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In this work we develop an hp-adaptive discontinuous Galerkin (DG) solver for aerodynamic flows. Previous work has focused on efficient solution techniques for discontinuous Galerkin discretizations. Recent work has focused on improving the robustness and efficiency of our discontinuous Galerkin solver for aerodynamic flows. Herein we propose an hp-adaptive approach which seeks to place degrees of freedom within the domain in the manner most appropriate for the nature of the solution. Near discontinuities the algorithm will refine the mesh while in regions where the solution is smooth it will enrich the discretization order. This has two effects, first of all degrees of freedom are placed where they are needed thus addressing the efficiency of the method and second we avoid attempting to use high-order polynomials to capture solutions which are discontinuous, addressing the robustness of the method. The adaptation procedure is driven via a discrete adjoint-based goaloriented error estimation technique. The method is evaluated using three test cases all of which are steady state flows. Two of these are laminar viscous flows and one is an inviscid transonic flow. In addition the transonic flow has been computed using an artificial diffusion method and the hp-adaptive approach is compared to the artificial diffusion shock capturing method.

The goal of this paper is to investigate and develop fast and robust solution techniques for high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured meshes. Previous work was focused on the development of hp-multigrid techniques for inviscid flows and the current work concentrates on the extension of these solvers to steady-state viscous flows including the effects of highly anisotropic hybrid meshes. Efficiency and robustness are improved through the use of mixed triangular and quadrilateral mesh elements, the formulation of local order-reduction techniques, the development of a line-implicit Jacobi smoother, and the implementation of a Newton-GMRES solution technique. The methodology is developed for the two-and three-dimensional Navier-Stokes equations on unstructured anisotropic grids, using linear multigrid schemes. Results are presented for a flat plate boundary layer and for flow over a NACA0012 airfoil and a two-element airfoil. Current results demonstrate convergence rates which are independent of the degree of mesh anisotropy, order of accuracy (p) of the discretization and level of mesh resolution (h). Additionally, preliminary results of ongoing work for the extension to the Reynolds Averaged Navier-Stokes(RANS) equations, the development of a Gas-Kinetic (BGK) inter-cell flux function implementation, and the extension to three dimensions are given.

This work considers the development of a robust discontinuous Galerkin (DG) solver for turbulent aerodynamic flows using the turbulence model of Spalart and Allmaras (SA). Previous work on this subject has demonstrated that applying DG discretizations to turbulent flows can be difficult, due to robustness issues related to non-smooth behavior of the turbulence model variable (or variables). This work presents two options for enhancing solver robustness. The first consists of employing a finite volume discretization with a first-order accurate convection term for the turbulence model, which is a standard practice in the low-order methods context. Computational results show that despite the first-order accurate discretization of the turbulence model there is still benefit to using higher-order discretizations for the mean flow equations and at the very least, discontinuous Galerkin solutions to the Reynolds Averaged Navier-Stokes (RANS) equations are obtained robustly. The second method of robustness enhancement considers modifications to the turbulence model equation. Numerical experiments have shown that the modifications to the turbulence model equation employed in this work are particularly effective at increasing solver robustness. Both robustness enhancement methods are applied to realistic aerodynamic flows including a subsonic turbulent airfoil flow and high-lift configurations at high angles of attack.