Low-Order Modelling of Thermoacoustic Limit Cycles

Abstract

Lean premixed prevaporised (LPP) combustion can reduce NOx emissions from gas turbines, but often leads to combustion instability. Acoustic waves produce fluctuations in heat release, for instance by perturbing the fuel-air ratio. These heat fluctuations will in turn generate more acoustic waves and in some situations self-sustained oscillations can form. The resulting limit cycles can have large amplitude causing structural damage. Thermoacoustic oscillations will have a low amplitude initially. Thus linear models can give stability predictions. An unstable linear mode will grow in amplitude until nonlinear effects become important and a limit cycle is achieved. While the frequency of the linear mode can provide a good approximation to that of the resulting limit cycle, linear theories give no prediction of its amplitude. A low-order model for thermoacoustic limit cycles in LPP combustors is described. The approach is based on the fact that the main nonlinearity is in the combustion response to flow perturbations. In LPP combustion, fluctuations in the inlet fuel-air ratio have been shown to be the dominant cause of unsteady combustion: these occur because velocity perturbations in the premix ducts cause a time-varying fuel-air ratio, which then convects downstream. If the velocity perturbation becomes comparable to the mean flow, there will be an amplitude-dependent effect on the equivalence ratio fluctuations entering the combustor and hence on the rate of heat release. A simple nonlinear flame model for this dependence is developed and is assumed to be the major non-linear effect on the limit cycle. Since the Mach number is low, the velocity perturbation can be comparable to the mean flow, with even reverse flow occurring, while the disturbances are still acoustically linear in that the pressure perturbation is still much smaller than the mean. Hence elsewhere the perturbations are treated as linear. In this nonlinear flame model, the flame transfer function describing the combustion response to changes in inlet flow is a function of both frequency and amplitude. The nonlinear flame transfer function is incorporated into a linear thermoacoustic network model for plane waves. Frequency, amplitude and modeshape predictions are compared with results from an atmospheric test rig. The approach is extended to circumferential waves in a thin annular geometry, where the nonlinearity leads to modal coupling.