Some dynamics of acoustic oscillations with nonlinear combustion and noise

The results given in this paper constitute a continuation of progress with nonlinear analysis of coherent oscillations in combustion chambers. We are currently focusing attention on two general problems of nonlinear behavior important to practical applications: the conditions under which a linearly unstable system will execute stable periodic limit cycles; and the conditions under which a linearly stable system is unstable to a sufficiently large disturbance. The first of these is often called 'soft' excitation, or supercritical bifurcation; the second is called 'hard' excitation, 'triggering,' or subcritical bifurcation and is the focus of this paper. Previous works extending over more than a decade have established beyond serious doubt (although no formal proof exists) that nonlinear gasdynamics alone does not contain subcritical bifurcations. The present work has shown that nonlinear combustion alone also does not contain subcritical bifurcations, but the combination of nonlinear gasdynamics and combustion does. Some examples are given for simple models of nonlinear combustion of a solid propellant but the broad conclusion just mentioned is valid for any combustion system. Although flows in combustors contain considerable noise, arising from several kinds of sources, there is sound basis for treating organized oscillations as distinct motions. That has been an essential assumption incorporated in virtually all treatments of combustion instabilities. However, certain characteristics of the organized or deterministic motions seem to have the nature of stochastic processes. For example, the amplitudes in limit cycles always exhibit a random character and even the occurrence of instabilities seems occasionally to possess some statistical features. Analysis of nonlinear coherent motions in the presence of stochastic sources is therefore an important part of the theory. We report here a few results of power spectral densities of acoustic amplitudes in the presence of a subcritical bifurcation associated with nonlinear combustion and gasdynamics