On the role of symmetry and degeneracy in nonlinear thermoacoustic eigenproblems with application to can-annular combustors

Abstract

The subject of this dissertation are thermoacoustic instabilities in combustion chambers with discrete rotational and reflective symmetries. Designing thermoacoustically stable combustors is one of the principal challenges in the engineering of modern gas turbines operating with low NOx emissions. This thesis contributes to the research in the field from three perspectives: numerical, theoretical and experimental. Combustion experiments are costly and hence the design phase needs accurate computational models to cut development costs. Frequency-domain models in the form of network models, or as finite-element discretizations of the thermoacoustic Helmholtz equation, have proven highly successful in predicting thermoacoustic stability. From a mathematical point of view, the models lead to nonlinear eigenvalue problems which need to be solved numerically. The first achievement of this thesis is to provide fast and reliable solution algorithms that are tailored to thermoacoustic problems. The methods have the capability to compute all relevant solutions –a crucial property to determine stability of a combustor. One of the methods based on contour integration proved essential in computing intrinsic thermoacoustic modes in annular geometries for the first time and it is explained why these modes appear in clusters. These modes cannot be computed with methods that were the de-facto standard in the thermoacoustic field at the beginning of this PhD, which is subsequently proven in work that is part of this thesis. Part of this thesis is also a strategy with which the high dimension of the discrete problems can be drastically reduced –permitting parameter studies of large-scale problems. Combustion chambers in modern industrial applications mainly come in two types: annular or can-annular layouts. Both designs exhibit discrete spatial symmetries, i.e. certain reflections and rotations leave the combustors invariant. The presence of the discrete symmetries has implications for the type of oscillations that occur. Most prominently, standing and spinning azimuthal modes originate from a degenerate mode pair. The second achievement of this work is to interpret the thermoacoustic nonlinear eigenvalue problem from the formal viewpoint of symmetry group theory for the first time. With this powerful framework it can be predicted which physical objects show degenerate modes and how these degeneracies split as the symmetries are lowered by perturbations. This has important implications for industrial combustors, which exhibit high symmetries and consequently a large number of degenerate modes. Moreover, a number of thermoacoustic configurations are discussed, which exhibit non-trivial symmetries with surprising degeneracies. In thermoacoustics, Bloch waves have been used successfully to reduce computational cost of solving the nonlinear eigenvalue problem. Bloch waves exploit exclusively the rotational symmetry. A major result of this work is that even when an additional mirror symmetry is present, a further reduction is only possible for simple eigenvalues. Related to this work on eigenvalue multiplicity is a result on so-called exceptional points in thermoacoustic spectra. For the first time it is shown that for certain parameter combinations two eigenvalues can coalesce and form a defective point. Can-annular combustors have received little experimental attention in academia. Unlike the annular design, the can-annular design exhibits socalled clustered modes, i.e. multiple modes within narrow frequency bands due to the presence of a weak coupling. From an engineering perspective there are many advantages to this design. However, the closely spaced modes lead to thermoacoustic effects that differ starkly from those observed in annular combustion chambers – which have been well-researched in the past two decades. The third achievement of this thesis is to establish a new can-annular model combustor to explore the complex dynamics of can-annular combustors in well-defined and accessible lab experiment. The design permits to adjust the can-to-can coupling and study its effect on clustered modes. Several tightly packed clusters are observed, which contain modes of different azimuthal mode orders. In addition, the frequencies and amplitudes of the observed limit cycle oscillations show a strong sensitivity to changes in the cross talk size. Thus, the results confirm recent theoretical work in the literature. The observed transient dynamics show that interactions between multiple unstable modes in a cluster are complex and it is difficult to predict which mode will form the limit cycle oscillation.