Flame balls in mixing layers

We present a study of flame balls in a two-dimensional mixing layer with one objective being to derive an ignition criterion (for triple-flames) in such a non-homogeneous reactive mixture. The problem is formulated within a thermo-diffusive single-reaction model and leads for large values of the Zeldovich number ββ to a free boundary problem. The free boundary problem is then solved analytically in the asymptotic limit of large values of the Damköhler number, which represents a non-dimensional measure of the (square of the) mixing layer thickness. The explicit solution, which describes a non-spherical flame ball generalising the classical Zeldovich flame balls (ZFB) to a non-uniform mixture, is shown to exist only if centred at a single location. This location is found to be precisely that of the leading-edge of a triple-flame in the mixing layer, and typically differs from the location of the stoichiometric surface by an amount of order β-1β-1 depending only on a normalised stoichiometric coefficient ΔΔ.The thermal energy of the burnt gas inside the flame ball is used to derive an expression for the minimum energy EminEmin (of an external spark say) required for successful ignition. In particular, it is found that the presence of the inhomogeneity increases EminEmin compared to the homogeneous case. For a stoichiometrically balanced mixture, corresponding to Δ=0Δ=0, the relative increase in the ignition energy is found to be proportional to β2/Daβ2/Da, i.e. to the square of the Zeldovich number and to the reciprocal of the Damköhler number Da. More generally, for arbitrary value of ΔΔ, the minimum ignition energy is found to correspond to that of the Zeldovich flame ball in a uniform mixture at the local conditions prevailing at the location of the leading edge of the triple-flame, plus a positive amount depending on ΔΔ which is again proportional to β2/Daβ2/Da. In short, the analysis provides a possible criterion for successful ignition in a non-homogeneous mixture by determining the minimum energy required (EminEmin) and the most favourable location (that of the leading-edge of a triple-flame) where it should be deposited.