Longitudinal low-frequency thermoacoustic unstable modes in combustion chambers have been intensively studied experimentally, numerically, and theoretically, leading to significant progress in both understanding and controlling these acoustic modes. However, modern annular gas turbines may also exhibit azimuthal modes, which are much less studied and feature specific mode structures and dynamic behaviors, leading to more complex situations. Moreover, dealing with 10–20 burners mounted in the same chamber limits the use of high fidelity simulations or annular experiments to investigate these modes because of their complexity and costs. Consequently, for such circumferential acoustic modes, theoretical tools have been developed to uncover underlying phenomena controlling their stability, nature, and dynamics. This review presents recent progress in this field. First, Galerkin and network models are described with their pros and cons in both the temporal and frequency framework. Then, key features of such acoustic modes are unveiled, focusing on their specificities such as symmetry breaking, non-linear modal coupling, forcing by turbulence. Finally, recent works on uncertainty quantifications, guided by theoretical studies and applied to annular combustors, are presented. The objective is to provide a global view of theoretical research on azimuthal modes to highlight their complexities and potential.

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