Application of Spectral Discontinuous Galerkin Methods to Hypersonic Flows

We apply a discontinuous Galerkin spectral element method (DG-SEM) to the threedimensional compressible Navier-Stokes equations. To achieve optimal accuracy in all variables, we use piecewise polynomial approximations of degree P for each of them. In order to capture discontinuities, we use a subcell shock-capturing method based on an artificial viscosity approach. To render the implementation more efficient, we use collocation- hence the name "spectral". We use general hexahedral elements with curved faces to fit the boundaries. Our main result is a series of numerical experiments on canonical verification cases which display the capability of the DG-SEM method for numerically solving hypersonic flows. First, we show that when the solution is smooth, the optimal order of P + 1 is obtained for all the variables. Then, we apply the method to the Shu-Osher method to show it can capture well the entropy-satisfying discontinuities. Finally, we apply it to a Mach 17.605 viscous flow impinging on a cylinder. We show that the method, with P = 2 polynomial degree, produces solutions comparable to one obtained by the finite volume code developed by Candler et al. [AIAA Paper 2015-1893].