On the use of operator-splitting methods for the equations of combustion

Abstract

A model problem is analyzed to illustrate the operator-splitting method for combustion problems with low Mach-number flows. In the limit of high Damköhler number, it is shown that the solution of the combustion equations can be separated at each time step into two parts, (1) a spatially homogeneous explosion calculation over the initial part of the time step and (2) a transport calculation over the remainder of the time step that redistributes the chemical energy released in (1). The limit of high activation energy (β → ∞) allows approximate analytical solution of the explosion equations. The method is applicable primarily for problems with two time scales, a short chemical time scale and a much longer convection or diffusion time scale. Nevertheless, it can still be profitably used for problems described only by the single chemical scale if certain restrictions are obeyed. The operator-splitting method is then applied to solve the problem of two initially separated coflowing streams of fuel and oxidizer that subsequently mix and react. Many advantages and disadvantages of operator splitting are well represented by this example; some, such as those for confined flames, are not. Error norms are evaluated, CPU times compared, and influences of the Damköhler number D and characteristic time step ratio χ analyzed. Extensions to more general cases are discussed.