Obtaining Bifurcation Diagrams with a Thermoacoustic Network Model (original) (raw)
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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.
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.
A two-way coupling for modeling thermoacoustic instabilities in a flat flame Rijke tube
Proceedings of The Combustion Institute, 2009
Thermoacoustic instabilities are a serious problem for lean premixed combustion systems. Due to different time and length scales associated with the flow field, combustion, and acoustics, numerical computations of thermoacoustic phenomena are conceptually challenging. This work presents a coupled method for the simulation of thermoacoustic instabilities in low Mach number reacting flows. The acoustics are represented by a reduced order model that can be obtained from network techniques or finite element computations. A detailed chemistry finite-difference zero Mach number solver is used for the small scale flame dynamics. Under the assumption that the pressure is continuous across the flame, the acoustic model can be reduced to a time-domain relation mapping the velocity perturbation downstream of the flame to that upstream. Closure is obtained by the flame code, which delivers the jump in velocity across the combustion zone. The method is applied to an experimental laminar premixed burner-stabilized flat flame Rijke tube, that exhibits strong thermoacoustic oscillations associated with the 5k=4 mode of the geometrical set-up. In addition to the fundamental oscillation, a significant subharmonic response of the flame is observed. Results from the coupled simulation are compared to the experimental data. Good qualitative and quantitative agreement is found.
Nonlinear modeling of thermoacoustic systems
2015
Thermoacoustic systems convert energy from heat to acoustic power and vice versa. These systems have commercial interest due to the high potential efficiency and low number of moving parts. To numerically predict the performance of a thermoacoustic device inherent nonlinearities in the system, such as thermoacoustic streaming and generation of harmonics need to be taken into account. We present a nonlinear frequency domain method with which these nonlinearities in thermoacoustic systems are modeled in a computationally efficient manner. Using this method, the nonlinear periodic steady state of a thermoacoustic engine can directly be computed, without computing the long initial transient of the system. In this publication, the developed method is applied to compute the periodic steady state of an experimental standing wave engine. The results obtained match well with experimental data.
Experimental investigation of bifurcations in a thermoacoustic engine
International Journal of Spray and Combustion Dynamics, 2015
In this study, variation in the characteristics of the pressure oscillations in a thermoacoustic engine is explored as the input heat flux is varied. A bifurcation diagram is plotted to study the variation in the qualitative behavior of the acoustic oscillations as the input heat flux changes. At a critical input heat flux (60 Watt), the engine begins to produce acoustic oscillations in its fundamental mode. As the input heat flux is increased, incommensurate frequencies appear in the power spectrum. The simultaneous presence of incommensurate frequencies results in quasiperiodic oscillations. On further increase of heat flux, the fundamental mode disappears and second mode oscillations are observed. These bifurcations in the characteristics of the pressure oscillations are the result of nonlinear interaction between multiple modes present in the thermoacoustic engine. The bifurcation diagram indicates hysteresis in forward and reverse path suggesting subcritical nature of the bifurcations. Further, the qualitative analysis of different dynamic regimes is performed using nonlinear time series analysis. The physical reason for the observed nonlinear behavior is discussed. Suggestions to avert the variations in qualitative behavior of the pressure oscillations in thermoacoustic engines are also provided.
ASME Turbo Expo 2013, San Antonio, Texas, U.S.A., 3-7 June 2013
In the recent years a great interest has been devoted to the understanding of the nonlinear dynamics characterizing the thermoacoustic combustion instabilities. Although linear techniques are able to predict whether the non-oscillating steady state of a thermoacoustic system is "asymptotically" stable (without oscillations) or unstable (increasing oscillations), a thermoacoustic system can reach a permanent oscillating state (the so called "limit cycle"), even when it is linearly stable, if a sufficiently large impulse occurs. A nonlinear analysis is able to predict the existence of this oscillating state and the nature of the bifurcation process. The aim of this work is to investigate the behavior of gas turbine combustion chambers in presence of nonlinear flame models. The bifurcation diagrams, obtained by using a "continuation" technique in the frequency domain, give the amplitude of the oscillations as a function of a chosen flame parameter. The Helmholtz equation is used to model the combustion chamber and nonlinear terms are introduced in the flame model, starting from the classical k-tau formulation. A three-dimensional finite element method (FEM) is used for discretization of the computational domain and a solver of quadratic eigenvalue problems is combined with Newton technique in order to identify the points of the bifurcation diagram. First, a simple Rijke tube configuration, as can be found in the literature, is examined in order to obtain bifurcation diagrams. Then, the nonlinear analysis is extended to simplified annular configurations. The obtained results show how the nonlinear behavior is influenced by varying some control parameters, such as the time delay, yielding useful indications to designers and experimentalists.
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.
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.
Non-linear phenomena in thermoacoustic engines
Journal of Non-Equilibrium Thermodynamics, 2004
A non-linear simplified model for a thermoacoustic engine is presented in this work, which assumes the form for the spatial dependence of the state variables and considers only e¤ects inside the stack. For such a model the limit cycle solution is found numerically using a shooting technique in the 'T > 'T crit region, where the stationary, fixed-point, trivial solution is unstable. With this solution a bifurcation diagram for thermoacoustic phenomena is built. Through such diagram an analysis of the non-linear nature of the generated oscillations is conducted, focusing on the presence of harmonics and the higher-order time-average pressure.
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.