Non-linear dynamics of thermoacoustic eigen-mode interactions

The natural eigen-modes of a combustion system are a strong function of its geometric design, temperature profile, and boundary conditions. Many practical systems have multiple, closely-spaced natural mode frequencies. While several prior studies have noted the nonlinear interactions that occur when multiple, linearly unstable modes are present, the nature of these interactions for closely-spaced modes has not been analytically treated. This paper treats this problem theoretically by utilizing a Galerkin expansion of the Euler equations, and the method of averaging to derive a set of amplitude equations for the modal amplitudes. In cases where the frequency spacing is large, many of the oscillatory terms have short time-scales and average out. However, in the case of closely-spaced modes, terms oscillating at the frequency difference between the closely-spaced frequencies correspond to long time-scales and, thus, remain after the averaging process. Results show that frequency spacing between the modes has significant impacts on limit-cycle amplitudes and their stability, even in the case of a static non-linearity. In addition, the conditions under which both modes can co-exist are a strong function of the frequency spacing. There are also certain conditions where one mode is completely suppressed by the other, even if both modes have positive linear growth rates; non-intuitively, the mode that is suppressed could be the one with the larger growth rate. The conditions under which one mode is suppressed are also a strong function of the frequency spacing parameter. Finally, for conditions in which one mode is suppressed, the limit cycle amplitude of the other mode is independent of the frequency spacing. Taken together these results show the difficulty in experimentally inferring the linear stability/instability of (non-dominant) combustor modes based upon “steady state” data taken during limit cycle conditions.