Physical Multidimensional Acoustics Convection Upstream Resolution Algorithm for the 3-D Euler and Navier-Stokes Equations

摘要

This project has synthesized an innovative implicit finite element CFD algorithm to advance the computational simulation of inviscid and viscous compressible flows. This report details for the 3-D Euler and Navier-Stokes equations an intrinsically infinite directional upstream formulation that rests on the mathematics and physics of multi-dimensional acoustics and convection. Based upon characteristic velocities, this formulation introduces the upstream bias directly at the differential equation level, before the spatial discretization, within a characteristics-bias governing system. A conventional centered discretization of this system on given grids directly yields an optimal discretely conservative and multi-dimensional upstream approximation for the Euler and Navier-Stokes equations. The induced dissipation is non-linearly minimized by way of a solution-dependent controller. Through a decomposition of the Beer flux divergence into multidimensional acoustics and convection acoustics components, the formulation induces consistent upstream bias along all directions of spatial wave propagation, with anisotropic variable-strength upstreaming that correlates with the spatial distribution of characteristic velocities.