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 β 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 - 1 depending only on a normalised stoichiometric coefficient Δ . ermal energy of the burnt gas inside the flame ball is used to derive an expression for the minimum energy E min (of an external spark say) required for successful ignition. In particular, it is found that the presence of the inhomogeneity increases E min compared to the homogeneous case. For a stoichiometrically balanced mixture, corresponding to Δ = 0 , the relative increase in the ignition energy is found to be proportional to β 2 / 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 Δ , 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 Δ which is again proportional to β 2 / Da . In short, the analysis provides a possible criterion for successful ignition in a non-homogeneous mixture by determining the minimum energy required ( E min ) and the most favourable location (that of the leading-edge of a triple-flame) where it should be deposited.