Acoustic response of a lean premixed flame propagating upward in a tube

An analysis is carried out of the response of a curved flame front rising in a vertical tube to acoustic perturbations of frequency small compared to the ratio of the speed of sound to the radius of the tube. The gas around the flame front, whose characteristic size is the radius of the tube, is then subjected to a oscillatory, nearly uniform acceleration that adds to the acceleration of gravity. As in the better studied case of a downward propagating flame, the response of the flame front to this oscillatory acceleration leads to fluctuations of the front area and the global thermal expansion of the gas, which may inject energy into the acoustic field and thus induce a thermoacoustic instability of the flame plus tube system. Acoustic temperature perturbations associated to adiabatic compression and expansion of the gas are assumed to be much smaller that the Frank-Kamenetskii temperature of the flame and are left out. The transfer function relating thermal expansion fluctuations to infinitesimal acoustic velocity perturbations is computed as a function of the frequency for conditions typical of a very lean flame in the standard flammability tube. The response of the flame front to finite amplitude acoustic perturbations has been also analyzed. At frequencies small compared to the inverse of the residence time of the gas around the flame front, the response is quasi-stationary until the oscillatory acceleration overcomes the acceleration of gravity during a part of the cycle sufficiently long for the front to develop wrinkles and eventually break down under a net upward acceleration. At higher frequencies, the curvature of the front decreases when the amplitude of the acoustic perturbation increases. A secondary instability of the oscillating flame that develops over many oscillation cycles and leads to breakdown appears at a certain value of the amplitude of the perturbation. The onset of this instability seems to be related to the oscillatory motion of the tube wall relative to the flame tip, which induces a flow that carries upwards a part of the front and causes negative velocities of the tip relative to the far fresh gas. This instability features period doubling in a certain range of frequencies. (C) 2018 The Combustion Institute. Published by Elsevier Inc. All rights reserved.