Modelling and analysis of nonlinear thermoacoustic systems using frequency and time domain methods
In this thesis, low-order nonlinear models for the prediction of the nonlinear behaviour of thermoacoustic systems are developed. These models are based on thermoacoustic networks, in which linear acoustics is combined with a nonlinear heat release model. The acoustic networks considered in this thesis can take into account mean flow and non-trivial acoustic reflection coefficients, and are cast in state-space form to enable analysis both in the frequency and time domains. Starting from linear analysis, the stability of thermoacoustic networks is investigated, and the use of adjoint methods for understanding the role of the system's parameters on its stability is demonstrated. Then, a nonlinear analysis using various state-of-the-art methods is performed, to highlight the strengths and weaknesses of each method. Two novel frameworks that fill some gaps in the available methods are developed: the first, called Flame Double Input Describing Function (FDIDF), is an extension of the Flame Describing Function (FDF). The FDIDF approximates the flame nonlinear response when it is forced simultaneously with two frequencies, whereas the FDF is limited to one frequency. Although more expensive to obtain, the FDIDF contains more nonlinear information than the FDF, and can predict periodic and quasiperiodic oscillations. It is shown how, in some cases, it corrects the prediction of the FDF about the stability of thermoacoustic oscillations. The second framework developed is a weakly nonlinear formulation of the thermoacoustic equations in the Rijke tube, in which the acoustic response is not limited to a single-Galerkin mode, and is embedded in a state-space model. The weakly nonlinear analysis is strictly valid only close to the expansion point, but is much cheaper than any other available method. The above methods are applied to relatively simple thermoacoustic configurations, in which the nonlinear heat release model is that of a laminar conical flame or an electrical heater. However, in real gas turbines more complex flame shapes are found, for which no reliable low-order models exist. Two models are developed in this thesis for turbulent bluff-body stabilised flames: one for a perfectly premixed flame, in which the modelling is focused on the flame-flow interaction, accounting for the presence of recirculation zones and temperature gradients; the second for imperfectly premixed flames, in which equivalence ratio fluctuations, modelled as a passive scalar field, dominate the heat release response. The second model was shown to agree reasonably well with experimental data, and was applied in an industrial modelling project.
Thermoacoustics, Nonlinear dynamics, Describing Function, Low-order modelling, Acoustics
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