Higher Order Upwind Schemes on Unstructured Grids for the Nonstationary Compressible Navier-Stokes Equations in Complex Timedependent Geometries in 3D

Abstract

SummaryThe aim of this project was to develop a higher order upwind finite volume scheme on unstructured grids of tetrahedra to solve the nonstationary compressible Navier-Stokes equations with high Reynolds numbers. The new aspects in this numerical scheme are a new definition of a limiter function for a special class of linear reconstruction function to get an upwind finite volume scheme of higher order on an unstructured grid of simplices, a new criterion to adapt the unstructured grid locally and a new kind of discretization of second derivatives. With several numerical tests we will show that we can obtain better results with these new approaches than with existing ones, which we have also used in numerical tests. So it turns out that the new upwind scheme of higher order for conservation laws is really of higher order in regions where the solution is smooth and has no oscillations at discontinuities. The quality of the new discretization of second derivatives will be shown also on grids with large aspect ratios. We will apply the solver for the compressible Euler equations to the flow in a simplified two-stroke engine with a moving piston in 3D. For the modelling of the moving boundary we developed a new technique which guarantees the conservation of mass during the calculations.