Thermoacoustic instability - a dynamical system and time domain analysis

This study focuses on the Rijke tube problem, which includes features relevant to the modeling of thermoacoustic coupling in reactive flows: a compact acoustic source, an empirical model for the heat source, and nonlinearities. This thermo-acoustic system features a complex dynamical behavior. In order to synthesize accurate time-series, we tackle this problem from a numerical point-of-view, and start by proposing a dedicated solver designed for dealing with the underlying stiffness, in particular, the retarded time and the discontinuity at the location of the heat source. Stability analysis is performed on the limit of low-amplitude disturbances by means of the projection method proposed by Jarlebring (2008), which alleviates the linearization with respect to the retarded time. The results are then compared to the analytical solution of the undamped system, and to Galerkin projection methods commonly used in this setting. This analysis provides insight into the consequences of the various assumptions and simplifications that justify the use of Galerkin expansions based on the eigenmodes of the unheated resonator. We illustrate that due to the presence of a discontinuity in the spatial domain, the eigenmodes in the heated case, predicted by using Galerkin expansion, show spurious oscillations resulting from the Gibbs phenomenon. By comparing the modes of the linear to that of the nonlinear regime, we are able to illustrate the mean-flow modulation and frequency switching. Finally, time-series in the fully nonlinear regime, where a limit cycle is established, are analyzed and dominant modes are extracted. The analysis of the saturated limit cycles shows the presence of higher frequency modes, which are linearly stable but become significant through nonlinear growth of the signal. This bimodal effect is not captured when the coupling between different frequencies is not accounted for.Comment: Submitted to Journal of Fluid Mechanic

Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube

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

Feedback Control of Thermoacoustic Instability Using Acoustic Actuator

Thermoacoustic instability is a phenomenom that appears in several technical applications, for instance rocket and jet engines, gas turbines, waste generators and industrial burners where it can cause heavy damage to the systems. As with most instabilities, they are undesired and this thesis' purpose is to identify and control a Rijke tube process (which has been used to stimulate the thermoacoustic instabilities) with the aid of microphones and a loudspeaker. A major part of the work has been done on setting up such a process, considering for instance the necessary high sample rates and make future experiments possible and also to show that the process can be controlled with very simple means in a very simple environment

A Green's function approach to predict nonlinear thermoacoustic instabilities in combustors

The prediction of thermoacoustic instabilities is fundamental for combustion systems such as domestic burners and industrial gas turbine engines. High-amplitude pressure oscillations cause thermal and mechanical stress to the equipment, leading to premature wear or even critical damage. In this paper we present a new approach to produce nonlinear (i.e. amplitude-dependent) stability maps of a combustion system as a function of various parameters. Our approach is based on the tailored Green’s function of the combustion system, which we calculate analytically. To this end, we assume that the combustor is one-dimensional, and we describe its boundary conditions through reflection coefficients. The heat release is modelled by a generalised law. This includes a direct-feedback term in addition to the usual time-lag term; moreover, its parameters (time lag, coupling coefficients) depend on the oscillation amplitude. The model provides new insight into the physical mechanism of the feedback between heat release rate and acoustic perturbations. It predicts the key nonlinear features of the thermoacoustic feedback, such as limit cycles, bistability and hysteresis. It also explains the frequency shift in the acoustic modes

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

Nonlinear Control of a Thermoacoustic System with Multiple Heat Sources and Actuators

Thermoacoustic instabilities can occur in thermal devices when unsteady heat release is coupled with pressure perturbations. This effect results in excitation of Eigen-acoustic modes of the system. These instabilities can lead to unpredictable behavior of the system. Gas-turbine combustion systems are especially prone to this phenomenon reducing their overall efficiency. Additionally, due to the nature of the combustion, the turbines end up releasing undesired amounts of harmful chemicals to the atmosphere, such as Nitrous Oxide (NOX). A Rijke tube, representing a resonator with a mean flow and a concentrated heat source, is a convenient system to study the thermoacoustic phenomena. Under certain conditions of the main system, a loud sound is generated through a process similar to that in devices prone to thermoacoustic instabilities. Rijke devices have been extensively studied and several models which can provide accurate representation of the system, already exists. These models often assume that the system is comprised of a single heat source which drives the instability. This may not be the case as combustors which can use more than one flame are common for engines and industrial burners. By using the aforementioned models, a nonlinear feedback control scheme is developed for a Rijke-type combustor system with n actuators and m heat sources. The performance of the controller is tested under different scenarios, assuring that is capable to exponentially stabilize the system despite any nonlinearities present in the heat release. Additionally, active control is studied in detail by analyzing the impact of the control parameters under different positioning of heat sources. The effect of the location for the actuators is also studied

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

Adaptive Nonlinear Regulation Control of Thermoacoustic Oscillations in Rijke-Type Systems

Adaptive nonlinear control of self-excited oscillations in Rijke-type thermoacoustic systems is considered. To demonstrate the methodology, a well-accepted thermoacoustic dynamic model is introduced, which includes arrays of sensors and monopole-like actuators. To facilitate the derivation of the adaptive control law, the dynamic model is recast as a set of nonlinear ordinary differential equations, which are amenable to control design. The control-oriented nonlinear model includes unknown, unmeasurable, nonvanishing disturbances in addition to parametric uncertainty in both the thermoacoustic dynamic model and the actuator dynamic model. To compensate for the unmodeled disturbances in the dynamic model, a robust nonlinear feedback term is included in the control law. One of the primary challenges in the control design is the presence of input-multiplicative parametric uncertainty in the dynamic model for the control actuator. This challenge is mitigated through innovative algebraic manipulation in the regulation error system derivation along with a Lyapunov-based adaptive control law. To address practical implementation considerations, where sensor measurements of the complete state are not available for feedback, a detailed analysis is provided to demonstrate that system observability can be ensured through judicious placement of pressure (and/or velocity) sensors. Based on this observability condition, a sliding-mode observer design is presented, which is shown to estimate the unmeasurable states using only the available sensor measurements. A detailed Lyapunov-based stability analysis is provided to prove that the proposed closed-loop active thermoacoustic control system achieves asymptotic (zero steady-state error) regulation of multiple thermoacoustic modes in the presence of the aforementioned model uncertainty. Numerical Monte Carlo-type simulation results are also provided, which demonstrate the performance of the proposed closed-loop control system under various sets of operating conditions

Experimental investigation of oscillatory heat release mechanisms and stability margin analysis in lean -premixed combustion

Lean-premixed combustion has become an acceptable means of achieving ultra-low NOx emissions from land-based gas turbines. Further reduction may be possible through the use of hydrogen augmented or syngas fuels. However, advanced combustor designs developed to utilize these technologies often encounter thermoacoustic instabilities that may significantly hamper engine performance and shorten component life-cycles. These dynamics, although not fully understood, occur through a complex interaction between variations in heat release rate and acoustic properties of the system, and can be exacerbated by variable fuel properties in natural gas and syngas applications.;Theoretical models of thermoacoustic instabilities have attempted to describe the coupling process through reduced-order models that represent mechanisms suspected of contributing to variations in the heat release rate such as variations in fuel/air mixing, fluctuations of heat release through vortex shedding and periodic changes in the flame structure. These reduced-order models have demonstrated only a modest ability at predicting instabilities even in relatively simple systems. This may be due to the inherent complexity from interacting processes, the use of over-simplifying assumptions and the lack of experimental verification.;In this study a simple conical flame, used to reduce the number of contributing mechanisms, is utilized to experimentally evaluate the relationship between the heat release rate and variations in the flame surface area. Results indicated that while area perturbations can adequately describe the magnitude of heat release fluctuations, the area perturbations are not a direct indicator of the phase of heat release needed for closed-loop stability analysis.;Time-resolved particle image velocimetry was used to quantify the near-field acoustics and the dilatation rate field in the pre- and post-flame regions of the flow. Measurements indicated that multi-dimensional acoustics dominate the pre-combustion flow field with radial and axial acoustic velocities of similar magnitudes. Variations in the flame structure potentially due to alternating regions of positive and negative flame stretch were also observed and may result in variations in the flame speed. As it is common to assume constant flame speed and one-dimensional acoustics, the experimental identification of these altered mechanisms may help to resolve discrepancies compared to a number of published reduced-order models

Physics-Based Statistical Learning in Thermoacoustics

Thermoacoustic oscillations arise because of the interaction between acoustic waves inside a duct or a combustion chamber, and heat release rate oscillations at the flame or heater location. When certain conditions are met, these oscillations may grow significantly in time and cause severe problems, particularly in gas turbines used for propulsion, i.e. in systems characterized by high powers. Thermoacoustic oscillations are extremely sensitive to small changes in the system geometry, parameters, and boundary conditions. For this reason, it is challenging to build quantitatively-accurate models that are general. In this thesis, we propose to generate physics-based qualitatively-accurate reduced-order models, which are general, and then tune their parameters so that they become quantitatively accurate to describe the system under investigation. To do this, we use statistical learning techniques in combination with an experimental dataset consisting of O(10^6) datapoints. The dataset is obtained from more than 210 hours of automated experiments on an electrically-heated vertical Rijke tube. We use the ensemble Kalman filter to infer the parameters of a conjugate heat transfer model driven by natural convection. Then we use the Markov Chain Monte Carlo (MCMC) method to infer the parameters of a linear acoustic model that is driven by the thermoacoustic mechanism and damped by visco-thermal dissipation and by radiation from the ends of the tube. We perform experiments only on the fully-assembled system, rather than on its individual components. We learn model parameters sequentially by using posterior values and uncertainties from early experiments as prior values and uncertainties for later experiments. With access to parameter uncertainties available with the MCMC, we quantitatively compare the marginal likelihood of the data for four tuned heat release rate models, thus finding the best performing model. Because it is physics-based, we find that the best model is quantitatively accurate, with known error bounds, significantly beyond the range of the training set. This process successfully combines physics-based modelling with data-driven methods in order to turn a qualitatively-accurate model into a quantitatively-accurate model, which is a significant challenge in thermoacoustics.The project has received funding from the European Union's Horizon 2020 research and innovation programme under Grant Agreement No. 766264