Prediction of limit cycle amplitudes over an operating domain and direct comparison with thermoacoustic instability experiments

One central challenge in combustion instability analysis is to reconcile experimental data and model predictions. In most cases, experimental observations are carried out at the limit cycle, i.e. when the regime of oscillation is established, while modeling generally provides growth rates which are used to infer whether the system is stable or unstable. A considerable effort has been made to develop models that describe linear and nonlinear features of the unstable process. However, there are not many contributions that are trying to obtain the layout of the unstable domain as a function of operating parameters. This article is aimed at showing that it is possible to calculate amplitudes at the limit cycle over a wide domain of operation with a reduced-order model, thus distinguishing stable and unstable regions and compare the calculated levels of oscillations to the amplitudes determined experimentally in a laboratory scale annular combustor (the MICCA facility). Calculations are carried out by projecting the wave equation onto acoustic modes to derive coupled evolution equations, which are then solved using slow-flow variables and time averaging. This model enables the determination at low computational cost of both limit-cycle amplitudes and effective growth rates, relying on velocity-based Flame Describing Function (FDF) data measured in a single-sector combustor equipped with the same injection unit, converted to pressure-based FDF and extended to actual oscillation conditions. The flame response (gain and phase of the FDF) is updated during integration to match with the observed frequency drift which is continuously monitored. The model successfully reproduces the experimental instability domain layout, as well as a frequency drift, highlighting its predictive capability for nonlinear dynamics in realistic annular combustor geometries. Novelty and significance statement This work extends a reduced-order model for thermoacoustic instability prediction in a multi-burner annular combustor [1] to a wide operating domain. The model includes several novel elements: an extension to annular conditions of velocity-based Flame Describing Functions using single-sector measurements; a frequency monitoring scheme based on dynamical equations to ensure consistent flame response; and a method to constrain admissible frequency shift using experimental knowledge and an extended Rayleigh criterion accounting for damping effects. This approach demonstrates, for the first time, that single-sector flame response data can be used to predict nonlinear instability behavior and limit-cycle amplitudes in multi-injector annular geometries across a broad operating range. The model captures experimental trends and stability boundaries with low computational cost and provides a practical framework for stability analysis and design optimization in future low-emission engines, offering a simple yet robust tool for predicting nonlinear thermoacoustic dynamics in annular multi-burner systems.