The Moving Discontinuous Galerkin Method with Interface Condition Enforcement for the Simulation of Hypersonic, Viscous Flows

Abstract

The moving discontinuous Galerkin method with interface condition enforcement (MDG-ICE) is a high-order, r-adaptive method that treats the grid as a variable and weakly enforces the conservation law, constitutive law, and corresponding interface conditions in order to implicitly fit high-gradient flow features. In this paper, we develop an optimization solver based on the Levenberg-Marquardt algorithm that features an anisotropic, locally adaptive penalty method to enhance robustness and prevent cell degeneration in the computation of hypersonic, viscous flows. Specifically, we incorporate an anisotropic grid regularization based on the mesh-implied metric that inhibits grid motion in directions with small element length scales, an element shape regularization that inhibits nonlinear deformations of the high-order elements, and a penalty regularization that penalizes degenerate elements. Additionally, we introduce a procedure for locally scaling the regularization operators in an adaptive, elementwise manner in order to maintain grid validity. We apply the proposed MDG-ICE formulation to two- and three-dimensional test cases involving viscous shocks and/or boundary layers, including Mach 17.6 hypersonic viscous flow over a circular cylinder and Mach 5 hypersonic viscous flow over a sphere, which are very challenging test cases for conventional numerical schemes on simplicial grids. Even without artificial dissipation, the computed solutions are free from spurious oscillations and yield highly symmetric surface heat-flux profiles.

Figures (35)

• Figure 1.1: Comparison of DG($\mathcal{P}_{1}$) (with artificial viscosity) (left) and MDG-ICE($\mathcal{P}_{4}$) (right) solutions to two-dimensional Mach 5 flow over a cylinder at $\mathrm{Re}=10^{4}$Ker20. The corresponding meshes are also displayed.
• Figure 1.2: Final mesh for the isoparametric MDG-ICE($\mathcal{P}_{4}$) solution to two-dimensional Mach 5 flow over a cylinder at $\mathrm{Re}=10^{5}$Ker20.
• Figure 5.1: The space-time Burgers initial condition and final solutions for $\mu=10^{-3}$, $\mu=10^{-4}$, and $\mu=10^{-5}$, with the corresponding grids superimposed. The initial grid consists of 200 triangular elements of quadratic geometric order with a regular topology. The initial condition is given in Equation (\ref{['eq:burgers_initial_condition']}).
• Figure 5.2: One-dimensional profiles at $t=0.025$ and $t=0.975$ for $\mu=10^{-3}$, $\mu=10^{-4}$, and $\mu=10^{-5}$ for space-time Burgers viscous shock formation. The initial grid consists of 200 triangular elements of quadratic geometric order with a regular topology. The initial condition is given in Equation (\ref{['eq:burgers_initial_condition']}).
• Figure 5.3: Nonlinear convergence history for space-time Burgers viscous shock formation. The initial grid consists of 200 triangular elements of quadratic geometric order with a regular topology. The initial condition is given in Equation (\ref{['eq:burgers_initial_condition']}).