We present an experimental and theoretical investigation of some aspects of the coupling between a premixed laminar quasi-planar flame front and acoustic standing waves in tubes. A multidimensional instability of the front arises from its interaction with the oscillating field of acceleration. This instability can be described by the Clavin–Williams laminar wrinkled flame theory in which the periodic acceleration created by the acoustic field is added to the acceleration due to gravity. As first suggested by Markstein, the resulting equation can be reduced to the Mathieu equation for a parametric oscillator. A cellular instability appears with a finite excitation threshold. This instability is responsible for the spontaneous generation of intense acoustic oscillations observed elsewhere. The value of the acoustic field at the threshold of instability and the wavelength of the cellular structures are measured experimentally for propane flames and are found to be in good agreement with the calculated values. It is also seen, both experimentally and theoretically, that for certain amplitudes of pumping, the parametric mechanism can also stabilize an initially unstable system.

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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