Application of dynamical systems theory to nonlinear combustion instabilities

Abstract

Two important approximations have been incorporated in much or the work with approximate analysis or unsteady motions in combustion chambers: 1) truncation of the series expansion to a finite number or modes, and 2) time-averaging. A major purpose or the present analysis is to investigate the limitations or those approximations. A continuation method Is used to determine the limit cycle behavior or the time-dependent amplitudes or the longitudinal acoustic modes in a combustion chamber. The results show that time-averaging works well only when the system Is slightly unstable. In addition, the stability boundaries predicted by the twomode approximation are shown to be artifacts of the truncation of the system. Systems of two, four. and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second-mode, two birurcations are found to exist: 1) a pitchfork bifurcation leading to a new branch of limit cycles, and 2) a torus bifurcation leading to quasiperiodic motions.