On spurious numerics in solving reactive equations

Authors

D. V. Kotov, Stanford University
Helen C. Yee, NASA Ames Research CenterFollow
W. Wang, California State University, Monterey Bay
C.-W. Shu, Brown University

Date of this Version

2012

Citation

Center for Turbulence Research Annual Research Briefs 2012.

Comments

U.S. government work.

Abstract

1. Motivation and objectives

Consider 3D reactive Euler equations of the form

U_t + F(U)_x + G(U)_y + H(U)_z = S(U), (1.1)

where U, F(U), G(U), H(U) and S(U) are vectors. Here, the source term S(U) is restricted to be homogeneous in U; that is, (x, y, z) and t do not appear explicitly in S(U). If physical viscosities are present, viscous flux derivative should be added. If the time scale of the ordinary differential equation (ODE) U_t = S(U) for the source term is orders of magnitude smaller than the time scale of the homogeneous conservation law U_t +F(U)_x +G(U)_y +H(U)_z = 0, then the problem is said to be stiff due to the source terms. In combustion or high speed chemical reacting flows the source term represents the chemical reactions which may be much faster than the gas flow, leading to problems of numerical stiffness. Insufficient spatial/temporal resolution may cause an incorrect propagation speed of discontinuities and nonphysical states for standard numerical methods that were developed for non-reacting flows. See Wang et al. (2012) for a comprehensive overview of the last two decades of development. Schemes designed to improve the prediction of propagation speed of discontinuities for systems of stiff reacting flows remain a challenge for algorithm development (Wang et al. 2012). Wang et al. also proposed a new high order finite difference method with subcell resolution for advection equations with stiff source terms for a single reaction for (1.1) to overcome this difficulty. Research for multi-species (or more species and multi-reactions) is forthcoming.

The objective of this study is to gain a deeper understanding of the behavior of high order shock-capturing schemes for problems with stiff source terms and discontinuities and on corresponding numerical prediction strategies. The studies by Yee et al. (2012) and Wang et al. (2012) focus only on solving the reactive system by the fractional step method using the Strang splitting (Strang 1968). It is a common practice by developers in computational physics and engineering simulations to include a cut off safeguard if densities are outside the permissible range. Here we compare the spurious behavior of the same schemes by solving the fully coupled reactive system without the Strang splitting vs. using the Strang splitting. Comparison between the two procedures and the effects of a cut off safeguard is the focus the present study. The comparison of the performance of these schemes is largely based on the degree to which each method captures the correct location of the reaction front for coarse grids. Here “coarse grids” means standard mesh density requirement for accurate simulation of typical non-reacting flows of similar problem setup. It is remarked that, in order to resolve the sharp reaction front, local refinement beyond standard mesh density is still needed.

For reacting flows there are different ways in formulating (1.1). The present study considers the following two commonly used formulations. These are using all the species variables vs. using the total density and N_s − 1 number of species variables (N_s is the total number of species).