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.

Onset mechanism of primary acoustic instability in downward-propagating flames

“…In this section, we review previously proposed mechanisms [8,11] for the generation of primary acoustic instability for propagating flames in tubes. Here, two different coupling mechanisms characterizing the feedback mechanism of the acoustic instability on flame fronts are introduced.…”

“…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.…”

“…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. More recently, Clavin et al [8] treated Dunlap's mechanism in detail using a one-step high-activation energy flame. They reported that the growth rate of the primary acoustic instability of planar flames is proportional to the coupling constant βM, and the flames may overcome the damping mechanism due to heat transfer and viscous friction at the tube wall and to acoustic radiation losses at the open end of the tube when the coupling constant is adequately large (βM ≈ 10 −2 ).…”

“…Thus, the violent acoustic instability is predicted to appear at sufficiently small Mach numbers, ≈ 10 −3 , for τ t /τ a = O(1) when β is sufficiently large. Unfortunately, this mechanism subjected to the coupling with pressure and temperature has limitations, which can be applied only for planar, rather than curved, flames, and the initial growth rate of the acoustic instability measured by experiments for nonplanar flames [12] is two to three orders of magnitude higher than that expected by the theoretical prediction of Clavin et al [8]. Thus, this analysis is very hard to apply to real combustion experiments since propagating premixed flames in an actual combustor do not take on a planar structure owing to exposure to hydrodynamic instabilities (e.g., D-L instability and gravity effects).…”

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

“…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).…”