Rankine–Hugoniot–Riemann Solver Considering Source Terms and Multidimensional Effects

Abstract

A new approach for a flux solver is introduced, which takes into account source terms, viscous terms, and multidimensional effects. The basic idea is to distribute the source terms, which also contain the viscous terms and multidimensional effects, from the cells to the cell interfaces. Then the fluxes on both sides of a cell interface are determined by the Rankine–Hugoniot conditions and a linearized Riemann solver. The resulting Rankine–Hugoniot–Riemann (RHR) solver yields much more accurate results than conventional Riemann solvers for steady premixed laminar flames in 1D and 2D and a steady 2D inviscid channel flow with injection. Unsteady flow simulations of two colliding flames producing sound and of acoustic oscillations flattening a 2D Bunsen flame demonstrate that the new flux solver is able to compute acoustic effects in flames accurately. This approach for a flux solver is more general and can also be applied to solve other partial differential equations which can be expressed as hyperbolic systems with source terms ex- or including higher spatial derivatives, e.g., for the shallow water equations and for the magnetohydrodynamical equations.