• Energy Research

Abstract Thermoacoustic oscillations in axisymmetric annular combustors are generally coupled by degenerate azimuthal modes, which can be of standing or spinning nature. Symmetry breaking due to the presence of a mean azimuthal flow splits the degenerate thermoacoustic eigenvalues, resulting in pairs of counter-spinning modes with close but distinct frequencies and growth rates. In this study, experiments have been performed using an annular system where the thermoacoustic feedback due to the flames is mimicked by twelve identical electro-acoustic feedback loops. The mean azimuthal flow is generated by fans. We investigate the standing/spinning nature of the oscillations as a function of the azimuthal Mach number for two types of initial states and how the stability of the system is affected by the mean azimuthal flow. It is found that spinning, standing, or mixed modes can be encountered at very low Mach number, but increasing the mean velocity promotes one spinning direction. At sufficiently high Mach number, only spinning modes are observed in the limit cycle oscillations. In some cases, the initial conditions have a significant impact on the final state of the system. It is found that the presence of a mean azimuthal flow increases the acoustic damping. This has a beneficial effect on stability: it often reduces the amplitude of the self-sustained oscillations, and can even suppress them in some cases. However, we observe that the suppression of a mode due to the mean flow may destabilize another one. We discuss our findings in relation to an existing low-order model.

Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally non-uniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact to the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz- and linearized Euler-equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function, which is solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal mode order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an “inclination rule” that explains the behavior of degenerate eigenvalues at first order is proven.

Abstract Self-excited pressure oscillations can occur in combustion systems due to the thermoacoustic coupling between the unsteady acoustics and flame heat release fluctuations. Usually, the knowledge of a Flame Transfer Function is used to predict the onset of thermoacoustic instabilities. However, it is also possible to take advantage of it to model a flame response and study experimentally thermoacoustic phenomena without flames. This is exploited in the present study, in which a novel annular setup for the study of thermoacoustics in annular combustors is presented. The thermoacoustic feedback is replaced by electroacoustic feedback. The pressure fluctuations, measured by a microphone, are delayed and filtered and then sent to a loudspeaker, which produces acoustic perturbations, closing the loop. Each flame model parameter can be varied in a flexible way, which allows to choose combinations of parameters that generate modal behaviours of interest in the experiments. For example, this setup can trigger on demand various configurations which exhibit multiple unstable modes, leading to diverse modal competition scenarios. This allows to assess the multi-input Describing Function method, which is used to predict the frequency and amplitude of each mode contributing to thermoacoustic oscillations, when multiple modes are linearly unstable. The experimental validation of predictions from this method, which can be somewhat cumbersome and expensive in the presence of flames, is facilitated by this setup, in which all parameters and boundary conditions are well known and the noise remains negligible. Prediction uncertainties connected to approximations intrinsic of this method when operating in the vicinity of bifurcation points are also discussed.

Abstract We propose a general classification of all the modes of a given thermoacoustic system into two sets: one of acoustic origin and one of intrinsic thermoacoustic (ITA) origin. To do this, the definition of intrinsic modes, traditionally based on anechoic boundary conditions, is reformulated in terms of the gain n of the Flame Transfer Function (FTF). As a consequence of this classification, we show how theoretical results for the estimation of all thermoacoustic modes can be derived in the limit n → 0, for both axial and annular combustors, independent of the acoustic boundary conditions. Starting from this limit and using standard continuation methods while increasing n, all the eigenvalues of interest in a given domain in the frequency space can be identified. We also discuss how thermoacoustic modes of acoustic and ITA origin can interact, and in some cases coalesce generating exceptional points (EPs). Although all EPs found have negative growth rates, in their vicinity thermoacoustic eigenmodes have very large sensitivities and exhibit strong mode veering. We demonstrate how, in some cases, mode veering is responsible for the occurrence of thermoacoustic instabilities, and propose a numerical method to identify EPs. All the theoretical results are numerically verified using two generic thermoacoustic configurations.

Abstract An effective impedance that models the aeroacoustic coupling between periodic arrays of ducts that communicate via apertures is derived. A lumped description of the interaction between the one-dimensional acoustics in the ducts and the transverse unsteady fluctuations in the connecting apertures is exploited in the derivation of the impedance. This configuration is of relevance for some industrial applications, for example to model the aeroacoustic coupling between cans in gas turbines equipped with can-annular combustors. The derived impedance can be used as a Bloch-type boundary condition to reduce theoretical and computational analysis on the dynamics of these arrays of ducts to a single duct. After the derivation of a general result, simpler limits of interest are discussed. For some of these limits, impedance models are available in the literature. It is shown how the effective impedance derived in this study reduces to already available models in these limits, but more generally embeds new features, including the explicit dependence of the impedance on the grazing flow Mach number and the finite extension of the apertures.

We propose a method to accelerate the solution of 3D FEM-discretized nonlinear eigenvalue problems by drastically reducing the problem dimension. Our method yields a reduced order model (ROM) via a projection onto a suitable subspace, with eigenpairs identical to the full problem in a region of the complex plane. The subspace is automatically constructed by solving the full problem at a few random points inside the region of interest. The method requires minimal user input and, although exemplified for with a thermoacoustic application, readily generalizes to applications dealing with other vibrational problems.

Abstract