Nonlinear coupling between azimuthal and axisymmetric modes in annular combustors is studied analytically. Based on the thermoacoustic wave equation, a model featuring three nonlinearly coupled oscillators is derived. Two oscillators represent the dynamics of an azimuthal mode, and the third accounts for the axisymmetric mode. A slow-time system for the evolution of the mode amplitudes and phases is obtained through the application of the method of averaging. The averaged system is shown to accurately reproduce the solutions of the full oscillator model. Analysis of this five-dimensional dynamical system shows that a standing azimuthal mode may synchronize with an axisymmetric mode, provided that their individual resonance frequencies and growth rates are similar. This phase-coupled two-mode oscillation corresponds to the so-called slanted mode, observed in recent experiments involving an annular model combustion chamber. Quantitative conditions for the occurrence of mode synchronization are derived in terms of the growth rate ratio and a frequency detuning parameter. The analysis results are found to be consistent with experimental observations of the slanted mode.

分析了环形燃烧器中方位角和轴对称模式之间的非线性耦合。基于热声波方程，推导了具有三个非线性耦合振荡器的模型。两个振荡器代表了方位角模式的动力学特性，而第三个则解释了轴对称模式。通过应用求平均值的方法，获得了一个用于模式幅度和相位演化的慢时系统。所示的平均系统可以准确地重现完整振荡器模型的解。对这个五维动力系统的分析表明，只要其各自的共振频率，站立的方位角模式可以与轴对称模式同步。和增长率相似。这种相耦合的双模式振荡对应于所谓的倾斜模式，在最近的涉及环形模型燃烧室的实验中观察到。根据增长率比和频率失谐参数，得出了发生模式同步的定量条件。发现分析结果与倾斜模式的实验观察结果一致。