Stability, sensitivity and optimisation of chaotic acoustic oscillations

Abstract

In an acoustic cavity with a heat source, such as a flame in a gas turbine, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation. If uncontrolled, these nonlinear acoustic oscillations, also known as thermoacoustic instabilities, can cause large vibrations up to structural failure. Numerical and experimental studies showed that thermoacoustic oscillations can be chaotic. It is not yet known, however, how to minimise such chaotic oscillations. We propose a strategy to analyse and minimise chaotic acoustic oscillations, for which traditional stability and sensitivity methods break down. We investigate the acoustics of a nonlinear heat source in an acoustic resonator. First, we propose covariant Lyapunov analysis as a tool to calculate the stability of chaotic acoustics making connections with eigenvalue and Floquet analyses. We show that covariant Lyapunov analysis is the most general flow stability tool. Second, covariant Lyapunov vector analysis is applied to a chaotic system. The time-averaged acoustic energy is investigated by varying the heat-source parameters. Thermoacoustic systems can display both hyperbolic and non-hyperbolic chaos, as well as discontinuities in the time-averaged acoustic energy. Third, we embed sensitivities of the time-averaged acoustic energy in an optimisation routine. This procedure achieves a significant reduction in acoustic energy and identifies the bifurcations to chaos.The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. The techniques presented can be used in other unsteady fluid dynamics problems with virtually no modification.

Learning Ergodic Averages in Chaotic Systems

“…We use a prototypical time-delayed thermoacoustic system composed of a longitudinal acoustic cavity and a heat source modelled with a nonlinear time-delayed model [8,12,16], which has been used to optimize ergodic averages in [8] with a dynamical systems approach. The non-dimensional governing equations are…”

“…where x f = 0.2 is the heat source location and ζ j = 0.1j + 0.06j 1/2 is the modal damping [8]. The heat release rate,q, is given by the modified King's law [8],…”

“…Using 10 Galerkin modes (N g = 10), β = 7.0 and τ = 0.2 results in a chaotic motion (Fig. 2), with the leading Lyapunov exponent being λ 1 ≈ 0.12 [8]. (The leading Lyapunov exponent measures the rate of (exponential) separation of two close initial conditions, i.e.…”

“…In this study, we use a hybrid echo state network (hESN) [14], originally proposed to time-accurately forecast the evolution of chaotic dynamical systems, to predict the long-term time averaged quantities, i.e., the ergodic averages. This is motivated by recent research in optimization of chaotic multiphysics fluid dynamics problems with applications to thermoacoustic instabilities [8]. The hESN is based on reservoir computing [10], in particular, conventional Echo State Networks (ESNs).…”

“…However, we stress that the present study is not focused on the accurate prediction of the time evolution of the system, but rather of its ergodic averages, which are obtained by the time averaging of a long time series (we implicitly assume that the system is ergodic, thus, the infinite time average is equal to the ergodic average [2].). Here, the physical system under study is a prototypical time-delayed thermoacoustic system, whose chaotic dynamics have been analyzed and optimized in [8].…”