A Semi-implicit Numerical Scheme for Reacting Flow

An additive semi-implicit projection scheme for the simulation of unsteady combustion in two dimensions is constructed. The scheme relies on a zero-Mach number formulation of the compressible conservation equations with detailed chemistry. The governing equations are discretized in space using second-order differences and integrated in time using a semi-implicit approach. Time integration of the evolution equations for species mass fraction, thermodynamic pressure, and density is performed using a semi-implicit, nonsplit scheme that combines a second-order predictor–corrector treatment of convection and diffusion terms, and a stiff integrator for the reaction source terms. Meanwhile, the momentum equations are integrated using a second-order projection scheme. The projection scheme is based on a predictor–corrector approach that couples the evolution of the velocity and density fields in order to stabilize computations of reacting flows with large density variations. A pressure Poisson equation is inverted following both the predictor and corrector steps using a fast solver. The advantages of the stiff integration of reaction source terms are analyzed by comparing the performance of the scheme to that of a predictor–corrector scheme in which reaction and diffusion are integrated in a similar nonstiff fashion. The comparison in based on both one-dimensional (1D) unsteady tests of a premixed methane–air flame, and unsteady two-dimensional tests of the same flame interacting with a counterrotating vortex pair. In both cases, the GRImech1.2 reaction mechanism with 32 species and 177 elementary reactions is used. Computed results show that the stiff reaction scheme enables selection of larger time steps and thus leads to substantial improvement in the performance of the computations. For the present reaction mechanism and flame conditions, speedup factors of about 10 are achieved in the 1D tests and about five in two dimensions. Possible extensions of the present scheme to further improve efficiency are also discussed.