On the dynamics of transition from propagating flame to stationary flame ball

Abstract

A theoretical and numerical study was conducted on expanding spherical flames in order to understand how stationary flame ball (SFB) can be attained. Numerical simulation of the full unsteady problem was first performed for mixtures with low Lewis numbers. Depending on the order of magnitude of the heat loss, three typical regimes were found: (i) when the heat loss is very small, the spherical flame expands outwardly and transforms asymptotically to a planar flame; (ii) when the heat loss is moderately large, the planar flame does not exist and the expanding flame quenches; and (iii) when the heat loss is large, the expanding spherical flame transforms to a stationary flame ball. A quasi-steady nonlinear relation between the instantaneous flame radius R and its velocity U was obtained via asymptotic analysis and numerical computations with constant density and one-step Arrhenius kinetics. It was found that there is a continuous variation of the flame velocity from zero to the planar flame velocity. When the heat loss is larger than a critical value, the velocity-radius relation exhibits a turning point which may correspond to either flame extinction or reversal of the direction of propagation. Introduction Studies have been performed on spherical flames with emphasis on different aspects of the phenomena, such as: i) Relation between the flame velocity, stretch, and curvature; ii) stationary flame balls (SFB); and iii) spherical flame ignition. Multi-dimensional instability of spherical flames and spherical turbulent flames have also been investigated. For an outwardly propagating spherical flame, both the curvature and the flow field characterized by stretch modify the propagation velocity. The linear velocity-stretch-curvature relation determined for the one-dimensional spherical flame [l] is similar to that determined in multi-dimensional instability analysis [2]. In fact, small-amplitude cellular flames can be approximated locally to a one-dimensional spherical flame surface. A recent study with detailed chemistry on this problem, together with a review of related studies, can be found in Ref. 3. The work mentioned above concerns spherical flames whose radius is much larger than the flame thickness such that both the structure and flame velocity are only slightly different from those of the planar flame. The structure of the SFB, however, is very different. For SFB, which was first proposed by Zeldovich [4] to explain certain existing experimental phenomena, convective transport is absent while the product and heat release are transported radially outward into the ambient via diffusion. Stability analyses subsequently carried out by Zeldovich et al. [5] and by Deshaies and Joulin [6] with a thermal-diffusion model show that the adiabatic SFB is unconditionally unstable at its equilibrium radius, either collapsing inwardly for a negative initial perturbation of the flame position, or propagating outwardly for a positive initial perturbation. Based on such an unstable nature of SFB, Zeldovich et al. [5] pointed out that the critical condition for flame ignition should be controlled by the radius of SFB. To explain phenomena similar to the SFB observed in experiments (see Ref. 5 and the references therein), it was [5] also suggested that heat loss could play a stabilizing role because the extent of heat loss increases with increasing flame -* Copyright