Rijke Tube Research Articles

Bifurcation and nonlinear analysis of a time-delayed thermoacoustic system

In this paper, of primary concern is a time-delayed thermoacoustic system, viz. a horizontal Rijke tube. A continuation approach is employed to capture the nonlinear behaviour inherent to the system. Unlike the conventional approach by the Galerkin method, a dynamic system is naturally built up by discretizing the acoustic momentum and energy equations incorporating appropriate boundary conditions using a finite difference method. In addition, the interaction of Rijke tube velocity with oscillatory heat release is modeled using a modified form of King's law. A comparison of the numerical results with experimental data and the calculations reported reveals that the current approach can yield very good predictions. Moreover, subcritical Hopf bifurcations and fold bifurcations are captured with the evolution of dimensionless heat release coefficient, generic damping coefficient and time delay. Linear stability boundary, nonlinear stability boundary, bistable region and limit cycles are thus determined to gain an understanding of the intrinsic nonlinear behaviours.

Stochastic bifurcations in a prototypical thermoacoustic system.

We study the influence of noise in a prototypical thermoacoustic system, which represents a nonlinear self-excited bistable oscillator. We analyze the time series of unsteady pressure obtained from a horizontal Rijke tube and a mathematical model to identify the effect of noise. We report the occurrence of stochastic bifurcations in a thermoacoustic system by tracking the changes in the stationary amplitude distribution. We observe a complete suppression of a bistable zone in the presence of high intensity noise. We find that the complete suppression of the bistable zone corresponds to the nonexistence of phenomenological (P) bifurcations. This is a study in thermoacoustics to identify the parameter regimes pertinent to P bifurcation using the stationary amplitude distribution obtained by solving the Fokker-Planck equation.

A new approach to treating pressure oscillations in combustion instability phenomena

Developing exact models of combustion instabilities is not an easy task to carry out and requires a great deal of time prior to obtaining success. The present study proposes a low-order model for pressure oscillations that does not require any knowledge of the systems, any new physical findings nor intricate details regarding its operating condition. This new approach is obtained using a Modified Van der Pol’s equation (MVDP) which is tuned by use of a Dual Extended Kalman Filter (DKEF) as a recursive estimator with perspectives in control by computer. This phenomenological model is used to predict the pressure signal from a variety of different combustors. Input data were taken from experimental cases such as a Rijke tube, a gas turbine and a liquid-fuel aero-engine combustor. Furthermore, a simulation considering high frequency oscillations to show the capability of the new approach is presented. In all cases, the results demonstrated the feasibility of applying the tractable model MVDP and DKEF running together to investigate pressure oscillations in practical cases.

Regularities of thermoacoustic oscillations in Lehmann’s plant with a coolant moving in reverse

An approximate method is developed for calculating the instability boundaries, as well as the frequencies and amplitudes, of acoolant’s autooscillations self-excited during its motion in a vertical channel exposed to locally applied heat from a constant power source. Unlike in the classical generators of thermoacoustic oscillations such as the Rijke tube and Lehmann’s plant, it is assumed that the main stationary flow is collinear with the gravity force. In order to exclude the surging nature of the self-oscillations, it is assumed that the flow is created by a supercharger with a monotonically decreasing head-capacity characteristic. The resulting mathematical model makes it possible to theoretically determine the self-excitation conditions of the considered autooscillations and also to evaluate the effect of various parameters on their shape.

Predicting the secondary dynamic mode interference phenomenon in thermoacoustic instability control

This paper brings a novel mathematical perspective in assessing the rise of the secondary dynamic modes to prominence during the suppression of thermoacoustic instability. This phenomenon is observed by many earlier investigators; however, without a complete analytical reasoning. We consider a Rijke tube with both a passive Helmholtz resonator and an active feedback control to suppress instabilities. The core dynamics is represented as a linear time-invariant multiple time-delay system of neutral type. Parametric stability of the resulting infinite-dimensional dynamics is investigated using a recent analytical tool: cluster treatment of characteristic roots paradigm. This tool reveals the stability outlook of such systems exhaustively and non-conservatively in the parameter space of the system. First, we examine the stability with and without the Helmholtz resonator. We then select an unstable operation for the resonator-mounted Rijke tube, impose a time-delayed integral feedback control over it and reveal the stabilizing controller parameters using the cluster treatment of characteristic roots methodology. When high control gains are inappropriately selected, the new analytical procedure declares how the secondary dynamic modes of the system exhibit instability although the initially unstable mode is now stabilized. All of these stability assessments are cross-validated using experimental results from a laboratory-scale Rijke tube set-up.

Experiments on heat content inside a Rijke tube with suppression of thermo-acoustics instability

The present work describes the occurrence of thermo-acoustic instability inside a horizontal Rijke tube and its suppression using an open loop active control technique. The Rijke tube is provided with a co-axial pre-mixed gas burner as the source of heat, which could be placed at any desired position. Radial injection of air (less than 3% of the total mass flow) through micro-jets into the flame is used as a control technique to suppress the thermo-acoustic instability. The rise in heat content inside the Rijke tube, estimated from the temperature mapping, clearly shows reduction in the heat loss as a result of complete suppression of the thermo-acoustic instability. However, the stability achieved passively by means of a slight shift in the burner position does not result in any change in the heat content. There is a visible change in the appearance of the burner flame when the above two methods are used to suppress the thermo-acoustic instability. The flame is seen to significantly shrink in length and spread radially when the control technique was applied. The flame dynamics is believed to determine the heat loss and hence the heat content inside the Rijke tube.

Suppression of thermo-acoustic instability using air injection in horizontal Rijke tube

The thermo-acoustic instability arising out of coupling between the pressure fluctuation and the unsteady heat release, when grows sufficiently, is known to cause serious structural damage thereby reducing the life span of systems having combustor for example, jet engines, gas turbines and industrial burners. The present work involves experimental study of thermo-acoustic instabilities occurring in a Rijke tube and their suppression by means of diverting a very small fraction of the incoming air flow in the form of radial injection from the wall of the tube through micro jets. A horizontal quartz tube with pre-mixed burner is used as the test model of Rijke tube. The coefficient Rayleigh Index (RI) is estimated from simultaneous measurement of chemiluminescence using a PMT and pressure in the plane of burner head. Experiments were carried out at the equivalence ratio of Φ = 1 for a range of the burner position inside the tube, velocity of micro jets, and total mass flow rate through the tube. The control technique of micro jets was found to completely suppress the thermo-acoustics in a range of jet velocity depending on the burner position and the total air mass flow rate. For complete suppression of the thermos-acoustics, the phase difference between the pressure and chemiluminescence waves was found to significantly increase and the RI was found to reduce to nearly zero. In such case, the flame luminescence was observed to significantly reduce. The proposed technique of air injection into the flame through radial micro jets is simple yet very effective in controlling the thermos-acoustic instability.

Effect of pointed and diffused air injection on premixed flame confined in a Rijke tube

AbstractThe coupling between pressure fluctuations and unsteady heat release in a combustion systems results in acoustic oscillations inside the combustion system. These acoustic oscillations, when grow sufficiently, may cause serious structural damage thereby reducing the lifespan of jet engines, gas turbines, and industrial burners. The aim of the first part of study is to define acoustically stable and unstable regions. The second part is focused on studying the effect of change in pressure field near the flame on the amplitude and frequency of the oscillations of instability. This study is carried out for three-burner positions and equivalence ratio of 0.7 by varying heat supply and total flow rate. The results show two acoustically unstable regions for 0.1 and 0.2 burner positions and only one acoustically unstable region for 0.25 burner position. The effect of pointed injection and diffused injection over a premixed flame on the sound pressure level was studied. The results show for burner position ...

Passive suppression of thermoacoustic instability in a Rijke tube

This article investigates passive stabilization of thermoacoustic dynamics in a Rijke tube by using a Helmholtz resonator. Under certain assumptions, the mathematical model of the dynamics is shown to fall into the linear time-invariant multiple time delay systems class of neutral type. To study the thermoacoustic instability phenomenon and its passive suppression, the cluster treatment of characteristic roots (CTCR) paradigm is implemented. This effort reveals the placement of Helmholtz resonator along the Rijke tube for stabilizing the unstable dynamics. The analytically obtained results are compared with experiments conducted on a laboratory scale Rijke tube. In addition, the effect of resonator’s design parameters on its stabilization capability is investigated.

Boundary Feedback Control of Unstable thermoacoustic Oscillations in the Rijke Tube

The problem of boundary stabilization of thermoacoustic oscillations is investigated. The Rijke tube is used as prototype system to study such phenomena. We consider that this system is modelled as a 2 x 2 linear first-order hyperbolic system that behaves like a wave equation with the control variable at one boundary condition. Our control approach is based on employing characteristic coordinates to convert the system into a system of two delay elements. Analysing the periodicity of the solution of these equations a discrete transfer function is obtained. This enable us to employ a discrete time domain control design to guarantee the exponential stability of the closed-loop system. The performance of the controller is evaluated through simulation results.

Experimental sensitivity analysis and control of thermoacoustic systems

In this paper, we report the results of an experimental sensitivity analysis on a thermoacoustic system – an electrically heated Rijke tube. We measure the change of the linear stability characteristics of the system, quantified as shifts in the growth rate and oscillation frequency, that is caused by the introduction of a passive control device. The control device is a mesh, which causes drag in the system. The rate of growth is slow, so the growth rate and frequency can be measured very accurately over many hundreds of cycles in the linear regime with and without control. These measurements agree qualitatively well with the theoretical predictions from adjoint-based methods of Magri & Juniper (J. Fluid Mech., vol. 719, 2013, pp. 183–202). This agreement supports the use of adjoint methods for the development and implementation of control strategies for more complex thermoacoustic systems.

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

An experimental investigation of the non-normal nature of thermoacoustic interactions in an electrically heated horizontal Rijke tube is performed. Since non-normality and the associated transient growth are linear phenomena, the experiments have to be confined to the linear regime. The bifurcation diagram for the subcritical Hopf bifurcation into a limit cycle behavior has been determined, after which the amplitude levels, for which the system acts linearly, have been identified for different power inputs to the heater. There are two main objectives for this experimental investigation. The first one deals with the extraction of the linear eigenmodes associated with the acoustic pressure from experimental data. This is accomplished by the Dynamic Mode Decomposition (DMD) technique applied in the linear regime. The non-orthogonality between the eigenmodes is determined for various values of heater power. The second objective is to identify evidence of transient perturbation growth in the system. The total acoustic energy in the duct has been monitored as the thermoacoustic system evolves from its initial condition. Transient growth, on the order of previous theoretical studies, has been found, and its parameteric dependence on amplitude ratio and phase angle of the initial eigenmode components has been determined. This study represents the first experimental confirmation of non-normality in thermoacoustic systems.

Principles of thermoacoustic energy harvesting

Thermoacoustics exploit a temperature gradient to produce powerful acoustic pressure waves. The technology has a key role to play in energy harvesting systems. A time-line in the development of thermoacoustics is presented from its earliest recorded example in glass blowing through to the development of the Sondhauss and Rijke tubes to Stirling engines and pulse-tube cryo-cooling. The review sets the current literature in context, identifies key publications and promising areas of research. The fundamental principles of thermoacoustic phenomena are explained; design challenges and factors influencing efficiency are explored. Thermoacoustic processes involve complex multi-physical coupling and transient, highly non-linear relationships which are computationally expensive to model; appropriate numerical modelling techniques and options for analyses are presented. Potential methods of harvesting the energy in the acoustic waves are also examined.

Thermoacoustic Instability in a Rijke Tube with a Distributed Heat Source

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.

Deployment of Time-Delayed Integral Control for Suppressing Thermoacoustic Instabilities

In this paper, active control of thermoacoustic instabilities is studied using a particular feedback control law and a mathematically novel perspective. The instabilities result from the complex dynamic exchange between the unsteady heat release and the pressure oscillations in a finite volume. A laboratory-scale Rijke tube, which is a simple thermoacoustic device, is considered in the paper. When linearized, its dynamics exhibits the neutral characteristics with multiple delays. To suppress the instabilities in this setting, a time-delayed integral control law is synthesized, and its performance is studied in detail. A recent mathematical paradigm, called the cluster treatment of characteristic roots is used to accomplish the stabilization of the time-delayed system. The paradigm declares exact stability conditions in the space of the system parameters. The analytical findings in this document are also validated with experimental results.

Effect of external noise on the hysteresis characteristics of a thermoacoustic system

We present the effect of noise on the hysteresis characteristics of a prototypical thermoacoustic system, a horizontal Rijke tube. As we increase the noise intensity, we find that the width of the hysteresis zone decreases. However, we find that the rate of decrease in hysteresis width is constant for all the mass flow rates considered in the present study. We also show that the subcritical transition observed in the absence of noise is no longer discernible once the intensity of noise is above a threshold value and the transition appears to be continuous. We compare our experimental observations with the results obtained from a numerical model perturbed with additive Gaussian white noise and we find a qualitative agreement between the experimental and the numerical results.

Thermoacoustics and the Rijke Tube: Experiments, Identification, and Modeling

The Rijke tube [1] is a classic experiment that is relatively simple and inexpensive to build in a typical university laboratory. Despite its construction simplicity, it can serve to illustrate a wide variety of mathematical modeling, empirical identification, verification, and feedback control techniques. As such, it is suitable for use in both advanced undergraduate and graduate control laboratory courses.

Time domain simulations of nonlinear thermoacoustic behaviour in a simple combustor using a wave-based approach

Lean premixed combustion chambers are susceptible to combustion instabilities arising from the coupling between the heat release rate perturbations and the acoustic disturbances. These instabilities are not desirable and knowledge of this complex mechanism is necessary in order to prevent or at least suppress them. A low order model is developed comprising a linear acoustic network and an improved analytical form for a flame describing function (FDF). The latter includes both the saturation of the amplitude of heat release rate perturbations and the change of phase lag relative to oncoming acoustic velocity fluctuations when the instability grows into a limit cycle. A stability map is constructed by moving the flame along the Rijke tube based on the eigenvalues resolved from the network. The acoustic model is then converted into the time domain and combined with the flame describing function to determine the evolutions of the heat release rate disturbances and velocity perturbations within the tube. It is shown that this method can be used to capture some quite intricate nonlinear behaviour of combustion instabilities and the results in the time domain are consistent with those predicted in the frequency domain.

Parametric Investigation of Thermoacoustic Instability (TAI) in a Rijke Tube: A Time-Delay Perspective

This paper presents a novel deployment of a recent mathematical paradigm for predicting the thermo-acoustic instability (TAI) of a Rijke tube in the relevant parametric space. This benchmark problem in combustion science has been studied for over 1½ centuries with phenomenal achievements both in theoretical and practical fronts. The new paradigm is called the Cluster Treatment of Characteristic Roots (CTCR), which is originally developed to assess the asymptotic stability of Linear Time Invariant (LTI) Time-delayed Systems (TDS). A notorious subcategory within LTI-TDS is called “Neutral TDS”, which matches the characteristics of the linearized dynamics of thermo-acoustic instability. The CTCR is shown to reveal a non-conservative and exhaustive linear stability map of the Rijke tube within the space of its geometric and operational parameters. We present a review of this paradigm as well as several case studies to demonstrate its capabilities and some encouraging comparison with the earlier literature. This paper is a concept document and it is prepared with the intent of providing a breeding ground for studies beyond its present coverage.

A New Perspective in Designing Delayed Feedback Control for Thermo-Acoustic Instabilities (TAI)

This article suggests the deployment of a unique mathematical tool for assessing the thermo-acoustic instability (TAI) in a Rijke tube and proposes an analytical design strategy for its feedback control. A widely accepted characteristic of TAI is its time-delayed dynamics, which originate from the regenerative acoustic coupling terms. Linear systems theory has also evolved on similar classes of problems especially in recent years. This document offers a bridge between the two veins of research. We first review the analytical model of the TAI phenomenon, which renders a set of delayed differential equations. Then, we apply a new mathematical tool called the cluster treatment of characteristic roots (CTCR) paradigm on this dynamics. CTCR provides non-conservative and exhaustive stability predictions for this class of systems. This capability is employed for both uncontrolled and feedback-controlled Rijke tube structures. The findings are unique from two angles: (i) stability declarations are made in the parametric space of the system, such as geometric dimensions (much differently from the peer studies that are at best point-wise evaluations), and (ii) these declared sets of stable operating parameters are exhaustive (i.e., for a given system no other parametric selection can provide stability). These capabilities become crucial when designing thermoacoustically stable combustors as well as determining their operating conditions. As a highlight contribution in this article, for those operating conditions that induce instability, we offer a methodology to synthesize a feedback control law that can recover stability, again utilizing the CTCR paradigm. Example case studies and analytical justifications of these novelties are provided.