High Accuracy Numerical Methods for Thermally Perfect Gas Flows with Chemistry

Abstract

The compressible Navier Stokes equations can be extended to model multi-species, chemically reacting gas flows. The result is a large system of convection-diffusion equations with stiff source terms. In this paper we develop the framework needed to apply modern high accuracy numerical methods from computational gas dynamics to this extended system. We also present representative computational results using one such method. The framework developed here is useful for many modern numerical schemes. We first present an enthalpy based form of the equations that is well suited both for physical modeling and for numerical implementation. We show how to treat the stiff reactions via time splitting, and in particular how to increase accuracy by avoidng the common practice of approximating the temperature. We derive simple, exact formulas for the characteristics of the convective part of the equations, which are essential for application of all characteristic-based schemes. We also show that the common practice of using approximate analytical expressions for the characteristics can potentially produce spurious oscillations in computations.We implement these developments with a particular high accuracy characteristic-based method, the finite difference ENO space discretization with the 3rd order TVD Runge–Kutta time discretization, combined with the second order accurate Strang time splitting of the reaction terms. We illustrate the capabilities of this approach with calculations of a 1-D reacting shock tube and a 2-D combustor.