Abstract
A simplified model is introduced to study finite-amplitude thermo-acoustic oscillations in N-periodic annular combustion devices. Such oscillations yield undesirable e↵ects and can be triggered by a positive feedback between heat-release and pressure fluctuations. The proposed model, comprised of the governing equations linearized in the acoustic limit, and with each burner modeled as a one-dimensional system with acoustic damping and a compact heat source, is used to study the instability caused by cross-sector coupling. The coupling between the sectors is included by solving the one-dimensional acoustic jump conditions at the locations where the burners are coupled to the annular chambers of the combustion device. The analysis takes advantage of the block-circulant structure of the underlying stability equations to develop an e cient methodology to describe the onset of azimuthally-synchronized motion. A modal analysis reveals the dominance of large-scale instabilities, while a non-modal analysis reveals a strong response to harmonic excitation at forcing frequencies far from the eigenfrequencies, when the overall system is linearly stable. In all presented cases, large-scale, azimuthally synchronized motion is observed. The relevance of the non-modal response is further emphasized by demonstrating the subcritical nature of the system's Hopf point via an asymptotic expansion of a nonlinear model representing the compact heat source within each burner.