Rijke Tube Research Articles

Subcritical bifurcation and bistability in thermoacoustic systems

AbstractThis paper analyses subcritical transition to instability, also known as triggering in thermoacoustic systems, with an example of a Rijke tube model with an explicit time delay. Linear stability analysis of the thermoacoustic system is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation. We then use the method of multiple scales to recast the model of a general thermoacoustic system near the Hopf point into the Stuart–Landau equation. From the Stuart–Landau equation, the relation between the nonlinearity in the model and the criticality of the ensuing bifurcation is derived. The specific example of a model for a horizontal Rijke tube is shown to lose stability through a subcritical Hopf bifurcation as a consequence of the nonlinearity in the model for the unsteady heat release rate. Analytical estimates are obtained for the triggering amplitudes close to the critical values of the bifurcation parameter corresponding to loss of linear stability. The unstable limit cycles born from the subcritical Hopf bifurcation undergo a fold bifurcation to become stable and create a region of bistability or hysteresis. Estimates are obtained for the region of bistability by locating the fold points from a fully nonlinear analysis using the method of harmonic balance. These analytical estimates help to identify parameter regions where triggering is possible. Results obtained from analytical methods compare reasonably well with results obtained from both experiments and numerical continuation.

Vibration combustion as a prospect for utilizing firewood contaminated with radiocesium

We present results of experimental investigation relevant to vibration combustion of solid fuel specimens in a Rijke tube and the interpretation of the combustion mechanism on the basis of “energy” approach. The decrease in the flame temperature is noted, which can be used to reduce the yield of low-boiling mineral components of ash, e.g., radioactive 137Cs isotope, in the volatile phase. The distribution of 137Cs in thermal conversions of firewood has been studied with the aid of computational-experimental methods, and the use of vibration combustion for utilizing the firewood contaminated with radiocesium has been suggested to decrease the emission of 137Cs into the surrounding medium.

Flame Shape and Spatially Resolved Rayleigh Criterion Using Proper Orthogonal Decomposition

Chemiluminescence imaging was performed for premixed flames with natural thermoacoustic instabilities in a Rijke tube. The images were phase locked to the natural pressure oscillations in the tube. The use of proper orthogonal decomposition (POD) to identify phase-resolved flame fronts and flame surface area is reported. The flame front results compare favorably to an Abel transform for axisymmetric cases but also work well for cases with non-axisymmetric contributions. The flame area perturbations are compared to previous work using OH planar laser induced fluorescence. The relatively-simpler chemiluminescence approach finds the same flame surface area correlations versus pressure. Additionally, the modes from POD are used to compute local acoustic coupling via the space and time resolved Rayleigh index. It is found that one mode in the decomposition is spatially the same as the Rayleigh index and can be used to characterize the thermoacoustic coupling. This similarity is due to the mode's time constant replicating the phase of the tube's natural pressure oscillations.

Spatial Averaging Combined with a Perturbation/Iteration Procedure

A matter of increasing interest is finding the best way to integrate the use of powerful computational facilities with the traditional practices of analysis and related disciplines. This is largely a problem in research and development, rather than fabrication of machines, or even development of codes. There is little question that modern manufacturing, for example, cannot be accomplished competitively without computing machinery – in fact, the more the merrier. However, there is a fairly widespread sense that in areas depending on the development and applications of new ideas, and perhaps especially in education, the general emphasis has been skewed too much to rely excessively on computers. This paper has been prepared partly to make the point with examples taken from the author's experiences with unsteady combustion. My claim here is that in many cases, analytical methods (necessarily approximate) offer a path often initially preferable to that presented by numerical methods, which in the cases at hand, mean computational fluid mechanics. The first simple example treated here is the Rijke tube, well known primarily for two reasons: The physical behavior is easy to produce, and for which data may be relatively easily collected; and the necessary analysis seems quite simple, at first glance. Some recently published experimental results will be cited, with an approximate theory based on the known differential equations for one-dimensional motions. The next section is a brief historical summary of Galerkin's method, followed by several sections summarizing the manner in which it may be combined with a perturbation/iteration method to give an effective approximate method. The general approach has been widely used to analyze practical problems of combustion instabilities arising in development of operational systems. Thus a large part of the paper is a review of previously published material, but with considerable clarifications of points that have caused some confusion. The paper ends with a brief discussion answering a serious criticism, of the method, nearly fifteen years old. The basis for the criticism, arising from solution to a relatively simple problem, is shown to be a result of an omission of a term that arises when the average density in a flow changes abruptly. Presently, there is no known problem of combustion instability for which the kind of analysis discussed here is not applicable. The formalism is general; much effort is generally required to apply the analysis to a particular problem. A particularly significant point, not elaborated here, is the inextricable dependence on expansion of the equations and their boundary conditions, in two small parameters, measures of the steady and unsteady flows. Whether or not those Mach numbers are actually ‘small’ in fact, is really beside the point. Work out applications of the method as if they were! Then maybe to get more accurate results, resort to some form of CFD. It is a huge practical point that the approach taken and advocated here cannot be expected to give precise results, but however accurate they may be, they will be obtained with relative ease and will always be instructive. In any case, the expansions must be carried out carefully with faithful attention to the rules of systematic procedures. Otherwise, inadvertent errors may arise from inclusion or exclusion of contributions. I state without proof or further examples that the general method discussed here has been quite well and widely tested for practical systems much more complex than those normally studied in the laboratory. Every case has shown encouraging results. Thus the lifetimes of approximate analyses developed before computing resources became commonplace seem to be very long indeed.

Heat transfer in a thermoacoustic process

Thermoacoustic instability is defined as the excitation of acoustic modes in chambers with heat sources due to the coupling between acoustic perturbations and unsteady heat addition. The major objective of this paper is to achieve accurate theoretical results in a thermoacoustic heat transfer process. We carry out a detailed heat transfer analysis aimed at determining the stability–instability border of the thermoacoustic system. In this paper, we present a project type of physical examination and modelling task. We employed an electrically heated Rijke tube in our thermoacoustic project work. The aim of our project is to help our students enlarge their knowledge about thermodynamics, mainly about thermoacoustics, and develop their applied information technology and mathematical skills.

Feedback control of unstable thermoacoustic modes in an annular Rijke tube

Simulation and experimental results from an annular Rijke tube are presented. This system is a thermoacoustic surrogate system of an annular gas turbine combustor which, despite its simplicity, possesses the basic mechanisms to feature unstable azimuthal modes. A thermoacoustic network model is set up and used to derive low-order models for modal control of the system. The derived controllers are successfully applied in simulation and experiment. With the modal controllers, all unstable acoustic modes can be eliminated individually. A simultaneous use of all controllers results in a complete stabilization of the system.

Control of weak perturbations

We define sound as being a weak perturbation in the properties of material consistent with the Navier-Stokes and continuity equations. Lighthill's pioneering paper on aerodynamic noise gives an exact theory that enables interesting connections to be made between flow and sound. Aerodynamic noise being caused by quadrupoles is a good point of view, but what caused the quadrupoles? Were they possibly initiated by sound? Conclusions deduced from such a theory are not necessarily helpful, but they are true and might be very helpful indeed. The linear perturbations we call sound obey linear rules and it can be suppressed by anti-sound, a subject now becoming both practical and useful. The same must apply to any weak perturbation of a dynamic system perturbed from rest. Some perturbations are unstable and grow exponentially in their early weak state. They might be eliminated altogether by suppressing their linear form. The Rijke tube experiment shows that to be practical and shows also the close similarity that exists between acoustics and control theory. The lecture will give more examples of that type and suggest others that have yet to be demonstrated.

Nonlinear Thermoacoustic Instability Dynamics in a Rijke Tube

We present a data-driven nonlinear and chaos theory–based analysis of thermoacoustic instabilities in a simple Rijke tube. Thermoacoustic instability modes in this simple Rijke tube display very rich nonlinear behavior because of the interaction of acoustic modes and unsteady heat-release processes during combustion. This approach of analyzing thermoacoustic instabilities, their evolution, and interactions differs from traditional linear time-series–based approaches, such as the Fourier transform or the autocorrelation function. The objectives of this work are to address the limitations of conventional linear and model-based nonlinear analyses and to describe the potential of data-driven nonlinear and chaos theory–based analyses to gain a deeper understanding of thermoacoustic instability dynamics for reacting flows. The Rijke tube permits investigation of the nonlinear dynamics of thermoacoustic instabilities in a very systematic way by using picosecond time-resolved laser-induced fluorescence (PITLIF) of the OH radical at a rate of 2500 Hz. The 2500-Hz measurement bandwidth is sufficient to capture all thermoacoustic instability modes encountered in the Rijke tube. Through the use of these nontraditional analyses, we find that thermoacoustic instabilities within the Rijke tube contain chaotic behavior. A simple low-order approximate model and a data-adaptive feedback controller are also introduced for the control of thermoacoustic instabilities.

Modular system for studying tonal sound excitation in resonators with heat addition and mean flow

An educational experimental system has been developed for studying tonal sound generation in acoustic resonators. Tones are excited by either heat addition or vortex shedding in the presence of mean flow. The system construction is straightforward and inexpensive. Several test arrangements and experimental data are described in this paper. The experimental setups include a modified Rijke tube, a standing-wave thermoacoustic engine, a baffled tube with mean flow, and an acoustic energy harvester with a piezoelement. Simplified mathematical models for interpreting data are discussed, and references are provided to literature with more advanced analyses. The developed system can assist both graduate and undergraduate students in understanding acoustic instabilities via conducting and analyzing interesting experiments.

Transient growth of flow disturbances in triggering a Rijke tube combustion instability

Combustion instabilities in a Rijke tube could be triggered by the transient growth of flow disturbances, which is associated with its non-normality. In this work, a Rijke tube with three different temperature configurations resulting from a laminar premixed flame are considered to investigate its non-normality and the resulting transient growth of flow disturbances in triggering combustion instabilities. For this, a general thermoacoustic model of a Rijke tube is developed. Unsteady heat release from the flame is assumed to be caused by its surface variations, which results from the fluctuations of the oncoming flow velocity. Coupling the flame model with a Galerkin series expansion of the acoustic waves present enables the time evolution of flow disturbances to be calculated, thus providing a platform on which to gain insights on the Rijke tube stability behaviors. Both eigenmodes orthogonality analysis and transient growth analysis of flow disturbances are performed by linearizing the flame model and recasting it into the classical time-lag N-τ formulation. It is shown from both analyses that Rijke tube is a non-normal thermoacoustic system and its non-normality depends strongly on the temperature configurations and the flame position. Furthermore, the most ‘dangerous’ position at which the flame is more susceptible to combustion instabilities are predicted by real-time calculating the maximum transient growth rate of acoustical energy.

Robust Output Model Predictive Control of an Unstable Rijke Tube

This work investigates the active control of an unstable Rijke tube using robust output model predictive control (RMPC). As internal model a polytopic linear system with constraints is assumed to account for uncertainties. For guaranteed stability, a linear state feedback controller is designed using linear matrix inequalities and used within a feedback formulation of the model predictive controller. For state estimation a robust gain-scheduled observer is developed. It is shown that the proposed RMPC ensures robust stability under constraints over the considered operating range.

Nonlinear Self-Excited Combustion Oscillations of a Premixed Laminar Flame

Self-excited combustion oscillations are caused by a coupling between acoustic waves and unsteady heat release. A premixed laminar flame in a Rijke tube, anchored to a metal gauze, is considered in this work. The flame response to flow disturbances is investigated by developing a nonlinear kinematic model based on the classical-equation, with the assumption of a time-invariant laminar flame speed. Unsteady heat release from the flame is assumed to be caused by its surface variations, which results from the fluctuations of the oncoming flow velocity. The flame is acoustically compact, and its presence causes the mean temperature undergoing a jump, whose effect on the dynamics of the thermo-acoustic system is discussed. Coupling the flame model with a Galerkin series expansion of the acoustic waves present enables the time evolution of the flow disturbances to be calculated. It was found that the model can predict the mode shape and the frequencies of the excited combustion oscillations very well. Moreover, the fundamental mode is found to be the easiest one to be triggered among all acoustic modes. To gain insight about the mode selection and triggering, further numerical investigation is conducted by linearizing the flame model and recasting into the popular formulation.

Triggering of self-excited thermo-acoustic oscillations in a Rijke tube with a premixed laminar flame

For a given Rijke tube, self-excited combustion oscillations could be caused by the transient growth of flow disturbances. A premixed laminar flame, anchored to a metal gauze, is considered to investigate the role of non-normality and the resulting transient growth in triggering such oscillations. The unsteady heat release is assumed to be caused by the flame surface variations, which results from the fluctuations of the oncoming flow. The flame is acoustically compact and its presence causes a jump in mean temperature. Coupling the flame model with a Galerkin series expansion of the acoustic waves enables the time evolution of the flow disturbances to be calculated. It was found that the nonlinear model can predict the mode shape and the frequencies of the excited oscillations very well. Moreover, the fundamental mode with the lowest frequency is the easiest one to be excited among all the acoustic modes. Linearizing the model and recasting it into the classical time-lag formulation provide insights on the mode selection and triggering. Finally, to gain insight about the stability behaviors of such non-normal Rijke tube, pseudo-spectra analysis is performed to obtain upper and lower bounds on the transient growth factor.

Experimental and theoretical investigation of thermoacoustic oscillations in natural gas metering stations

In natural gas flow metering and pressure regulation stations flowmeters, heat exchangers and control valves are usually connected in series. Especially in the case of two or more measuring bars sometimes at minor flow rates untypical pipe vibration together with flow metering faults are observed. Based on field investigations the dependencies between the pipe vibration level, the gas pulsation inside the pipe and the operating conditions of the heat exchanger are analysed. It turns out that with increasing heat flow rates into the natural gas the pulsation and hence the metering faults as well as the pipe vibrations are amplified.In order to understand the physical dependencies of this phenomenon besides experiment a theoretical study is performed. The origin of the vibration turns out to be a thermoacoustic instability. According to the Rayleigh criterion gas pulsations are amplified if heat is given to the gas at the moment of greatest condensation. Based on detailed theoretical investigations by means of the method of characteristics using a Rijke tube model the physical dependencies are analysed. Finally potential solutions to avoid this vibration problem at natural gas metering stations are introduced.

Triggering in a Thermoacoustic System with Stochastic Noise

This paper explores the mechanism of triggering in a simple thermoacoustic system, the Rijke tube. It is demonstrated that additive stochastic perturbations can cause triggering before the linear stability limit of a thermoacoustic system. When triggering from low noise amplitudes, the system is seen to evolve to self-sustained oscillations via an unstable periodic solution of the governing equations. Practical stability is introduced as a measure of the stability of a linearly stable state when finite perturbations are present. The concept of a stochastic stability map is used to demonstrate the change in practical stability limits for a system with a subcritical bifurcation, once stochastic terms are included. The practical stability limits are found to be strongly dependent on the strength of noise.

Transient Growth and Triggering in the Horizontal Rijke Tube

This theoretical paper examines a non-normal and non-linear model of a horizontal Rijke tube. Linear and non-linear optimal initial states, which maximize acoustic energy growth over a given time from a given energy, are calculated. It is found that non-linearity and non-normality both contribute to transient growth and that, for this model, linear optimal states are only a good predictor of non-linear optimal states for low initial energies. Two types of non-linear optimal initial state are found. The first has strong energy growth during the first period of the fundamental mode but loses energy thereafter. The second has weaker energy growth during the first period but retains high energy for longer. The second type causes triggering to self-sustained oscillations from lower energy than the first and has higher energy in the fundamental mode. This suggests, for instance, that low frequency noise will be more effective at causing triggering than high frequency noise.

사운드카드를 이용한 음향학 탐구학습 사례

PC의 내장 사운드 카드를 이용하여 음향학 탐구학습을 수행하였다. 사운드 카드는 함수발생기와 오실로스코프에 비해 가격이 저렴하고 학생들의 접근이 용이하다는 장점이 있다. 학생들은 다양한 음향 현상에서 발생하는 소리를 사운드 카드를 이용하여 녹음하고 주파수 스펙트럼을 분석하였다. 소리굽쇠의 맥놀이, Rijke 관에서의 음의 발생, 물을 따를 때에 나는 소리 변화, 소리로 포도주잔 깨기 및 포도주 병마개를 딸 때에 발생하는 소리 등의 현상을 정량적으로 분석하면서 소리의 중첩, 공진 및 정상파에 대한 탐구를 수행하였다. Inquiring activities on the acoustic phenomena have been carried out by using a sound card installed in a personal computer. A sound card is cheaper and more accessible to the students than the precision equipment such as a function generator or an oscilloscope. The students record the sounds from various acoustic phenomena to the sound card. Then they analyze the frequency spectrums of that sounds by using a program. Inquired phenomena include beat by two tuning forks, sound from Rijke tube, pouring sound, breaking of a wine glass and pop-up sound of a wine bottle. Through these activities students perform quantitative analysis of various phenomena due to superposition, resonance and standing wave.

Modelling nonlinear thermoacoustic instability in an electrically heated Rijke tube

An analysis of thermoacoustic instability is performed for a horizontal Rijke tube with an electrical resistance heater as the heat source. The governing equations for this fluid flow become stiff and are difficult to solve by the computational fluid dynamics (CFD) technique, as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore, an asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. The unknown variables are expanded in powers of Mach number. Two systems of governing equations are obtained: one for the acoustic field and the other for the unsteady flow field in the hydrodynamic zone around the heater. In this analysis, the coupling between the acoustic field and the unsteady heat release rate from the heater appears from the asymptotic analysis. Furthermore, a non-trivial additional term, referred to as the global-acceleration term, appears in the momentum equation of the hydrodynamic zone, which has serious consequences for the stability of the system. This term can be interpreted as a pressure gradient applied from the acoustic onto the hydrodynamic zone. The asymptotic stability of the system with the variation of system parameters is presented using the bifurcation diagram. Numerical simulations are performed using the Galerkin technique for the acoustic zone and CFD techniques for the hydrodynamic zone. The results confirm the importance of the global-acceleration term. Bifurcation diagrams obtained from the simulations with and without the above term are different. Acoustic streaming is shown to occur during the limit cycle and its effect on the unsteady heat release rate is discussed.

Experiments with heat- and vortex-driven acoustic instabilities in tube resonators.

Self-excited tonal sound often appears in engineering systems with mean flow and heat release. To assist students in learning about acoustic instabilities in resonators, a modular low-cost system was constructed and a series of experiments was developed. The experimental setup consists of pipe sections with baffles, damping chamber, air blower, nichrome-wire heater, and piezoelements. The system can operate as a Rijke tube, thermoacoustic engine, vortex-excited resonator, and harvester of acoustic power. Tests carried out with this system include (i) identification of instability domains in the controllable parametric space of the system geometry, mean flow rates, and supplied heat; (ii) investigation of transient phenomena, such as growth and attenuation of sound amplitude; (iii) demonstration of nonlinear effects, such as hysteresis in the system behavior and frequency locking; and (iv) harvesting of energy of self-excited sound using piezoelements. The observed phenomena are interpreted with help of simplified theoretical models. [Work supported by the NSF Grant No. 0853171.]

Prediction of thermoacoustic instability in Rijke tube using CFD-CAA numerical method

The coupling between the emitted energy and the acoustic energy in a thermal system, e.g. combustor, may result in system instability accompanying severe vibration. In this study, the limit curve for thermo-acoustic instability is numerically obtained by using the CFD- CAA combined method. The response function for pulsating release of heat in the Rijke tube and the distribution of mean density in the tube are computed accurately by the CFD analysis and used as inputs to CAA predictions. To verify the accuracy of the prediction method, the thermo-acoustics inside the horizontal, one meter long Rijke tube of square cross-section, which has the heating part at the location of one quarter length, is simulated. The predicted instability curve agrees well with the experimental one except the low flow- rate region of ReDo < 1.