A diffusion-flame analog of forward smolder waves: (II) stability analysis

Abstract

We proceed to examine the stability of the adiabatic and non-adiabatic structures of forward smolder waves elaborated in Part (I) of this series. The dispersion relation for adiabatic forward smolder waves with a reaction trailing structure turns out to take a form similar to that for premixed flames, thereby strengthening the analogy of the reaction trailing structure with the premixed flame regime of diffusion flames. According to the dispersion relation, corresponding to each Damköhler number there exists a marginal oxygen Lewis number, below which cellular instability occurs. In particular, similar to the Burke–Schumann limit of diffusion flames, the stoichiometric limit at infinite Damköhler number is unconditionally stable. Such unconditional stability is found to further extend to the entire Damköhler number range for adiabatic forward smolder waves with a reaction leading structure. Linear stability analysis of non-adiabatic forward smolder waves indicates that, for both reaction trailing and reaction leading structures, the low smolder temperature (or high reactant leakage) solution branch is physically unrealistic, whereas on the high smolder temperature (or low reactant leakage) branch different kinds of instabilities may develop near the quenching limit. Under a fixed Damköhler number, the range of the heat loss coefficient corresponding to these instabilities shows a trend to grow with decreasing oxygen Lewis number. 2-D time-dependent numerical simulations of unstable non-adiabatic forward smolder waves confirm that fingering or cellular instability occurs exclusively for the reaction trailing structure, whereas traveling wave instability prevails for the reaction leading structure. A comparison is made between the current stability analysis results of non-adiabatic forward smolder waves and results from a concurrent flame spread experiment. Agreement is achieved not only on the existence of reaction front instabilities near the quenching limit, but also on the conditions determining the type of these instabilities.