Adjoint-based mean-flow uncertainty and feedback-forcing analyses of a thermoacoustic model system

Clean combustion, particularly premixed hydrogen combustion aimed at reducing NOx emissions, is prone to thermoacoustic instabilities that can cause structural vibrations and equipment failures. This study focuses on a low-order model for a thermoacoustic prototype, a simple quasi-one-dimensional combustor comprising a plenum, premixing duct, and combustion chamber. Resonant modes of the combustor are identified by solving a nonlinear eigenvalue problem. Using an adjoint-based sensitivity analysis, the impact of uncertainties in base flow parameters on resonant frequencies and linear growth rates is assessed. The results obtained highlight the significant influence of variations in cold gas density within the plenum and premixing duct on the linear growth rates, potentially explaining discrepancies with literature data. Additionally, structural sensitivities in both the base and the perturbation flow are examined to evaluate the effects of a generic feedback mechanism on the eigenvalues. Structural sensitivities at the base-flow level are evaluated as a function of the flame position, identifying effective stabilizing mechanisms such as heat addition and mass flow rate reduction at duct intersections. The most stabilizing feedback mechanism is identified as mass fluctuations proportional to pressure perturbation at the end of the plenum, an effect achievable with Helmholtz resonators. Adjoint analyses permit uncertainty quantification of base-state parameters and gradient information for optimization strategies aimed at mitigating thermoacoustic instabilities through efficient and low-cost calculations.