The moving discontinuous Galerkin finite element method with interface condition enforcement for compressible viscous flows

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AbstractThe moving discontinuous Galerkin finite element method with interface condition enforcement is applied to the case of viscous flows. This method uses a weak formulation that separately enforces the conservation law, constitutive law, and the corresponding interface conditions in order to provide the means to detect interfaces or underresolved flow features. To satisfy the resulting overdetermined weak formulation, the discrete domain geometry is introduced as a variable, so that the method implicitly fits a priori unknown interfaces and moves the grid to resolve sharp, but smooth, gradients, achieving a form of anisotropic curvilinear r‐adaptivity. This approach avoids introducing low‐order errors that arise using shock capturing, artificial dissipation, or limiting. The utility of this approach is demonstrated with its application to a series of test problems culminating with the compressible Navier–Stokes solution to a Mach 5 viscous bow shock for a Reynolds number of 105 in two‐dimensional space. Time accurate solutions of unsteady problems are obtained via a space‐time formulation, in which the unsteady problem is formulated as a higher dimensional steady space‐time problem. The method is shown to accurately resolve and transport viscous structures without relying on numerical dissipation for stabilization.