An analytical theory is developed for the stability properties of planar fronts of premixed laminar flames freely propagating downwards in a uniform reacting mixture. The coupling between the hydrodynamics and the diffusion process is described for an arbitrary expansion of the gas across the flame. Viscous effects are included with an arbitrary Prandtl number. The flame structure is described for a large value of the reduced activation energy and for a Lewis number close to unity. The flame thickness is assumed to be small compared with the wavelength of the wrinkles of the front, this wavelength being also the characteristic lengthscale of the perturbations of the flow field outside the flame. A two-scale method is then used to solve the problem. The results show that the acceleration of gravity associated with the diffusion mechanisms inside the front can counterbalance the hydrodynamical instability when the laminar-flame velocity is low enough. The theory provides predictions concerning the instability threshold. In particular, the dimensions of the cells are predicted to be large compared with the flame thickness, and thus the basic assumption of the theory is verified. Furthermore, the quantitative predictions are in good agreement with the existing experimental data.The bifurcation is shown to be of a different nature than predicted by the purely diffusive–thermal model.The viscous diffusivities are supposed to be independent of the temperature, and then the viscosity is proved to have no effect at all on the dynamical properties of the flame front.

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A complete analysis of the one-dimensional vibratory instability of planar flames of premixed gases propagating in tubes is provided. The driving mechanism results from unsteady coupling between flame structure and acoustic waves through temperature fluctuations. In certain conditions, the strength of such an instability will be proved to be sufficiently strong to produce large-amplitude fluctuations as soon as the flame has travelled a distance of the order of the acoustic wavelength. Stability limits and total amplification of an initial perturbation are computed in the framework of the simple flame mode of a one-step exothermic reaction governed by an Arrhenius law with an activation energy much larger than the thermal energy. Diffusive and thermal effects within the flame are included with a Lewis number different from unity. Damping mechanisms associated with viscous and thermal dissipation at the walls, as well as with loss of acoustic energy by sound radiation from the open end of the tube, are retained. In ordinary conditions, for a reactive mixture with an effective Lewis number close to unity, the predicted instability is weak. In the framework of the simplified flame model used here, islands of strong instabilities are predicted to occur at low Mach numbers for Lewis numbers larger than unity.

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In this paper, we study the vibratory instability of a cellular flame, propagating downwards in a tube, which results from the coupling between the longitudinal acoustic modes of the tube and the modification of the cellular flame structure by the acceleration of the acoustic field. We assume that the wrinkling of the flame is of small amplitude a0, which is the case when the flame burning velocity is just above the critical velocity characterizing the Darrieus–Landau instability threshold. We demonstrate that, in this case, the growth rate of the corresponding thermoacoustic instability, non-dimensionalized with the acoustic frequency, is proportional to (kca0)2, where kc is the critical wavenumber of the cellular instability. If one extends the result up to amplitudes of the same order as the wavelength, then one obtains a relative growth rate of order unity which is much larger than the one obtained from the study of the vibratory instability of the planar flame. As is observed in experiments, the theory predicts that the primary sound is generated when the amplitude of the cells is sufficiently large that the fundamental tone becomes unstable first and that the vibratory instability for the fundamental tone occurs in the lower half of the tube. This suggests that the coupling between cellular flame and acoustic field studied here is the mechanism for primary sound generation.

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