Vibratory instability of cellular flames propagating in tubes

Abstract

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.

Onset mechanism of primary acoustic instability in downward-propagating flames

“…Then, the increasing velocity gradients induce change in the flame surface area, which is in phase with pressure fluctuation, and the primary acoustic sound is generated. Finally, the variation in the flame surface area caused by the acoustic velocity field forces the amplification of the primary acoustic sound and leads to the stabilization of D-L instability [1,2,11]. Their calculation indicates that the maximum growth rate takes place in the lower half of the tube, which is in good agreement with Searby's experimental observations [1].…”

“…An approach toward resolving this issue has generally been to consider the Rayleigh criterion [5]: an acoustic wave will be amplified if a time integral of the product of pressure and heat release fluctuations is positive over a pressure cycle. Considering this criterion, many previous researchers [6][7][8][9][10][11][12] have proposed various mechanisms on primary acoustic instability. We can classify them into two large groups depending whether they are related to coupling of the acoustic Initially, Dunlap [7] theoretically showed that the effect of the pressure and adiabatic temperature of an incoming acoustic wave modulates local heat release rates of a flame with the correct phase relationship.…”

“…An analytical analysis of this mechanism was further performed by Pelcé and Rochwerger [11] in connection with the experiments conducted by Searby [1]. They studied the coupling with variation in the flame surface area produced by the presence of an initial curved structure due to D-L instability and demonstrated that the growth rate of the primary acoustic instability is proportional to (ak) 2 , where a is an amplitude and k is a wave number of the curved flame.…”

Large-activation-energy theory for premixed combustion under the influence of enthalpy fluctuations

“…They formulated and solved the stability problem of a flame subjected to an externally prescribed acoustic pressure, and showed that as the acoustic acceleration exceeds a threshold, it induces a violent subharmonic parametric instability. Pelce & Rochwerger (1992) analyzed the acoustic instability of a (slightly) curved stationary flame, and showed that the unsteady heat release due to the flame surface-area change drives exponential growth of acoustic modes. The back effect of the latter on the flame was however not considered.…”

“…A general asymptotic theory was presented by Wu, Wang, Moin & Peters (2003) (referred to as WWMP hereafter) to describe the acoustic-flow-flame coupling in the 'corrugated flamelet' regime. Using this general formulation, they provide a unified description of the flame-acoustic coupling mechanisms of Clavin et al (1990) and Pelce & Rochwerger (1992).…”