Lambert W function for effect of gas compression on premixed flames with ignition temperature approximation

Abstract

The assumption that the gradient of temperature at the burned-side edge of a premixed flame front is zero has been usually imposed in investigating the adiabatic profile of flames for quasi-isobaric flow or in the zero-Mach-number limit. However, the non-monotonic behaviour of temperature is also observed for flames, or deflagrations, in the process of deflagration-to-detonation transition inside tubes or in the context of laser ablation or thermonuclear explosions in the universe. This non-adiabatic profile of a flame front is mainly due to the effect of gas compression, which is different from the conductive or volumetric heat loss. To clarify this point, we derive the non-zero condition of temperature gradient by considering the additional region, or the compression zone, behind a freely propagating flame front. The method of singular perturbations, or matched asymptotic expansions, is employed to calculate the inner and outer solutions of the compression zone of thickness with the squared Mach number assumed to be small, . The inner solutions are expressed by use of Lambert W function, and the pressure and dissipation terms in the heat-conduction equation play an important role to capture the non-adiabaticity of a flame front for non-zero values of Mach number. The strong ignition temperature approximation is imposed to obtain exact solutions in the outer region, composed of the reaction and preheat zones. The adiabatic value of ignition temperature is determined. The non-adiabatic feature is categorised into two cases. One case shows the non-monotonic behaviour of temperature. The other case presents the monotonic profile of temperature, but its maximum value is beyond the adiabatic value. The boundary, which determines whether the compressibility effect increases or decreases the flame temperature, is given with respect to the values of ignition temperature, heat release and Prandtl number.