A Space-Time Conservation Element and Solution Element Method for Solving the Two- and Three-Dimensional Unsteady Euler Equations Using Quadrilateral and Hexahedral Meshes

Abstract

In this paper, we report a version of the space-time conservation element and solution element (CE/SE) method in which the 2D and 3D unsteady Euler equations are simulated using structured or unstructured quadrilateral and hexahedral meshes, respectively. In the present method, mesh values of flow variables and their spatial derivatives are treated as independent unknowns to be solved for. At each mesh point, the value of a flow variable is obtained by imposing a flux conservation condition. On the other hand, the spatial derivatives are evaluated using a finite-difference/weighted-average procedure. Note that the present extension retains many key advantages of the original CE/SE method which uses triangular and tetrahedral meshes, respectively, for its 2D and 3D applications. These advantages include efficient parallel computing, ease of implementing nonreflecting boundary conditions, high-fidelity resolution of shocks and waves, and a genuinely multidimensional formulation without the need to use a dimensional-splitting approach. In particular, because Riemann solvers—the cornerstones of the Godunov-type upwind schemes—are not needed to capture shocks, the computational logic of the present method is considerably simpler. To demonstrate the capability of the present method, numerical results are presented for several benchmark problems including oblique shock reflection, supersonic flow over a wedge, and a 3D detonation flow.