Thermoacoustic instability – a dynamical system and time domain analysis (original) (raw)

Non-normal and nonlinear dynamics of thermoacoustic instability in a horizontal Rijke tube

The paper focusses on the non-normal and nonlinear effects of thermoacoustic interaction in a horizontal electrically heated Rijke tube. The analysis starts with the governing equations for the fluid flow. The governing equations become stiff as the Mach number of the steady flow and the thickness of the heat source (compared to the acoustic wavelength) are small. Therefore asymptotic analysis is performed in the limit of small Mach number and compact heat source to eliminate the above stiffness problem. 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. A theoretical framework is developed to understand the non-normal nature of the thermoacoustic interaction in the Rijke tube. The role of non-normality in the subcritical transition to instability regime is explored.

Thermoacoustic instabilities in a Rijke tube with heating and cooling elements

2017

Thermoacoustic instabilities are investigated in a Rijke tube with both heating and cooling elements. The effects of the time delays of the elements, magnitudes of the temperature jumps, and location and compactness of the cooling element are investigated. The results show that for small temperature jumps, the system behaves periodically with changing source time delay. However, for larger temperature jumps corresponding to realistic combustion systems, various stable and unstable situations can occur, which simultaneously depend on the source and sink time delays. This dependence in some configurations is to the extent that the system is predicted unstable or stable, depending on whether or not the heat exchanger is included. In addition, the location or compactness of the heat sink has opposing effects depending on the source time delay. This investigation shows that a correct prediction of the system stability requires taking into account the coupled effects of the heat source an...

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

Journal of Mechanical Science and Technology, 2011

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 flowrate region of Re Do < 1.

Time domain modelling and stability analysis of complex thermoacoustic systems

Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 2007

A methodology allowing for a modular setup of complex acoustic systems is developed. The transfer behaviour of the individual subsystems is formulated in time domain. Subsystem descriptions can be obtained by analytical considerations, numerical methods, or experimental data. Once the complex subsystems have been characterized experimentally, changes in system geometry can be implemented easily by exchanging or adding subsystems. To validate the modelling approach, experiments are conducted in an acoustic test rig with a combustor-type geometry. Results are compared to predictions from the model, demonstrating accuracy in frequency and time domain. Application to thermoacoustic instabilities arising in lean-premixed combustion is given. The influence of a modified fuel distribution on an unstable operating point of a lean-premixed combustor is studied and validated with experimental data. Additionally, a study on the parameters governing the flame transfer function is performed to generate a stability map of a model combustor. An advantage of the state-space approach is that stability of a thermoacoustic system can be determined by simply solving a matrix eigenvalue problem. This is in strong contrast to the traditional approach, where the complete model is formulated in frequency domain with infinite-dimensional transfer functions. The time domain approach is based on the methodology presented by Schuermans et al. [1]. In contrast to their work, however, subsystems are not obtained from modal expansions but are characterized by using system identification techniques. Additionally, accuracy of the time domain model is verified by experiments.

Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube

2010

A bifurcation analysis of the dynamical behavior of a horizontal Rijke tube model is performed in this paper. The method of numerical continuation is used to obtain the bifurcation plots, including the amplitude of the unstable limit cycles. Bifurcation plots for the variation of nondimensional heater power, damping coefficient and the heater location are obtained for different values of time lag in the system. Subcritical bifurcation was observed for variation of parameters and regions of global stability, global instability and bistability are characterized. Linear and nonlinear stability boundaries are obtained for the simultaneous variation of two parameters of the system. The validity of the small time lag assumption in the calculation of linear stability boundary has been shown to fail at typical values of time lag of system. Accurate calculation of the linear stability boundary in systems with explicit time delay models, must therefore, not assume a small time lag assumption. Interesting dynamical behavior such as co-existing multiple attractors, quasiperiodic behavior and period doubling route to chaos have been observed in the analysis of the model. Comparison of the linear stability boundaries and bifurcation behavior from this reduced order model are shown to display trends similar to experimental data. Prof. Wolfgang Polifke served as the sole independent editor for this paper.

NUMERICAL SIMULATION OF SELF-EXCITED THERMOACOUSTIC INSTABILITIES IN A RIJKE TUBE

Self-excited thermoacoustic instabilities or oscillations occur in con"ned geometries and result from a feedback loop between the heat transferred to the #uid from a heat source and the acoustics of the geometry. If the heat input is at times of high pressure, a self-ampli"cation of acoustic #uctuations may lead to high pressure amplitudes. The e!ect can be observed in a Rijke tube, a straight tube with a heating element made from hot wires or gauze that provides the heat input. In the presence of a gas #ow, pressure oscillations are excited at one of the tube's natural frequencies. Two di!erent kinds of Rijke tubes are modelled by using a control volume based "nite di!erence method to solve iteratively the unsteady conservation equations for mass, momentum and energy. The obtained results are in good agreement with experiments. Besides the general behaviour of the oscillating system, non-linear e!ects are also accounted for by the simulations. The non-linearities in the heat transferred to the #uid from the heat source were investigated. These determine the limit cycle amplitudes of the self-excited oscillations.

Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations' amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

One-Dimensional Acoustic Modeling of Thermoacoustic Instabilities

In this paper the acoustic stability of a premixed turbulent natural gas ame conned in a combustor is investigated. Specically when the ame is operated in a lean premixed mode, the thermoacoustic system is known to exhibit instabilities. These arise from a feedback mechanism between the oscillatory o w and heat release rate perturbations in the ame and often lead to large amplitude pressure and velocity perturbations in the combustor. The acoustics of the combustor are described with a one-dimensional trans- fer matrix method. The feedback mechanisms that can cause instabilities are included in this method. The (complex) frequency for which the determinant of the transfer ma- trix goes to zero indicates an instability. An important factor in the one-dimensional acoustic model is the transfer function between the oscillatory o w and heat release rate perturbations. This transfer function is obtained from a well-stirred reactor dy- namic combustion model. Results show that the one-dim...

A linear model for control of thermoacoustic instabilities on annular domain

2003

We present a distributed linear model of thermoacoustic instability in form of a set of coupled PDEs including an acoustic model based on Potential Euler formulation, a fully distributed fuel transport model based on advection equation, and a fuel-sensitive heat release model based on assumption of fixed flame location. The damping in the distributed model is provided on the acoustic boundaries using local acoustic impedance models. The model is suitable for analysis and control of multiple acoustic modes in annular combustors with bluff body stabilized flames and for optimization of fuel control architecture. We also derive a low order model for control using Galerkin projection of the Potential Euler equations on finite number of acoustic basis functions and analytically solving the linearized fuel advection equation. The resulting frequency domain model has a form of coupled system involving undamped oscillators representing acoustic modes, distributed delays representing effect of acoustic perturbation on the fuel transport and combustion, and positive real transfer functions representing acoustic impedances of the boundaries. A simple control algorithm to suppress pressure oscillations is derived using the reduced order model.

Boundary Feedback Control of Unstable thermoacoustic Oscillations in the Rijke Tube

IFAC-PapersOnLine, 2016

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 × 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.