A Fully Implicit, Compact Finite Difference Method for the Numerical Solution of Unsteady Laminar Flames

Forced laminar diffusion flames form an important class of problems that can help to bridge the significant gap between steady laminar flames in simple burner configurations and the turbulent flames found in many practical combustors. Such flames offer a much wider range of interactions between convection, diffusion, and chemical reaction than can be examined under steady-state conditions, and yet detailed simulations of them should be feasible without having to resort to modeling any of the relevant physics, above all without having prematurely to reduce the large kinetic mechanisms typical of hydrocarbon fuels. Nevertheless, the computation of time-dependent laminar diffusion flames with conventional numerical methods is hindered by technical challenges that, while not new, are more troublesome to surmount than in the calculation of otherwise similar, unforced flames. First, the intricate spatiotemporal coupling between fluid dynamics and combustion thermochemistry ensures that spurious numerical diffusion or spatial under-resolution of the mixing process at any stage of the computation can lead to inaccurate prediction of flame characteristics for the remainder thereof. Second, relatively long simulated flow times and extremely short chemical time scales make many standard time integration algorithms impractical on all but the largest parallel computer clusters. This paper introduces a new numerical approach for time-varying laminar flames that addresses these challenges through the use of high order compact finite difference schemes within a robust, fully implicit solver based on a Jacobian-free Newton-Krylov method. The capabilities of this implicit-compact solver are demonstrated on a periodically forced axisymmetric laminar jet diffusion flame with one-step Arrhenius chemistry, and the results are compared to those of a conventional low order finite difference solver.