A Reduced Model for Flame-Flow Interaction

Abstract

Abstract. The paper is concerned with an extension of the classical relation between the flame speed and the curvature-flow stretch, valid only for high Lewis numbers (diffusively stable flames). At low Lewis numbers the corresponding flame-flow system suffers shortwavelength instability, making the associated initial value problem ill-posed. In this study the difficulty is resolved by incorporation of higher-order effects. As a result one ends up with a reduced model based on a coupled system of second-order dynamic equations for the flame interface and its temperature. As an illustration the new model is applied for description of diffusively unstable stagnation-point flow flames.

Potential-flow models for channelled two-dimensional premixed flames around near-circular obstacles

“…Carrying the differentiation needed in (8) and specialising the result to the entrance of the front yields the contribution to fresh gas velocity ahead of it induced by the sources:…”

“…This w sup (z) is analytic across the front, because (3) is already accounted for by (8), and must also satisfy the slip condition ( 4) along all impenetrable boundaries. w sup (z)/w sup (−i∞) will only depend on the geometry of the channel and/or obstacle(s) under consideration and is closely related to the conformal map from the physical channel to an auxiliary straight one (endowed with cuts for each obstacle(s) present); in particular, w sup (z) ≡ w sup (−i∞) for straight channels.…”

“…where the symbol = ′ means 'equal up to additive functions of the primed quantities only', here z ′ ; these are indeed annihilated by ∂ z , and hence would play no role in (8). The cosine term in (11) accounts for the images in the lateral walls, and G 0 is 4L-periodic in z.…”

“…A way out of the difficulties in many such situations is to idealise flames as hydrodynamic discontinuitiesfronts -separating chemically inert and inviscid fresh or burned media of markedly different densities, and equipped with a propagation law to fix the local burn-ing speed and with Hugoniot jumps, both amended by analytically derived corrections resulting from a finite ℓ; the needed studies of local flame structure are now essentially complete (see [6][7][8]). Anyway, the number of nodes involved in numerical d-dimensional simulations of an interface embedded in the now chemistry-free flows still scales like (L/ℓ) d , d =2 or 3, which often is the limiting step of simulations since L exceeds ℓ by far in most situations.…”