Three-dimensional interaction of thermoacoustic modes in a circular tube

This study applies the Green's function method to investigate the modal interaction during thermoacoustic instability specifically in the afterburner. The afterburner is modelled as a cylindrical tube with a compact flame. Nonlinear effects are accounted for by employing the flame describing function (FDF). An integral governing equation for the acoustic velocity at the flame is derived. This is solved by an iteration method to obtain the time history of the acoustic velocity at the flame. The coupling mechanism, which is nonlinear due to the amplitude-dependence of the FDF, is explored using a two-mode analysis as an illustrative example. Different scenarios are observed when the initial amplitude is varied: the long-term behaviour of the time history may be dominated by one of the modes, which forms a limit cycle and squeezes out the other mode, i.e. there is a mutually inhibitory effect; however, it is also possible, for both modes to coexist. This dependence on the initial condition is a consequence of the amplitude-dependent heat release rate, and it is clearly a nonlinear effect. The time history calculation is supplemented by a phase analysis, which is based on the Rayleigh criterion and reveals the stability behaviour and limit cycles of the individual modes. In order to simulate changing operating conditions in a real afterburner, the coupling coefficient and the time-lag in the heat release rate are changed abruptly during the time history calculation. The change in coupling coefficient has no dramatic effect, while the change in time-lag can lead to mode switch. This is examined in detail by the phase analysis, which reveals that mode switch is also a nonlinear effect.