Nonlinear stability of combustion-driven acoustic oscillations in resonance tubes

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Abstract
The leading-order fluid motions and frequencies in resonance tubes coupled to a combustion-driven flow source, such as occurs in various types of pulse combustors, are usually strongly related to those predicted by linear acoustics. However, in order to determine the amplitudes of the infinite number of classical acoustic modes predicted by linear theory alone, and hence the complete solution, a nonlinear analysis is required. In the present work, we adopt a formal perturbation approach based on the smallness of the mean-flow Mach number which, as a consequence of solvability conditions at higher orders in the analysis, results in an infinitely coupled system of nonlinear evolution equations for the amplitudes of the linear acoustic modes. An analysis of these amplitude equations then shows that the combination of driving processes, such as combustion, that supply energy to the acoustic oscillations and those, such as viscous effects, that dampen such motions, in conjunction with the manner in which the resonance tube is coupled to its flow source, provides an effective mode-selection mechanism that inhibits the (linear) growth of all but a few of the lower-frequency modes. For the common case of long resonance tubes, the lowest frequencies correspond to purely longitudinal modes, and we analyse in detail the solution behaviour for a typical situation in which only the first of these has a positive linear growth rate. Basic truncation strategies for the infinitely coupled amplitudes are discussed, and we demonstrate, based on analyses with both two and three modes, the stable bifurcation of an acoustic oscillation, or limit cycle, at a critical value of an appropriate bifurcation parameter. In addition, we show that the bifurcated solution branch has a turning point at a second critical value of the bifurcation parameter beyond which no stable bounded solutions exist.

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