A New Numerical Method for Simulation of Low-Mach Combustion

The detailed numerical simulation of unsteady phenomena where there is strong coupling between momentum, mass an d energy transport, with source terms adding to the stiffness of the problem, generally requires huge computing resources. Many different methods were developed so far, some valid for steady-state problems [1,2], some suitable for unsteady simulations as well [3,4,5]. Progress in this field is achieved by development and refinement of models and methods, and by advances in computing speed. The latter are essentially foreseen through parallel architectures, and this certainly must influence the former. In this work, a new method is presented that is meant to be especially efficient in the numerica! simulation of low-Mach,low-Reynolds flows with strong thermal expansion due to heat transfer and/or chemical reactions. The mode! is based on the classic Navier-Stokes equations for reacting flows, with few simplifying assumptions which certainly apply to the cases mentioned. The equations are closed with proper constitutive equations determined by the nature of the gas mixture, and with proper initial and boundary conditions. The method is then derived by manipulating the balance equations following the approach originally developed by Harlow [6] for incompressible, non-reacting flows, here extended to compressible (in a restricted sense), reacting flows. Conventional methods developed for compressible flows solve for density as a main dependent variable (whose variatiqn explicitly appears in the continuity equation), and subsequently calculate pressure from the gas state equation. In our case, by expressing the pressure as p(x,t) = p0 (t )+ p1 (x,t) where p0 (t) =-/i J0 p(x,t)d0., it can be shown that, in the limit of Mach <<l, it is p1 << p0 o Moreover, in the same limit, from the ideai gas mixture state equation it appears that density gradients are essentially due to temperature gradients, while remaining practically decoupled from p1 (the sole part of p playing a role in the momentum balance). For this reason, conventional compressible methods become criticai in this limit. Therefore we chose to compute composition and temperature first, by explicitly solving species and energy balance, then density directly from the gas state equation. Continuity is implicitly satisfied when combined with equilibrium to derive the elliptic equation V2 p1 = R(pv, v)+ a2pja t2 , whence finally pressure gradients are known to update the velocity field through the momentum balance equation. The method is fully explicit for the parabolic equations, while fast elliptic solvers are available for the pressure equation. The organisation of the method is similar to that of time-splitting methods, thus making it possible to independently adopt the best available schemes for each piece of the equations. The robustness of the method was successfully tested for both a non-reacting and a reacting case in an axisymmetric mixing layer. The stability limits in the tested cases allow time steps larger than those suggested by accuracy considerations in the description of the dynamic evolution of the physical phenomena. For example, the simulation of the ignition of a stationary laminar fuel jet, ali through the achievement of a steady state diffusion flame having a Burke-Schumann-type configuration, only takes few hours on a dedicated RISC workstation. The method was also tested on various parallel architectures, exhibiting good parallel speed-up with little or no implementation effort.