Tracking of acoustic and intrinsic modes for thermoacoustic systems with a general flame model

Abstract

In a thermoacoustic feedback loop, the flame gain parameter can be used to measure the coupling strength between the acoustic field and the flame response. In this study, it is shown for an arbitrary flame model that the thermoacoustic solutions in the zero-coupling limit split into two distinct sets: modes of (i) acoustic and (ii) intrinsic (ITA) origin. This result was previously shown in a rigorous manner only for $n$–$\tau$ flame models, which are special in the sense that they have ITA poles only at infinity. Consequently, all thermoacoustic eigenvalues can generally be calculated from the acoustic and intrinsic poles using continuation methods. In this study, we provide an explicit eigenvalue tracking scheme based on the integration of the local eigenvalue sensitivity to the flame gain parameter. The initial conditions required for integration are considered in detail. While the acoustic poles can be determined directly via Helmholtz solvers, the intrinsic poles are less trivial since they depend on the flame model. An asymptotic expansion of a generic transfer function is derived that is representative of all common flame models. It provides the necessary estimates for the intrinsic poles as the flame gain approaches zero in terms of the Lambert $W$ function. This approach represents an explicit scheme that guarantees to find all thermoacoustic eigenvalues. The methodology is demonstrated using a simple Rijke tube network model and an experimentally determined state-space model of an annular setup.