Thermoacoustic instability poses a significant challenge to the operational safety and stability of multi-nozzle combustors in both aero-engines and industrial gas turbines. To achieve a comprehensive understanding and effective control of this phenomenon, this paper employs a three-dimensional Green's function method to characterize the geometric and thermal response differences among multiple nozzles, aiming to reveal the key parameters influencing three-dimensional thermo-acoustic instability and the control effects of combining different types of nozzles. The study focuses on analyzing the impact of the average temperature of the combustion chamber, axial length of nozzles, and inlet boundary conditions of nozzles on the azimuthal, radial, and axial modal thermoacoustic instability. The results indicate that when the axial length of nozzles approximates an odd multiple of a quarter wavelength, the thermoacoustic instability evolves gradually; in con-trast, when it approaches an integer multiple of a half wavelength, an abrupt shift in the instability state occurs. Moreover, the inlet boundary conditions of nozzles affect the stability of thermoacoustic modes by altering the acoustic energy dis-sipation at the inlet and the phase difference between sound pressure and unsteady heat release rate. By leveraging the identified patterns of key parameters, adjusting the geometric structure and heat source response of some nozzles can effectively control three-dimensional unstable modes. Moreover, coordinated adjustments of multiple parameters can significantly enhance the control of first-order axial mode thermoacoustic instability across a broader parameter spectrum.
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