Guide Theory of Nonlinear Acoustics in Fluids (Fluid Mechanics and Its Applications)

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Structure and bonding - order and disorder; types of bonding and structure; electrons in periodic potentials; the Bloch theorem; tight-binding models; 1D chain models; band structures of real materials; optical transitions and photoemission. Interactions — effective medium approximations for electron-electron interactions; Hartree-Fock theory; exchange and correlation energy; electron fluids and electrostatic screening; the exclusion principle and quasiparticles.

Transport and scattering — crystal momentum; neutron scattering; electron-phonon scattering; optical conductivity; Drude theory, plasmons; transport in electric and magnetic fields; quantization of orbits, cyclotron resonance; the de Haas-van Alphen effect; Fermi surfaces; magnetoresistance oscillation; the quantum Hall effect. Semiconductors — thermal equilibrium of quasiparticles; field effect transistor; p-n junctions, LED; excitons; semiconductor heterostrutures; quantum wells; semiconductor lasers.

Magnetism — origin of magnetic moments and interactions; ferromagnetism; itinerant magnetism; the Stoner model; strongly interacting systems; Mott insulators. PH - Surfaces and Interfaces. Thermodynamics of surface phenomena — electronic structures; phase transitions; elementary excitations; physisorption and chemisorption; energy transfer. Schottky barrier and band offsets in semiconductors; band engineering. Analytical techniques — scanning tunneling microscopy; electron diffraction methods; photoemission; ballistic electron emission microscopy.

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PH - Nuclear Physics. Properties of nuclei — nuclear radii, masses, and abundances; binding energies; spins and electromagnetic moments. Nuclear structure — deuterons; nucleon-nucleon scattering and exchange forces; the semi-empirical mass formula; the Fermi gas model; the shell model; liquid drop models with vibrational and rotational excitations; collective structure. Selection rules for alpha, beta and gamma decay processes. Nuclear lifetimes; applications of nuclear physics including fusion and fission processes.

Nuclear reactors and nuclear power — neutron difussion and moderation; radiation protection and radiation shielding; safety and the environment.

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Classical linear and nonlinear optics; statistical optics. Quantum optics — physics of photoncs; semiclassical and quantum models of light-matter interaction; lasers; generation of short laser pulses. PH - Econophysics. Introductory concepts — basic concepts in probability and statistics; low- and high-frequency data in economics and finance; Gaussian and fat-tailed return distributions.

Time series — autocorrelations, memory, and nonstationarity; cross correlations in financial markets; time series clustering. Random matrix theory. Correlation filtering and minimal spanning trees. Agent-based models of financial markets. Common models of nuclear and high energy particle physics — the Standard Model of quarks, leptons and the fundamental interactions; tests of conservation laws; indications of physics beyond the Standard Model.


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Current and future experimental challenges — data volume and computing problems; online collision selection; data analysis. Magnetic recording — components of magnetic recording media; recent developments. Prerequisites: PH and PH or equivalent. Data structures for scientific programming — arrays; runtime and memory scaling analysis; numerical linear algebra; numerical eigenvalue problem solvers. Monte Carlo method for statistical mechanics simulation. Optimization and data analysis.

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Discuss cutting-edge theories and research papers in fundamental theoretical physics 2. Analyze problems in physics using advanced theorectical and mathematical skilss and techniques 3. Conduct research on the latest topics in theorectical physics Prerequisite: division approval. Semiconductor physics — electronic band structures of semiconductors; electronic properties of defects; charge carrier concentrations; drift of carriers in electric and magnetic fields; diffusion and recombination of excess carriers; p-n junction physics; junction diodes; tunnel diodes; bipolar junction transistors; metal-semiconductor contacts; metal-insulator-semiconductor interfaces; MOSFET and advanced FinFET.

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Introduction to vector calculus and partial differential equations. MH - Complex Methods for the Sciences. The book differs from mathematical books on nonlinear wave equations by its stress on their origin in physical principles and their use for physical applications. It differs from books on applications of nonlinear acoustics by its ambition to explain all steps in mathematical derivations of physical results. It is useful for practicians and researchers in acoustics feeling the need for more theoretical understanding.

It can be used as a textbook for graduate or advanced undergraduate students with an adequate background in physics and mathematical analysis, specializing in acoustics, mechanics or applied mathematics. Please wait English Svenska Norsk. Cite Export. 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. 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.

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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.

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Philip L. Marston

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.

Toggle navigation Matthew Juniper. Automated databases. Google Scholar. Adjoints and Passive Control in Thermoacoustics M. Modal Stability Theory M. Juniper, A. Hanifi, and V. Theofilis Applied Mechanics Review 66 , , , doi It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier-Stokes equations, the Orr-Sommerfeld equation, the Rayleigh equation, the Briggs-Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability.

It also covers the Parabolized Stability Equation PSE , temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes.

Sensitivity and nonlinearity in Thermoacoustics M. Juniper, R. Sujith Annual Review of Fluid Mechanics 50 , , These changes can, however, be ruinously expensive to devise. This review explains why linear and nonlinear thermoacoustic behaviour is so sensitive to parameters such as operating point, fuel composition, and injector geometry. It shows how non-periodic behaviour arises in experiments and simulations and discusses how fluctuations in thermoacoustic systems with turbulent reacting flow, which are usually filtered or averaged out as noise, can reveal useful information.

Finally, it proposes tools to exploit this sensitivity in the future: adjoint-based sensitivity analysis to optimize passive control designs, and complex systems theory to warn of impending thermoacoustic oscillations and to identify the most sensitive elements of a thermoacoustic system. Non-normality and nonlinearity in thermoacoustic instabilities R. Sujith, M. Juniper, P. Recent progress in nonmodal stability analysis allows us to study the problem from a different perspective, by quantitatively describing the short-term behavior of disturbances.

The short term evolution has a bearing on subcritical transition to instability, known popularly as triggering instability in thermoacoustic parlance. We provide a review of the recent developments in the context of triggering instability. A tutorial for non-modal stability analysis is provided. The applicability of the tools from non-modal stability analysis are demonstrated with the help of a simple model of a Rjike tube. The paper closes with a brief description of how to characterize bifurcations in thermoacoustic systems. Review of Unsteady Combustor Physics, by T.

Lieuwen M. Juniper Journal of Fluid Mechanics , , , doi Triggering in thermoacoustics M. Experimental studies show that these large oscillations can sometimes be triggered by very low levels of background noise. This theoretical paper sets out the conditions that are necessary for triggering to occur. It uses a weakly nonlinear analysis to show when these conditions will be satisfied for cases where the heat release rate is a function of the acoustic velocity. The role played by non-normality is investigated.

It is shown that, when a state triggers to sustained oscillations from the lowest possible energy, it exploits transient energy growth around an unstable limit cycle. The positions of these limit cycles in state space is determined by nonlinearity, but the tangled-ness of trajectories in state space is determined by non-normality. When viewed in this dynamical systems framework, triggering in thermoacoustics is seen to be directly analogous to bypass transition to turbulence in pipe flow.

Combined state and parameter estimation in level-set methods H. Yu, M. Juniper, and L. Magri Journal of Computational Physics xxx xxx , xxx-xxx, Yu, T. Jaravel, M. Ihme, M. The reduced-order model learns the state and the parameters of the premixed flame on the fly with the ensemble Kalman filter, which is a Bayesian filter used, for example, in weather forecasting. The proposed method and algorithm are applied to two test cases with relevance to reacting flows and instabilities.

First, the capabilities of the framework are demonstrated in a twin experiment, where the assimilated data is produced from the same model as that used in prediction. Second, the assimilated data is extracted from a high-fidelity reacting-flow direct numerical simulation DNS , which provides the reference solution. The results are analyzed by using Bayesian statistics, which robustly provide the level of confidence in the calculations from the reduced-order model.

The versatile method we propose enables the optimal calibration of computationally inexpensive reduced-order models in real time when experimental data becomes available, for example, from gas-turbine sensors. Adjoint based shape optimization of the microchannels in an inkjet printhead P. Kungurtsev, M. Juniper Journal of Fluid Mechanics xxx , xxx--xxx, When an actuator pulses, a droplet is forced through the nozzle, after which acoustic oscillations reverberate within the channel. Manufacturers would like to damp the residual reverberations, without increasing the pressure drop required to drive the steady flow.

In this paper we show that this can be achieved by constricting the channel where the acoustic velocity is largest and enlarging the channel where the acoustic velocity is smallest. This increases the viscothermal dissipation of the acoustics without changing the viscous dissipation of the steady flow. We separate the compressible Navier--Stokes equations into equations for a steady flow with no oscillations and equations for oscillations with no steady flow. We define two objective functions: the viscous dissipation of the base flow and the viscous dissipation of the oscillations.

We then derive the adjoints for both sets of equations, and obtain expressions for the gradient of each objective function with respect to boundary deformations in Hadamard form. We combine these with a gradient-based optimization algorithm, incorporating constraints such as the shapes of the actuator and nozzle. We show that this design is nearly optimal.

It is a shape that inkjet manufacturers, using physical insight and trial and error, have not yet considered. We also show how the adjoint fields provide physical insight into the mechanisms affecting each objective function. The main requirements of this method are that the steady flow Mach number and oscillating flow Mach number are small, that the oscillations have relatively small amplitude, and that dissipation is dominated by thermo-viscous mechanisms.

These requirements are often satisfied in microfluidics, so the method in this paper could be applied to many other applications. Juniper Physical Review Fluids 3 , , , doi These are known as thermoacoustic oscillations and they often occur late in the design process. Their elimination usually requires expensive tests and re-design. Full scale tests and laboratory scale experiments show that these oscillations can usually be stabilized by making small changes to the system. The complication is that, while there is often just one unstable natural oscillation eigenmode , there are very many possible changes to the system.

The challenge is to identify the optimal change systematically, cheaply, and accurately. This paper shows how to evaluate the sensitivities of a thermoacoustic eigenmode to all possible system changes with a single calculation by applying adjoint methods to a thermoacoustic Helmholtz solver. These sensitivities are calculated here with finite difference and finite element methods, in the weak form and the strong form, with the discrete adjoint and the continuous adjoint, and with a Newton method applied to a nonlinear eigenvalue problem and an iterative method applied to a linear eigenvalue problem.

This is the first detailed comparison of adjoint methods applied to thermoacoustic Helmholtz solvers. Matlab codes are provided for all methods and all figures so that the techniques can be easily applied and tested. This paper explains why the finite difference of the strong form equations with replacement boundary conditions should be avoided and why all of the other methods work well.

Of the other methods, the discrete adjoint of the weak form equations is the easiest method to implement; it can use any discretization and the boundary conditions are straightforward.

Applications of Fluid Mechanics

The continuous adjoint is relatively easy to implement but requires careful attention to boundary conditions. The Summation by Parts finite difference of the strong form equations with a Simultaneous Approximation Term for the boundary conditions SBP--SAT is more challenging to implement, particularly at high order or on non-uniform grids. Physical interpretation of these results shows that the well-known Rayleigh criterion should be revised for a linear analysis.

This criterion states that thermoacoustic oscillations will grow if heat release rate oscillations are sufficiently in phase with pressure oscillations. In fact, the criterion should contain the adjoint pressure rather than the pressure. In self-adjoint systems the two are equivalent.

In non-self-adjoint systems, such as all but a special case of thermoacoustic systems, the two are different. Finally, the sensitivities of the growth rate of oscillations to placement of a hot or cold mesh are calculated, simply by multiplying the feedback sensitivities by a number.

These sensitivities are compared successfully with experimental results. With the same technique, the influence of the viscous and thermal acoustic boundary layers is found to be negligible, while the influence of a Helmholtz resonator is found, as expected, to be considerable. Passive control of global instability in low-density jets U. Qadri, G. Chandler, M. Juniper European Journal of Mechanics B 72 , , , doi It is formed from the overlap of the flow's direct and adjoint global modes.

These global modes are usually calculated with 2D or 3D global stability analyses, which can be very computationally expensive. For weakly non-parallel flows the direct global mode can also be calculated with a local stability analysis, which is orders of magnitude cheaper. In this theoretical paper we show that, if the direct global mode has been calculated with a local analysis, then the adjoint global mode follows at little extra cost. As expected, we find that the local analysis becomes less accurate when the flow becomes less parallel.

Forced synchronization of periodic and aperiodic thermoacoustic oscillations: lock-in, bifurcations, and open-loop control K. Kashinath, L. Li, M. The effect of the flame phase on thermoacoustic instabilities G. Ghirardo, M. Juniper, M. Bothien Combustion and Flame , , , doi Experimental sensitivity analysis of a linearly stable thermoacoustic system via a pulsed forcing technique N. Jamieson, M.

1. Introduction

Juniper Experiments in Fluids 58 , , , doi We examine the shift in linear decay rates and frequencies of thermoacoustic oscillations, with and without control devices. To measure the decay rate we wait for the system to reach a steady state and then excite it with an acoustic pulse from a loudspeaker. We identify the range of amplitudes over which the amplitude decays exponentially with time. In this range, the rate of change of the amplitude is linearly proportional to the amplitude, and we calculate the constant of proportionality, the linear decay rate, which can be compared with model predictions.

The aim of this work is i to improve the experimental techniques implemented by Rigas et al. Fluid Mech. Spray and Comb. Experimental sensitivity analysis via a secondary heat source in an oscillating thermoacoustic system N. Jamieson, G. Rigas, M. We examine the stability characteristics of the system due to the introduction of a secondary heat source.

The experimental sensitivity analysis is quantified by measuring the shift in linear growth and decay rate as well as the shift in the linear frequency during periods of growth and decay of thermoacoustic oscillations. A discrepancy in the linear frequency measurements highlight deficiencies in the model used for those predictions and shows that the experimental measurement of sensitivities is a stringent test of any thermoacoustic model.

The findings suggest that adjoint-based methods are, in principle, capable of providing industry with a cheap and efficient tool for developing optimal control strategies for more complex thermoacoustic systems. G-equation modelling of thermo-acoustic oscillations of partially-premixed flames B.

Semlitsch, A. Orchini, A. Dowling, M. Nonlinear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. To calculate an FDF, a wide range of forcing amplitudes and frequencies needs to be considered. For this reason, we present a computationally inexpensive level-set approach, which accounts for equivalence ratio perturbations on flames with arbitrarily-complex shapes.

The influence of flame parameters and modelling approaches on flame describing functions and time delay coefficient distributions are discussed in detail. The numerically-obtained flame describing functions are compared with experimental data and used in an acoustic network model for limit cycle prediction. A reasonable agreement of the heat release gain and limit cycle frequency is achieved even with a simplistic, analytical velocity fluctuation model.

However, the phase decay is over-predicted. For sophisticated flame shapes, only the realistic modelling of large-scale flow structures allows the correct phase decay predictions of the heat release rate response. Adjoint-based sensitivity analysis of low-order thermoacoustic networks using a wave-based approach J. Aguilar, L. Magri, M.

Juniper Journal of Computational Physics , , , doi Methods to predict and control unstable modes inside combustion chambers have been developed in the last decades but, in some cases, they are computationally expensive. Sensitivity analysis aided by adjoint methods provides valuable sensitivity information at a low computational cost. This paper introduces adjoint methods and their application in wave-based low order network models, which are used as industrial tools, to predict and control thermoacoustic oscillations.

Two thermoacoustic models of interest are analysed. First, in the zero Mach number limit, a nonlinear eigenvalue problem is derived, and continuous and discrete adjoint methods are used to obtain the sensitivities of the system to small modifications. Sensitivities to base-state modification and feedback devices are presented. Second, a more general case with non-zero Mach number, a moving flame front and choked outlet, is presented. The influence of the entropy waves on the computed sensitivities is shown. Local linear stability analysis of cyclone separators T. Grimble, A.

Agarwal, M. The results of the stability analysis are compared with experimental measurements of the vortex oscillations using high speed photography with particle seeding, and hot wire anemometry. The experiments reveal distinct spatial variation in the oscillation behaviour within the cyclones. The unsteady motion is focused at each end of the device, at both the narrow cone tip and just below the exhaust duct at the top of the cone, which is known as a vortex finder.

The local stability analysis shows that an absolute instability is present throughout the flow for some non-zero azimuthal wavenumbers. The unsteady flow is observed to be driven by coupling between the shear layer and inertial waves confined within the vortex core. Comparing the stability analysis with experiments shows the same frequency and mode shape behaviour and suggests that the local analysis accurately predicts the unstable modes of the system. The precessing vortex core is responsible for a narrow-band acoustic noise. Comparisons are also drawn with acoustic measurements made on cyclones in which the system is defined by key non-dimensional parameters, such as the swirl number and outlet diameter ratio.

The results in this study demonstrate the applicability of local stability analysis to a complex swirling system and yield credible details about the underlying mechanisms of the unstable flow inside the cyclone. Multiple-scale thermo-acoustic stability analysis of a coaxial jet combustor L. Magri, Y-C. See, O. Tammisola, M. Juniper Proceedings of the Combustion Institute 36 , , , doi The resulting sets of nonlinear equations for hydrodynamics and acoustics are two-way coupled.

The coupling strength depends on which multiple scales are used. The double-time-double-space 2T-2S , double-time-single-space 2T-1S and single-time-double-space 1T-2S limits are revisited, derived and linearized. It is shown that only the 1T-2S limit produces a two-way coupled linearized system.

Therefore this limit is adopted and implemented in a finite-element solver.


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  • The methodology is applied to a coaxial jet combustor. By using an adjoint method and introducing the intrinsic sensitivity, i the interaction between the acoustic and hydrodynamic subsystems is calculated and ii the role of the global acceleration term, which is the coupling term from the acoustics to the hydrodynamics, is analysed. For the confined coaxial jet diffusion flame studied here, i the growth rate of the thermo-acoustic oscillations is found to be more sensitive to small changes in the hydrodynamic field around the flame and ii increasing the global acceleration term is found to be stabilizing in agreement with the Rayleigh Criterion.

    Weakly nonlinear analysis of thermoacoustic instabilities in annular combustors G. Juniper, J. Moeck Journal of Fluid Mechanics , , , doi We focus on azimuthal thermoacoustic oscillations and model the fluctuating heat release rate as being dependent only on the local pressure in the combustion chamber. We study the dynamics of the annular combustor with a finite number of compact flames equispaced around the annulus, and characterize the flames?

    We discuss the existence, amplitude and the stability of standing and spinning waves, as a function of: 1 the number of the burners; 2 the acoustic damping in the chamber; 3 the flame response. We present the implications for industrial applications and the future direction of investigations. We then present as an example the first theoretical study of thermoacoustic triggering in annular combustors, which shows that rotationally symmetric annular chambers that are thermoacoustically unstable do not experience only stable spinning solutions, but can also experience stable standing solutions.

    We finally test the theory on one experiment with good agreement. Stability analysis of thermoacoustic nonlinear eigenproblems in annular combustors. Part 2: Uncertainty Quantification L. Bauerheim, F. Nicoud, M. This method is applied to evaluate the risk factor, i. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by as much as the Monte Carlo samples. Part 1: sensitivity L. Bauerheim, M. This method is applied to a thermo-acoustic annular combustor network, the stability of which is governed by a nonlinear eigenproblem.

    We calculate the first- and second-order sensitivities of the growth rate and frequency to geometric, flow and flame parameters. Three different configurations are analysed. The benchmark sensitivities are obtained by finite difference, which involves solving the nonlinear eigenproblem at least as many times as the number of parameters. By solving only one adjoint eigenproblem, we obtain the sensitivities to any thermo-acoustic parameter, which match the finite-difference solutions at much lower computational cost.

    Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube A. Orchini, G. 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? 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.