On the Numerical Solution of the Compressible Navier-Stokes Equations for Reacting and Non-Reacting Gas Mixtures

To study acoustic effects in premixed laminar flames the compressible Navier-Stokes equations with stiff source terms for the chemistry are numerically solved by a cell centered finite volume method. Three basic numerical problems are discussed which deal with gas mixtures, convergence acceleration for subsonic simulations and the computation of the inviscid fluxes at the volume interfaces taking the viscous terms, the source terms and 2D effects into account. Conservative Euler solvers for gas mixtures produce numerical errors, if the temperature and the ratio of specific heats are not constant. For mixtures of calorically perfect gases, a simple correction of the total energy per unit volume is proposed to avoid these errors. This is done in a physical way and only the total energy looses some of its conservativity. Numerical simulations of contact discontinuity convection, a shock tube problem and shock-interface interactions in 1D and 2D yield much more accurate solutions, if the correction is applied. The straightforward extension to 3D is outlined. As the ratio of the acoustic and entropy wave speeds is large for low Mach number flames, a lot of time steps are necessary with an explicit scheme to simulate a contact discontinuity crossing the computational domain. An easy way is shown how one can use much larger time steps with an explicit code to obtain the steady state solution. The method is based on the idea that the ratio of the acoustic and entropy wave speeds gets closer to one by subtracting a constant value from the pressure in the whole field. Only the inviscid terms of the energy equation are influenced by that pressure decrease. As long as compressibility effects remain small, the error remains small. Moreover, the error can be corrected by solving a scalar equation after each time step such that the steady state solution of the modified scheme is equal to the steady state of the non-modified scheme. Applying a conventional Riemann solver for flame simulations and even for 2D Euler simulations without source terms can lead to dramatic inaccuracies. A new approach for a flux solver is introduced, which takes viscous terms, source terms and 2D effects into account. The basic idea is to distribute the source terms, which also contain the viscous terms and 2D effects, to the corresponding volume interfaces. The price is a nonlinear algebraic system for six unknowns instead of a linear system for three unknowns to evaluate the fluxes. Simulations of premixed laminar flames in 1D and 2D and a 2D Euler simulation without source terms yield much more accurate results, if the new solver is applied. Unsteady simulations of two colliding flames producing sound show results which correspond almost precisely to the analytic solution. Thus, opposed to conventional Riemann solvers, our new flux solver is able to compute acoustic effects in flames accurately. Finally numerical results of acoustic interaction with a 2D Bunsen flame show a flattened flame shape which is at least qualitatively comparable with experimental measurements. The present approach for a flux solver is more general and can be applied to solve other systems of partial differential equations which contain inviscid terms, e.g. for the shallow water equations.