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 linear oscillations grow into large amplitude self-sustained oscillations. The resulting limit cycles can cause structural damage. Thermoacoustic oscillations will have a low amplitude initially. Thus linear models can describe the initial growth and hence give stability predictions. An unstable linear mode will grow in amplitude until nonlinear effects become sufficiently important to achieve a limit cycle. While the frequency of the linear mode can often provide a good approximation to that of the resulting limit cycle, linear theories give no prediction of its resulting amplitude. In previous work, we developed a low-order frequency-domain method to model thermoacoustic limit cycles in LPP combustors. This was based on a ‘describing function’ approach and is only applicable when there is a dominant mode and the main nonlinearity is in the combustion response to flow perturbations. In this paper that method is extended into the time domain. The main advantage of the time-domain approach is that limit-cycle stability, the influence of harmonics, and the interaction between different modes can be simulated. 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. 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 while the combustion response to flow velocity and equivalence ratio fluctuations must be modelled nonlinearly, the flow perturbations generated as a result of the unsteady combustion can be treated as linear. In developing a time-domain network model for nonlinear thermoacoustic oscillations an initial frequency-domain calculation is performed. The linear network model, LOTAN, is used to categorise the combustor geometry by finding the transfer function for the response of flow perturbations (at the fuel injectors, say) to heat-release oscillations. This transfer function is then converted into the time domain through an inverse Fourier transform to obtain the Green’s function, which thus relates unsteady flow to heat release at previous times. By combining this with a nonlinear flame model (relating heat release to unsteady flow at previous times) a complete time-domain solution can be found by stepping forward in time. If an unstable mode is present, its amplitude will initially grow exponentially (in accordance with linear theory) until saturation effects in the flame model become significant, and eventually a stable limit cycle will be attained. The time-domain approach enables determination of the limit-cycle. In addition, the influence of harmonics and the interaction and exchange of energy between different modes can be simulated. These effects are investigated for longitudinal and circumferential instabilities in an example combustor system and results are compared to frequency-domain limit-cycle predictions.
Analytical approximations for heat release rate laws in the time- and frequency-domains