The paper examines the existence and stability of axisymmetric ﬂame balls in a non-uniformreactive mixturecorresponding toamixing layertaking intoaccount preferential diffusion and volumetric heat-loss. The mixture’s non-uniformity is measured with a non-dimensional parameter 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 , and numerically in the general case. New simple formulas accounting for preferential diffusion are derived which determine in particularthethermalenergyandlocationoftheﬂameball;thesemaybearguedtorepresenttheminimumignitionenergyandoptimumsparklocationforasuccessfulforced 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 subjecttovolumetricheat-lossfromtheburntgas.Forsmall ,theanalyticalsolutiontothe 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 Req which is plotted versus a heat-loss parameter κ. Speciﬁcally, the curve Req(κ) is the same inverse-C shaped curve found in the literature in the uniform case ( =0) but shifted to the left by an amount, proportional to 2, 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- 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 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 .Furthermore,anincreasein wasfoundtoplayastabilisingeffect,atleast for the cases considered.
|Journal||Combustion Theory and Modelling|
|Early online date||17 Mar 2019|
|Publication status||Published - 3 Sep 2019|