Thermoacoustic Modes of Quasi-One-Dimensional Combustors in the Region of Marginal Stability

It may be generally believed that the thermoacoustic eigenfrequencies of a combustor with fully acoustically reflecting boundary conditions depend on both flame dynamics and geometry of the system. In this work, we show that there are situations where this understanding does not strictly apply. The purpose of this study is twofold. In the first part, we show that the resonance frequencies of two premixed combustors with fully acoustically reflecting boundary conditions in the region of marginal stability depend only on the parameters of the flame dynamics but do not depend on the combustor's geometry. This is shown by means of a parametric study, where the time delay and the interaction index of the flame response are varied and the resulting complex eigenfrequency locus is shown. Assuming longitudinal acoustics and a low Mach number, a quasi-1D Helmholtz solver is utilized. The time delay and interaction index of the flame response are parametrically varied to calculate the complex eigenfrequency locus. It is found that all the eigenfrequency trajectories cross the real axis at a resonance frequency that depends only on the time delay. Such marginally stable frequencies are independent of the resonant cavity modes of the two combustors, i.e., the passive thermoacoustic modes. In the second part, we exploit the aforementioned observation to evaluate the critical flame gain required for the systems to become unstable at four eigenfrequencies located in the marginally stable region. A computationally efficient method is proposed. The key ingredient is to consider both direct and adjoint eigenvectors associated with the four eigenfrequencies. Hence, the sensitivity of the eigenfrequencies to changes in the gain at the region of marginal stability is evaluated with cheap and accurate calculations. This work contributes to the understanding of thermoacoustic stability of combustors. In the same manner, the understanding of the nature of distinct resonance frequencies in unstable combustors may be enhanced by employing the analysis of the eigenfrequency locus here reported.