Thermoacoustic instability – a dynamical system and time domain analysis

Abstract

This study focuses on the Rijke tube problem, which includes features relevant to the modelling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source and nonlinearities. This thermoacoustic system features both linear and nonlinear flow regimes with complex dynamical behaviour. In order to synthesize accurate time series, we tackle this problem from a numerical point of view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness -in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by using the projection method proposed by Jarlebring (PhD thesis, Technische Universität Braunschweig, 2008), which alleviates the problems arising from linearization with respect to the retarded time. The results are then compared with the analytical solution of the undamped system and with the results obtained from Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We demonstrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case predicted by using Galerkin expansion show spurious oscillations resulting from the Gibbs phenomenon. Finally, time series in the fully nonlinear regime, where a limit cycle is established, are analysed and dominant modes are extracted. By comparing the modes of the linear regime to those of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching, which appear as the nonlinearities become significant and ultimately affect the form of the limit cycle. Analysis of the saturated limit cycles shows the presence of higher-frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not exhibited when the coupling between different frequencies is not accounted for. In conclusion, a dedicated solver for capturing thermoacoustic instability is proposed and methods for analysing linear and nonlinear regions of the resulting time series are introduced.

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

“…To proceed along this line, we require a representation of the system as a temporal evolution problem. We choose the model used by Sayadi et al (2014) which accurately captures the velocity jump condition across the flame. The governing equations are identical to (2.1), and the flame model is implemented directly as the inverse Laplace transform of expression (3.5).…”

Global modes, receptivity, and sensitivity analysis of diffusion flames coupled with duct acoustics

“…Other numerical discretizations, such as Chebyshev polynomials used by Illingworth et al (2013), are numerically more efficient but make the implementation of the adjoint problem much more difficult because of the way that the integration of the heat release is handled. It is worth pointing out that Sayadi et al (2014) have developed a new numerical method for the acoustics that, amongst other things, prevents the Gibbs' phenomenon across the discontinuity, which arises with a fine Galerkin discretization of the acoustics (Magri & Juniper 2013b).…”

Experimental Investigation of Non-Normality of Thermoacoustic Interaction in an Electrically Heated Rijke Tube

“…Due to its simplicity and representative character, the study of thermoacoustic instabilities in a Rijke tube has a long history, going back to the original investigation of Rijke [1] who observed acoustic amplifications in a vertical tube as a coiled electrical heating filament is turned on. Over the past decades, this observation has given rise to a significant body of literature exploring this phenomenon by numerical simulations [2,3,4,5,6] or by experimental efforts [7,8]. The goal of these studies was a better understanding of the basic mechanisms linking the acoustic field in the chamber to the unsteady heat release rates.…”