations shows that the best performance is achieved when the $ L<^>\infty $ L infinity-SSED indicator is used. Finally, the developed adaptation algorithm is applied to the LES of the transitional flow past a NACA0012 airfoil at $ {\rm Re}=50\,000 $ Re=50000 and $ \alpha =5<^>\circ $ alpha=5 degrees using the $ L<^>\infty $ L infinity-SSED and the $ L<^>\infty $ L infinity-SSL indicators. It is shown that the use of the SSL indicator provides improved results as compared to the SSED indicator. The results presented in this paper demonstrate that the use of p-adaption improves the quality of under-resolved turbulent flow simulations for a similar computational cost as compared to p-uniform simulations.