Journal of Thermodynamics

Volume 2015, Article ID 949384, 9 pages

http://dx.doi.org/10.1155/2015/949384

## Thermoacoustic Instability in a Rijke Tube with a Distributed Heat Source

^{1}School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, P.O. Box 88, Manchester M60 1QD, UK^{2}Thermal Management Research Group, Efficient Energy Transfer Department (*η*ET), Bell Labs, Alcatel-Lucent, Dublin 15, Ireland

Received 7 July 2015; Revised 8 October 2015; Accepted 12 October 2015

Academic Editor: Philip De Goey

Copyright © 2015 Xiaochuan Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A Rijke tube with a distributed heat source is investigated. Driven by the widely existing thermoacoustic instability in lean premixed gas turbine combustors, this work aims to explore the physicochemical underpinning and assist in the elucidation and analysis of this problem. The heat release model consists of a row of distributed heat sources with individual heat release rates. The integrated heat release rate is then coupled with the acoustic perturbation for thermoacoustic analysis. A continuation approach is employed to conduct the bifurcation analysis and capture the nonlinear behaviour inherent in the system. Unlike the conventional approach by the Galerkin method, the acoustic equations are originally discretized using the Method of Lines (MOL) to build up a dynamic system. The model is first validated and shown to yield good predictions with available experimental data. Influences of multiple heat sources, time delay, and heat release distribution are then studied to reveal the extensive nonlinear characteristics involved in the case of a distributed heat source. It is found that distributed heat source plays an important role in determining the stability of a thermoacoustic system.

#### 1. Introduction

Thermoacoustic instability has been a serious impediment to develop NOx tolerant combustion systems for both aircraft propulsion and power generation gas turbines including rocket motors and industrial burners [1–6]. It results from the interaction between the heat release and acoustic pressure or velocity oscillations within the combustion system. Rijke tube, a typical time-delayed thermoacoustic system, has been a classical tool employed for the study of thermoacoustic instability. It usually consists of an open-end tube and heat source inside it. When the heat source is placed in certain positions along the tube, sound would emit from the tube. The sound is generated as a result of the transfer from unsteady heat release to acoustic energy. In spite of its simple structure, it could demonstrate abundant nonlinear phenomena, such as fixed point, bifurcation, limit cycle, quasiperiodicity, and chaos. Comparing with the expensive and complicated experiments, its simplicity and operability have made it an excellent example to investigate thermoacoustic instability in practical problems [7–10].

Rijke tube has been broadly studied to understand the intrinsic nonlinear behaviors of the thermoacoustic instability during the past decades. Several review works have been conducted by [11–13] regarding the theoretical, experimental, and numerical research on Rijke tube perturbations. The theoretical work by Culick [14, 15] focused on the nonlinear behavior of acoustic waves within a combustion chamber and was extensively referred to by a number of subsequent works on the study of Rijke tube. Heckl [16] developed an empirical heat release model based on experimental results and predicted limit cycle amplitudes and nonlinear effects in the tube. Hantschk and Vortmeyer [17] investigated self-excited thermoacoustic instabilities in the Rijke tube using a commercial CFD code and nonlinearity in the heat flux from the heating source to the flow was found to determine the limit cycle amplitudes. Matveev [18, 19] introduced a nonlinear heat transfer function for the nonlinear stability analysis of a horizontal Rijke tube. Hysteresis phenomenon was reported in the stability boundary and limit cycles were predicted as observed in experiments. Ananthkrishnan et al. [20] captured the dynamics of the nonlinear acoustic waves with reduced-order models via truncating the modal expansions and determined the number of modes required for accurate predictions. Balasubramanian and Sujith [21] studied the role of nonnormality and nonlinearity in thermoacoustic system in a Rijke tube and found out that the nonnormality could result in transient growth of oscillations which can trigger nonlinearities in the system. Subramanian et al. [22] conducted bifurcation analysis of the dynamic behaviors of a horizontal Rijke tube as a function of different system parameters. Nonlinear stability boundaries and other nonlinear phenomena were also observed in the analysis of the thermoacoustic system. Juniper [23] employed flame describing function, adjoint-based approach, and matrix-free continuation to predict the limit cycles and bifurcations in the Rijke tube.

Most of the previous research has focused on the classical Rijke tube with a single compact heat source, either a flame or a hot-wire gauze, allowing for the fact that the heat source is small enough compared with the acoustic wavelength. However, in practical combustion systems, the flame scale is usually considerable under most circumstances, especially with larger fuel flow or output power load, smaller excess air ratio, smaller nozzle spray angle, and so forth [24–26]. In this case, it is not quite proper to deal with the heat source as a single point source.

There have been various previous studies on the effect of distributed heat source which have extended the study of thermoacoustic instability to a wider scope. For example, Dowling’s theoretical work [2] proves that distributed heat input over an axial distance can lead to a significantly different frequency of oscillation from that when the heat input is concentrated. Kim and his coworkers [3] investigated the influence of the spatial heat release distribution using local flame transfer function (FTF) measurements. Heckl [27] employed Green’s function to model the behaviour of a Rijke tube with a distributed heat source and concluded that the heat source distribution has a first order influence on the stability of the thermoacoustic system and could not be ignored.

However, most of them are centred on the frequency-domain analysis. This paper focuses specifically on the time domain and conducts the bifurcation analysis regarding the thermoacoustic system, which has not been attempted previously. In this study, a horizontal Rijke tube with distributed heat source is employed for stability study. The distributed heat source consists of a row of single heat source. A modified form of Heckl’s model is utilized to couple the heat release with acoustics. Method of Lines (MOL) is adopted to discretize the governing equations and a linear multistep method (LMS-method) and Newton iteration method [28, 29] are used for linear stability study. A numerical continuation method is then employed to obtain bifurcation diagrams for investigating nonlinear behavior including the Hopf bifurcation, fold bifurcation, and limit cycles. This model is shown to be accurate and could contribute to understanding the physical nature of the thermoacoustic instability and provide guidance concerning the design and operation of a practical thermoacoustic system.

#### 2. Physical Model

Rijke tube is often oriented vertically in practice, in which the base flow is driven by natural convection. With the view of neglecting this complicated convection, in this paper, a horizontal Rijke tube is of major concern for the instability study of the thermoacoustic system.

Figure 1 shows a schematic of a horizontal Rijke tube with a distributed heat source. The base flow driven by an external fan passes through the tube and is heated up by hot wire gauzes (as discrete heat sources) placed along the tube at locations , where . Naturally, the tube displays an infinite number of acoustic modes. According to Rayleigh criterion, the thermal energy could be transferred to acoustic energy as long as they are in phase [30]. The influence of the mean flow and mean temperature gradients is excluded.