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 noncompact gas turbine combustion systems from time domain data. Knowledge of these quantities is of high technical relevance as a required input for the design of damping devices for high frequency (HF) 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; and (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 (SDE), 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 HF 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.