VIBRATORY INSTABILITY OF CELLULAR FLAMES PROPAGATING IN TUBES

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 (k(c)a0)2, where k(c) 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.