Degenerate high order perturbation theory of thermoacoustic eigenmodes

Thermoacoustic instabilities are due to the coupling between the acoustic response of a cavity and the heat release rate of a heat source. In gas turbines, they are detrimental because they cause large amplitude oscillations of the structure, limiting operating conditions and the life span of the gas turbine components. The instabilities can be predicted by identifying the thermoacoustic eigenvalues for a given complex geometry. These eigenvalues are strong nonlinear functions of the system parameters, and obtaining them for many different configurations is numerically demanding. However, by using once large order perturbation theory on the operator governing the dynamics, one can build power series expansions to a desired order that approximate well both eigenvalues and eigenvectors of the system for any value of a parameter of interest within a certain threshold. In particular, we discuss the complications in the perturbation method arising when dealing with degenerate modes, which are often observed in gas turbines due to their discrete rotational symmetry. The theory is tested on an annular atmospheric combustor for various choices of perturbation parameters.