A long-wave approximation of the dispersion relation encompassing hydrodynamic and diffusional-thermal instabilities for premixed flames

Flame instabilities play a central role in combustion science. Since practical systems are continually disrupted by small disturbances, the onset of instabilities promotes spontaneous creation of highly corrugated flames with large surface area, and consequently an enhancement in their propagation speed and overall burning rates. The two prominent intrinsic instabilities of premixed flames are the hydrodynamic instability that results from gas expansion caused from the heat released by the chemical reactions, and the diffusional-thermal instability that results from the competing rates of heat conduction and reactant diffusion occurring inside the flame zone. Their combined description has been elucidated by a hydrodynamic model whereby the flame is treated as a thin layer in a flow field that exhibits the realistic density change encountered in real systems. The dispersion relation derived in this context provides a clear distinction between the two modes of instability in terms of an effective Lewis number of the mixture. It is a long-wave approximation that provides a good description of the hydrodynamic instability observed in super-critical Lewis number flames, specifically when hydrodynamic effects cause an amplification of the long waves and diffusion influences act to stabilize the short-wavelength disturbances. However, below criticality, this dispersion relation fails; diffusion effects become predominant, further amplifying the hydrodynamically unstable long waves without offering short-wavelength stabilization. In this paper, we extend the stability analysis to higher orders in wavenumber and provide a correction that combines hydrodynamic and diffusional-thermal effects for pertinent Lewis numbers of combustible mixtures. The results are exemplified for hydrogen-air flames showing a good agreement when compared to numerical simulations, particularly for lean flames which are characterized by a relatively low Lewis number. Novelty and significance statement This study presents for the first time an extension of the stability analysis of premixed flames within the hydrodynamic framework to higher orders in wavenumber. The derived dispersion relation emerges as a satisfactory approximation for both super- and sub-critical Lewis numbers in the long-wave limit. Specifically, it accurately characterizes not only hydrodynamic instabilities but also diffusional-thermal instabilities that manifest in sub-critical Lewis number flames, such as in lean hydrogen air mixtures.