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Show ible systems and for reacting systems with a large number of species equations. Also, due to the stiffness inherent in systems involving chemical reactions of many species, each of which react on different time scales, convergence using the SIMPLE based schemes becomes difficult and often impossible. Research into high speed flow where high temperatures due to viscous and compressive heating result in the dissociation of air has spawned the development of more sophisticated numerical algorithms for solving the Navier-Stokes equations with fully coupled chemistry. These compressible Navier-Stokes methods have been significantly advanced to successfully include the effects of non-equilibrium chemistry. However, at very low Mach numbers, these compressible Navier-Stokes methods become inefficient or inaccurate; the cause of which is due to the increasingly different convective and acoustical signal speeds as the Mach number approaches zero. These differences in signal speeds result in large differences in the magnitudes of the eigenvalues of the system. As the Mach number tends to zero, the matrix system becomes singular and numerical solution becomes impossible using traditional density based Navier-Stokes algorithms. Time derivative preconditioning methods have become common as a method of controlling system eigenvalues to maintain effective convergence for flows as the convective speed drops appreciably below that of the acoustical speed. Some recent work on this subject includes Turkel (1987), Feng and Merkle (1990), Choi and Merkle (1993), Weiss and Smith (1994) and Turkel et al. (1997) . For reacting systems of interest in the chemical manufacturing industry, the flow speed is usually very low. However, because of the release of heat due to chemical reactions, the density of the fluid can vary appreciably and therefore, an incompressible formulation is not possible. Preconditioning methods have recently been applied to chemical reacting flows by Shuen et al. (1993) and Shuen (1992) Quite recently Edwards and Roy (1998) have combined low speed preconditioning with a full multigrid solution algorithm. In this work, convective fluxes are treated using the second-order flux-splitting technique introduced by Edwards (1997) and Edwards and Liou (1998). Viscous terms are treated using standard central differencing. This method of solving the chemical non-equilibrium Navier-Stokes equations is adopted in the present work and is outlined in detail in the next section. The successful numerical solution of complex reacting flows depends on providing a computational grid with adequate resolution in high gradient regions of the flow (large error regions). To provide this same resolution uniformly throughout the entire domain would result in overly expensive computational requirements. To minimize the number of grid points, a solution-adaptive grid algorithm is presented which redistributes grid points during the course of the Navier- Stokes solution process. cpShtreoemkmieiTssxht eredfy o.mr pemTrtuhehlceaao ntndeide-oitanati irloi endsdei dfa1f,p7u ps lsisiopoeneldc fu ilteiatsomo/n5e -8aa w di ratelphaat cmfituivilenola nyr Nmc,ao evucipnheloare- nd-nism of Bilger et al. (1990) is used. This reacting flow was first studied experimentally and computationally by Mitchell (1975) and Smooke et al. (1989) using a detailed chemistry mechanism and a stream function-vorticity solution technique. Xu et al. (1993) solved the same problem using a primitive variables formulation with a implicit Newton's method. The present results for the laminar flame with detailed chemistry illustrate that the Navier-Stokes methods outlined below provides an inexpensive means of computing complex reacting flow systems. Governing Equations Compressible Reacting Formulation The steady Navier-Stokes and species transport equations for a chemically reacting mixture of N (species) ideal gases can be written in a generalized coordinate system as: |(E-E„) + |(F-F„; + -(G-Gv) = Hg (1) where the inviscid and viscous flux vectors in the curvilinear coordinate system are written in terms of the corresponding flux vectors in a Cartesian frame as E= I^E + ^F + kG) E, J F = -feE + T}yF + rjzG) G = j(rxE + CyF + CzG) = j(GE„ +^F„ +&GV) -(r)TEv + T]yFv + T]:Gv) (2a) F = (26) Gv = -(CxEv + <;zFv + CzGv) In the above expressions, £, n and C are the spatial coordinates in the generalized frame of reference. The Cartesian fluxes are defined by the following expressions: Q = / pYl \ pYN pu pv pw \E< I ( pvYi \ E = F = pvYi N puv pv. 2 +p pvw \(Et+p)vJ / puY\ \ puYN pu2 + p puv puw \(Et +p)uJ ( pwY\ \ (3,4a) G = pwYN puw pvw pw2 + p \(Et +p)wJ (4b Ac) 2 |
