Extraction of Linear Growth and Damping Rates of High-Frequency Thermoacoustic Oscillations From Time Domain Data

Abstract

This paper presents a set of methodologies for the extraction of linear growth and damping rates associated with transversal eigenmodes at screech level frequencies in thermoacoustically non-compact gas turbine combustion systems from time domain data. Knowledge of these quantities is of high technical relevance as an required input for the design of damping devices for high frequency oscillations. In addition, validation of prediction tools and flame models as well as the thermoacoustic characterization of a given unstable/stable operation point in terms of their distance from the Hopf bifurcation point occurs via the system growth/damping rates. The methodologies solely rely on dynamic measurement data (i.e. unsteady heat release and/or pressure recordings) while avoiding the need of any external excitation (e.g. via sirens), and are thus in principle suitable for the employment on operational engine data. Specifically, the following methodologies are presented: 1) The extraction of pure acoustic damping rates (i.e. without any flame contribution) from oscillatory chemiluminescence and pressure recordings. 2) The obtainment of net growth rates of linearly stable operation points from oscillatory pressure signals. 3) The identification of net growth rates of linearly unstable operation points from noisy pressure envelope data. The fundamental basis of these procedures is the derivation of appropriate stochastic differential equations, which admit analytical solutions that depend on the global system parameters. These analytical expressions serve as objective functions against which measured data are fitted to yield the desired growth or damping rates. Bayesian methods are employed to optimize precision and confidence of the fitting results. Numerical test cases given by time domain formulations of the acoustic conservation equations including high-frequency flame models as well as acoustic damping terms are set up and solved. The resulting unsteady pressure and heat release data are then subjected to the proposed identification methodologies to present corresponding proof of principles and grant suitability for employment on real systems.

Imperfect symmetry of real annular combustors: beating thermoacoustic modes and heteroclinic orbits

“…The regularity of this phenomenon suggests that the alternation of the spinning direction is not caused by turbulence-induced stochastic perturbations of the limit cycle, as was shown to be the case in some recent studies (e.g. Noiray & Schuermans 2013;Hummel et al 2018;Mazur et al 2019;, but by an underlying deterministic mechanism, which we aim to explain in the present work. We will achieve this goal with a model that includes small, but not negligible, resistive and reactive asymmetries.…”

Symmetry breaking of azimuthal waves: Slow-flow dynamics on the Bloch sphere

“…Indeed, the approach consisting in projecting the acoustic field onto the orthogonal acoustic eigenmodes of the annulus [15,4,43] captures rotational symmetry breaking induced by spatial thermoacoustic non-uniformities, while it is not appropriate to describe the reflectional symmetry breaking induced by a mean swirl. In contrast, the approach that is based on a representation of the acoustic field using clockwise and counterclockwise spinning waves [16,17] cannot capture the former symmetry breaking, while it is very suited for describing the latter. As shown already in [15] with the model based on Helmholtz eigenmode pairs, the quaternion-based framework predicts i) spinning waves limit-cycles for a uniform distribution of thermoacoustic sources without mean flow, where the direction of the spinning wave depending on the initial conditions, ii) mixed-modes for non-zero 2n component c 2n of the Fourier series for the thermoacoustic source distribution and ultimately standing modes when c 2n exceeds (β 0 −α)/β 0 , where the damping coefficient α characterizes uniformly distributed acoustic losses around the annular combustor, and where β 0 is the zero order component of the Fourier series.…”

“…They emphasize on the fact that this representation overcomes the problem of ill-posedness of the phase in previously adopted projections on standing modes e.g. [14,4] or on counter-rotating modes [16,17]. In the quaternion representation, it is always possible to properly define a temporal phase, making this representation very suited to model wave propagation phenomena in annular or cylindrical elements, and we will show that another advantage of the quaternion embedding is that it captures both rotational and reflectional symmetry breaking bifurcations.…”

“…In previous studies using this 1D simplified description, the acoustic field has been projected onto orthogonal basis formed by pairs of degenerate standing modes [14,4,15] or spinning modes [16,17], which we now briefly explain. Regarding the former modelling strategy, Noiray et al [4,15] proposed a nonlinear theoretical model of the spinning/standing/mixed mode dynamics by projecting the azimuthal thermoacoustic eigenmodes on orthogonal standing waves.…”

“…In [14,15], these theoretical findings are confirmed by time-domain simulations thanks to a reduced-order network model of the thermoacoustic system, including experimentally measured flame transfer functions and three-dimensional (3D) azimuthal modes computed with a finiteelement Helmholtz solver. Hummel et al [16,17] proposed to use a different 1D nonlinear formalism where the azimuthal mode is described as the sum of counterclockwise and a clockwise-spinning waves. Performing spatial averaging and time averaging of the wave equation, they also conclude that limit cycle spinning modes are stable for uniformly distributed thermoacoustic feedback, while standing modes are unstable.…”