On the dynamics of the collapse of a diffusion-flame hole

Abstract

The collapse dynamics of a diffusion-flame hole in the presence of a counterflow are studied. We construct unsteady solutions of the one-dimensional edge-flame model of Buckmaster (1996), in which heat and mass transverse losses are algebraic. The flame structure is determined in the classical limit of large activation energy. Solutions for both planar and axisymmetric strain geometry are considered for the particular case of unity Lewis number. It is shown that the final stage of the edge-flame collapse is determined by a dominant balance between the time rate of change of the mass fractions (and temperature) and diffusion, giving a self-similar structure in which the size of the edge-flame hole approaches zero, to leading (zeroth) order, as a 1/2-power of time. This solution suggests an expansion of the full model equations in 1/2-powers of time that allows detailed analysis of the effects of side losses and flow distribution in the edge-flame collapse process. It is found that side loss effects are apparent at the first order, whereas convection by the counterflow is first felt during collapse at the second order in the fractional-time expansion. Numerical integrations of the governing equations are found to verify the analytic results.