Pocket formation and the flame surface density equation

Abstract

The occurrence and properties of singularities in the equation for the surface density function σ≡|∇| are analyzed analytically and numerically using data from two-dimensional direct numerical simulation (DNS) of pocket formation in a premixed methane-air flame. The various stages and the relevant time scales associated with pocket formation were determined in a previous study. It was found that isolated pockets form if and only if a nondegenerate critical point of a saddle point type appears. The appearance of a singularity in the isoline representing the flame front may have implications to modeling of the terms in the surface density function (SDF) approach during such transient events as pocket formation. The sink and source terms in SDF are evaluated in the neighborhood of a critical point using DNS data during pocket formation and an analytic representation of a scalar in the vicinity of the critical point that allows for the computation of all kinematic properties. The analytic and computational results show that the normal restoration and dissipation terms in the SDF become singular at the critical point when the pocket emerges. Furthermore, the analytic results show that the singularities exactly cancel, and therefore, the main conclusion is that it is unnecessary to model the singular behavior of these terms at critical points. However, closure of their sum is recommended.