Flame Balls in a non-uniform reactive mixture: preferential diffusion, heat-loss and stability

Abstract

The paper examines the existence and stability of axisymmetric flame balls in a non-uniform reactive mixture corresponding to a mixing layer taking into account preferential diffusion and volumetric heat-loss. The mixture's non-uniformity is measured with a non-dimensional parameter ε which is inversely proportional to the square root of the Damköhler's number. The investigation is carried out analytically in the limit of large activation energy and small values of ε, and numerically in the general case. New simple formulas accounting for preferential diffusion are derived which determine in particular the thermal energy and location of the flame ball; these may be argued to represent the minimum ignition energy and optimum spark location for a successful forced ignition of the diffusion flame in the mixing layer. A new free boundary problem (FBP) with two dependent variables is derived which describes non-adiabatic flame balls subject to volumetric heat-loss from the burnt gas. For small ε, the analytical solution to the FBP shows that the main effect of weak non-uniformity can be understood in a simple way if the volume of the distorted flame ball is characterised by an equivalent radius which is plotted versus a heat-loss parameter κ. Specifically, the curve is the same inverse-C shaped curve found in the literature in the uniform case () but shifted to the left by an amount, proportional to , which explicitly accounts for all parameters. The numerical investigation addresses the existence of the axisymmetric flame balls and their stability within two models familiar in studies on flame balls in uniform mixtures, namely the ‘far-field losses model’ where heat-losses from the burnt and unburnt gas are accounted for, and the ‘near-field losses model’ adopted in our analytical investigation, where heat-loss from the unburnt gas is neglected. Typically four regions are determined in the κ-ε plane for fixed Lewis numbers which identify conditions for the existence of either the flame ball, the diffusion flame or of both. This subdivision is argued to provide useful insight regarding the possible modes of burning in the mixing layer. A particularly interesting type of solutions identified for moderate values of ε corresponds to ring-shaped flame balls, termed ‘flame rings’, in regions where the diffusion flame cannot exist. As for the stability of the flame balls, we have found these to be typically unstable, as expected for their spherical counterparts. However, we have also determined these to be stable in special circumstances requiring low Lewis numbers and the presence of heat-losses and depending on the non-uniformity parameter ε. Furthermore, an increase in ε was found to play a stabilising effect, at least for the cases considered.