Synchronization is a universal concept in nonlinear science but has received little attention in thermoacoustics. In this numerical study, we take a dynamical systems approach to investigating the influence of harmonic acoustic forcing on three different types of self-excited thermoacoustic oscillations: periodic, quasiperiodic and chaotic.

When the periodic system is forced, we find that (i) at low forcing amplitudes, it responds at both the forcing frequency and the natural (self-excited) frequency, as well as at their linear combinations, indicating quasiperiodicity; (ii) above a critical forcing amplitude, the system locks into the forcing; (iii) the bifurcations leading up to lock-in and the critical forcing amplitude required for lock-in depend on the proximity of the forcing frequency to the natural frequency; (iv) the response amplitude at lock-in may be larger or smaller than that of the unforced system and the system can exhibit hysteresis and the jump phenomenon owing to a cusp catastrophe; and (v) at forcing amplitudes above lock-in, the oscillations can become unstable and transition to chaos, or switch between different stable attractors depending on the forcing amplitude.

When the quasiperiodic system is forced at a frequency equal to one of the two characteristic frequencies of the torus attractor, we find that lock-in occurs via a saddle node bifurcation with frequency pulling. When the chaotic system is forced at a frequency close to the dominant frequency of its strange attractor, we find that it is possible to destroy chaos and establish stable periodic oscillations.

These results show that the open-loop application of harmonic acoustic forcing can be an effective strategy for controlling periodic or aperiodic thermoacoustic oscillations. In some cases, we find that such forcing can reduce the response amplitude by up to 90%, making it a viable way to weaken thermoacoustic oscillations.

This theoretical paper concerns the influence of the phase of the heat release response on thermoacoustic systems. We focus on one pair of degenerate azimuthal acoustic modes, with frequency omega_0. The same results apply for an axial acoustic mode. We show how the value ph_0 and the slope ?tau of the flame phase at the frequency omega_0 affects the boundary of stability, the frequency and amplitude of oscillation, and the phase phi_qp between heat release rate and acoustic pressure. This effect depends on phi_0 and on the nondimensional number tau_omega_0, which can be quickly calculated. We find for example that systems with large values of tau_omega_0 are more prone to oscillate, i.e. they are more likely to have larger growth rates, and that at very large values of tau_omega_0 the value phi_0 of the flame phase at omega_0 does not play a role in determining the system?s stability. Moreover for a fixed flame gain, a flame whose phase changes rapidly with frequency is more likely to excite an acoustic mode.

We propose ranges for typical values of nondimensional acoustic damping rates, frequency shifts and growth rates based on a literature review. We study the system in the nonlinear regime by applying the method of averaging and of multiple scales. We show how to account in the time domain for a varying frequency of oscillation as a function of amplitude, and validate these results with extensive numerical simulations for the parameters in the proposed ranges. We show that the frequency of oscillation omega_B and the flame phase phi_qp at the limit cycle match the respective values on the boundary of stability. We find good agreement between the model and thermoacoustic experiments, both in terms of the ratio omega_B/omega_0 and of the phase phi_qp, and provide an interpretation of the transition between different thermoacoustic states of an experiment. We discuss the effect of neglecting the component of heat release rate not in phase with the pressure p as assumed in previous studies. We show that this component should not be neglected when making a prediction of the system?s stability and amplitudes, but we present some evidence that it may be neglected when identifying a system that is unstable and is already oscillating.

The Flame Describing Function (FDF) is a useful and relatively cheap approximation of a flame's nonlinearity with respect to harmonic velocity fluctuations. When embedded into a linear acoustic network, it is able to predict the amplitude and stability of harmonic thermoacoustic oscillations through the harmonic balance procedure. However, situations exist in which these oscillations are not periodic, but their spectrum contains peaks at several incommensurate frequencies. If one assumes that two frequencies dominate the spectrum, these oscillations are quasiperiodic, and the FDF concept can be extended by forcing the flame with two amplitudes and two frequencies. The nonlinearity is then approximated by a Flame Double Input Describing Function (FDIDF), which is a more expensive object to calculate than the FDF, but contains more information about the nonlinear response.

In this study, we present the calculation of a non-static flame's FDIDF. We use a G-equation-based laminar conical flame. We embed the FDIDF into a thermoacoustic network and we predict the nature and amplitude of thermoacoustic oscillations through the harmonic balance method. A criterion for the stability of these oscillations is outlined. We compare our results with a classical FDF analysis and self-excited time domain simulations of the same system. We show how the FDIDF improves the stability prediction provided by the FDF. At a numerical cost roughly equivalent to that of two FDFs, the FDIDF is capable to predict the onset of Neimark-Sacker bifurcations and to identify the frequency of oscillations around unstable limit cycles. At a higher cost, it can also saturate in amplitude these oscillations and predict the amplitude and stability of quasiperiodic oscillations.

Hydrodynamically self-excited flames are often assumed to be insensitive to low-amplitude external forcing. To test this assumption, we apply acoustic forcing to a range of jet diffusion flames. These flames have regions of absolute instability at their base and this causes them to oscillate at discrete natural frequencies. We apply the forcing around these frequencies, at varying amplitudes, and measure the response leading up to lock-in. We then model the system as a forced van der Pol oscillator.

Our results show that, contrary to some expectations, a hydrodynamically self-excited flame oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, it responds at both frequencies as well as at several nearby frequencies, indicating quasiperiodicity. When forced at high amplitudes, it locks into the forcing. The critical forcing amplitude for lock-in increases both with the strength of the self-excited instability and with the deviation of the forcing frequency from the natural fre- quency. Qualitatively, these features are accurately predicted by the forced van der Pol oscillator. There are, nevertheless, two features that are not predicted, both concerning the asymmetries of lock-in. When forced below its natural frequency, the flame is more resistant to lock-in, and its oscillations at lock-in are stronger than those of the unforced flame. When forced above its natural frequency, the flame is less resistant to lock-in, and its oscillations at lock-in are weaker than those of the unforced flame. This last finding suggests that, for thermoacoustic systems, lock-in may not be as detrimental as it is thought to be.

The large-scale coherent motions in a realistic swirl fuel injector geometry are analysed by direct numerical simulations (DNS), proper orthogonal decomposition (POD), and linear global modes. The aim is to identify the origin of instability in this turbulent flow in a complex internal geometry.

The flow field in the nonlinear simulation is highly turbulent, but with a distinguishable coherent structure: the precessing vortex core (a spiraling mode). The most energetic POD mode pair is identified as the precessing vortex core. By analysing the FFT of the time coefficients of the POD modes, we conclude that the first four POD modes contain the coherent fluctuations. The remaining POD modes (incoherent fluctuations) are used to form a turbulent viscosity field, using the Newtonian eddy model.

The turbulence sets in from convective shear layer instabilities even before the nonlinear flow reaches the other end of the domain, indicating that equilibrium solutions of the Navier?Stokes are never observed. Linear global modes are computed around the mean flow from DNS, applying the turbulent viscosity extracted from POD modes. A slightly stable discrete m = 1 eigenmode is found, well separated from the continuous spectrum, in very good agreement with the POD mode shape and frequency. The structural sensitivity of the precessing vortex core is located upstream of the central recirculation zone, identifying it as a spiral vortex breakdown instability in the nozzle. Furthermore, the structural sensitivity indicates that the dominant instability mechanism is the Kelvin-Helmholtz instability at the inflection point forming near vortex breakdown. Adjoint modes are strong in the shear layer along the whole extent of the nozzle, showing that the optimal initial condition for the global mode is localized in the shear layer.

We analyse the qualitative influence of turbulent dissipation in the stability problem (eddy viscosity) on the eigenmodes by comparing them to eigenmodes computed without eddy viscosity. The results show that the eddy viscosity improves the complex frequency and shape of global modes around the fuel injector mean flow, while a qualitative wavemaker position can be obtained with or without turbulent dissipation, in agreement with previous studies.

This study shows how sensitivity analysis can identify which parts of the flow in a complex geometry need to be altered in order to change its hydrodynamic stability characteristics.

This paper explores the analogy between triggering in thermoacoustics and bypass transition to turbulence in hydrodynamics. These are both mechanisms through which a small perturbation causes a system to develop large self-sustained oscillations, despite the system being linearly stable. For example, it explains why round pipe flow (Hagen-Poiseuille flow) can become turbulent, even though all its eigenvalues are stable at all Reynolds numbers.

In hydrodynamics, bypass transition involves transient growth of the initial perturbation, which arises due to linear non-normality of the stability operator, followed by attraction towards a series of unstable periodic solutions of the Navier-Stokes equations, followed by repulsion either to full turbulence or re-laminarization. This paper shows that the triggering process in thermoacoustics is directly analogous to this. In thermoacoustics, the linearized stability operator is also non-normal and also gives rise to transient growth. The system then evolves towards an unstable periodic solution of the governing equations, followed by repulsion either to a stable periodic solution or to the zero solution. The paper demonstrates that initial perturbations that have higher amplitudes at low frequencies are more effective at triggering self-sustained oscillations than perturbations that have similar amplitudes at all frequencies.

This paper then explores the effect that different types of noise have on triggering. Three types of noise are considered: pink noise (higher amplitudes at low frequencies), white noise (similar amplitudes at all frequencies) and blue noise (higher amplitudes at high frequencies). Different amplitudes of noise are applied, both as short bursts and continuously. Pink noise is found to be more effective at causing triggering than white noise and blue noise, in line with the results found in the first part of the paper.

In summary, this paper investigates the triggering mechanism in thermoacoustics and demonstrates that some types of noise cause triggering more effectively than others.

In this experimental and numerical study, two types of round jet are examined under acoustic forcing. The first is a non-reacting low density jet (density ratio 0.14). The second is a buoyant jet diffusion flame at a Reynolds number of 1100 (density ratio of unburnt fluids 0.5). Both jets have regions of strong absolute instability at their base and this causes them to exhibit strong self-excited bulging oscillations at well-defined natural frequencies. This study particularly focuses on the heat release of the jet diffusion flame, which oscillates at the same natural frequency as the bulging mode, due to the absolutely unstable shear layer just outside the flame.

The jets are forced at several amplitudes around their natural frequencies. In the non-reacting jet, the frequency of the bulging oscillation locks into the forcing frequency relatively easily. In the jet diffusion flame, however, very large forcing amplitudes are required to make the heat release lock into the forcing frequency. Even at these high forcing amplitudes, the natural mode takes over again from the forced mode in the downstream region of the flow, where the perturbation is beginning to saturate non-linearly and where the heat release is high. This raises the possibility that, in a flame with large regions of absolute instability, the strong natural mode could saturate before the forced mode, weakening the coupling between heat release and incident pressure perturbations, hence weakening the feedback loop that causes combustion instability.

Hydrodynamic oscillations in gas turbine fuel injectors help to mix the fuel and air but can also contribute to thermoacoustic instability. Small changes to some parts of a fuel injector greatly affect the frequency and amplitude of these oscillations. These regions can be identified efficiently with adjoint-based sensitivity analysis. This is a linear technique that identifies the region of the flow that causes the oscillation, the regions of the flow that are most sensitive to external forcing, and the regions of the flow that, when altered, have most influence on the oscillation. In this paper, we extend this to the flow from a gas turbine?s single stream radial swirler, which has been extensively studied experimentally (GT2008-50278).

The swirling annular flow enters the combustion chamber and expands to the chamber walls, forming a conical recirculation zone along the centreline and an annular recirculation zone in the upstream corner. In this study, the steady base flow and the stability analysis are calculated at Re 200-3800 based on the mean flow velocity and inlet diameter. The velocity field is similar to that found from experiments and LES, and the local stability results are close to those at higher Re (GT2012-68253).

All the analyses (experiments, LES, uRANS, local stability, and the global stability in this paper) show that a helical motion develops around the central recirculation zone. This develops into a precessing vortex core. The adjoint-based sensitivity analysis reveals that the frequency and growth rate of the oscillation is dictated by conditions just upstream of the central recirculation zone (the wavemaker region). It also reveals that this oscillation is very receptive to forcing at the sharp edges of the injector. In practical situations, this forcing could arise from an impinging acoustic wave, showing that these edges could be influential in the feedback mechanism that causes thermoacoustic instability.

The analysis also shows how the growth rate and frequency of the oscillation change with either small shape changes of the nozzle, or additional suction or blowing at the walls of the injector. It reveals that the oscillations originate in a very localized region at the entry to the combustion chamber, which lies near the separation point at the outer inlet, and extends to the outlet of the inner pipe. Any scheme designed to control the frequency and amplitude of the oscillation only needs to change the flow in this localized region.

Linear techniques can predict whether the non-oscillating (steady) state of a thermoacoustic system is stable or unstable. With a sufficiently large impulse, however, a thermoacoustic system can reach a stable oscillating state even when the steady state is also stable. A nonlinear analysis is required to predict the existence of this oscillating state. Continuation methods are often used for this but they are computationally expensive.

In this paper, an acoustic network code called LOTAN is used to obtain the steady and the oscillating solutions for a horizontal Rijke tube. The heat release is modelled as a nonlinear function of the mass flow rate. Several test cases from the literature are analysed in order to investigate the effect of various nonlinear terms in the flame model. The results agree well with the literature, showing that LOTAN can be used to map the steady and oscillating solutions as a function of the control parameters. Furthermore, the nature of the bifurcation between steady and oscillating states can be predicted directly from the nonlinear terms inside the flame model.

Hydrodynamic instabilities in gas turbine fuel injectors help to mix the fuel and air but can sometimes lock into acoustic oscillations and contribute to thermoacoustic instability. This paper describes a linear stability analysis that predicts the frequencies and strengths of hydrodynamic instabilities and identifies the regions of the flow that cause them. It distinguishes between convective instabilities, which grow in time but are convected away by the flow, and absolute instabilities, which grow in time without being convected away. Convectively unstable flows amplify external perturbations, while absolutely unstable flows also oscillate at intrinsic frequencies. As an input, this analysis requires velocity and density fields, either from a steady but unstable solution to the Navier--Stokes equations, or from time-averaged numerical simulations. In the former case, the analysis is a predictive tool. In the latter case, it is a diagnostic tool. This technique is applied to three flows: a swirling wake at Re = 400, a single stream swirling fuel injector at Re ~ 10^6, and a lean premixed gas turbine injector with five swirling streams at Re ~ 10^6.

Its application to the swirling wake demonstrates that this technique can correctly predict the frequency, growth rate and dominant wavemaker region of the flow. It also shows that the zone of absolute instability found from the spatio-temporal analysis is a good approximation to the wavemaker region, which is found by overlapping the direct and adjoint global modes. This approximation is used in the other two flows because it is difficult to calculate their adjoint global modes.

Its application to the single stream fuel injector demonstrates that it can identify the regions of the flow that are responsible for generating the hydrodynamic oscillations seen in LES and experimental data. The frequencies predicted by this technique are within a few percent of the measured frequencies. The technique also explains why these oscillations become weaker when a central jet is injected along the centreline. This is because the absolutely unstable region that causes the oscillations becomes convectively unstable.

Its application to the lean premixed gas turbine injector reveals that several regions of the flow are hydrodynamically unstable, each with a different frequency and a different strength. For example, it reveals that the central region of confined swirling flow is strongly absolutely unstable and sets up a precessing vortex core, which is likely to aid mixing throughout the injector. It also reveals that the region between the second and third streams is slightly absolutely unstable at a frequency that is likely to coincide with acoustic modes within the combustion chamber. This technique, coupled with knowledge of the acoustic modes in a combustion chamber, is likely to be a useful design tool for the passive control of mixing and combustion instability.

Thermoacoustics is a branch of fluid mechanics, and is as such governed by the conservation laws of mass, momentum, energy and species. While computational fluid dynamics (CFD) has entered the design process of many applications in fluid mechanics, its success in thermoacoustics is limited by the multi-scale, multi-physics nature of the subject. In his influential monograph from 2006, Prof. Fred Culick writes about the role of CFD in thermoacoustic modeling:

"The main reason that CFD has otherwise been relatively helpless in this subject is that problems of combustion instabilities involve physical and chemical matters that are still not well understood. Moreover, they exist in practical circumstances which are not readily approximated by models suitable to formulation within CFD. Hence, the methods discussed and developed in this book will likely be useful for a long time to come, in both research and practice. [...] It seems to me that eventually the most effective ways of formulating predictions and theoretical interpretations of combustion instabilities in practice will rest on combining methods of the sort discussed in this book with computational fluid dynamics, the whole confirmed by experimental results." (Culick, Fred: Unsteady Motions in Combustion Chambers for Propulsion Systems. NATO Research and Technology Organisation, 2006)

Despite advances in CFD and large-eddy simulation (LES) in particular, unsteady simulations for more than a few selected operating points are computationally infeasible. The 'methods discussed in this book' refer to reduced-order models of thermoacoustic oscillations. Whether intentional or not, the last sentence anticipates the advent of data-driven methods, and encapsulates the philosophy behind this work.

This work brings together two workhorses of the design process: physics-informed reduced-order models and data from higher-fidelity sources such as simulations and experiments. The three building blocks to all our statistical inference frameworks are: (i) a hierarchical view of reduced-order models consisting of states, parameters and governing equations; (ii) probabilistic formulations with random variables and stochastic processes; and (iii) efficient algorithms from statistical learning theory and machine learning. While leveraging advances in statistical and machine learning, we demonstrate the feasibility of Bayes? rule as a first principle in physics-informed statistical inference. In particular, we discuss two types of inverse problems in thermoacoustics: (i) implicit reduced-order models representative of nonlinear eigenproblems from linear stability analysis; and (ii) time-dependent reduced-order models used to investigate nonlinear dynamics. The outcomes of statistical inference are improved predictions of the state, estimates of the parameters with uncertainty quantification and an assessment of the reduced-order model itself.

This work highlights the role that data can play in the future of combustion modeling for thermoacoustics. It is increasingly impractical to store data, particularly as experiments become automated and numerical simulations become more detailed. Rather than store the data itself, the techniques in this work optimally assimilate the data into the parameters of a physics-informed reduced-order model. With data-driven reduced-order models, rapid prototyping of combustion systems can feed into rapid calibration of their reduced-order models and then into gradient-based design optimization. While it has been shown, e.g. in the context of ignition and extinction, that large-eddy simulations become quantitatively predictive when augmented with data, the reduced-order modeling of flame dynamics in turbulent flows remains challenging. For these challenging situations, this work opens up new possibilities for the development of reduced-order models that adaptively change any time that data from experiments or simulations becomes available.

The reader in mind is a scientist or engineer with an interest in data-driven methods. For readers mostly interested in the results, we provide references to our ideally more self-contained publications where available. For the more methodological chapters, we provide JUPYTER notebooks so that inclined readers are able to familiarize themselves with the statistical and numerical concepts of this work. They are either available on GITLAB2 for download or as a BINDER3 executed within the browser. More information on JUPYTER notebooks are found online.

Thesis embargoed until 2025

In many engineering applications it is desirable to have a steady flow field that is stable to perturbations. This can be described through a global stability analysis. This is an eigenvalue problem which gives a series of mode shapes (the eigenfunctions) and their growth rates and frequencies (the eigenvalues).

This thesis calculates and interprets the shape sensitivity of the eigenvalue of a global stability analysis. The shape sensitivity allows quick calculation of the change in a mode?s growth rate and frequency when the flow geometry is changed. The shape sensitivity is calculated using an adjoint method. This is derived both for flows governed by the incompressible Navier?Stokes equations and also for flows governed by the URANS equations using the Spalart?Allmaras and k-omega turbulence models. Using the laminar and turbulent flow past a cylinder as a model problem, the shape sensitivities of the laminar and turbulent vortex shedding modes are presented. For both cases, control of the eigenvalue by shape deformation is shown to occur primarily by changing the steady base-flow and not by changing the mode?s unsteady feedback. A technique for finding the deformations with the greatest influence on the base-flow is then demonstrated. For the laminar flow, this shows that the shape sensitivity of the growth rate and frequency is primarily due to a single deformation that causes large widespread base-flow changes. This deformation increases the growth rate but decreases the frequency, leading the shape sensitivities to have similar shapes but opposite signs.

The thesis then applies this framework to the cyclone separator. This is an industrial application which has been shown to have a helical instability that reduces the cyclone's performance. Helical and double-helical modes with similar Strouhal numbers to those seen in experiment are found. As with the cylinder, control of the eigenvalue by shape deformation is shown to occur primarily by changing the steady base-flow and not by changing the mode?s unsteady feedback. The shape sensitivities are shown to be concentrated at the cyclone's cone tip and vortex finder. Small changes of these parts of the geometry are shown to cause large widespread changes to the base-flow. Finally, the shape sensitivities are used in a gradient based method to show that small changes in the cyclone geometry can significantly reduce the growth rate of the unstable modes. This is done in conjunction with a separation metric to show that the stability of the flow can be improved without reducing separation performance.

The techniques demonstrated in this thesis for finding the shape sensitivity of the eigenvalue and for identifying influential deformations can be easily applied to a wide range of different flow geometries, governing equations and cost functions. This highlights the benefits of the use of adjoint-based methods in engineering design.

Lean combustion technologies in gas turbines reduce the generation of NOx but increase the susceptibility to thermoacoustic oscillations. These oscillations can produce structural damage and need to be eliminated. The stability of a given configuration can be examined with a thermoacoustic model. In this thesis a wave-based network model is used. Using adjoint methods the gradients of the eigenvalue with respect to system parameters can be obtained at a low computational cost. This information is used as an input to an optimization routine to find stable thermoacoustic configurations.

In this thesis thermocaoustic oscillations are analysed using a linear low order network model. This modelling approach is used to predict the unstable modes of five different configurations: a Rijke tube, a choked combustor, a longitudinal combustor, a generic lean premix prevaporized annular combustor and the laboratory scale annular combustor built in Cambridge University Engineering Department. The continuous and discrete adjoint equations for the low order network model are derived. Using the adjoint equations the sensitivities of the eigenvalues to changes in base state parameters such as time delays, areas, lengths and mean radii are computed. Similarly, the sensitivity of the eigenvalues to the introduction of a feedback device such as a drag mesh or a secondary heat source is investigated. By fitting experimental data to a low Mach number model of the Rijke tube, the predictions of the growth rate and frequency shifts due to the presence of these mechanisms are improved. Finally, using the sensitivity information, two different optimization algorithms are developed to stabilize the thermoacoustic systems. Different stabilization scenarios are presented, showing the changes required in each section of the configurations to eliminate thermoacoustic oscillations.

The techniques presented as part of this thesis are readily scalable to more complex models and geometries and the inclusion of further constraints. This demonstrates that adjoint-based sensitivity analysis and optimization could become an indispensable tool for the design of thermoacoustically-stable combustors.

Thermoacoustic instability is one of the most significant problems faced in the design of some combustion systems. Thermoacoustic oscillations arise due to feedback between acoustic waves and unsteady heat release rate when the fluctuating heat release rate is sufficiently in phase with the unsteady pressure. The primary aim of designers is to design linearly stable thermoacoustic systems in which these dangerous oscillations do not arise. In thermoacoustics, adjoint-based sensitivity analysis has shown promise at predicting the parameters which have the most influence on the linear growth and decay rates as well as oscillation frequency observed during periods of linear growth and decay. Therefore, adjoint-based methods could prove to be a valuable tool for developing optimal passive control solutions. This thesis aims to develop novel experimental sensitivity analysis techniques and provide a first comparison with the predictions of adjoint-based sensitivity analysis. In this thesis experimental sensitivity analysis is performed on (i) a vertical electrically-driven Rijke tube, and (ii) a vertical flame-driven Rijke tube. On the electrically-driven Rijke tube, the feedback sensitivity is studied by investigating the shift in linear growth and decay rates and oscillation frequency observed during periods of linear growth and decay due to the introduction of a variety of passive control devices. On the flame-driven Rijke tube, the base-state sensitivity is studied by investigating how the linear growth and decay rates as well as oscillation frequency during periods of linear growth and decay change as the convective time delay of the flame is modified. Adjoint-based sensitivity analysis gives the shift in linear growth and decay rate and the oscillation frequency when parameters are changed. This thesis provides experimental measurements of the same quantities, for comparison with the numerical sensitivity analysis, opening up new avenues for the development, implementation and validation of optimal passive control strategies for more complex thermoacoustic systems.

Thermoacoustic oscillations may arise in combustion chambers when unsteady heat release and acoustic fluctuations constructively interfere. These oscillations generally lead to undesired consequences, and need to be avoided. Linear stability analysis can be used to investigate the linear stability of a thermoacoustic system, by calculating the frequencies and growth rates of thermoacoustic modes. Adjoint methods can then be used to understand what parameters in the configuration under investigation have to be changed to make it less susceptible to thermoacoustic oscillations. Linear stability is, however, not sufficient in general to ensure safe operability conditions. This is because nonlinear and non-normal effects may trigger finite amplitude oscillations when the system is subject to finite amplitude perturbations. A thorough fully nonlinear investigation of thermoacoustic systems is prohibitively expensive both experimentally and numerically, and one needs to approximate the nonlinear response of the system.

In this thesis, low-order nonlinear models for the prediction of the nonlinear behaviour of thermoacoustic systems are developed. These models are based on thermoacoustic networks, in which linear acoustics is combined with a nonlinear heat release model. The acoustic networks considered in this thesis can take into account mean flow and non-trivial acoustic reflection coefficients, and are cast in state-space form to enable analysis both in the frequency and time domains.

Starting from linear analysis, the stability of thermoacoustic networks is investigated, and the use of adjoint methods for understanding the role of the system's parameters on its stability is demonstrated. Then, a fully nonlinear analysis using various state-of-the-art methods is performed, to highlight the strengths and weaknesses of each method. Two novel frameworks that fill some gaps in the available methods are developed: the first, called Flame Double Input Describing Function (FDIDF), is an extension of the Flame Describing Function (FDF). The FDIDF approximates the flame nonlinear response when it is forced simultaneously with two frequencies, whereas the FDF is limited to one frequency. Although more expensive to obtain, the FDIDF contains more nonlinear information than the FDF, and can predict periodic and quasiperiodic oscillations. It is shown how, in some cases, it corrects the prediction of the FDF about the stability of thermoacoustic oscillations. The ?second framework developed is a general weakly nonlinear formulation of the thermoacoustic equations in the Rijke tube, in which the acoustic response is not limited to a single-Galerkin mode, and is embedded in a state-space model. The weakly nonlinear analysis is strictly valid only close to the expansion point, but is much cheaper than any other available method.

The above methods are applied to relatively simple thermoacoustic configurations, in which the nonlinear heat release model is that of a laminar conical flame or an electrical heater. However, in real gas turbines more complex flame shapes are found, for which no reliable low-order models exist. Two models are developed in this thesis for turbulent bluff-body stabilised flames: one for a perfectly premixed flame, in which the modelling is focused on the flame-flow interaction, accounting for the presence of recirculation zones and temperature gradients; the second for imperfectly premixed flames, in which equivalence ratio fluctuations, modelled as a passive scalar field, dominate the heat release response. The second model was shown to agree reasonably well with experimental data, and was applied in an industrial modelling project. When embedded in a thermoacoustic network, it is capable of predicting the value of the frequency at which thermoacoustic oscillations are prone to grow.

Thermoacoustic oscillations in annular combustors are often of azimuthal type, with two distinct thermoacoustic modes sharing the same frequency of oscillation. The two modes interact because each flame responds to the sum of the two modes and acts as a source term for both modes. In the nonlinear regime the system converge to a limit-cycle solution, which is an acoustic wave that is either spinning around the annulus, or a standing wave with pressure and velocity nodes fixed in space. This thesis answers some questions regarding these two types of solutions, and provides tools to analyse azimuthal modes.

A flame in the annular combustion chamber is subject to an axial acoustic field through the burner, and a transverse acoustic field sweeping it sideways. We show that the effect of this transverse acoustic field on the flame response can make the system prefer standing solutions instead of spinning solutions.

We present a tool to map a flame response from the frequency domain, where it is often described, to the time domain, where it is needed to discuss azimuthal instabilities.

We then carry out a weakly nonlinear analysis of the system taking into account the number of equispaced identical burners, the level of linear acoustic damping, the geometry and the nonlinear flame response to axial forcing. This leads to a low-order model of azimuthal instabilities that is ready to be used for the purpose of system identification. We provide conditions for the existence and stability of standing and spinning waves and the orientation and amplitudes of these solutions, and then discuss their physical interpretation.

We finally apply two mathematical methods, the method of averaging and the method of multiple scales, to predict the solutions of the system. This allows a validation of the methods, of which the first is used extensively in the rest of the manuscript, and a study of the effect of the delay between acoustic forcing and flame response, both in the linear and nonlinear regime.

Coherent structures in turbulent flows provide a means of understanding turbulence in terms of large organised motions. Understanding the mechanism of formation of coherent structures can be helpful in suppressing or enhancing the turbulence in a flow by means of active or passive control devices. Knowledge of the Reynolds number scaling of the size and energy content of coherent structures can extend the knowledge to high Reynolds number flows, which are out of reach of the present computational and experimental facilities.

In this thesis, linear amplification and eigenvalue stability analyses are performed by linearising the Navier?-Stokes or Reynolds-averaged Navier-?Stokes (RANS) equations over the mean flow profiles in several wall-bounded turbulent shear flows. It is investigated whether the linear optimal modes or the leading eigenmodes approximate the coherent structures in fully nonlinear turbulent flows. This is done by comparing various kinematic properties of the optimal modes, such as the shape and energy spectra, with those of the observed coherent structures in turbulent channel and pipe flows in the first half of the thesis. The use of the linearised Navier-?Stokes equations in the regions of high mean shear in the flows is justified based on rapid distortion theory. In the linearised RANS equations-based analysis, turbulence models are used to account for the effect of wave-induced perturbations in the Reynolds stress on the behaviour of small external wave motions. The turbulence models used in this thesis are the eddy viscosity model (EVM) and the explicit algebraic Reynolds stress model (EARSM). The focus of this thesis is to investigate whether this effect of wave-induced perturbations in the Reynolds stress needs to be included in stability analysis of wall-bounded turbulent flows.

The linear amplification analysis based on the Navier?Stokes equations finds three main types of structures in turbulent channel flows. The first type are the small streamwise wavelength (lambda x + = 200 ? 800) structures, which are found to scale in inner units and have preferred spanwise wavelength equal to around one hundred wall-units. These properties match well with those of observed near-wall structures. The second type are the intermediate streamwise wavelengths (from lambda x + > 800 to lambda x < 3) structures which correspond to hairpin vortical and large-scale streaky like structures. The peak in energy amplification in this wavelength range found from the analysis matches well with that from DNS. Various kinematic properties, such as the inclination angle of streaks with the wall, also match with those of large-scale-motions (LSMs) observed in experiments. The third type are the large streamwise wavelength (lambda x >= 6) structures. The preferred spanwise wavelength of these structures (lambda z peak ~ 2), their scaling in outer units, and the fact that they extend to the wall match with the observed features of very-large-scale-motions (VLSMs). All these results show that the most optimal modes obtained from the linearised Navier?Stokes equations, without any turbulence model or eddy viscosity, share many important features with those of observed coherent structures in turbulent channel flows.

In comparison, the results from the EVM- and EARSM-based linear amplification analyses find only two types of coherent structures. One type are of the small wavelengths, which correspond to the near-wall structures, and the other type are of the large wavelengths, which correspond to the VLSMs. These analyses, however, find minima in energy spectra in the intermediate wavelength region, where DNS and the Navier?Stokes equations-based analysis find maxima in energy spectra.

In axially rotating turbulent pipe flows, it is found from the linearised Navier?-Stokes equations-based analysis that rotation causes the widening of streaks and prevents the formation of quasi-streamwise vortices. These results match well with observations from DNS, which further shows the usefulness of the linearised Navier?Stokes equations.

In the second part of the thesis, stability analyses based on the linearised Navier-?Stokes and RANS equations are applied in more complex flows. Based on the results from the stability analyses for flows in gas-turbine systems, it is found that for such flows the inclusion of turbulence models in stability analysis has no significant qualitative effect on the results. This is because these instabilities are driven by regions of high mean shear for which analysis based on the linearised Navier?-Stokes equations is sufficient. It is also found from stability analysis that an expansion at the nozzle exit and swirl in the flow are destabilising, and therefore increase hydrodynamic instability.

Based on the preliminary comparisons of stability results and observations from DNS in Taylor-Couette flows, it is again concluded that the linearised Navier?-Stokes equations-based analysis is better at capturing intermittent coherent structures as compared to the linearised RANS equations-based analysis.

It is concluded in this thesis that the linearised Navier?-Stokes equations-based analysis, which does not require any turbulence model, can be used to find information about coherent structures in high mean shear flows, such as the flows in gas-turbine fuel injectors or wall-bounded turbulent flows.

Finding limit cycles and their stability is one of the central problems of nonlinear thermoacoustics. However, a limit cycle is not the only type of self-excited oscillation in a nonlinear system. Nonlinear systems can have quasi-periodic and chaotic oscillations. This thesis examines the different types of oscillation in a numerical model of a ducted premixed flame, the bifurcations that lead to these oscillations and the influence of external forcing on these oscillations.

Criteria for the existence and stability of limit cycles in single mode thermoacoustic systems are derived analytically. These criteria, along with the flame describing function, are used to find the types of bifurcation and minimum triggering amplitudes. The role that the gain and the phase of the flame describing function play in determining the growth or decay of perturbations is identified. The choice of model for the velocity perturbation field around the flame is shown to have a strong influence on the types of bifurcation in the system. Therefore, a reduced order model of the velocity perturbation field in a forced laminar premixed flame is obtained from Direct Numerical Simulation. This model has a perturbation convection speed that is frequency-dependent and different from the mean flow, unlike the model commonly used in the literature. Limit cycles and bifurcations are found with both these models and it is shown that the model currently used in the literature precludes subcritical bifurcations and multi-stability in single mode thermoacoustic systems.

The self-excited thermoacoustic system is simulated in the time domain with several modes in the acoustics and analysed using methods from nonlinear dynamical systems theory. The transitions to the periodic, quasiperiodic and chaotic oscillations are via sub/supercritical Hopf, Neimark-Sacker and period-doubling bifurcations. The trajectory of the system involves transient attraction and repulsion by one or more unstable attractors before the system reaches a stable state. Two routes to chaos are established in this system: the period-doubling route and the Ruelle-Takens-Newhouse route.

It is shown that the single mode system, which gives the same results as a describing function approach, fails to capture the period-2, period-k, quasi-periodic and chaotic oscillations or the bifurcations and multi-stability seen in the multi-modal case, and underpredicts the amplitude of period-1 oscillations.

Instantaneous flame images reveal that the wrinkles on the flame surface and pinch off of flame pockets are regular for periodic oscillations, while they are irregular and have multiple time and length scales for quasi-periodic and chaotic oscillations. Cusp formation, their destruction by flame propagation normal to itself, and pinch-off and rapid burning of pockets of reactants are shown to be responsible for generating a heat release rate that is a highly nonlinear function of the velocity perturbations. It is also shown that for a given acoustic model of the duct, several modes are required to capture the influence of this highly nonlinear unsteady heat release rate on the acoustics and the interactions between the acoustic modes via the unsteady heat release rate. Both of these are required to simulate the rich dynamics seen in experiments.

The infuence of external harmonic forcing, at different frequencies and amplitudes, on self-excited periodic, quasi-periodic and chaotic oscillations are examined. The bifurcations that lead to lock-in are either saddle-node or inverse Neimark-Sacker bifurcations. The transition to lock-in, the forcing amplitude required for lock-in and the system response at lock-in depend on the proximity of the forcing frequency to the natural frequency and whether the forcing frequency is above or below the natural frequency. At certain frequencies, even low-amplitude forcing is sufficient to suppress period-1 oscillations to amplitudes that are 90% lower than that of the unforced state. Therefore, open-loop forcing can be an effective strategy for the suppression of thermoacoustic oscillations.

This thesis shows that a ducted premixed flame behaves similarly to low-dimensional chaotic systems and that methods from nonlinear dynamical systems theory are superior to the describing function approach in the frequency domain and time domain analysis currently used in nonlinear thermoacoustics.

Large-scale unsteady flow structures play an influential role in the dynamics of many practical flows, such as those found in gas turbine combustion chambers. This thesis is concerned primarily with large-scale unsteady structures that arise due to self-sustained hydrodynamic oscillations, also known as global hydrodynamic instability. Direct numerical simulation (DNS) of the Navier?-Stokes equations in the low Mach number limit is used to obtain a steady base flow, and the most unstable direct and adjoint global modes. These are combined, using a structural sensitivity framework, to identify the region of the flow and the feedback mechanisms that are responsible for causing the global instability. Using a Lagrangian framework, the direct and adjoint global modes are also used to identify the regions of the flow where steady and unsteady control, such as a drag force or heat input, can suppress or promote the global instability.

These tools are used to study a variety of reacting and non-reacting flows to build an understanding of the physical mechanisms that are responsible for global hydrodynamic instability in swirling diffusion flames. In a non-swirling lifted jet diffusion flame, two modes of global instability are found. The first mode is a high-frequency mode caused by the instability of the low-density jet shear layer in the premixing zone. The second mode is a low-frequency mode caused by an instability of the outer shear layer of the flame. Two types of swirling diffusion flames with vortex breakdown bubbles are considered. They show qualitatively similar behaviour to the lifted jet diffusion flames. The first type of flame is unstable to a low-frequency mode, with wavemaker located at the flame base. The second type of flame is unstable to a high-frequency mode, with wavemaker located at the upstream edge of the vortex breakdown bubble. Feedback from density perturbations is found to have a strong influence on the unstable modes in the reacting flows. The wavemaker of the high-frequency mode in the reacting flows is very similar to its non-reacting counterpart. The low-frequency mode, however, is only observed in the reacting flows. The presence of reaction increases the influence of changes in the base flow mixture fraction profiles on the eigenmode. This increased influence acts through the heat release term.

These results emphasize the possibility that non-reacting simulations and experiments may not always capture the important instability mechanisms of reacting flows, and highlight the importance of including heat release terms in stability analyses of reacting flows.

This thesis examines the nonlinear behaviour of thermoacoustic systems by using approaches from the field of nonlinear dynamics. The behaviour of a nonlinear system is determined by two things, which are the focus of this thesis: first, by the mechanism that the system transitions from one attractor to another, and second, by the type and form of the attractors in phase space.

In the first part of the thesis, a triggering mechanism is presented for a Rijke tube model, whereby the system transitions from a stable fixed point to a stable limit cycle, via an unstable limit cycle. Using this mechanism, low levels of stochastic noise result in triggering much before the linear stability limit. Stochastic stability maps are introduced to visualise the practical stability of a thermoacoustic system. These theoretical results match well with those from experiments.

In the second part of the thesis, two time domain methods are presented for finding limit cycles in large thermoacoustic systems: matrix-free continuation methods and gradient methods.

Most continuation methods are too computationally expensive for finding limit cycles in large thermoacoustic systems. For dissipative systems, matrix-free continuation methods are shown to converge quickly to limit cycles by preferentially using the influential bulk motions of the system, whilst ignoring the features that are quickly dissipated in time. These matrix-free methods are demonstrated on a model of a ducted 2D diffusion flame and a model of a ducted axisymmetric premixed flame (with G-equation solver). Rich nonlinear behaviour is found: fixed points, sub/supercritical Hopf bifurcations, limit cycles, period-2 limit cycles, fold bifurcations, period-doubling bifurcations and Neimark-Sacker bifurcations. Physical information about the flame-acoustic interaction is found from the limit cycles and Floquet modes. Invariant subspace preconditioning, higher order prediction techniques, and multiple shooting techniques are all shown to reduce the time required to generate bifurcation surfaces.

Gradient methods define a scalar cost function that describes the proximity of a state to a limit cycle. The gradient of the cost function is calculated using adjoint equations and then used to iteratively converge to a limit cycle (or fixed point). The gradient method is demonstrated on a model of a horizontal Rijke tube. This thesis describes novel nonlinear analysis techniques that can be applied to coupled systems with both advanced acoustic models and advanced flame models. The techniques can characterise the rich nonlinear behaviour of thermoacoustic models with a level of detail that was not previously possible.

In the analysis of thermoacoustic systems, a flame is usually characterised by the way its heat release responds to acoustic forcing. This response depends on the hydrodynamic stability of the flame. Some flames, such as a premixed bunsen flame, are hydrodynamically globally stable. They respond only at the forcing frequency. Other flames, such as a jet diffusion flame, are hydrodynamically globally unstable. They oscillate at their own natural frequencies and are often assumed to be insensitive to low-amplitude forcing at other frequencies.

If a hydrodynamically globally unstable flame really is insensitive to forcing at other frequencies, then it should be possible to weaken thermoacoustic oscillations by detuning the frequency of the natural hydrodynamic mode from that of the natural acoustic modes. This would be very beneficial for industrial combustors.

In this thesis, that assumption of insensitivity to forcing is tested experimentally. This is done by acoustically forcing two different self-excited flows: a non-reacting jet and a reacting jet. Both jets have regions of absolute instability at their base and this causes them to exhibit varicose oscillations at discrete natural frequencies. The forcing is applied around these frequencies, at varying amplitudes, and the response examined over a range of frequencies (not just at the forcing frequency). The overall system is then modelled as a forced van der Pol oscillator.

The results show that, contrary to some expectations, a hydrodynamically self-excited jet oscillating at one frequency is sensitive to forcing at other frequencies. When forced at low amplitudes, the jet responds at both frequencies as well as at several nearby frequencies, and there is beating, indicating quasiperiodicity. When forced at high amplitudes, however, it locks into the forcing. The critical forcing amplitude required for lock-in increases with the deviation of the forcing frequency from the natural frequency. This increase is linear, indicating a Hopf bifurcation to a global mode.

The lock-in curve has a characteristic V-shape, but with two subtle asymmetries about the natural frequency. The first asymmetry concerns the forcing amplitude required for lock-in. In the non-reacting jet, higher amplitudes are required when the forcing frequency is above the natural frequency. In the reacting jet, lower amplitudes are required when the forcing frequency is above the natural frequency. The second asymmetry concerns the broadband response at lock-in. In the non-reacting jet, this response is always weaker than the unforced response, regardless of whether the forcing frequency is above or below the natural frequency. In the reacting jet, that response is weaker than the unforced response when the forcing frequency is above the natural frequency, but is stronger than it when the forcing frequency is below the natural frequency.

In the reacting jet, weakening the global instability - by adding coflow or by diluting the fuel mixture - causes the flame to lock in at lower forcing amplitudes. This finding, however, cannot be detected in the flame describing function. That is because the flame describing function captures the response at only the forcing frequency and ignores all other frequencies, most notably those arising from the natural mode and from its interactions with the forcing. Nevertheless, the flame describing function does show a rise in gain below the natural frequency and a drop above it, consistent with the broadband response. Many of these features can be predicted by the forced van der Pol oscillator. They include (i) the coexistence of the natural and forcing frequencies before lock-in; (ii) the presence of multiple spectral peaks around these competing frequencies, indicating quasiperiodicity; (iii) the occurrence of lock-in above a critical forcing amplitude; (iv) the V-shaped lock-in curve; and (v) the reduced broadband response at lock-in. There are, however, some features that cannot be predicted. They include (i) the asymmetry of the forcing amplitude required for lock-in, found in both jets; (ii) the asymmetry of the response at lock-in, found in the reacting jet; and (iii) the interactions between the fundamental and harmonics of both the natural and forcing frequencies, found in both jets.

This work represents the initial steps in a wider project that aims to map out the sensitive areas in fuel injectors and combustion chambers. Direct numerical simulation (DNS) using a Low-Mach-number formulation of the Navier?-Stokes equations is used to calculate direct-linear and adjoint global modes for axisymmetric low-density jets and lifted jet diffusion flames. The adjoint global modes provide a map of the most sensitive locations to open-loop external forcing and heating. For the jet flows considered here, the most sensitive region is at the inlet of the domain.

The sensitivity of the global-mode eigenvalues to force feedback and to heat and drag from a hot-wire is found using a general structural sensitivity framework. Force feedback can occur from a sensor-actuator in the flow or as a mechanism that drives global instability. For the lifted flames, the most sensitive areas lie between the inlet and flame base. In this region the jet is absolutely unstable, but the close proximity of the flame suppresses the global instability seen in the non-reacting case. The lifted flame is therefore particularly sensitive to outside disturbances in the non-reacting zone.

The DNS results are compared to a local analysis. The most absolutely unstable region for all the flows considered is at the inlet, with the wavemaker slightly downstream of the inlet. For lifted flames, the region of largest sensitivity to force feedback is near to the location of the wavemaker, but for the non-reacting jet this region is downstream of the wavemaker and outside of the pocket of absolute instability near the inlet.

Analysing the sensitivity of reacting and non-reacting variable-density shear flows using the low-Mach-number approximation has up until now not been done. By includ- ing reaction, a large forward step has been taken in applying these techniques to real fuel injectors.

This dissertation investigates the stability of injector flows. This is carried out both theoretically and numerically.

In injector flows three main features are identified which affect the stability of the flow. These are: shear, geometry and density and are given in the relative order of im- portance for the consideration of this dissertation.

Shear is the primary instability mechanism within an injector flow. In order to capture this physical mechanism the simplest flow with shear is considered: the inviscid single vortex sheet. This is unstable due to the Kelvin--Helmholtz instability and forms the building block with which to construct various models of injector flows. Variants of this construct include the inclusion of surface tension at the interface and a finite thickness shear layer. Injector flows are most simply modelled by considering two shear layers interacting. Depending upon the relative velocity of the different streams the flow can describe a jet or a wake.

The second feature, geometry, is introduced into the model by placing confining walls either side of the two shear layers. It is shown that the configuration of these confining walls has a profound effect on the instability of the flow and can in some case make the flow much more unstable. Further realism is added by introducing curvature by considering a round geometry. Many of the results in the planar case are carried over into the round case.

The third feature, density, is explored briefly in this dissertation and is found to also have a profound effect on the stability. In particular low density jets and high density wake configurations are found to be strongly unstable. Density does not receive nearly as much attention as does shear and geometry since in practical terms it is largely fixed with little scope for wide-scale variation. The other two parameters by comparison can be chosen over a wide range of values in a practical setup.

Even these simple models are still capable of producing very complex stability characteristics. These models, however, represent the limit of the theoretical studies. In order to progress any further and add more realism to the model, either in the form of viscosity or smooth velocity profiles it was necessary to adopt a numerical approach. This has led to the develop of FLOWTOOL, a piece of software capable of calculating a spatio-temporal analysis of a given velocity profile and determining the local stability properties. The code is successfully demonstrated on a real injector flow. Excellent agreement is found between the predicted frequencies and those obtained from global methods, namely a Large Eddy Simulation.

The success of a satellite launcher depends to a great extent on its efficiency and reliability. Engines using cryogenic fuels, such as liquid oxygen and hydrogen, are used for most missions since they combine high performance with a relatively light structure. The design of such motors has, until recently, been based on empirical results from systematic tests. Future design will rely on numerical simulations and will envisage alternative reactant combinations, such as methane and oxygen. The definition of entry conditions to these numerical simulations requires a knowledge of the flame structure, particularly of the region near the fuel injectors. These practical considerations motivate this investigation.

As well as discussion on the overall flame shape under subcritical and supercritical conditions, two aspects are given special attention: (1) the injector geometry, (2) stabilization of the flame. The latter question is critical for the system's reliability and is particularly important when considering fuels which are less reactive than hydrogen and oxygen.

Systematic experiments are performed at up to 70 bar pressure on a coaxial fuel injector similar to those used in current mototrs. Optical diagnostics combined with image processing yield the flame structure. Models are then developed regarding the effect of injector geometry and tested against experimental results from this and other coaxial injectors. In this manner the physical mechanisms controlling flame shape are deduced. A result of scientific interest is that a wake flow, consisting of a slow stream within a faster stream, is more unstable when enclosed within a duct. This provides one possible mechanism for the effect of recess on the cryogenic flame.

The question of stabilization is approached in carefully-defined stages so that model problems from the field of combustion science can be applied. First the base of the flame is divided into two parts and one is treated as a counterflow diffusion flame above a condensed surface. Numerical simulations performed here add new results to the study of this configuration. The second part of the base is treated initially as a corner flame, a model problem which has been investigated only recently. Two parameters controlling the shape of the flame are defined and the relationship between them is deduced from nuerical simulations. This approach permits a simple progression to more complex geometries. The flames above a porous plate with fuel injection and then above a vaporizing reactant are considered. Finally, the situation of a flame behind a step over a vaporizing reactant is analysed. This is a realistic model of the base of a cyrogenic spray flame. Through this progression the non-dimensional parameters governing behaviour are introduced successively and the most influential parameters are identified. The final result will aid design both of the engine and the control sequences of ignition, leading to enhanced reliability.