THE NONLINEAR DYNAMICS OF INTRINSIC ACOUSTIC-OSCILLATIONS IN A MODEL PULSE COMBUSTOR

The appearance of nonlinear acoustic oscillations in pulse combustors and other unsteady combustion devices arises from combustion-driven in stabilities that excite one or more classical acoustic modes of the sy stem. For sufficiently strong acoustic driving relative to various dam ping processes, these disturbances grow to finite amplitudes and, owin g to the nonlinear coupling between linearly unstable and stable modes , a stable limit-cycle oscillation is typically established. The nonli near dynamics of these oscillations are formally governed by an infini tely coupled system of evolution equations for the complex mode amplit udes. The linear terms determine the relative growth and decay rates o f infinitesimal perturbations, while the nonlinear coupling terms dete rmine the ultimate amplitude of each mode. The present work describes the nonlinear acoustic response of the system by obtaining approximate analytical and numerical solutions of the dynamical system of amplitu de equations. In particular, it is shown that, depending on the value of a reduced driving parameter, various finite-mode approximations to the full infinitely coupled system may be employed to describe the aco ustic response of the model. The nature of acoustic mode interactions is also investigated, and it is formally demonstrated that a resonance -like coupling exists between any growing mode, whose frequency is ome ga(j), and its first resonant harmonic, which is the mode whose freque ncy is 3 omega(j). Thus, the first acoustic bifurcation of the system occurs at a critical value of the driving parameter for which some mod e achieves a positive linear growth rate, and is governed by a decoupl ed subsystem for the two modes corresponding to these frequencies. At larger values of the driving parameter, a second mode may achieve a po sitive linear growth rate, which in turn may produce a secondary trans ition in the acoustic response. The results for two cases, correspondi ng to whether or not one growing mode is the first resonant harmonic o f the other, are discussed in terms of the minimal number of modes req uired to correctly predict the nonlinear acoustic behavior of the syst em.