Compact Direct Flux Reconstruction for the Navier-Stokes Equations on Dynamic Meshes

Abstract

In this study, the high-order discontinuous compact direct flux reconstruction (CDFR) method is used to solve the two-dimensional (2D) Navier-Stokes equations on quadrilateral unstructured dynamic meshes. Within a standard element, the CDFR method employs com-pact finite difference (FD) techniques to directly construct the nodal spatial derivatives on Gauss-Legendre solution points. In the procedure of constructing an arbitrary CDFR method, the spatial derivatives are approximated with local fluxes on solution points and common fluxes on element interfaces (flux points) in FD forms. No polynomial reconstruction needs to be employed explicitly. It is observed that the CDFR method is identical with the direct flux re-construction (DFR) method and the nodal flux reconstruction-discontinuous Galerkin (FR-DG) method if Gauss-Legendre points are selected as solution points. For simulations with dynamic meshes, the geometric conservation law (GCL) has been incorporated into the Navier-Stokes equations. The performance of CDFR methods has been verified with various test cases, in-cluding the Euler vortex propagation on deformable meshes, and the Couette flow. Laminar flows of Ma = 0.2 over a static circular cylinder (Re = 100, 185) and an oscillating circular cylinder (Re = 185) have been studied to demonstrate the capability of the solver developed in this study.