Compressibility effect on volumetric heat loss and its influence on the Darrieus-Landau instability of a planar front of premixed flame

The effect of compressibility on the Darrieus-Landau instability (DLI) of a plane front of a premixed flame is investigated in the form of the M^2 expansion for small Mach number M. The method of matched asymptotic expansions is used to derive jump conditions for hydrodynamic variables across the flame front which itself is separated into the preheat and the reaction zonez sandwiched by the former. With this jump conditions across the flame front, we obtain the correction to the growth rate of the DLI to first order in M^2. If the Prandtl number and the heat release are sufficiently large, the compressibility effect can suppress the DLI. Our treatment accounts for the volumetric heat-loss effect, without having to add an ad hoc sink term in the heat-conduction equation. We show that the compressibility raises the temperature in the upstream (unburned) side of the reaction zone and decreases it in the downstream (burned) side and that the maximum value of the temperature is attained at some position located inside the reaction zone. This is peculiar to the compressibility effect, which is brought by pressure variation terms.