One-dimensional vibratory instability of planar flames propagating in tubes

Abstract

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.