The intrinsic thermoacoustic (ITA) feedbackloop constitutes a coupling between flow, flame and acoustics that does not involve the natural acoustic modes of the system. One recent study showed that ITA modes in annular combustors come in significant number and with the peculiar behavior of clusters, i.e. several modes with close frequencies. In the present work an analytical model of a typical annular combustor is derived via Riemann invariants and Bloch theory. The resulting formulation describes the full annular system as a longitudinal combustor with an outlet reflection coefficient that depends on frequency and the azimuthal mode order. The model explains the underlying mechanism of the clustering phenomena and the structure of the clusters associated with ITA modes of different azimuthal orders. In addition, a phasor analysis is proposed, which enclose the conditions for which the 1D model remains valid when describing the thermoacoustic behavior of an annular combustor.

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Thermo-acoustic eigenmodes of annular or can-annular combustion chambers, which typically feature a discrete rotational symmetry, may be computed in an efficient manner by utilizing the Bloch-wave theory. Unfortunately, the application of the Bloch-wave theory to combustion dynamics has hitherto been limited to the frequency domain. In this study, we present a time-domain formulation of Bloch boundary conditions (BBC), which allows to employ them in time domain simulations, e.g., computational fluid dynamics (CFD) simulations. The BBCs are expressed as acoustic scattering matrices and translated to complex-valued state-space systems. In a hybrid approach an unsteady, compressible CFD simulation of the burner-flame zone is coupled via characteristic-based state-space boundary conditions to a reduced order model of the combustor acoustics that includes BBCs. The acoustic model with BBC accounts for cross-can acoustic coupling and the discrete rotational symmetry of the configuration, while the CFD simulation accounts for the nonlinear flow–flame acoustic interactions. This approach makes it possible to model limit cycle oscillations of (can-)annular combustors at drastically reduced computational cost compared to CFD simulations of the full configuration and without the limitations of weakly nonlinear approaches that utilize a flame describing function. In this study, the suggested approach is applied to a generic multican combustor. Results agree well with a fully compressible CFD simulation of the complete configuration.

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Thermo-acoustic eigenmodes of annular or can-annular combustion chambers, which typically feature a discrete rotational symmetry, may be computed in an efficient manner by utilizing the Bloch-wave theory. Unfortunately, the application of the Bloch-wave theory to combustion dynamics has hitherto been limited to the frequency domain. In this study we present a time domain formulation of Bloch boundary conditions (BBC), which allows to employ them in time domain simulations, e.g. CFD simulations. The BBCs are expressed as acoustic scattering matrices and translated to complex-valued state-space systems. In a hybrid approach an unsteady, compressible CFD simulation of the burner-flame zone is coupled via characteristic-based state-space boundary-conditions to a reduced order model of the combustor acoustics that includes BBCs. The acoustic model with BBC accounts for cross-can acoustic coupling and the discrete rotational symmetry of the configuration, while the CFD simulation accounts for the nonlinear flow-flame-acoustic interactions. This approach makes it possible to model limit cycle oscillations of (can-)annular combustors at drastically reduced computational cost compared to CFD simulations of the full configuration, and without the limitations of weakly nonlinear approaches that utilize a flame describing function. In the current study the suggested approach is applied to a generic multi-can combustor. Results agree well with a fully compressible CFD simulation of the complete configuration.

Thermoacoustic properties of can-annular combustors are commonly investigated by means of single-can test-rigs. To obtain representative results, it is crucial to mimic can-can coupling present in the full engine. However, current approaches either lack a solid theoretical foundation or are not practicable for high-pressure rigs. In the present study we employ Bloch-wave theory to derive reflection coefficients that correctly represent can-can coupling. We propose a strategy to impose such reflection coefficients at the acoustic terminations of a single-can test-rig by installing passive acoustic elements, namely straight ducts or Helmholtz resonators. In an iterative process, these elements are adapted to match the reflection coefficients for the dominant frequencies of the full engine. The strategy is demonstrated with a network model of a generic can-annular combustor and a 3D model of a realistic can-annular combustor configuration. For the latter we show that can-can coupling via the compressor exit plenum is negligible for frequencies sufficiently far away from plenum eigenfrequencies. Without utilizing previous knowledge of relevant frequencies or flame dynamics, the test-rig models are adapted within a few iterations and match the full engine with good accuracy. Using Helmholtz resonators for test-rig adaption turns out to be more viable than using straight ducts.

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Thermoacoustic systems can exhibit self-excited instabilities of two nature, namely cavity modes or intrinsic thermoacoustic (ITA) modes. In heavy-duty land-based gas turbines with canannular combustors, the cross-talk between cans causes the cavity modes of various azimuthal order to create clusters, i.e. ensembles of modes with close frequencies. Similarly, in systems exhibiting rotational symmetry, ITA modes also have the peculiar behavior of forming clusters. In the present study, we investigate how such clusters interplay when they are located in the same frequency range. We first consider a simple Rijke tube configuration and derive a general analytical low-order network model using only dimensionless numbers. We investigate the trajectories of the eigenmodes when changing the downstream length and the flame position. In particular, we show that ITA and acoustic modes can switch nature and their trajectories are strongly influenced by the presence of exceptional points. We then study a generic can-annular combustor. We show that such configuration can be approximated by an equivalent Rijke tube. We demonstrate that, in the absence of mean flow, the eigenvalues of the system necessarily lie on specific trajectories imposed by the upstream conditions.