On third order density contrast expansion of the evolution equation for wrinkled unsteady premixed flames

The dynamics of flat-on-average wrinkled flame front propagating through gaseous premixtures is considered. Leading the asymptotic expansions in powers of the burnt to unburned fractional density contrast (0 < gamma < 1) to third order, an evolution equation (called 53) is obtained for the instantaneous front shapes. It reduces to Sivashinsky's original equation (called Si) as It also modifies a previous attempt by Sivashinsky and Clavin (called S2) to improve it. Numerical integrations of the S3 equation reveals that the new quadratic and cubic non-linearities featured at 3rd order happen to mutually compensate partially one another for realistic gamma's, and are negligible at gamma << 1. As a result, the flame shape and speed solutions to S3 nearly coincide with those of a S1/S2 type of equation, even for a 10-fold density variation (gamma = 0.9) and for unsteady situations, provided a single O(1) coefficient a(gamma) be adjusted therein, once for all for each gamma. The O(gamma(2)) (and small) correction to it mainly originates from a quartic non-linearity of geometrical origin. The agreement carries over to comparisons with some DNS of 2D steady wrinkled fronts. A phenomenological (yet asymptotically correct at gamma << 1 and exact in the linear limit) interpolating model equation is finally proposed to try and account for inertia effects associated with fast transients (e.g. acoustics related) while reproducing the above results on steady patterns. (C) 2011 Elsevier Ltd. All rights reserved.