Nonlinear Themoacoustic In/Stability of a Rijke Tube with a Distributed Heat Source

In this paper, thermoacoustic dynamic behaviour patterns of a Rijke tube with a distributed heat source have been investigated. 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. Unlike the conventional approach utilizing the Galerkin method to simulate the acoustic field in time domain, a dynamic system is developed by spatially discretizing the acoustic equations using standard finite difference schemes in a Method of Lines (MOL) manner. In addition, a continuation approach is employed to analyze the nonlinear characteristics inherent to the heat-acoustic interaction. Hopf bifurcations and limit cycles are captured to delineate the nonlinear stability of the system. This methodology is first validated and shown to yield good predictions. 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.  Corresponding author: [email protected] Proceedings of the European Combustion Meeting 2015 Introduction Thermoacoustic instability has been a serious impediment to develop NOx tolerant combustion systems both for aircraft propulsion and power generation gas turbines including rocket motors, industrial burners etc[1-3]. It arises 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, is 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 due to the transfer from unsteady heat release to acoustic energy. Despite the simplicity in structure, it contains rich nonlinear behaviors, which make it an excellent example for the study of thermoacoustic instability [4-6]. Most previous research, has focused on the classical Rijke tube with a single compact heat source, either a flame or a hot-wire gauze for the fact that the heat source is small enough compared with the acoustic wavelength. Heckl [7] developed empirical models for the nonlinear behavior of both heat release and the reflection coefficients based on experimental results. Limit cycle amplitudes were predicted and flow reversal at the heat source and nonlinear effects at the tube ends were demonstrated to be important in limiting the amplitude. Matveev [8-9] combined linear theory and thermal analysis to predict the linear stability boundaries in a horizontal Rijke tube. A special form of the nonlinear heat transfer function was introduced to extend the method to nonlinear stability analysis. Hysteresis phenomenon was reported in the stability boundary and limit cycles were predicted as observed in experiments. Heckl and Howe [10] conducted stability analysis of the Rijke tube by making use of a Green’s function. Oscillations were described in terms of the eigenmodes of an integral equation derived using the Green’s function and the predictions of stability behavior were in line with Rayleigh’s criterion. Balasubramanian and Sujith [11] studied the role of non-normality and nonlinearity in thermoacoustic system in a Rijke tube using the heat release model from Heckl [7]. It was shown that the non-normality inherent in the thermoacoutic system could result in transient growth of oscillations which can trigger nonlinearities in the system. Noble et al. [12] described a data-driven nonlinear and chaos theory–based analysis of thermoacoustic instabilities in a simple Rijke tube. It only relied on experimental data with no implicit assumptions. PLIFH measurement of OH radical at the rate of 2500 Hz was used to capture the thermoacoustic instability modes appeared in the Rijke tube. Chaotic behavior was identified in the thermoacoustic instability. Juniper [13] employed adjoint looping of the nonlinear governing equations as well as an optimization routine to find the lowest initial energy of a Rijke tube. This state could trigger self-sustained oscillations and was known as the ‘most dangerous’ initial state. It was found that self-sustained oscillations can be reached over approximately half the linearly stable domain. 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 etc. In this case, it is not quite appropriate to deal with the heat source as a single point source. Moreover, Heckl [14] reported that the heat source distribution has a first order influence on the stability of