A new class of Galerkin expansion models for the study of thermoacoustic instabilities

This study proposes a new formulation of the one-mode Galerkin expansion and compares it against a classical one found in the literature, while considering a velocity-sensitive flame response. In the classical formulation, the fluctuations of upstream velocity perturbations uˆ at a reference location x_ref, required for modeling the flame response, are related to a corresponding pressure pˆ through an acoustic impedance at a given frequency. The formulation proposed in this study assumes that an accurate modeling of the relative phase between uˆ and pˆ is more important than a corresponding gain. Whereas the latter is set to a constant value, the phase difference at the reference location, written as ϕ=∠uˆ_ref−∠pˆ_ref, is approximated as ϕ≈±∠s, where s represents the Laplace variable. This approximation, which accounts for frequency and growth rate dependence of ϕ, stems from the low-angle approximation, which is a good assumption for small values of a characteristic time. The latter may be defined as the time required for an acoustic wave to travel from the inlet of the system to a defined location x_ref. We demonstrate that the proposed model outperforms the classical counterpart, as it accurately predicts acoustic and intrinsic thermoacoustic eigenfrequencies within a range close to the system's natural eigenfrequency, at which the Galerkin expansion is performed. Subsequently, this formulation is extended to a two-mode Galerkin expansion, leading to the derivation of a characteristic equation. Solutions to this straightforward algebraic equation accurately represent the thermoacoustic spectrum across a wide range of frequencies and growth rates for the three configurations considered: two Rijke burners and one swirled combustor.