NONLINEAR STABILITY OF COMBUSTION-DRIVEN ACOUSTIC-OSCILLATIONS IN RESONANCE TUBES

The leading-order fluid motions and frequencies in resonance tubes cou pled to a combustion-driven flow source, such as occurs in various typ es of pulse combustors, are usually strongly related to those predicte d by linear acoustics. However, in order to determine the amplitudes o f 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 approac h based on the smallness of the mean-flow Mach number which, as a cons equence of solvability conditions at higher orders in the analysis, re sults 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 proces ses, such as combustion, that supply energy to the acoustic oscillatio ns and those, such as viscous effects, that dampen such motions, in co njunction with the manner in which the resonance tube is coupled to it s flow source, provides an effective mode-selection mechanism that inh ibits the (linear) growth of all but a few of the lower-frequency mode s. 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 demonstrat e, based on analyses with both two and three modes, the stable bifurca tion of an acoustic oscillation, or limit cycle, at a critical value o f an appropriate bifurcation parameter. In addition, we show that the bifurcated solution branch has a turning point at a second critical va lue of the bifurcation parameter beyond which no stable bounded soluti ons exist.