An Application of Dynamical Systems Theory to Nonlinear Combustion Instabilities

Abstract

Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode 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 bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.