JOURNAL OF COMPUTATIONAL PHYSICS 132, 175±190 (1997) ARTICLE NO. CP965622 High Accuracy Numerical Methods for Thermally Perfect Gas Flows with Chemistry Ronald P. Fedkiw, Barry Merriman, and Stanley Osher1 Department of Mathematics, University of California, 405 Hilgard Avenue, Los Angeles, California 90095-1555 Received July 25, 1996 sider the total mixture as a single compressible ¯uid, with The compressible Navier Stokes equations can be extended to the species-averaged density, momentum, and energy model multi-species, chemically reacting gas ¯ows. The result is a evolving according to the corresponding conservation laws. large system of convection-diffusion equations with stiff source In addition, the mass fraction of each species evolves ac- terms. In this paper we develop the framework needed to apply cording to a separate continuity equation. These continuity modern high accuracy numerical methods from computational gas dynamics to this extended system. We also present representative equations are strongly coupled through the chemical reac- computational results using one such method. The framework de- tions, and they also couple strongly to the equations for veloped here is useful for many modern numerical schemes. We the mixture via the effect of reactions on temperature ®rst present an enthalpy based form of the equations that is well and pressure. suited both for physical modeling and for numerical implementa- Since chemical reactions can cause large localized tem- tion. We show how to treat the stiff reactions via time splitting, and in particular how to increase accuracy by avoidng the common perature variations during combustion, it is important to practice of approximating the temperature. We derive simple, exact accurately include the temperature dependencies in the formulas for the characteristics of the convective part of the equa- equations of state used for the gas species. The most tracta- tions, which are essential for application of all characteristic-based ble model that includes a realistic temperature dependence schemes. We also show that the common practice of using approxi- is that of a thermally perfect gas, for which the heat capaci- mate analytical expressions for the characteristics can potentially produce spurious oscillations in computations. ties can be general functions of temperature. In practice We implement these developments with a particular high ac- these functions are based on experimental data and they curacy characteristic-based method, the ®nite difference ENO differ signi®cantly from the ideal gas law at the higher space discretization with the 3rd order TVD Runge±Kutta time dis- temperatures encountered during combustion. cretization, combined with the second order accurate Strang time By considering the mathematical and physical character 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 of the problem, we can pose some general requirements for combustor. Q 1997 Academic Press suitable numerical methods. The resulting model equations form a large system of nonlinear conservation laws with both ®rst and second order derivative terms (from convec- 1. INTRODUCTION tive and diffusive transport) and zeroth order source terms (from reactions). Because the diffusive terms are weak, Chemically reacting, high speed gas ¯ows arise in a vari- we expect that the spatial transport terms will result in ety of combustion problems, such as the fueling of a scram- the development of steep fronts. Because the reactions jet engine or the incineration of waste in a dump combus- proceed rapidly once they are triggered, we expect that tor. The combination of energetic chemical reactions and the source terms will be stiff in time. Thus any numerical compressible gas dynamics yields the unique phenomena approach must effectively handle stiff time integration and of detonation and de¯agration. The basic properties of steep spatial fronts. these effects can be understood via the Chapman± Since the stiff source terms require specialized and costly Jouget theory. time integration, it is most practical to use a time splitting For theoretical modeling or numerical simulation of such to isolate their treatment from the rest of the problem. To ¯ows, the compressible Navier Stokes equations can be handle the steep spatial fronts, it is natural to apply modern extended to include multiple gas species and the appro- shock-capturing numerical methods for the convective part priate chemical reactions. The standard approach is to con- of the conservation laws. These methods typically require complete analytic expressions for the characteristic data, 1 Research supported in part by ARPA URI-ONR-N00014-92-J-1890, i.e., the eigenvalues and eigenvectors of the linearized con- NSF DMS 94-04942, and ARO DAAL03-91-G-0162. vective ¯ux matrix. 175 0021-9991/97 $25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved. 176 FEDKIW, MERRIMAN, AND OSHER Based on these general considerations, we expect many approach. We implement this framework using thermody- numerical approaches will have a common need for a namic and chemistry data tables from CHEMKIN, and proper time split formulation and analytic expressions for numerics consisting of second order Strang time splitting the characteristic data. Obtaining both these things would with a stiff ODE integrator (LSODE) for the reaction seem routine, but in fact the complexity of the equations equations, 3rd order TVD Runge±Kutta time integration makes both potentially dif®cult and has led to the use for the convection-diffusion terms, central differencing for of a variety of simplifying procedures which may cause the diffusive terms, and 3rd order ®nite difference ENO unanticipated errors in the computations, as some of our for the characteristic based discretization of the convection examples will illustrate. Our primary goal here is to show terms. We apply this to a one-dimensional Sod shock tube that, with the equations properly formulated, both the time in the presence of combustion reactions, and to a two- splitting and characteristic data can be obtained without dimensional model of a toxic waste combustor, and discuss simplifying assumptions in an unambiguous and practically the results. useful form. We also show that with these in hand, modern characteristic based methods do an excellent job of captur- 2. MODEL EQUATIONS ing the phenomena present in chemically reacting gas ¯ows. 2.1. Multiple Species We develop our framework as follows: ®rst, we present The 2-D Euler equations can be modi®ed to account for an enthalpy based formulation of the governing equations, compressible ¯ows with more than one species. The 2-D i.e., the energy equation for the mixture is written in terms Euler equations for multi-species ¯ow are of the enthalpy. Various other equivalent forms are possi- ble, such as using temperature or internal energy as the U 1 [F(U)] 1 [G(U)] 5 0 (1) explicit variable, but the enthalpy formulation is advanta- t x y geous for two reasons: it is convenient for physical model- r ru ing, and it results in a system for which the characteristics can be determined analytically in a compact and relatively ru ru2 1 p simple form. rv ruv Then, we show how to apply time splitting to these U 5 E , F(U) 5 (E 1 p)u , equations in order to isolate the time evolution of the stiff reaction terms. In the previous work there has been some rY1 ruY1 ambiguity regarding what terms should be held constant _ _ in the reaction portion of this time split evolution. For 1rYN212 1 ruYN21 2 example, it has been a common practice to freeze the (2) temperature during this step, but this is not a true time splitting of the model equations. Given the strong tempera- rv ture dependence of the reaction rates, this is also a physi- ruv cally questionable practice. Others have considered adding rv2 1 p an additional ODE for the simultaneous evolution of tem- perature with the reaction ODEs, but this approach adds G(U) 5 (E 1 p)v , unnecessary complication and also requires a decision rvY1 about which thermodynamic quantities are being held con- _ stant during the step. In contrast, we show that a proper time splitting of the stiff reaction terms unambiguously 1 rvYN21 2 requires that certain thermodynamic quantities (not tem- r(u21v2) perature) be held constant during the solution of the reac- E 52p1 1rh, (3) 2 tion ODEs, and further we show that a simple scalar root ®nding procedure, such as Newton's method, is all that is E h required to implement this proper time splitting. where is the energy per unit volume, is enthalpy per N Next, we derive simple expressions for the characteristic unit mass, is the number of species being considered, and Yi is the mass fraction of the ith species [16]. Note data, i.e., Jacobian matrix of the convective ¯uxes, and the N21 Y 5 2 Y associated eigenvalues and eigenvectors. These are the that N 1 oi51 i . primary ingredients needed to apply a variety of modern 2.1.1. Energy and Enthalpy high accuracy characteristic based methods developed for gas dynamics. The total energy per unit volume is designated by E. Finally, we illustrate the capabilities of this numerical We can write THERMALLY PERFECT GAS FLOWS 177 r(u2 1 v2) h (T) 5 h f 1 c T, (12) E 5 re 1 , (4) i i pi 2 f where h i 5 hi(0) is the enthalpy per unit mass at 0K for where e is the internal energy per unit mass. We write the the ith species. This is also sometimes called the heat of enthalpy per unit mass as formation. The heat of formation for a gas is a constant and can be found in the JANAF Thermochemical Tables N N N N [13].