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.

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No. of bitstreams: 1 DissAMSe.pdf: 3200404 bytes, checksum: 6e7b3b24a5f41a42da15de0b498e3cf7 (MD5) Previous issue date: 2015-04-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / To contribute to overcoming the difficulties related to the teaching of the concept of function and its applications, build an interdisciplinary didactic proposal. I delimited my job choosing the theme affine function, which is usually presented to students in the 1st year of high school. To validate the pedagogical product presented, I used the basic guidelines proposed by the Didactic Engineering. I developed an experimental activity of chemistry in which the resulting function is the type order. During the experiment of applying students have organized themselves into groups. Each group performed the experiment, collected the figures composing a table. The data analysis begins with the layout of the data as points on a Cartesian plane. The linear pattern was observed and these points using algebraic resources, students obtained an expression of affine function. In a second phase the same data is handled with the help of a computational tool. The application has ended with socializing the results of each group. I noticed that these activities give students a chance to better build the concept of function. I saw that they had to see for themselves what were the variables involved, and what was the independent variable and which was dependent. Also they had transit of opportunity between various types of representation in an interdisciplinary context, noting the reason for each passage. Another difficulty faced by students was handling not whole greatness and who had different absolute values and therefore built a chart that required care in relation to the adopted scale. I concluded that the application of this pedagogical product achieved the desired goals and that students were very interested in participating in this activity. I realize also that, after application of the experiment, my relationship with the class became closer and spontaneous, greatly improving our dialogue. To fellow teachers who want to apply this product in their teaching classes, I note that it is not easy to conduct a chemistry experiment, especially for those who have not been trained for this. This requires a mathematics teacher a lot of determination and patience. So I believe that the ideal situation is to combine with the teachers of the experimental subjects so that they can experiment and get the data. The math teacher would get the part to analyze this data. I conclude that my didactic proposal is validated and is very enriching for teaching functions. / Para contribuir com a superação das dificuldades relativas ao ensino do conceito de função e suas aplicações, construí uma proposta didática interdisciplinar. Delimitei meu trabalho escolhendo o tema função afim, que geralmente é apresentado aos estudantes no 1º ano do Ensino Médio. Para a validação do produto pedagógico apresentado, utilizei as orientações básicas propostas pela Engenharia Didática. Desenvolvi uma atividade experimental da área de Química da qual a função resultante é do tipo afim. No decorrer da aplicação do experimento os estudantes se organizaram em grupos. Cada grupo executou o experimento, coletou os dados numéricos compondo uma tabela. A análise desses dados se inicia com a disposição dos dados como pontos em um plano cartesiano. Foi observado o padrão linear desses pontos e, usando recursos algébricos, os estudantes obtiveram a expressão de uma função afim. Numa segunda fase os mesmos dados foram manipulados com o auxílio de uma ferramenta computacional. A aplicação foi encerrada com a socialização dos resultados de cada grupo. Notei que essas atividades proporcionaram aos estudantes uma oportunidade de construir melhor o conceito de função. Vi que eles tiveram que perceber por si mesmos quais eram as variáveis envolvidas, qual era a variável independente e qual era a dependente. Além disso tiveram oportunidade de transitar entre vários tipos de representação em um contexto interdisciplinar, observando o motivo de cada passagem. Outra dificuldade encontrada pelos estudantes foi a manipulação de grandezas não inteiras e que apresentavam valores absolutos diferentes e, portanto, construíram um gráfico que exigiu cuidados em relação à escala adotada. Concluí que a aplicação desse produto pedagógico alcançou os objetivos desejados e que os estudantes demonstraram muito interesse em participar dessa atividade. Percebo também que, após a aplicação do experimento, o meu relacionamento com a turma se tornou mais próximo e espontâneo, melhorando muito o nosso diálogo. Aos colegas professores que queiram aplicar esse produto pedagógico em suas classes, observo que não é fácil conduzir um experimento de Química, principalmente por quem não foi treinado para isso. Isso exige de um professor de Matemática muita determinação e paciência. Por isso considero que a situação ideal é combinar com os professores das disciplinas experimentais de forma que eles possam fazer experimentos e obter os dados. O professor de Matemática ficaria com a parte de analisar esses dados. Concluo que minha proposta didática está validada e é muito enriquecedora para o ensino de funções.