### Automated databases

### Movies of conference presentations

K. Kashinath, I. Waugh, S. Hemchandra, M. Juniper

*APS Fluid Dynamics Division, San Diego*, (2012)

L. Magri, M. Juniper

*APS Fluid Dynamics Division, San Diego*, (2012)

### Tutorials

M. P. Juniper

*21st CISM-IUTAM International Summer School on Measurement, Analysis, and Passive Control of Thermoacoustic Oscillations*, , (2015)

M. P. Juniper, A. Hanifi, and V. Theofilis

*Applied Mechanics Review*

**66**, 024804, (2014), doi:10.1115/1.4026604

doi: https://doi.org/10.1115/1.4026604

Section 2.8 Plug flow Matlab tutorial

Section 2.9 to 2.10 Planar Poiseuille Flow Matlab tutorial

Section 4.6 Parabolized Stability Equation Matlab tutorial

Section 5.3.1 2D Helmholtz Equation Matlab tutorial

Section 5.3.2 2D Eigenvalue problem Matlab tutorial

Section 6.1 Linearized Navier-Stokes Equation Matlab tutorial

This article contains a review of modal stability theory. 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.

### Review papers

M. Juniper, R. I. Sujith

*Annual Review of Fluid Mechanics*

**50**, 661--689, (2018)

Open Access

Tutorial 1: Obtaining thermoacoustic eigenvalue sensitivities with adjoint methods

Tutorial 1 Matlab files

Tutorial 2: Tools from nonlinear dynamics

Tutorial 2 Matlab files

Nine decades of rocket engine and gas turbine development have shown that thermoacoustic oscillations are difficult to predict but can usually be eliminated with relatively small ad hoc design changes. 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.

R. I. Sujith, M. P. Juniper, P. J. Schmid

*International Journal of Spray and Combustion Dynamics*

**8**(2), 119--146, (2016), doi:10.1177/1756827716651571

doi: https://doi.org/10.1177/1756827716651571

Analysis of thermoacoustic instabilities were dominated by modal (eigenvalue) analysis for many decades. 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.

M. Juniper

*Journal of Fluid Mechanics*

**766**, 656--658, (2015), doi:10.1017/jfm.2015.67

M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**4**(3), 217--238, (2012)

Under certain conditions, the flow in a combustion chamber can sustain large amplitude oscillations even when its steady state is linearly stable. 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.

### Journal papers

K. Kashinath, L. K. B. Li, M. P. Juniper

*Journal of Fluid Mechanics*

**xxx**, xxx--xxx, (2017)

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.

G. Ghirardo, M. P. Juniper, M. R. Bothien

*Combustion and Flame*

**187**, 165--184, (2017), doi:10.1016/j.combustflame.2017.09.007

Open Access

doi: https://doi.org/10.1016/j.combustflame.2017.09.007

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.

N. Jamieson, M. P. Juniper

*Experiments in Fluids*

**58**, 123, (2017), doi:10.1007/s00348-017-2402-2

Open Access

doi: https://doi.org/10.1007/s00348-017-2402-2

In this paper we present the results of an experimental sensitivity analysis on a vertical electrically-heated Rijke tube. 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. (J. Fluid Mech., 2016, vol. 787, R1), Jamieson et al. (Int. J. Spray and Comb. Dyn., Accepted, 2016 ), using a technique inspired by Mejia et al. (Comb. and Flame, 169:287-296, 2016), and (ii) to provide experimental data for future comparison with adjoint-based sensitivity analysis. Our experimental set up is automated and we can obtain thousands of decay rates in 1/12 the time of our previous method.

N. Jamieson, G. Rigas, M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**9**(4), 230--240, (2017), doi:10.1177/1756827717696325

Open Access

doi: https://doi.org/10.1177/1756827717696325

In this paper, we report the results of an experimental sensitivity analysis on a vertical electrically-heated Rijke tube. 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. Linear growth and decay rate measurements agree qualitatively well with the theoretical predictions from adjoint-based methods of Magri & Juniper (

*J. Fluid Mech.*, vol.

**719**, 2013, pp. 183--202). 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.

B. Semlitsch, A. Orchini, A. P. Dowling, M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**9**(4), 260--276, (2017), doi:10.1177/1756827717711405

Open Access

doi: https://doi.org/10.1177/1756827717711405

Numerical simulations aid combustor design to avoid and reduce thermo-acoustic oscillations. Nonlinear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. The heat release dynamics of flames can be approximated by a Flame Describing Function (FDF). 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.

J. G. Aguilar, L. Magri, M. P. Juniper

*Journal of Computational Physics*

**341**, 163--181, (2017), doi:10.1016/j.jcp.2017.04.013

Open Access

doi: https://doi.org/10.1016/j.jcp.2017.04.013

Strict pollutant emission regulations are pushing gas turbine manufacturers to develop devices that operate in lean conditions, with the downside that combustion instabilities are more likely to occur. 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.

T. Grimble, A. Agarwal, M. P. Juniper

*Journal of Fluid Mechanics*

**816**, 507--538, (2017), doi:10.1017/jfm.2017.89

Open Access

doi: https://doi.org/10.1017/jfm.2017.89

Local linear stability analysis is applied to the flow inside a cyclone separator to investigate the unsteady precession of the vortex core. 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.

L. Magri, Y-C. See, O. Tammisola, M. Ihme, M. P. Juniper

*Proceedings of the Combustion Institute*

**36**, 3863--3871, (2016), doi:10.1016/j.proci.2016.06.009

Open Access

doi: https://doi.org/10.1016/j.proci.2016.06.009

In this paper, asymptotic multiple-scale methods are used to formulate a mathematically consistent set of thermo-acoustic equations in the low-Mach number limit for linear stability analysis. 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. 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.

G. Ghirardo, M. P. Juniper, J. Moeck

*Journal of Fluid Mechanics*

**805**, 52--87, (2016), doi:10.1017/jfm.2016.494

Open Access

doi: https://doi.org/10.1017/jfm.2016.494

Rotationally symmetric annular combustors are of practical importance because they generically resemble combustion chambers in gas turbines, in which thermoacoustically driven oscillations are a major concern. 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? response with a describing function. 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.

L. Magri, M. Bauerheim, F. Nicoud, M. P. Juniper

*Journal of Computational Physics*

**325**, 411--421, (2016), doi:10.1016/j.jcp.2016.08.043

Open Access

doi: https://doi.org/10.1016/j.jcp.2016.08.043

Monte Carlo and Active Subspace Identification methods are combined with first- and second-order adjoint sensitivities to perform (forward) uncertainty quantification analysis of the thermo-acoustic stability of two annular combustor configurations. This method is applied to evaluate the risk factor, i.e., the probability for the system to be unstable. It is shown that the adjoint approach reduces the number of nonlinear-eigenproblem calculations by as much as the Monte Carlo samples.

L. Magri, M. Bauerheim, M. P. Juniper

*Journal of Computational Physics*

**325**, 395--410, (2016), doi:10.1016/j.jcp.2016.07.032

Open Access

doi: https://doi.org/10.1016/j.jcp.2016.07.032

We present an adjoint-based method for the calculation of eigenvalue perturbations in nonlinear, degenerate and non self-adjoint eigenproblems. 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.

A. Orchini, G. Rigas, M. P. Juniper

*Journal of Fluid Mechanics*

**805**, 523--550, (2016), doi:10.1017/jfm.2016.585

Open Access

doi: https://doi.org/10.1017/jfm.2016.585

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

A. Orchini, M. P. Juniper

*Combustion and Flame*

**171**, 87--102, (2016), doi:10.1016/j.combustflame.2016.06.014

Open Access

doi: https://doi.org/10.1016/j.combustflame.2016.06.014

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.

O. Tammisola, Juniper, M. P.

*Journal of Fluid Mechanics*

**792**, 620--657, (2016), doi:10.1017/jfm.2016.86

Open Access

doi: https://doi.org/10.1017/jfm.2016.86

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.

B. Emerson, T. C. Lieuwen, M. P. Juniper

*Journal of Fluid Mechanics*

**788**, 549-575, (2016), doi:10.1017/jfm.2015.724

Open Access

doi: https://doi.org/10.1017/jfm.2015.724

This paper presents an experimental and theoretical investigation of high Reynolds number, low density reacting wakes near a hydrodynamic Hopf bifurcation. This configuration is applicable to the wake flows that are commonly used to stabilize flames in high velocity flows. First, an experimental study is conducted to measure the limit cycle oscillation of this reacting bluff body wake. The experiment is repeated while independently varying the bluff body lip velocity and the density ratio across the flame. In all cases, the wake exhibits a sinuous oscillation. Linear stability analysis is performed on the measured time-averaged velocity and density fields. In the first stage of this analysis, a local, spatio-temporal stability analysis is performed on the measured, time averaged velocity and density fields. The stability analysis results are compared to the experimental measurement, and demonstrate that the local stability analysis correctly captures the influence of the lip velocity and density ratio parameters on the sinuous mode. In the second stage of this analysis, the linear direct and adjoint global modes are estimated by combining the local results. The sensitivity of the eigenvalue to changes in intrinsic feedback mechanisms is found by combining the direct and adjoint global modes. This is referred to as the eigenvalue sensitivity throughout the paper for reasons of brevity. The predicted global mode frequency is consistently within 10 % of the measured value, and the linear global mode shape closely resembles the measured nonlinear oscillations. The adjoint global mode reveals that the oscillation is strongly sensitive to open loop forcing in the shear layers. The eigenvalue sensitivity identifies a wavemaker in the recirculation zone of the wake. A parametric study shows that these regions change little when the density ratio and lip velocity change. In the third stage of the analysis, the stability analysis is repeated for the varicose hydrodynamic mode. Although not physically observed in this unforced flow, the varicose mode can lock into longitudinal acoustic waves and cause thermoacoustic oscillations to occur. This paper shows that the local stability analysis successfully predicts the global hydrodynamic stability characteristics of this flow and shows that experimental data can be post-processed with this method in order to identify the wavemaker regions and the regions that are most sensitive to external forcing, for example from acoustic waves.

G. Rigas, N. Jamieson, L. K. B. Li, M. Juniper

*Journal of Fluid Mechanics*

**787**, R1, (2016), doi:10.1017/jfm.2015.715

C-Y Lee, L. K. B. Li, M. Juniper, R. S. Cant

*Combustion Theory and Modelling*

**20**(1), 131--153, (2016), doi:10.1080/13647830.2015.1118555

Open Access

doi: https://doi.org/10.1080/13647830.2015.1118555

Turbulent premixed flames often experience thermoacoustic instabilities when the combustion heat release rate is in phase with acoustic pressure fluctuations. Linear methods often assume a priori that oscillations are periodic and occur at a dominant frequency with a fixed amplitude. Such assumptions are not made when using nonlinear analysis. When an oscillation is fully saturated, nonlinear analysis can serve as a useful avenue to reveal flame behaviour far more elaborate than period-one limit cycles, including quasi-periodicity and chaos in hydrodynamically or thermoacoustically self-excited system. In this paper, the behaviour of a bluff-body stabilised turbulent premixed propane/air flame in a model jet-engine afterburner configuration is investigated using computational fluid dynamics. For the frequencies of interest in this investigation, an unsteady Reynolds-averaged Navier?Stokes approach is found to be appropriate. Combustion is represented using a modified laminar flamelet approach with an algebraic closure for the flame surface density. The results are validated by comparison with existing experimental data and with large eddy simulation, and the observed self-excited oscillations in pressure and heat release are studied using methods derived from dynamical systems theory. A systematic analysis is carried out by increasing the equivalence ratio of the reactant stream supplied to the premixed flame. A strong variation in the global flame structure is observed. The flame exhibits a self-excited hydrodynamic oscillation at low equivalence ratios, becomes steady as the equivalence ratio is increased to intermediate values, and again exhibits a self-excited thermoacoustic oscillation at higher equivalence ratios. Rich nonlinear behaviour is observed and the investigation demonstrates that turbulent premixed flames can exhibit complex dynamical behaviour including quasiperiodicity, limit cycles and period-two limit cycles due to the interactions of various physical mechanisms. This has implications in selecting the operating conditions for such flames and for devising proper control strategies for the avoidance of thermoacoustic instability.

A. Orchini, M. P. Juniper

*Combustion and Flame*

**165**, 97--108, (2015), doi:10.1016/j.combustflame.2015.10.011

Open Access

doi: https://doi.org/10.1016/j.combustflame.2015.10.011

We analyse the linear response of laminar conical premixed flames modelled with the linearised front-track kinematic G-equation. We start by considering the case in which the flame speed is fixed, and travelling wave velocity perturbations are advected at a speed different from the mean flow velocity. A previous study of this case contains a small error in the Flame Transfer Function (FTF), which we correct. We then allow the flame speed to depend on curvature. No analytical solutions for the FTF exist for this case so the FTF has to be calculated numerically as its parameters -- aspect ratio, convection speed and Markstein length -- are varied. Then we consider the stability and sensitivity of thermoacoustic systems containing these flames. Traditionally, the stability of a thermoacoustic system is found by embedding the FTF within an acoustic network model. This can be expensive, however, because the FTF must be re-calculated whenever a flame parameter is varied. Instead, we couple the linearised G-equation directly with an acoustic network model, creating a linear eigenvalue problem without explicit knowledge of the FTF. This provides a simple and quick way to analyse the stability of thermoacoustic networks. It also allows us to use adjoint sensitivity analysis to examine, at little extra cost, how the system?s stability is affected by every parameter of the system.

J. Samuelson, O. Tammisola, M. P. Juniper

*Physics of Fluids*

**27**, 104103, (2015), doi:10.1063/1.4934530

Open Access

doi: https://doi.org/10.1063/1.4934530

Flow through a sinuous stenosis with varying degrees of non-axisymmetric shape variations and at Reynolds number ranging from 250 to 750 is investigated using direct numerical simulation (DNS) and global linear stability analysis. At low Reynolds numbers (Re < 390), the flow is always steady and symmetric for an axisymmetric geometry. Two steady state solutions are obtained when the Reynolds number is increased: a symmetric steady state and an eccentric, non-axisymmetric steady state. Either one can be obtained in the DNS depending on the initial condition. A linear global stability analysis around the symmetric and non-axisymmetric steady state reveals that both flows are linearly stable for the same Reynolds number, showing that the first bifurcation from symmetry to antisymmetry is subcritical. When the Reynolds number is increased further, the symmetric state becomes linearly unstable to an eigenmode, which drives the flow towards the non-axisymmetric state. The symmetric state remains steady up to Re = 713, while the non-axisymmetric state displays regimes of periodic oscillations for Re >= 417 and intermittency for Re >~ 525. Further, an offset of the stenosis throat is introduced through the eccentricity parameter E. When eccentricity is increased from zero to only 0.3% of the pipe diameter, the bifurcation Reynolds number decreases by more than 50%, showing that it is highly sensitive to non-axisymmetric shape variations. Based on the resulting bifurcation map and its dependency on E, we resolve the discrepancies between previous experimental and computational studies. We also present excellent agreement between our numerical results and previous experimental results.

G. Ghirardo, B. Cosic, M. P. Juniper, J. Moeck

*Nonlinear Dynamics*

**82**, 9--28, (2015), doi:10.1007/s11071-015-2134-x

Open Access

doi: https://doi.org/10.1007/s11071-015-2134-x

The describing function is a powerful tool for characterising nonlinear dynamical systems in the frequency domain. In some cases, it is the only available description of a nonlinear operator characterising a certain subcomponent of the system. This paper presents a methodology to provide a state-space realization of one given describing function, in order to allow the study of the system in the time domain as well. The realization is based on Hammerstein models and Fourier?Bessel series. It can be embedded in time domain simulations of complex configurations with many nonlinear elements interacting, accurately describing the nonlinear saturation of the system. The technique is applied to an example application in the field of combustion instability, featuring self-excited thermoacoustic oscillations. We benchmark the performance of the tool comparing the results with a frequency domain analysis of the same system, obtaining good agreement between the two formulations.

U. A. Qadri, G. J. Chandler, M. P. Juniper

*Journal of Fluid Mechanics*

**775**, 201--222, (2015), doi:10.1017/jfm.2015.297

Open Access

doi: https://doi.org/10.1017/jfm.2015.297

We use direct numerical simulation (DNS) of the Navier?Stokes equations in the low-Mach-number limit to investigate the hydrodynamic instability of a lifted jet diffusion flame. We obtain steady solutions for flames using a finite rate reaction chemistry, and perform a linear global stability analysis around these steady flames. We calculate the direct and adjoint global modes and use these to identify the regions of the flow that are responsible for causing oscillations in lifted jet diffusion flames, and to identify how passive control strategies might be used to control these oscillations. We also apply a local stability analysis to identify the instability mechanisms that are active. We find that two axisymmetric modes are responsible for the oscillations. The first is a high-frequency mode with wavemaker in the jet shear layer in the premixing zone. The second is a low-frequency mode with wavemaker in the outer part of the shear layer in the flame. We find that both of these modes are most sensitive to feedback involving perturbations to the density and axial momentum. Using the local stability analysis, we find that the high-frequency mode is caused by a resonant mode in the premixing region, and that the low-frequency mode is caused by a region of local absolute instability in the flame, not by the interaction between resonant modes, as proposed in Nichols et al. (Phys. Fluids, vol. 21, 2009, article 015110). Our linear analysis shows that passive control of the low-frequency mode may be feasible because regions up to three diameters away from the fuel jet are moderately sensitive to steady control forces.

A. Orchini, S. Illingworth, M. P. Juniper

*Journal of Fluid Mechanics*

**775**, 387--414, (2015), doi:10.1017/jfm.2015.139

Open Access

doi: https://doi.org/10.1017/jfm.2015.139

Many thermoacoustic systems exhibit rich nonlinear behaviour. Recent studies show that this nonlinear dynamics can be well captured by low-order time domain models that couple a level set kinematic model for a laminar flame, the G-equation, with a state-space realization of the linearized acoustic equations. However, so far the G-equation has been coupled only with straight ducts with uniform mean acoustic properties, which is a simplistic configuration. In this study, we incorporate a wave-based model of the acoustic network, containing area and temperature variations and frequency-dependent boundary conditions. We cast the linear acoustics into state-space form using a different approach from that in the existing literature. We then use this state-space form to investigate the stability of the thermoacoustic system, both in the frequency and time domains, using the flame position as a control parameter. We observe frequency-locked, quasiperiodic, and chaotic oscillations. We identify the location of Neimark?Sacker bifurcations with Floquet theory. We also find the Ruelle?Takens?Newhouse route to chaos with nonlinear time series analysis techniques. We highlight important differences between the nonlinear response predicted by the frequency domain and the time domain methods. This reveals deficiencies with the frequency domain technique, which is commonly used in academic and industrial studies of thermoacoustic systems. We then demonstrate a more accurate approach based on continuation analysis applied to time domain techniques.

K. Kashinath, I. C. Waugh, M. P. Juniper

*Journal of Fluid Mechanics*

**761**, 399--430, (2014), doi:10.1017/jfm.2014.601

Open Access

doi: https://doi.org/10.1017/jfm.2014.601

Thermoacoustic systems can oscillate self-excitedly, and often non-periodically, owing to coupling between unsteady heat release and acoustic waves. We study a slot-stabilized two-dimensional premixed flame in a duct via numerical simulations of a G-equation flame coupled with duct acoustics. We examine the bifurcations and routes to chaos for three control parameters: (i) the flame position in the duct, (ii) the length of the duct and (iii) the mean flow velocity. We observe period-1, period-2, quasi-periodic and chaotic oscillations. For certain parameter ranges, more than one stable state exists, so mode switching is possible. At intermediate times, the system is attracted to and repelled from unstable states, which are also identified. Two routes to chaos are established for this system: the period-doubling route and the Ruelle?Takens?Newhouse route. These are corroborated by analyses of the power spectra of the acoustic velocity. 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 aperiodic oscillations. This study complements recent experiments by providing a reduced-order model of a system with approximately 5000 degrees of freedom that captures much of the elaborate nonlinear behaviour of ducted premixed flames observed in the laboratory.

I. C. Waugh, K. Kashinath and M. P. Juniper

*Journal of Fluid Mechanics*

**759**, 1--27, (2014), doi:10.1017/jfm.2014.549

Open Access

doi: https://doi.org/10.1017/jfm.2014.549

Many experimental studies have demonstrated that ducted premixed flames exhibit stable limit cycles in some regions of parameter space. Recent experiments have also shown that these (period-1) limit cycles subsequently bifurcate to period-2n, quasiperiodic, multiperiodic or chaotic behaviour. These secondary bifurcations cannot be found computationally using most existing frequency domain methods, because these methods assume that the velocity and pressure signals are harmonic. In an earlier study we have shown that matrix-free continuation methods can efficiently calculate the limit cycles of large thermoacoustic systems. This paper demonstrates that these continuation methods can also efficiently calculate the bifurcations from the limit cycles. Furthermore, once these bifurcations are found, it is then possible to isolate the coupled flame?acoustic motion that causes the qualitative change in behaviour. This information is vital for techniques that use selective damping to move bifurcations to more favourable locations in the parameter space. The matrix-free methods are demonstrated on a model of a ducted axisymmetric premixed flame, using a kinematic G-equation solver. The methods find limit cycles and period-2 limit cycles, and fold, period-doubling and Neimark?Sacker bifurcations as a function of the location of the flame in the duct, and the aspect ratio of the steady flame.

O. Tammisola, F. Giannetti, V. Citro and M. P Juniper

*Journal of Fluid Mechanics*

**755**, 314--335, (2014), doi:10.1017/jfm.2014.415

Open Access

doi: https://doi.org/10.1017/jfm.2014.415

Sensitivity analysis has successfully located the most efficient regions in which to apply passive control in many globally unstable flows. As is shown here and in previous studies, the standard sensitivity analysis, which is linear (first order) with respect to the actuation amplitude, predicts that steady spanwise wavy alternating actuation/modification has no effect on the stability of planar flows, because the eigenvalue change integrates to zero in the spanwise direction. In experiments, however, spanwise wavy modification has been shown to stabilize the flow behind a cylinder quite efficiently. In this paper, we generalize sensitivity analysis by examining the eigenvalue drift (including stabilization/destabilization) up to second order in the perturbation, and show how the second-order eigenvalue changes can be computed numerically by overlapping the adjoint eigenfunction with the first-order global eigenmode correction, shown here for the first time. We confirm the prediction against a direct computation, showing that the eigenvalue drift due to a spanwise wavy base flow modification is of second order. Further analysis reveals that the second-order change in the eigenvalue arises through a resonance of the original (2-D) eigenmode with other unperturbed eigenmodes that have the same spanwise wavelength as the base flow modification. The eigenvalue drift due to each mode interaction is inversely proportional to the distance between the eigenvalues of the modes (which is similar to resonance), but also depends on mutual overlap of direct and adjoint eigenfunctions (which is similar to pseudoresonance). By this argument, and by calculating the most sensitive regions identified by our analysis, we explain why an in-phase actuation/modification is better than an out-of-phase actuation for control of wake flows by spanwise wavy suction and blowing. We also explain why wavelengths several times longer than the wake thickness are more efficient than short wavelengths.

L. Magri, M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**6**(3), 225--246, (2014), doi:10.1260/1756-8277.6.3.225

Open Access

doi: https://doi.org/10.1260/1756-8277.6.3.225

arXiv

This paper presents the linear theory of adjoint equations as applied to thermo-acoustics. The purpose is to describe the mathematical foundations of adjoint equations for linear sensitivity analysis of thermo-acoustic systems, recently developed by Magri and Juniper (J. Fluid Mech. (2013), vol. 719, pp. 183?202). This method is applied pedagogically to a damped oscillator, for which analytical solutions are available, and then for an electrically heated Rijke tube with a mean-flow temperature discontinuity induced by the compact heat source. Passive devices that most affect the growth rate/frequency of the electrical Rijke-tube system are presented, including a discussion about the effect of modelling the mean-flow temperature discontinuity.

I. Lashgari, O. Tammisola, V. Citro, M. P. Juniper, L. Brandt

*Journal of Fluid Mechanics*

**753**, 1--28, (2014), doi:10.1017/jfm.2014.364

Open Access

doi: https://doi.org/10.1017/jfm.2014.364

The bifurcations and control of the flow in a planar X-junction are studied via linear stability analysis and direct numerical simulations. This study reveals the instability mechanisms in a symmetric channel junction and shows how these can be stabilized or destabilized by boundary modification. We observe two bifurcations as the Reynolds number increases. They both scale with the inlet speed of the two side channels and are almost independent of the inlet speed of the main channel. Equivalently, both bifurcations appear when the recirculation zones reach a critical length. A two-dimensional stationary global mode becomes unstable first, changing the flow from a steady symmetric state to a steady asymmetric state via a pitchfork bifurcation. The core of this instability, whether defined by the structural sensitivity or by the disturbance energy production, is at the edges of the recirculation bubbles, which are located symmetrically along the walls of the downstream channel. The energy analysis shows that the first bifurcation is due to a lift-up mechanism. We develop an adjustable control strategy for the first bifurcation with distributed suction or blowing at the walls. The linearly optimal wall-normal velocity distribution is computed through a sensitivity analysis and is shown to delay the first bifurcation from Re = 82.5 to Re = 150. This stabilizing effect arises because blowing at the walls weakens the wall-normal gradient of the streamwise velocity around the recirculation zone and hinders the lift-up. At the second bifurcation, a three-dimensional stationary global mode with a spanwise wavenumber of order unity becomes unstable around the asymmetric steady state. Nonlinear three-dimensional simulations at the second bifurcation display transition to a nonlinear cycle involving growth of a three-dimensional steady structure, time-periodic secondary instability and nonlinear breakdown restoring a two-dimensional flow. Finally, we show that the sensitivity to wall suction at the second bifurcation is as large as it is at the first bifurcation, providing a possible mechanism for destabilization.

L. Magri, M. P. Juniper

*Journal of Fluid Mechanics*

**752**, 237--265, (2014), doi:10.1017/jfm.2014.328

Open Access

doi: https://doi.org/10.1017/jfm.2014.328

In this theoretical and numerical paper, we derive the adjoint equations for a thermo-acoustic system consisting of an infinite-rate chemistry diffusion flame coupled with duct acoustics. We then calculate the thermo-acoustic system's linear global modes (i.e. the frequency/growth rate of oscillations, together with their mode shapes), and the global modes' receptivity to species injection, sensitivity to base-state perturbations and structural sensitivity to advective-velocity perturbations. Some of these could be found by finite difference calculations but the adjoint analysis is computationally much cheaper. We then compare these with the Rayleigh index. The receptivity analysis shows the regions of the flame where open-loop injection of fuel or oxidizer will have the greatest influence on the thermo-acoustic oscillation. We find that the flame is most receptive at its tip. The base-state sensitivity analysis shows the influence of each parameter on the frequency/growth rate. We find that perturbations to the stoichiometric mixture fraction, the fuel slot width and the heat-release parameter have most influence, while perturbations to the Peclet number have the least influence for most of the operating points considered. These sensitivities oscillate, e.g. positive perturbations to the fuel slot width either stabilizes or destabilizes the system, depending on the operating point. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. The structural-sensitivity analysis shows the influence of perturbations to the advective flame velocity. The regions of highest sensitivity are around the stoichiometric line close to the inlet, showing where velocity models need to be most accurate. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.

S. Balusamy, L. K. B. Li, Z. Han, M. P. Juniper, S. Hochgreb

*Proceedings of the Combustion Institute*

**35**, 426--437, (2014), doi:10.1016/j.proci.2014.05.029

M. P. Juniper, B. Pier

*European Journal of Mechanics B*

**49**, 426--437, (2014), doi:10.1016/j.euromechflu.2014.05.011

Open Access

doi: https://doi.org/10.1016/j.euromechflu.2014.05.011

The structural sensitivity shows where an instability of a fluid flow is most sensitive to changes in internal feedback mechanisms. 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. We also show that the maximum of the structural sensitivity is the location at which the local k+ and k- branches have the same imaginary value. Finally, we use the local analysis to derive the structural sensitivity of two flows: a confined co-flow wake at Re = 400, for which it works very well, and the flow behind a cylinder at Re = 50, for which it works reasonably well. As expected, we find that the local analysis becomes less accurate when the flow becomes less parallel.

L. K. B. Li, M. P. Juniper

*Journal of Fluid Mechanics*

**735**, R5, (2013), doi:10.1017/jfm.2013.533

doi: https://doi.org/10.1017/jfm.2013.533

In a recent study on a coupled laser system, Thevenin et al. (Phys. Rev. Lett., vol. 107, 2011, 104101) reported the first experimental evidence of phase trapping, a partially synchronous state characterized by frequency locking without phase locking. To determine whether this state can arise in a hydrodynamic system, we reanalyse the data from our recent experiment on a periodically forced self-excited low-density jet (J. Fluid Mech., vol. 726, 2013, pp. 624?655). We find that this jet exhibits the full range of phase dynamics predicted by model oscillators with weak nonlinearity. These dynamics include (i) phase trapping between phase drifting and phase locking when the jet is forced far from its natural frequency and (ii) phase slipping during phase drifting when it is forced close to its natural frequency. This raises the possibility that similar phase dynamics can be found in other similarly self-excited flows. It also strengthens the validity of using low-dimensional nonlinear dynamical systems based on a universal amplitude equation to model such flows, many of which are of industrial importance.

D. Rodriguez, E. Gennaro, M. P. Juniper

*Journal of Fluid Mechanics*

**734**, R4, (2013), doi:10.1017/jfm.2013.504

doi: https://doi.org/10.1017/jfm.2013.504

The self-excited global instability mechanisms existing in flat-plate laminar separation bubbles are studied here, in order to shed light on the causes of unsteadiness and three-dimensionality of unforced, nominally two-dimensional separated flows. The presence of two known linear global mechanisms, namely an oscillator behaviour driven by local regions of absolute inflectional instability and a centrifugal instability giving rise to a steady three-dimensionalization of the bubble, is studied in a series of model separation bubbles. These results indicate that absolute instability, and consequently a global oscillator behaviour, does not exist for two-dimensional bubbles with a peak reversed-flow velocity below 12% of the free-stream velocity. However, the three-dimensional instability becomes active for recirculation levels as low as urev = 7 %. These findings suggest a route to the three-dimensionality and unsteadiness observed in experiments and simulations substantially different from that usually found in the literature of laminar separation bubbles, in which two-dimensional vortex shedding is followed by three-dimensionalization.

L. Magri, K. Balasubramanian, R. I. Sujith, M. P. Juniper

*Journal of Fluid Mechanics*

**733**, 681--683, (2013), doi:10.1017/jfm.2013.468

doi: https://doi.org/10.1017/jfm.2013.468

arXiv

Perturbations in a non-normal system can grow transiently even if the system is linearly stable. If this transient growth is sufficiently large, it can trigger self-sustained oscillations from small initial disturbances. This has important practical consequences for combustion-acoustic oscillations, which are a persistent problem in rocket and aircraft engines. Balasubramanian and Sujith (Journal of Fluid Mechanics 2008, 594, 29--57) modelled an infinite-rate chemistry diffusion flame in an acoustic duct and found that the transient growth in this system can amplify the initial energy by a factor, G_{max}, of order 10^5 to 10^7. However, recent investigations by L. Magri M. P. Juniper have brought to light certain errors in that paper. When the errors are corrected, G_{max} is found to be of order 1 to 10, revealing that non-normality is not as influential as it was thought to be.

L. K. B. Li, M. P. Juniper

*Journal of Fluid Mechanics*

**726**, 624--655, (2013), doi:10.1017/jfm.2013.223

doi: https://doi.org/10.1017/jfm.2013.223

The ability of hydrodynamically self-excited jets to lock into strong external forcing is well known. Their dynamics before lock-in and the specific bifurcations through which they lock in, however, are less well known. In this experimental study, we acoustically force a low-density jet around its natural global frequency. We examine its response leading up to lock-in and compare this to that of a forced van der Pol oscillator. We find that, when forced at increasing amplitudes, the jet undergoes a sequence of two nonlinear transitions: (i) from periodicity to T2 quasiperiodicity via a torus-birth bifurcation; and then (ii) from T2 quasiperiodicity to 1:1 lock-in via either a saddle-node bifurcation with frequency pulling, if the forcing and natural frequencies are close together, or a torus-death bifurcation without frequency pulling, but with a gradual suppression of the natural mode, if the two frequencies are far apart. We also find that the jet locks in most readily when forced close to its natural frequency, but that the details contain two asymmetries: the jet (i) locks in more readily and (ii) oscillates more strongly when it is forced below its natural frequency than when it is forced above it. Except for the second asymmetry, all of these transitions, bifurcations and dynamics are accurately reproduced by the forced van der Pol oscillator. This shows that this complex (infinite-dimensional) forced self-excited jet can be modelled reasonably well as a simple (three-dimensional) forced self-excited oscillator. This result adds to the growing evidence that open self-excited flows behave essentially like low-dimensional nonlinear dynamical systems. It also strengthens the universality of such flows, raising the possibility that more of them, including some industrially relevant flames, can be similarly modelled.

L. Magri, M. P. Juniper

*Journal of Engineering for Gas Turbines and Power*

**135**, 091604-1, (2013), doi:10.1115/1.4024957

doi: https://doi.org/10.1115/1.4024957

In this paper, we develop a linear technique that predicts how the stability of a thermoacoustic system changes due to the action of a generic passive feedback device or a generic change in the base state. From this, one can calculate the passive device or base state change that most stabilizes the system. This theoretical framework, based on adjoint equations, is applied to two types of Rijke tube. The first contains an electrically heated hot wire, and the second contains a diffusion flame. Both heat sources are assumed to be compact, so that the acoustic and heat release models can be decoupled. We find that the most effective passive control device is an adiabatic mesh placed at the downstream end of the Rijke tube. We also investigate the effects of a second hot wire and a local variation of the cross-sectional area but find that both affect the frequency more than the growth rate. This application of adjoint sensitivity analysis opens up new possibilities for the passive control of thermoacoustic oscillations. For example, the influence of base state changes can be combined with other constraints, such as that the total heat release rate remains constant, in order to show how an unstable thermoacoustic system should be changed in order to make it stable

G. Ghirardo, M. P. Juniper

*Proceedings of the Royal Society A.*

**469**, 20130232, (2013), doi:10.1098/rspa.2013.0232

doi: https://doi.org/10.1098/rspa.2013.0232

This theoretical study investigates spinning and standing modes in azimuthally symmetric annular combustion chambers. Both modes are observed in experiments and simulations, and an existing model predicts that spinning modes are the only stable state of the system. We extend this model to take into account the effect that the acoustic azimuthal velocity, u, has on the flames, and propose a phenomenological model based on experiments performed on transversely forced flames. This model contains a parameter, delta, that quantifies the influence that the transversal excitation has on the fluctuating heat release. For small values of delta, spinning modes are the only stable state of the system. In an intermediate range of delta, both spinning and standing modes are stable states. For large values of delta, standing modes are the only stable state. This study shows that a flame?s response to azimuthal velocity fluctuations plays an important role in determining the type of thermoacoustic oscillations found in annular combustors.

K. Kashinath, S. Hemchandra, M. P. Juniper

*Combustion and Flame*

**160**(12), 2856--2865, (2013), doi:10.1016/j.combustflame.2013.06.019

doi: https://doi.org/10.1016/j.combustflame.2013.06.019

When a premixed flame is placed within a duct, acoustic waves induce velocity perturbations at the flame?s base. These travel down the flame, distorting its surface and modulating its heat release. This can induce self-sustained thermoacoustic oscillations. Although the phase speed of these perturbations is often assumed to equal the mean flow speed, experiments conducted in other studies and Direct Numerical Simulation (DNS) conducted in this study show that it varies with the acoustic frequency. In this paper, we examine how these variations affect the nonlinear thermoacoustic behaviour. We model the heat release with a nonlinear kinematic G-equation, in which the velocity perturbation is modelled on DNS results. The acoustics are governed by linearised momentum and energy equations. We calculate the flame describing function (FDF) using harmonic forcing at several frequencies and amplitudes. Then we calculate thermoacoustic limit cycles and explain their existence and stability by examining the amplitude-dependence of the gain and phase of the FDF. We find that, when the phase speed equals the mean flow speed, the system has only one stable state. When the phase speed does not equal the mean flow speed, however, the system supports multiple limit cycles because the phase of the FDF changes significantly with oscillation amplitude. This shows that the phase speed of velocity perturbations has a strong influence on the nonlinear thermoacoustic behaviour of ducted premixed flames.

K. Kashinath, S. Hemchandra, M. P. Juniper

*Journal of Engineering for Gas Turbines and Power*

**135**, 061502, (2013), doi:10.1115/1.4023305

doi: https://doi.org/10.1115/1.4023305

Nonlinear analysis of thermoacoustic instability is essential for the prediction of the frequencies, amplitudes, and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper, an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, however, it can be used to calculate the amplitudes of limit cycles and their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modeled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearized momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, along with twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact, resulting in a larger limit cycle amplitude. Multimode simulations show that, in some situations, the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

U. Qadri, D. Mistry and M. P. Juniper

*Journal of Fluid Mechanics*

**720**, 558--581, (2013), doi:10.1017/jfm.2013.34

doi: https://doi.org/10.1017/jfm.2013.34

data for fig 3: base flow, eigenvalues, and most unstable eigenfunction for Re=200, Sw=0.915

Previous numerical simulations have shown that vortex breakdown starts with the formation of a steady axisymmetric bubble and that an unsteady spiralling mode then develops on top of this. We investigate this spiral mode with a linear global stability analysis around the steady bubble and its wake. We obtain the linear direct and adjoint global modes of the linearized Navier?Stokes equations and overlap these to obtain the structural sensitivity of the spiral mode, which identifies the wavemaker region. We also identify regions of absolute instability with a local stability analysis. At moderate swirls, we find that the m = ?1 azimuthal mode is the most unstable and that the wavemaker regions of the m = ?1 mode lie around the bubble, which is absolutely unstable. The mode is most sensitive to feedback involving the radial and azimuthal components of momentum in the region just upstream of the bubble. To a lesser extent, the mode is also sensitive to feedback involving the axial component of momentum in regions of high shear around the bubble. At an intermediate swirl, in which the bubble and wake have similar absolute growth rates, other researchers have found that the wavemaker of the nonlinear global mode lies in the wake. We agree with their analysis but find that the regions around the bubble are more influential than the wake in determining the growth rate and frequency of the linear global mode. The results from this paper provide the first steps towards passive control strategies for spiral vortex breakdown.

I. C. Waugh, S. Illingworth, M. Juniper

*Journal of Computational Physics*

**240**, 183--202, (2013), doi:10.1016/j.jcp.2012.12.034

doi: https://doi.org/10.1016/j.jcp.2012.12.034

In order to define the nonlinear behaviour of a thermoacoustic system, it is important to find the regions of parameter space where limit cycles exist. Continuation methods find limit cycles numerically in the time domain, with no additional assumptions other than those used to form the governing equations. Once the limit cycles are found, these continuation methods track them as the operating condition of the system changes.

Most continuation methods are impractical for finding limit cycles in large thermoacoustic systems because the methods require too much computational time and memory. In the literature, there are therefore only a few applications of continuation methods to thermoacoustics, all with low-order models.

Matrix-free shooting methods efficiently calculate the limit cycles of dissipative systems and have been demonstrated recently in fluid dynamics, but are as yet unused in thermoacoustics. These matrix-free methods are shown to converge quickly to limit cycles by implicitly using a ?reduced order model? property. This is because the methods preferentially use the influential bulk motions of the system, whilst ignoring the features that are quickly dissipated in time.

The matrix-free methods are demonstrated on a model of a ducted 2D diffusion flame, and the stability limits are calculated as a function of the Peclet number and the heat release parameter. Both subcritical and supercritical Hopf bifurcations are found. 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. Two types of shooting are compared, and two types of matrix-free evaluation are compared.

L. Magri and M. P. Juniper

*Journal of Fluid Mechanics*

**719**, 183--202, (2013), doi:10.1017/jfm.2012.639

doi: https://doi.org/10.1017/jfm.2012.639

We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat-release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release time-delay coefficient, the heat-release parameter, and the hot-wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates.

S. Illingworth, I. Waugh, M. Juniper

*Proceedings of the Combustion Institute*

**34**, 911--920, (2012), doi:10.1016/j.proci.2012.06.017

doi: https://doi.org/10.1016/j.proci.2012.06.017

This paper examines nonlinear thermoacoustic oscillations of a ducted Burke?Schumann diffusion flame. The nonlinear dynamics of the thermoacoustic system are studied using two distinct approaches. In the first approach, a continuation analysis is performed to find limit cycle amplitudes over a range of operating conditions. The strength of this approach is that one can characterize the coupled system?s nonlinear behaviour over a large parameter space with relative ease. It is not able to give physical insight into that behaviour, however. The second approach uses a Flame Describing Function (FDF) to characterize the flame?s response to harmonic velocity fluctuations over a range of forcing frequencies and forcing amplitudes, from which limit cycle amplitudes can be found. A strength of the FDF approach is that it reveals the physical mechanisms responsible for the behaviour observed. However, the calculation of the FDF is time consuming, and it must be recalculated if the flame?s operating conditions change. With the strengths and shortcomings of the two approaches in mind, this paper advocates combining the two to provide the dynamics over a large parameter space and, furthermore, physical insight into that behaviour at judiciously-chosen points in the parameter space. Further physical insight concerning the flame?s near-linear response at all forcing amplitudes is given by studying the forced flame in the time domain. It is shown that, for this flame model, the limit cycles arise because of the flame?s nonlinear behaviour when it is close to the inlet.

L. K. B. Li, M. P. Juniper

*Proceedings of the Combustion Institute*

**34**, 947--954, (2012), doi:10.1016/j.proci.2012.06.022

doi: https://doi.org/10.1016/j.proci.2012.06.022

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.

B. Emerson, J. O'Connor, M. P. Juniper, T. Lieuwen

*Journal of Fluid Mechanics*

**706**, 219--250, (2012), doi:doi:10.1017/jfm.2012.248

doi: https://doi.org/doi:10.1017/jfm.2012.248

The wake characteristics of bluff-body-stabilized flames are a strong function of the density ratio across the flame and the relative offset between the flame and shear layer. This paper describes systematic experimental measurements and stability calculations of the dependence of the flow field characteristics and flame sheet dynamics upon flame density ratio, rho_u/rho_b, over the Reynolds number range of 1000?3300. We show that two fundamentally different flame/flow behaviours are observed at high and low rho_u/rho_b values: a stable, noise-driven fixed point and limit-cycle oscillations, respectively. These results are interpreted as a transition from convective to global instability and are captured well by stability calculations that used the measured velocity and density profiles as inputs. However, in this high-Reynolds-number flow, the measurements show that no abrupt bifurcation in flow/flame behaviour occurs at a given rho_u/rho_b value. Rather, the flow field is highly intermittent in a transitional rho_u/rho_b range, with the relative fraction of the two different flow/flame behaviours monotonically varying with rho_u/rho_b. This intermittent behaviour is a result of parametric excitation of the global mode growth rate in the vicinity of a supercritical Hopf bifurcation. It is shown that this parametric excitation is due to random fluctuations in relative locations of the flame and shear layer.

G. J. Chandler, M. P. Juniper, J. W. Nichols, P. J. Schmid

*Journal of Computational Physics*

**231**, 1900--1916, (2012), doi:10.1016/j.jcp.2011.11.013

doi: https://doi.org/10.1016/j.jcp.2011.11.013

This paper describes a derivation of the adjoint low Mach number equations and their implementation and validation within a global mode solver. The advantage of using the low Mach number equations and their adjoints is that they are appropriate for flows with variable density, such as flames, but do not require resolution of acoustic waves. Two versions of the adjoint are implemented and assessed: a discrete-adjoint and a continuous-adjoint. The most unstable global mode calculated with the discrete-adjoint has exactly the same eigenvalue as the corresponding direct global mode but contains numerical artifacts near the inlet. The most unstable global mode calculated with the continuous-adjoint has no numerical artifacts but a slightly different eigenvalue. The eigenvalues converge, however, as the timestep reduces. Apart from the numerical artifacts, the mode shapes are very similar, which supports the expectation that they are otherwise equivalent. The continuous-adjoint requires less resolution and usually converges more quickly than the discrete-adjoint but is more challenging to implement. Finally, the direct and adjoint global modes are combined in order to calculate the wavemaker region of a low density jet.

M. P. Juniper, O. Tammisola

*Journal of Fluid Mechanics*

**686**, 218--238, (2011), doi:10.1017/jfm.2011.324

doi: https://doi.org/10.1017/jfm.2011.324

Base flow data (Matlab format)

Global mode data (Matlab format)

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.

I. C. Waugh, M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**3**(3), 225--242, (2011)

This paper explores the mechanism of triggering in a simple thermoacoustic system, the Rijke tube. It is demonstrated that additive stochastic perturbations can cause triggering before the linear stability limit of a thermoacoustic system. When triggering from low noise amplitudes, the system is seen to evolve to self-sustained oscillations via an unstable periodic solution of the governing equations. Practical stability is introduced as a measure of the stability of a linearly stable state when finite perturbations are present. The concept of a stochastic stability map is used to demonstrate the change in practical stability limits for a system with a subcritical bifurcation, once stochastic terms are included. The practical stability limits are found to be strongly dependent on the strength of noise.

M. P. Juniper

*International Journal of Spray and Combustion Dynamics*

**3**(3), 209--224, (2011)

This theoretical paper examines a non-normal and nonlinear model of a horizontal Rijke tube. Linear and non-linear optimal initial states, which maximize acoustic energy growth over a given time from a given energy, are calculated. It is found that non-linearity and non-normality both contribute to transient growth and that, for this model, linear optimal states are only a good predictor of non-linear optimal states for low initial energies. Two types of non-linear optimal initial state are found. The first has strong energy growth during the first period of the fundamental mode but loses energy thereafter. The second has weaker energy growth during the first period but retains high energy for longer. The second type causes triggering to self-sustained oscillations from lower energy than the first and has higher energy in the fundamental mode. This suggests, for instance, that low frequency noise will be more effective at causing triggering than high frequency noise.

I. C. Waugh, M. Geuss, M. P. Juniper

*Proceedings of the Combustion Institute*

**33**, 2945--2952, (2011), doi:10.1016/j.proci.2010.06.018

doi: https://doi.org/10.1016/j.proci.2010.06.018

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.

M. P. Juniper

*Journal of Fluid Mechanics*

**667**, 272--308, (2011), doi:10.1017/S0022112010004453

doi: https://doi.org/10.1017/S0022112010004453

With a sufficiently large impulse, a thermoacoustic system can reach self-sustained oscillations even when it is linearly stable, a process known as triggering. In this paper, a procedure is developed to find the lowest initial energy that can trigger self-sustained oscillations, as well as the corresponding initial state. This is known as the ?most dangerous? initial state. The procedure is based on adjoint looping of the nonlinear governing equations, combined with an optimization routine. It is developed for a simple model of a thermoacoustic system, the horizontal Rijke tube, and can be extended to more sophisticated thermoacoustic models. It is observed that the most dangerous initial state grows transiently towards an unstable periodic solution before growing to a stable periodic solution. The initial energy required to trigger these self- sustained oscillations is much lower than the energy of the oscillations themselves and slightly lower than the lowest energy on the unstable periodic solution. It is shown that this transient growth arises due to non-normality of the governing equations. This is analogous to the sequence of events observed in bypass transition to turbulence in fluid mechanical systems and has the same underlying cause. The most dangerous initial state is calculated as a function of the heat-release parameter. It is found that self-sustained oscillations can be reached over approximately half the linearly stable domain. Transient growth in real thermoacoustic systems is 10^5?10^6 times greater than that in this simple model. One practical conclusion is that, even in the linearly stable regime, it may take very little initial energy for a real thermoacoustic system to trigger to high-amplitude self-sustained oscillations through the mechanism described in this paper.

The Web of Science ranked this paper 373 out of 82732 articles (top 0.45%) published between 2010 to 2014 in the field of Mechanics.

S. J. Rees, M. P. Juniper

*Journal of Fluid Mechanics*

**656**, 309--336, (2010), doi:10.1017/S0022112010001060

doi: https://doi.org/10.1017/S0022112010001060

This theoretical study examines confined viscous planar jet/wake flows with continuous velocity profiles. These flows are characterized by the shear, confinement, Reynolds number and shear-layer thickness. The primary aim of this paper is to determine the effect of confinement on viscous jets and wakes and to compare these results with corresponding inviscid results. The secondary aim is to consider the effect of viscosity and shear-layer thickness. A spatio-temporal analysis is performed in order to determine absolute/convective instability criteria. This analysis is carried out numerically by solving the Orr?Sommerfeld equation using a Chebyshev collocation method. Results are produced over a large range of parameter space, including both co-flow and counter-flow domains and confinements corresponding to 0.1 < h2/h1 < 10, where the subscripts 1 and 2 refer to the inner and outer streams, respectively. The Reynolds number, which is defined using the channel width, takes values between 10 and 1000. Different velocity profiles are used so that the shear layers occupy between 1/2 and 1/24 of the channel width. Results indicate that confinement has a destabilizing effect on both inviscid and viscous flows. Viscosity is found always to be stabilizing, although its effect can safely be neglected above Re = 1000. Thick shear layers are found to have a stabilizing effect on the flow, but infinitely thin shear layers are not the most unstable; having shear layers of a small, but finite, thickness gives rise to the strongest instability.

P. J. Schmid, L. K. B. Li, M. P. Juniper, O. Pust

*Theoretical and Computational Fluid Dynamics*

**25**, 0935-4964, (2010), doi:10.1007/s00162-010-0203-9

doi: https://doi.org/10.1007/s00162-010-0203-9

The decomposition of experimental data into dynamic modes using a data-based algorithm is applied to Schlieren snapshots of a helium jet and to time-resolved PIV-measurements of an unforced and harmonically forced jet. The algorithm relies on the reconstruction of a low-dimensional inter-snapshot map from the available flow field data. The spectral decomposition of this map results in an eigenvalue and eigenvector representation (referred to as dynamic modes) of the underlying fluid behavior contained in the processed flow fields. This dynamic mode decomposition allows the breakdown of a fluid process into dynamically revelant and coherent structures and thus aids in the characterization and quantification of physical mechanisms in fluid flow.

S. J. Rees, M. P. Juniper

*Journal of Fluid Mechanics*

**633**, 71--97, (2009), doi:10.1017/S0022112009007186

doi: https://doi.org/10.1017/S0022112009007186

In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined inviscid jet/wake flows with surface tension in order to determine convective/absolute instability criteria. There is a single mode that is due to surface tension and many modes that are due to the jet/wake column. In the unconfined case, the full impulse response is considered in the entire outer flow. On the one hand, the surface tension mode propagates slowly in the cross-stream direction but dominates at the front and back of the wavepacket. On the other hand, the jet/wake column modes propagate more quickly in the cross-stream direction and therefore define the boundaries of the central region of the wavepacket. The flow is particularly unstable when these modes interact. For unconfined flows, it is found that at low and intermediate surface tensions the flow can be more absolutely unstable than that without surface tension but at high surface tensions the flow is stabilized. The effect of confinement has previously been studied but not with the inclusion of surface tension. Confinement and surface tension combined cause the transition from convective to absolute instability to occur even with significant coflow. This effect is examined over an infinite domain of density ratios and confinement.

M. Juniper, L. Li, J. Nichols

*Proceedings of the Combustion Institute*

**32**(1), 1191--1198, (2008), doi:10.1016/j.proci.2008.05.065

doi: https://doi.org/10.1016/j.proci.2008.05.065

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.

M. Juniper

*Journal of Fluid Mechanics*

**605**, 227--252, (2008), doi:10.1017/S0022112008001547

doi: https://doi.org/10.1017/S0022112008001547

It has recently been shown that the instability characteristics of planar jets and wakes change when the flows are confined between two flat plates. This is due to constructive interaction between modes with zero group velocity in the inner and outer flows. In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined round jets and wakes in order to discover whether the same effect is observed. There are several similarities between the planar case and the round case as well as some significant differences. Nevertheless, the effect of confinement on round flows is found to be very similar to that on planar flows and to act via the same physical mechanism. This paper examines density ratios from 0.001 to 1000 and has important implications for the design of fuel injectors, which often employ confined shear flows at high Reynolds numbers and large density ratios to generate strong mixing in combustion chambers.

M. Juniper

*Journal of Fluid Mechanics*

**590**, 163--185, (2007), doi:10.1017/S0022112007007975

doi: https://doi.org/10.1017/S0022112007007975

In this theoretical study, a linear spatio-temporal analysis is performed on unconfined and confined inviscid jet/wake flows in order to determine whether they are absolutely or convectively unstable. The impulse response is evaluated in the entire outer fluid, rather than just at the point of impulse, over a wide range of density ratios. This confirms that the dominant saddle point can validly migrate into the plane of diverging eigenfunctions. This reveals that, at certain density ratios and shear numbers, the response can grow upstream in some directions with a cross-stream component, even though it decays directly upstream of, and at, the point of impulse. This type of flow is convectively unstable when unconfined, but becomes absolutely unstable when confined. Other effects of confinement are described in a previous paper. Together, these articles have important implications for the design of fuel injectors, which often employ confined shear flows at high Reynolds number and large density ratios to generate strong mixing in combustion chambers.

M. Juniper

*Journal of Fluid Mechanics*

**565**, 171--195, (2006), doi:10.1017/S0022112006001558

doi: https://doi.org/10.1017/S0022112006001558

It has been shown recently that the instability of a two-dimensional wake increases when it is confined in the transverse direction by two flat plates. Confinement causes the transition from convective to absolute instability to occur at lower values of shear. This paper examines this effect comprehensively and concludes that it is due to the constructive interaction of modes with zero group velocity in the wake (or jet) and in the surrounding flow. Maximum instability occurs when the wavenumber of the fundamental mode in the wake (or jet) matches that of the funda- mental mode in the surrounding flow. Other regions of high instability occur when the harmonics of one mode interact with the fundamental of the other. This effect is examined at density ratios from 0.001 to 1000. At each density ratio, the confinement which causes maximum absolute instability can be predicted. This study also shows that it is vital to examine the wavenumber of absolutely unstable modes in order to avoid over-predicting the absolute instability. In some situations this wavenumber is vanishingly small and the mode must be discounted on physical grounds.

S. Candel, M. Juniper, G. Singla, P. Scouflaire, J.-C. Rolon

*Combustion Science and Technology*

**178**, 161--192, (2006), doi:10.1080/00102200500292530

doi: https://doi.org/10.1080/00102200500292530

A detailed understanding of liquid propellant combustion is necessary for the development of improved and more reliable propulsion systems. This article describes experimental investigations aimed at providing such a fundamental basis for design and engineering of combustion components. It reports recent applications of imaging techniques to cryogenic combustion at high pressure. The flame structure is investigated in the transcritical range where the pressure exceeds the critical pressure of oxygen (5.04 MPa) but the temperature of the injected liquid oxygen is below its critical value (154 K). Data obtained from imaging of OH radicals emission, CH radicals emission in the case of LOx/GCH4 flames and backlighing provide a detailed view of the flame structure for a set of injection conditions. The data may be used to guide numerical modelling of transcritical flames and the theoretical and numerical analysis of the stabilization process. Calculations of the flame edge are used to illustrate this aspect. Results obtained may also be employed to devise engineering modelling tools and methodologies for component development aimed at improved efficiency and augmented reliability.

M. P. Juniper, S. M. Candel

*Journal of Fluid Mechanics*

**482**, 257--269, (2003), doi:10.1017/S0022112003004075

doi: https://doi.org/10.1017/S0022112003004075

A two-dimensional wake-like compound flow, formed by a low-speed stream embedded within a high-speed flow, is examined in this article. It is shown that the range of absolutely unstable flow in parameter space greatly increases when such a flow is confined within a duct. Parameters studied here are: the density ratio, which is from 0.1 to 1000; the velocity ratio, which varies from co-flow to counter-flow; and the ratio of the duct width to the width of the central jet. Absolutely unstable flows permit perturbations to propagate upstream, and can lead to self-sustained global oscillations similar to the vortex shedding process which takes place in the wake of a bluff body. This theoretical situation models the wake-like behaviour of a two-fluid coaxial injector with a recessed central tube. The aerodynamic destabilizing mechanism is of primary importance whereas the stabilizing mechanisms, which are not considered here, are of secondary importance. The conclusions from this analysis of a ducted compound flow can explain why one observes self-sustained oscillations in recessed coaxial injectors. The presence of a recirculation bubble in the central flow, which is the basis of other proposed explanations, is not required.

M. P. Juniper, S. M. Candel

*AIAA Journal of Propulsion and Power*

**19**(3), 332--341, (2003), doi:10.2514/2.6134

doi: https://doi.org/10.2514/2.6134

The stabilization of a flame behind a step over a liquid reactant is discussed. Dimensional analysis is performed to highlight the parameters that could be influential. Simpler configurations are studied first, including a crossflow flame and the flame in a boundary layer over a liquid fuel without a step. From systematic numerical calculations it is found that the most effective parameter regarding stabilization is the height of the step with respect to the flame's thickness. If the flame is thin, it tucks behind the step and is insensitive to the other parameters. If the flame is thick, it cannot remain in the slow-moving zone behind the step and is exposed to the freestream. It is then very sensitive to the Damkohler number and is readily blown off. The numerical simulations are performed on a configuration that represents a cryogenic coaxial injector between gaseous hydrogen flowing at high speed and a stream of low-speed liquid oxygen. Nevertheless, the non-dimensional results should be valid over a wide range of scales and reactant combinations.

M. P. Juniper, N. Darabiha, S. M. Candel

*Combustion and Flame*

**135**(1--2), 87--96, (2003), doi:10.1016/S0010-2180(03)00149-4

doi: https://doi.org/10.1016/S0010-2180(03)00149-4

A counterflow diffusion flame between gaseous hydrogen and liquid oxygen (LOx) is studied numerically at 1 and 2 bar pressures. Conditions at the liquid interface are modelled using the Clausius-Clapeyron relation and the species profiles are evaluated with a one-dimensional numerical code. Complex chemistry and multicomponent transport are employed. Thermodynamic and transport properties are taken from Chemkin and the corresponding Transport packages. Typical species and temperature profiles are presented. The extinction strain rate is evaluated as a function of the inlet hydrogen temperature. This varies from 1.2 x 10^5 s^{-1} at a hydrogen temperature of 20K to 6.0 x 10^5 s^{-1} at a temperature of 310K, indicating that hydrogen/LOx flames are extremely resistant to strain rate. The effect of the temperature gradient on the liquid side of the interface is examined and found to be negligible. When applied to one aspect of the flame-holding in cryogenic rocket motors, these results may be used to infer that extinction by strain rate is improbable in the injector near-field, even for very low hydrogen stream temperatures.

M. P. Juniper, S. M. Candel

*Combustion Theory and Modelling*

**7**(3), 563--577, (2003), doi:10.1088/1364-7830/7/3/307

doi: https://doi.org/10.1088/1364-7830/7/3/307

Cross-flow flames, formed between two reactants approaching at 90 degrees, have many similarities to edge flames formed between parallel-flowing reactants. Two types can be distinguished: one whose inlet velocity profiles have a uniform strain rate and another whose inlet velocity profile is flat. Dimensional analysis suggests that the distance between the flame head and the confluence point, Lc ,is affected by a Damkohler number. A simplified solution for the relationship between the non-dimensional stand-off distance Pi and the Damkohler number is determined here by correlating the results of several hundred numerical simulations. For a cross-flow flame that is controlled by the strain rate, it is found that L ~ A D^(1/2) tau^(1/2), where A is the strain rate, D is the diffusivity and tau is the chemical time. For a convection-controlled flame, the expression is: L ~ U^3 tau^2 D^{-1}, where U is the entry velocity

M. P. Juniper, A. Tripathi, P. Scouflaire, J.-C. Rolon, S. M. Candel

*Proceedings of the Combustion Institute*

**28**, 1103--1109, (2000), doi:10.1016/S0082-0784(00)80320-3

doi: https://doi.org/10.1016/S0082-0784(00)80320-3

This paper presents new experimental results on cryogenic jet flames formed by a coaxial injector at a pressure of 70 bar, which approaches the pressures found in rocket engines. This element, fed with liquid oxygen and gaseous hydrogen, is placed in a square combustion chamber equipped with quartz windows. The flame is examined via spectroscopy, OH* emission, and backlighting, the aim being to provide basic information on the flame structure. It is found that some of the OH* emission is absorbed by the OH radicals present in the flame. A detailed examination of this effect is presented, in which it is shown that, for this turbulent flame, the Abel transform gives the position of the intense reaction region, whether or not absorption is signficant. The flame is attached to the oxygen injector, as at low pressure. At high pressure, flame expansion is reduced compared with low pressure and is also less dependent on the momentum flux ratio between the hydrogen and the oxygen streams. An analysis of the relevant Damkohler numbers suggests that this is because the rate of combustion is mainly controlled by large-scale turbulent mixing at high pressure, and it is dominated by jet break-up, atomization, and vaporization at low pressures. Jet break-up is particularly dependent on the momentum flux ratio. Finally, the mean volumetric heat release rates and flame surface density in the experimental facility are estimated.

### Selected conference proceedings

N. Jamieson and M. P. Juniper

*ASME Turbo Expo, Charlotte, NC, USA*, GT2017-63441, (2017)

We examine the shifts in linear decay rates and growth rates, with and without control devices on a simple thermoacoustic system. When the system is stable, we measure the decay rate by pulsing with a loudspeaker. This improves on the experimental techniques developed by Rigas et al. (J. Fluid Mech., 2016, vol. 787, R1 [1]) and Jamieson et al. (Int. J. Spray and Comb. Dyn., accepted, 2016 [2]), to develop a more efficient method of experimental sensitivity analysis to apply in thermoacoustic systems. When the system is unstable, we use feedback control to bring it to a non-oscillating state. We then switch off the feedback control and measure the growth rate. The results suggest that both methods are suitable for use in the experimental sensitivity analysis of thermoacoustic systems. Our experimental set up is automated and we find that we can obtain thousands of decay rates in 1/12 the time compared with the previous work.

O. Tammisola and M. P. Juniper

*ASME Turbo Expo, Montreal Canada*, GT2015-42736, (2015)

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.

B. Semlitsch, A. Orchini, A. P. Dowling, and M. P. Juniper

*Thermoacoustic instabilities in gas turbines and rocket engines: industry meets academia, Munich, Germany, 30 May - 02 June 2016*, (2016)

Self-excited oscillations of unsteady heat release and acoustic waves cannot be entirely avoided for all operating conditions of gas turbines equipped with lean-burn technology. Nevertheless, safe operability can be guaranteed as long as thermo-acoustic limit cycles are diminished to tolerable amplitudes. Numerical simulations aid combustor design to avoid and reduce thermo-acoustic oscillations. Non-linear heat release rate estimation and its modelling are essential for the prediction of saturation amplitudes of limit cycles. The heat release dynamics of arbitrarily-complex flames can be approximated by a Flame Describing Function (FDF). 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 while modelling the velocity fluctuations analytically. 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. However, the phase decay is over-predicted, which can be attributed to the fact that turbulence is neglected. The lack of turbulent dispersion causes a highly correlated heat release response, which is artificial.

S. Illingworth, M. P. Juniper

*21st International Congress on Sound and Vibration, Beijing, China, 13-17 July 2014*, (2014)

A wave-based analysis is a convenient method for generating acoustic models, both for simple and for more complex geometries. However, there is no straightforward method to describe the resulting system in state-space form. This prevents powerful, well-established methods from dynamics and control being applied to a wave-based acoustic model because these methods require a state-space description of the system to which they are applied. This paper presents a simple method for generating a state-space description of an acoustic model when that model has been derived using a wave-based approach. The utility of the method is demonstrated by applying it to a simple open-ended duct with a temperature jump across the flame, and the resulting state-space model is validated in both the time domain and the frequency domain. The method is sufficiently general that it can also be applied to more complex geometries.

U. A. Qadri, M. P. Juniper

*UKACC International Conference on Control, Cardiff, UK, 3-5 Sept 2012*, (2012)

Previous numerical simulations have shown that vortex breakdown starts with the formation of a steady axisymmetric bubble and that an unsteady spiralling mode then develops on top of this. We study how this spiral mode of vortex breakdown might be suppressed or promoted. We use a Lagrangian approach to identify regions of the flow which are sensitive to small open-loop steady and unsteady (harmonic) forces. We find these regions to be upstream of the vortex breakdown bubble. We investigate passive control using a small axisymmetric control ring. In this case, the steady and unsteady control forces are caused by the drag force on the control ring. We find a narrow region upstream of the bubble where the control ring will stabilise the flow and we verify this using numerical simulations.

M. P. Juniper

*Int. Conf. on Sound and Vibration, Vilnius, Lithuania, 08-12 July 2012*, (2012)

In this theoretical study, a weakly nonlinear analysis is performed on a generic thermoacoustic system. The velocity and heat release are assumed to be periodic. The heat release fluctuations are characterized by their phase, phi, relative to the velocity fluctuations, and by their amplitude, A. Both phi and A are functions of the velocity amplitude, r. Around r = 0, phi must be known up to the second derivative with respect to r, and A up to the third derivative. A standard linear analysis shows that the point of linear instability (the Hopf bifurcation point) is determined only by zeta, the first derivative of A, and the zeroth derivative of phi (i.e. the value of phi when r = 0). The weakly nonlinear analysis shows that the type of bifurcation (super- critical or subcritical) is determined by a simple expression containing the first, second, and third derivatives of A, and the zeroth, first, and second derivatives of phi. The functions phi(r) and A(r), which characterize a flame?s response to forcing, can be measured from experiments or numerical simulations. They are called the Flame Describing Function. This analysis quickly reveals the type of bifurcation that this flame will cause and whether this behaviour is due to phase-dependence, amplitude-dependence, or some combination of the two.

S. Illingworth and M. P. Juniper

*Int. Conf. on Sound and Vibration, Vilnius, Lithuania, 08-12 July 2012*, (2012)

In any thermoacoustic analysis, it is important not only to predict linear frequencies and growth rates, but also the amplitude and frequencies of any limit cycles. The Flame Describing Function (FDF) approach is a quasi-linear analysis which allows the prediction of both the linear and nonlinear behaviour of a thermoacoustic system. This means that one can predict linear growth rates and frequencies, and also the amplitudes and frequencies of any limit cycles. The FDF achieves this by assuming that the acoustics are linear and that the flame, which is the only nonlinear element in the thermoacoustic system, can be adequately described by considering only its response at the frequency at which it is forced. Therefore any harmonics generated by the flame's nonlinear response are not considered. This implies that these nonlinear harmonics are small or that they are sufficiently filtered out by the linear dynamics of the system (the low-pass filter assumption). In this paper, a flame model with a simple saturation nonlinearity is coupled to simple duct acoustics, and the success of the FDF in predicting limit cycles is studied over a range of flame positions and acoustic damping parameters. Although these two parameters affect only the linear acoustics and not the nonlinear flame dynamics, they determine the validity of the low-pass filter assumption made in applying the flame describing function approach. Their importance is highlighted by studying the level of success of an FDF-based analysis as they are varied. This is achieved by comparing the FDF?s prediction of limit-cycle amplitudes to the amplitudes seen in time domain simulations.

G. Campa and M. P. Juniper

*ASME Turbo Expo, Copenhagen, Denmark, 11-15 June 2012*, GT2012-68241, (2012)

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.

M. P. Juniper

*ASME Turbo Expo, Copenhagen, Denmark, 11-15 June 2012*, GT2012-68253, (2012)

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.

### Group theses

A Orchini

*University of Cambridge*, (2016), examined by N. Noiray and E. Mastorakos

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.

G. Ghirardo

*University of Cambridge*, (2015), examined by F. Nicoud and R. S. Cant

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.

L. Magri

*University of Cambridge*, (2015), examined by A. Bottaro and A. Agarwal

not yet available

V. Gupta

*University of Cambridge*, (2014), examined by A. Sharma and R. S. Cant

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.

K. Kashinath

*University of Cambridge*, (2013), examined by A. Morgans and E. Mastorakos

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.

U. A. Qadri

*University of Cambridge*, (2013), examined by C. Caulfield and L. Lesshafft

Open Access

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.

I. C. Waugh

*University of Cambridge*, (2013), examined by W. Polifke and R. S. Cant

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.

L. K. B. Li

*University of Cambridge*, (2011), examined by Y. Hardalupas and S. Hochgreb

Open Access

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.

G. J. Chandler

*University of Cambridge*, (2010), examined by J-M Chomaz and C. Caulfield

Open Access

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.

S. J. Rees

*University of Cambridge*, (2009), examined by B. Pier and N. Peake

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.

M. P. Juniper

*Ecole Centrale de Paris*, (2001), examined by S. Candel, N. Darabiha, E. Hopfinger, G. Searby, J-L Thomas, P. Vuillermoz, V. Yang, S. Zaleski

Open Access

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.