On the Behaviour of Numerical Schemes in the Low Mach Number Regime: an Analysis of the Dissipation Mechanism of Upwind Flux Functions on Different Cell Geometries

Computational fluid dynamics (CFD) was for a long time rigidly divided between simulating compressible and incompressible flows, but a variety of important flow phenomena like atmospheric flows are quasi-incompressible with significant density varations. Because of the good properties of schemes for compressible flows, one asks the question: can these methods cope with low Mach numbers? For decades the answer was somewhat fuzzy: Yes, in principle, but with deteriorating results for decreasing Mach numbers. In this book we shed light on this phenomenon, showing that there are two sources of error: The flux-function and the grid cell geometry. In the first part we demonstrate that flux-functions fall into two classes: one resolves all characteristic waves of the Riemann problem while the other becomes more and more diffusive for lower Mach numbers. In the second part, we present an intriguing new result: first-order upwind schemes can manage small Mach number flows but only if the grid is made up of triangular cells. Using graph theory we show that the number of degrees of freedom for the velocity field on cells with more than three edges is reduced to zero.