From Shock-Capturing to High-Order Shock-Fitting Using an Unfitted Discontinuous Galerkin Method

In industry and research, CFD methods play an essential role in the study of compressible flows which occur, for example, around airplanes or in jet engines, and complement experiments as well as theoretical analysis. In transonic flows, the flow speed may already exceed the speed of sound locally, giving rise to discontinuous flow phenomena, such as shock waves. These phenomena are numerically challenging due to having a size of only a few mean free paths and featuring a large gradient in physical quantities. The application of traditional low-order approaches, such as the FEM or the FVM, is usually limited by their immense computational costs for large three-dimensional problems with complex geometries when aiming for highly accurate solutions. By contrast, high-order methods, such as the DG method, inherently enable a deep insight into complex fluid flows due to their high-order spatial convergence rate for smooth problems while requiring comparatively few. This work presents two different numerical approaches in the context of unfitted DG discretizations of the Euler equations for inviscid compressible flow. In these approaches, we employ a sharp interface description by means of the zero iso-contour of a level-set function for the treatment of immersed boundaries and shock fronts, respectively. Thus, we omit the elaborate and computationally expensive generation of boundary-fitted grids. The robustness, stability, and accuracy of the presented numerical approaches are tested against a variety of benchmarks. The presented shock-capturing approach makes use of a DG IBM. It features a cell-agglomeration strategy in order to avoid ill-conditioned system matrices and a severe explicit time-step restriction both caused by small and ill-shaped cut cells. In high Mach number flows, the polynomial approximation oscillates in the vicinity of discontinuous flow phenomena, degrading the accuracy and the stability of the numerical method. As a remedy, we adapt a two-step shock-capturing strategy consisting of a modal-decay detection and a smoothing based on artificial viscosity for the application on an agglomerated cut-cell grid. However, the second-order artificial viscosity term drastically restricts the globally admissible time-step size. We address this issue by means of an adaptive LTS scheme, which we extend by a dynamic rebuild of the cell clustering for an efficient application in unsteady flows. The presented shock-fitting approach employs an XDG method, which we enhance by verifying the implementation of two level-set functions. Their zero iso-contours describe a solid body and a shock front, respectively. We present a novel sub-cell accurate reconstruction procedure of the shock front. In particular, we show a one-dimensional proof of concept for a stationary normal shock wave by applying an implicit pseudo-time-stepping procedure in order to correct the interface position inside a cut background cell. Thus, this work builds a fundamental basis on the way towards a high-order XDG method for supersonic compressible flow in three dimensions