Flame balls in non-uniform mixtures: existence and finite activation energy effects

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The paper's broad motivation, shared by a recent theoretical investigation [Daou and Daou, “Flame balls in mixing layers,” Combustion and Flame, Vol. 161 (2014), pp. 2015–2024], is a fundamental but apparently untouched combustion question; specifically, ‘What are the critical conditions insuring the successful ignition of a diffusion flame by means of an external energy deposit (spark), after mixing of cold reactants has occurred in a mixing layer?’ The approach is based on a generalisation of the concept of Zeldovich flame balls, well known in premixed reactive mixtures, to non-uniform mixtures. This generalisation leads to a free boundary problem (FBP) for axisymmetric flame balls in a two-dimensional mixing layer in the distinguished limit β → ∞ with εL = O(1); here β is the Zeldovich number and εL is a non-dimensional measure of the stoichiometric premixed flame thickness. The existence of such flame balls is the main object of current investigation. Several original contributions are presented. First, an analytical contribution is made by carrying out the analysis of Daou and Daou (2014) in the asymptotic limit εL → 0 to higher order. The results capture, in particular, the dependence of the location of the flame ball centre (argued to represent the optimal ignition location which differs from the stoichiometric location) on εL. Second, two detailed numerical studies of the axisymmetric flame balls are presented for arbitrary values of εL. The first study addresses the infinite-β FBP and the second one the original finite-β problem based on the constant density reaction–diffusion equations. In particular, it is shown that solutions to the FBP exist for arbitrary values of εL while actual finite-β flame balls exist in a specific domain of the β–εL plane, namely for εL less than a maximum value proportional to ; this scaling is consistent with the existence of solutions to the FBP for arbitrary εL. In fact, the flame ball existence domain is found to have little dependence on the stoichiometry of the reaction and to coincide, to a good approximation, with the domain of existence of the positively-propagating two-dimensional triple flames in the mixing layer. Finally, we confirm that the flame balls are typically unstable, as one expects in the absence of heat losses.