Modelling and analysis of nonlinear
thermoacoustic systems using frequency
and time domain methods
Alessandro Orchini
Department of Engineering
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Robinson College September 2016

To my friends.
Mi fan patir costoro il grande stento,
Che vanno il sommo bene investigando,
E per ancor non v’hanno dato drento.
E mi vo col cervello immaginando,
Che questa cosa solamente avviene
Perché non è dove lo van cercando.
Questi dottor non l’han mai intesa bene,
Mai son entrati per la buona via,
Che gli possa condurre al sommo bene.
Perché, secondo l’opinion mia,
A chi vuol una cosa ritrovare,
Bisogna adoperar la fantasia,
E giocar d’invenzione, e ’ndovinare;
E se tu non puoi ire a dirittura,
Mill’altre vie ti posson aiutare.
– Galileo, 1590

Declaration
I hereby declare that, except where specific reference is made to the work of others, the
contents of this dissertation are original and have not been submitted in whole or in part
for consideration for any other degree or qualification in this, or any other University. This
dissertation is my own work and contains nothing which is the outcome of work done in
collaboration with others, except as specified in the text and Acknowledgements. This
dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes,
tables and equations and has fewer than 150 figures.
Alessandro Orchini
September 2016

Acknowledgements
I would like to thank Professor Matthew Juniper, who offered me the opportunity to study in
Cambridge. I have (hopefully) learnt from him the art of writing a scientific paper, being less
impulsive and more critic with my research/myself. As a general rule, PhD students tend to
treat their supervisor as a scapegoat in difficult situations. I likely have been no exception.
Sorry for that.
Thanks to Dr. Simon Illingworth and Dr. Iain Waugh for their support during the first
year of my PhD, and for having helped me in understanding LOTAN, LSGEN, and many
other British acronyms. Thanks to Peter Benie for his invaluable help with (any sort of)
programming problems. Thanks to Professors Jonas Moeck and Nicolas Noiray, who showed
interest in my research and have advised me multiple times.
I have been lucky enough to meet and work with many great people in the last four years.
Thanks to Giulio for everything, ranging from endless discussions on thermoacoustics to the
nights spent in a caravan on the Dolomites. And thanks to George(ios) for another set of
everlasting discussions on fluid mechanics, and for summer barbecues.
Thanks to all the other friends/colleagues – Vik, Karthik, José, Francesca, Caro, Chris,
Outi, Ewa, Jack, Nick, Juan, Bernhard, Luca, Pau, Ubaid and Chiara – for the formals, hiking
trips, work discussions, robot’s brain programming, tango and salsa classes, drinks, chess
games, punting and kayaking afternoons, picnics and language exchange we had together.
Thanks to my climbing mates, particularly Alex and Lisa, for all the fun we had at and
around the wall.
Thanks to my family – Susy, Fede, Fra – and friends – Nuccia e Fede, Ema, Simone,
Momo e Dado, Ari, Silvina, Bute, and the “physicists’ gang” – who constantly checked that
I am doing fine here, even if it always rains.
And thanks to Carolina for having shared with me part of this adventure. Unfortunately
it had some collateral effects.
A PhD is a long and perilous path, which I would not have been able to walk on alone.
Thank you all :)
Ale

Abstract
Thermoacoustic oscillations may arise in combustion chambers when unsteady heat release
and acoustic fluctuations constructively interfere. These oscillations generally lead to un-
desired 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 ampli-
tude 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
xsecond 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.
Table of contents
1 Introduction 1
1.1 Nonlinear phenomena in thermoacoustics . . . . . . . . . . . . . . . . . . 2
1.1.1 Flame Describing Function . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Harmonic balance . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Acoustic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Flame modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Thesis scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Linear stability and adjoint sensitivity analysis of thermoacoustic networks 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Linear acoustics: wave-based approach and state-space representation . . . 20
2.2.1 Wave-based modelling approach . . . . . . . . . . . . . . . . . . . 21
2.2.2 Finding a state-space model . . . . . . . . . . . . . . . . . . . . . 24
2.3 Linear flame dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Forced response to harmonic fluctuations . . . . . . . . . . . . . . . . . . 28
2.5 Curvature effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Self-excited problem: eigenvalue problem formulation . . . . . . . . . . . 36
2.7 Linear analysis of a thermoacoustic network . . . . . . . . . . . . . . . . . 39
2.7.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7.2 Linear stability of a ducted flame . . . . . . . . . . . . . . . . . . . 42
2.7.3 Adjoint-based sensitivity . . . . . . . . . . . . . . . . . . . . . . . 46
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 Frequency and time domain nonlinear analysis of thermoacoustic oscillations 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Nonlinear flame modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Effect of the temperature jump on the acoustic response . . . . . . . . . . . 56
xii Table of contents
3.4 Nonlinear analysis in the frequency domain . . . . . . . . . . . . . . . . . 58
3.4.1 Flame Describing Function . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 Harmonic balance . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Nonlinear dynamics in the time domain . . . . . . . . . . . . . . . . . . . 64
3.5.1 Time-marching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.2 Numerical continuation of limit cycles . . . . . . . . . . . . . . . . 67
3.6 Comparison between frequency and time domain results . . . . . . . . . . 69
3.6.1 Quasi-linear regime: frequency of oscillations . . . . . . . . . . . . 69
3.6.2 Fully nonlinear regime: limits of the FDF method . . . . . . . . . . 70
3.6.3 Numerical continuation results . . . . . . . . . . . . . . . . . . . . 73
3.7 Nonlinear dynamical behaviour . . . . . . . . . . . . . . . . . . . . . . . . 74
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Flame Double Input Describing Function analysis 79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Review of the FDF analysis approximations . . . . . . . . . . . . . . . . . 81
4.3 FDIDF assumptions and calculation . . . . . . . . . . . . . . . . . . . . . 87
4.3.1 FDIDF definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 FDIDF amplitude saturation . . . . . . . . . . . . . . . . . . . . . 89
4.3.3 FDIDF calculation and validation . . . . . . . . . . . . . . . . . . 90
4.4 FDIDF analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.1 Linear stability of limit cycles: Neimark–Sacker bifurcations . . . . 93
4.4.2 Prediction and stability of quasiperiodic oscillations . . . . . . . . 97
4.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Weakly nonlinear analysis of thermoacoustic bifurcations 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Thermoacoustic modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.1 Acoustic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2.2 Heat release model . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.3 Damping models . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4 Weakly nonlinear analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.1 O(ε): eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . 118
5.4.2 O(ε2): mean shift and second harmonic . . . . . . . . . . . . . . . 119
5.4.3 O(ε3): third harmonic and saturation . . . . . . . . . . . . . . . . 121
5.4.4 O(ε4): mean shift and fourth harmonic . . . . . . . . . . . . . . . 124
Table of contents xiii
5.4.5 O(ε5): fifth harmonic and saturation . . . . . . . . . . . . . . . . . 125
5.5 Results validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.6 Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Heat release response to forced flow oscillations of a low-order modelled com-
bustor 135
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.1 Flow field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2.2 Numerical implementation and mean flame shape . . . . . . . . . . 142
6.3 Forced acoustic response . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.3.1 Qualitative comparison . . . . . . . . . . . . . . . . . . . . . . . . 145
6.3.2 Heat release response . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4 Convection speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7 Modelling of equivalence ratio fluctuations 155
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.1.1 Mean flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.1.2 Velocity and equivalence ratio perturbations . . . . . . . . . . . . . 158
7.1.3 Flame dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2 FDF calculation and comparison with the experiments . . . . . . . . . . . 160
7.2.1 Perfectly premixed response . . . . . . . . . . . . . . . . . . . . . 161
7.2.2 Sum of time delays approximation . . . . . . . . . . . . . . . . . . 162
7.2.3 Imperfectly premixed response . . . . . . . . . . . . . . . . . . . . 166
7.3 Harmonic balance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8 Conclusions and future work 177
Appendix A Non-dimensional variables 181
Appendix B Eigenvalue problem 182
Appendix C Growth rate variations by implicit function theorem 185
Appendix D Acoustic matrix coefficients 187
xiv Table of contents
Appendix E Nonlinear expansion terms 188
Appendix F Forcing terms 191
F.1 O(ε3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
F.2 O(ε4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
F.3 O(ε5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
References 195
Chapter 1
Introduction
Thermoacoustic oscillations may arise in systems in which a heat source is enclosed in a
confined environment, which acts as an acoustic resonator (Culick, 2006; Lieuwen, 2012).
Although the phenomenon was first reported two centuries ago by Higgins (1802), and
qualitative understood in terms of coupling between unsteady heat release and acoustic waves
by Rayleigh (1878), it became a problem of academic and industrial relevance only in the
last few decades, with the advent of rocket engine and gas turbine technologies.
Gas turbines play a major role in the conversion of energy from burning fuel. They are
used to create thrust for aircraft, and provide more than 20% of the world’s electrical energy
production. Over the past decades more and more stringent regulations on pollution emissions
have been imposed on gas turbine manufacturers. In particular, the demand of reducing
NOx emissions – which are responsible for acid rain, and react to produce ozone, which
in turn causes breathing problems and is a strong greenhouse gas – led to the development
of combustors working in lean, premixed regimes. These configurations, however, often
exhibit thermoacoustic oscillations. In almost all situations, thermoacoustic oscillations are
undesirable: they produce noise, structural vibrations, and impose limits to the gas turbines
operating conditions, which may reduce the system’s efficiency and power output (Culick,
2006; Lieuwen et al., 2010; Lieuwen, 2012).
Thus, one of the manufacturer’s goals is to design an engine that is not susceptible to
thermoacoustic oscillations over the entire operating range. This is currently achieved by
extensive experimental testing, repeated re-design, and sometimes by the retro-fitting of
damping devices such as Helmholtz resonators. Experiments and high-fidelity numerical
simulations, such as compressible Large Eddy Simulations (LES) of reacting flows (Pitsch,
2002; Moin & Apte, 2006; Moureau et al., 2009), are the most accurate ways to investigate
thermoacoustic oscillations, as they include all physical phenomena involved. They, however,
face some major shortcomings (Poinsot, 2013). First, because the stability of a thermoacoustic
2 Introduction
system depends on a large number of parameters (for example air/fuel composition, flow
rate, and acoustic reflection coefficients), a very large number of experiments or simulations
has to be performed to identify safe operating conditions. It is prohibitively expensive to
use them to perform a thorough investigation on the system stability. Secondly, although
these approaches provide a detailed description of the flow/flame dynamics, they typically
provide little insight on the causes of thermoacoustic instabilities. As well as being of
fundamental interest, this insight is crucial for the designing of strategies for passive control
of thermoacoustic instabilities.
There is therefore considerable industrial motivation in developing low-order analytical
and numerical tools that can predict whether thermoacoustic oscillations will occur in a
system and, if so, understand how to change the system in order to damp them, or predict
their amplitude. This thesis deals with developing and improving such tools.
1.1 Nonlinear phenomena in thermoacoustics
Linear stability analysis is the primary analytical tool that is applied to thermoacoustic
systems to investigate their stability. Under the assumption that the heat source is compact
– i.e., that the extension of the heat source is much smaller than the acoustic wavelengths
of interest – the acoustic and heat release linear responses can be decoupled and measured
separately. The linear dynamic response of a flame to perturbations in the acoustic field or
equivalence ratio mixture is known as Flame Transfer Function (FTF) (Candel, 1992). The
FTF and the combustor acoustic response form a positive feedback loop (see Figure 1.2). The
closed-loop system is referred to as a thermoacoustic network. Its stability can be determined
by looking at the location of the closed-loop system’s poles in the complex plane. This,
together with the aid of adjoint-based sensitivity analysis (Magri & Juniper, 2013a,b), gives
information on the linear behaviour of the system and is helpful in understanding to what
parameters the system’s stability is most susceptible.
There exist situations, however, in which linear stability may fail to predict the real
behaviour of the system, as illustrated in Figure 1.1. This is because noise or non-normal
effects may trigger finite amplitude oscillations, even when the thermoacoustic system is
linearly stable, and lead to a variety of complex behaviour (Balasubramanian & Sujith,
2008; Juniper, 2011; Waugh & Juniper, 2011). Nonlinear effects determine the nature of
the final state of a thermoacoustic oscillation, which can be a fixed point, a limit cycle or a
more complex (non-periodic) solution. All these type of oscillations have been observed
experimentally in thermoacoustic systems (Noiray et al., 2008; Gotoda et al., 2011; Kabiraj
& Sujith, 2012; Jegadeesan & Sujith, 2013; Kashinath et al., 2014).
1.1 Nonlinear phenomena in thermoacoustics 3
Control parameter (a.u.)
Ac
ou
st
ic
 e
ne
rg
y 
(a.
u.)
Bistable region
Stable fixed points Unstable fixed points
Unstable limit cycles
Stab
le lim
it cyc
les
Tr
ig
ge
rin
g 
le
ve
ls
Fold point
Hopf point
Fig. 1.1 Stable (solid) and unstable (dashed) states that a thermoacoustic system may exhibit
close to a subcritical Hopf bifurcation. Linear stability analysis calculates the stability of
fixed point solutions only with respect to small perturbations. In the bistable region, however,
finite amplitude perturbations can trigger the state from the linearly stable fixed points to
stable limit cycles, or vice versa.
1.1.1 Flame Describing Function
It is therefore important to gather information about and model the nonlinear behaviour of
a thermoacoustic system, in order to predict the non-trivial effects of nonlinearities. An
existing method able to predict part of the nonlinear thermoacoustic features is known as
Flame Describing Function (FDF) analysis (Noiray et al., 2008). The FDF generalises the
concept of the FTF to perturbations with finite amplitude, but is a more expensive object to
obtain from experiments or simulations than an FTF. It can be used to study the amplitude
and stability of limit cycle oscillations, approximated as harmonic solutions (Noiray et al.,
2008; Boudy et al., 2011; Illingworth et al., 2013). However, the FDF analysis assumes that
the dynamics of a given thermoacoustic mode is decoupled from the dynamics of the others.
Situations exist in which this is not true (Moeck & Paschereit, 2012). The nonlinear coupling
between the modes can lead to significant changes in the behaviour of a thermoacoustic
system with respect to the one predicted by the FDF. For example, one can observe mode-
switching – i.e., although the FDF approximation predicts thermoacoustic oscillations at a
certain frequency, the system actually converges to oscillations with a different frequency –
or non-periodic thermoacoustic vibrations (Noiray et al., 2008; Kabiraj et al., 2012a), which
the FDF cannot predict.
4 Introduction
Z
(acoustic
network)
F
(flame jump)
p−
ρ−
u−
u−
FDF
+
u−
q′
uext
p+ ρ+ u+ H
FDF
+
u′
q′
uext
1
Fig. 1.2 Overview of a closed-loop thermoacoustic network. ρ , p and u denote the flow
density, acoustic pressure and velocity variables. (a) The jump across the flame element is
highlighted, using superscripts− and + for the acoustic properties upstream and downstream
the flame, respectively. All the remaining acoustic information is embedded into the acoustic
block Z. (b) The same closed-loop thermoacoustic network simplified across the FDF
element; H contains all the linear acoustic response with respect to heat release perturbations.
Lastly, in a situation such as the one depicted in Figure 1.1, multiple stable states may exist.
Because the system is subject to finite amplitude perturbations – such as the noise generated
by the intrinsic combustion process or external causes – its behaviour may switch between
these states in a non-deterministic way. If the effect of noise is modelled as a stochastic
component in the governing equations, a statistical description of the system’s behaviour in
terms of a Probability Density Function (PDF) can be obtained. By extracting the stochastic
modelling parameters from experimental data, this has been shown to accurately describe the
transitions observed between different states in azimuthal thermoacoustic oscillations (Noiray
& Schuermans, 2013b).
1.1.2 Harmonic balance
The calculation of an FDF can be expensive, but if the flame model is not changed, the same
FDF can be used to test many acoustic configurations with a low-cost procedure, known as
harmonic balance (Khalil, 2002).
Under the assumption that the flame is acoustically compact, a generic thermoacoustic
network can be drawn as a block diagram as in Figure 1.2a. The acoustic jump conditions
1.1 Nonlinear phenomena in thermoacoustics 5
across the flame (see Dowling (1995) and Chapter 2) have been highlighted. Their inputs
are the acoustic variables upstream of the flame and the instantaneous heat release fluctu-
ations, q′. The remaining acoustic response is contained in the acoustic block Z. For the
simple configuration composed of two straight ducts interconnected by a flame, it contains
information about the mean flow, end reflection coefficients, and wave time delays (Dowling,
1995). Lastly, the FDF converts velocity disturbances upstream of the flame into heat release
fluctuations.
The feedback loop in Figure 1.2a can be simplified by choosing an external input signal
uext just upstream of the FDF and the heat release as an output, so that the open-loop heat
release response with respect to velocity fluctuations is given by:
q′ = FDF
(
u′+uext
)
. (1.1)
Furthermore, the entire open-loop acoustic response with respect to heat release fluctuations
can be embedded into a single transfer function H so that, for velocity fluctuations upstream
of the flame, we can write (see Figure 1.2b):
u′ = Hq′. (1.2)
For simple acoustic networks, the expression for the transfer function H can be found
analytically (Heckl, 1988; Dowling, 1995, 1997). It becomes rather complicated for complex
networks, and numerical methods are used in these cases to evaluate H over a certain range
of frequencies.
Closing the feedback loop between the velocity at the reference point and the heat release
fluctuations yields:
q′ =
FDF(A,s)
1−FDF(A,s)H(s)uext . (1.3)
Equation (1.3) represents a Single Input Single Output system: if no input velocity is
prescribed, the system will be linearly unstable if and only if it has poles in the r.h.s. of the
complex plane in the zero amplitude limit. The harmonic balance extends this concept to
perturbations with a finite amplitude. The system dynamics when the oscillations have an
amplitude A is determined by the growth rates σ of the Laplace variable s ≡ σ + iω that
satisfy the harmonic balance dispersion relation:
FDF(A,s)H(s) = 1. (1.4)
6 Introduction
The dispersion relation (1.4) is also able to identify poles that have a negative growth rate
at small amplitudes, but become unstable at finite amplitudes. This is a characteristic of
subcritical Hopf bifurcations, and phenomena such as bistability and triggering may be
observed (see Figure 1.1).
Solving the dispersion relation (1.4) at various amplitudes leads to harmonic limit cycle
solutions of the closed-loop system, for which the growth rate σ is equal to zero. Their
stability may be analysed by investigating the change in growth rate across the saturated
amplitude (Noiray et al., 2008; Boudy et al., 2011).
1.2 Acoustic modelling
The compressible Navier–Stokes equations, together with mass and energy conservation
laws, govern the air and fuel dynamics in a combustor. Assuming that viscous effects are
negligible, and ignoring thermal diffusivity (Chu & Kovásznay, 1958; Dowling & Stow,
2003; Schuermans, 2003; Culick, 2006), they read:
Dρ
Dt
+ρ∇ ·u = 0 (1.5a)
Du
Dt
=− 1
ρ
∇p (1.5b)
ρT
Ds
Dt
= Q, (1.5c)
where D/Dt ≡ ∂/∂ t+u ·∇ is the material derivative, u the flow velocity, p the pressure, ρ
the density, T the temperature and Q the heat release rate per unit of volume. The energy
equation is expressed in terms of the entropy, s = cv log(p/ργ).
Because we are interested in investigating the stability of thermoacoustic modes, the flow
equations will be analysed using perturbation theory. In this context, the flow is decomposed
into a mean, stationary component (denoted with overlines) on top of which small unsteady
perturbations (denoted with primes) can grow. Because combustor geometries are typically
axisymmetric, and the flow develops mostly in the axial direction, the conservation laws
can be expanded in cylindrical coordinates (rˆ, θˆ , xˆ) on top of a uniform axial mean flow u,
yielding:
∂ρ ′
∂ t
+u
∂ρ ′
∂x
+ρ
∂u′
∂x
= 0 (1.6a)
∂u′
∂ t
+u
∂u′
∂x
=− 1
ρ
∂ p
∂x
(1.6b)
1.2 Acoustic modelling 7
∂ p′
∂ t
+u
∂ p′
∂x
+ γ p
∂u′
∂x
= 0, (1.6c)
where the entropy equation has been replaced with an equivalent expression for the pressure
field, making use of the state equation p = ρRT .
By assuming that solutions have the form eiωt+ikx, where ω and k are the angular
frequency and axial wavenumber, the linearised equations can be rewritten in matrix form as: −uk −ρk 00 −uk − kρ
0 −kρc2 −uk

 ρˆ ′uˆ′
pˆ′
= ω
 ρˆ ′uˆ′
pˆ′
 , (1.7)
where the speed of sound c≡√γRT has been defined. (1.7) is an eigenvalue problem with
eigenmodes:
Eigenvalue Eigenvector
(ω)
−ku
−k(u+ c)
−k(u− c)
(ρˆ ′, uˆ′, pˆ′)
(+ 1c2 , 0, 0)
(+ 1c2 , +
1
ρc , 1)
(+ 1c2 , − 1ρc , 1)
entropy wave αˆ
acoustic wave fˆ
acoustic wave gˆ
The eigenvalue ω =−ku reflects the convective property of the mean flow. It vanishes
in the case of zero mean flow. The relative eigenmode affects only density and entropy
fluctuations, and it is therefore labelled as an entropy mode. The other two eigenmodes
are acoustic modes, and exist even in the case of zero mean flow. Their eigenvectors have
pressure, velocity and density components, but they do not affect the entropy. Azimuthal
modes can be straightforwardly included in the analysis by accounting for fluctuations in
the azimuthal velocity, uθ , decomposed into Fourier modes (Stow & Dowling, 2001; Noiray
et al., 2011).
By taking linear combinations of the eigenvectors, the fluctuating flow variables can be
decomposed in terms of travelling waves:
p′(x, t) = fˆ e−
xs
u+c + gˆe−
xs
u−c (1.8a)
u′(x, t) =
1
ρc
(
fˆ e−
xs
u+c − gˆe− xsu−c
)
(1.8b)
ρ ′(x, t) =
1
c2
(
αˆe−
xs
u + fˆ e−
xs
u+c + gˆe−
xs
u−c
)
. (1.8c)
The decomposition of the acoustic variables into waves as described above constitutes
the basic idea underlying acoustic network models. An acoustic network is formed by
a series of axisymmetric, straight modules in which acoustic and entropy waves travel
8 Introduction
upstream and downstream. Modules with different properties (for example cross-sectional
area or temperature) are matched together by imposing mass, momentum and energy fluxes
conservation through the Rankine-Hugoniot jump conditions (Dowling, 1995; Dowling &
Stow, 2003). When an unsteady heat release model is embedded in the acoustic model, a
low-order framework for a thermoacoustic system is obtained, referred to as a thermoacoustic
network.
Thermoacoustic networks have been developed and are used for research purposes both
by academia and industry: LOTAN was developed first in Cambridge University then by
Rolls-Royce (Stow & Dowling, 2001), taX in TU Munich (Emmert et al., 2013), OSCILOS
in Imperial College London (Li et al., 2014), Ta3 by (former) ALSTOM (Schuermans, 2003;
Bellucci et al., 2005), and LOMTI by Ansaldo Energia (Campa et al., 2011). In this thesis,
some of the acoustic functionalities of LOTAN have been exploited, as described in Chapter 2.
Also, a simple acoustic network coded in MATLAB/Python and a way to cast the frequency
response of an acoustic network into state-space form have been developed with Dr. Simon
Illingworth, as described in (Orchini et al., 2015).
1.3 Flame modelling
An FTF measured from experiments can be fed into a thermoacoustic network to investigate
the linear stability of a combustor. However, when one is interested in investigating nonlinear
effects, an FDF has to be measured, which can be expensive. Often, low-order models
containing the most relevant features of the flame dynamics are used in place of an accurate
FDF to obtain a cheaper tool to assess the thermoacoustic system’s behaviour. As an example,
simple static relations between the heat release rate and the acoustic pressure and velocity
have been exploited for the analysis of azimuthal thermoacoustic oscillations (Noiray et al.,
2011; Ghirardo & Juniper, 2013). These models, however, tend to be oversimplistic, as they
are static and therefore cannot capture the dependence of the heat release response on the
frequency, which may contain relevant damping or source mechanisms.
For premixed flames, the kinematic G-equation is a common dynamic model used to
determine the position and evolution of a thin flame front, as well as the amount of heat
released by the flame, as functions of the oscillation frequency and amplitude. The G-
equation is based on the assumption that chemical reactions between fuel and oxidisers are
fast, and occur in a very thin region that defines the flame surface, which is found to be
much smaller than laminar flow length scales. Under these conditions, the flame front can
be treated as a surface discontinuity between the unburned and the burned region (Emmons,
1958; Markstein, 1964; Williams, 1985).
1.3 Flame modelling 9
To identify the flame surface, it is convenient to define a scalar field G which takes
negative values inside the flame (reactants zone), positive values outside (products zone),
and is zero on the flame surface. The values of the G-field away from the interface have no
physical meaning. Indeed, the identification of flame’s surface with the G = 0 contour is
just a convention, and any other isoline could have been chosen to locate the discontinuity.
The field is assumed to be continuous and smooth, such that its first two derivatives in all
spatial dimensions can be uniquely defined. The flame surface is thus defined in an implicit
way, and pinch-offs, pockets of fuel that detaches from the main flame body, can be easily
described. Also, geometrical quantities can be defined as functions of G. For example, the
unit vector nˆ normal to the interface can be expressed as:
nˆ ≡− ∇G|∇G| . (1.9)
The direction of the normal is chosen such that it points in the direction of the unburned gas.
This is the direction along which the flame burns, at a speed sL, which can be non-uniform
along the interface.
Assume that an underlying unsteady, non-uniform flow-field u is prescribed. In a La-
grangian sense, we can follow a particle that at time t = 0 lies on the flame surface G(x, t) = 0
by means of the material derivative:
DG
Dt
=
∂G
∂ t
+
∂x
∂ t
·∇G = 0. (1.10)
The term ∂x∂ t in Eq. (1.10) represents the velocity experienced by a particle at position x,
which is the superposition of the flow and flame combustion velocities:
∂x
∂ t
= u+ sLnˆ = u− sL ∇G|∇G| . (1.11)
Substituting in Eq. (1.10), the kinematic G-equation governing the evolution of the flame
surface is obtained:
∂G
∂ t
+u ·∇G = sL|∇G|. (1.12)
The right hand side of equation (1.12) is highly nonlinear. Because the flame burns
towards the reactants, when non-uniform perturbations in the velocity field or flame speed are
considered, the unsteady flame shape becomes very structured, and pinch-offs may separate
10 Introduction
from the flame. These pockets of fuel cannot be easily treated with an explicit formulation1,
and it is more convenient to use an implicit one. The latter allows for a fully nonlinear
description of the flame surface dynamics, and naturally takes into account complex shapes
and topological changes.
Calculations on the G-field in the entire computational domain require O(N2) operations
in 2-dimensions, where N is the number of grid points per dimension. If a high resolution is
required, e.g., to capture the formation of small structures or short wavelength perturbations
on the interface, the computational cost can become high, in contrast with the G-equation
low-order model characteristics. Because we, however, are interested in tracking the evo-
lution of the flame interface only, there is no need to solve for the G-field in the entire
computational domain. Instead, one can solve for the evolution of the G-field only in a
small tube surrounding the flame interface, as shown in Figure 1.3. If k is the number of
grid points in a section of the tube, the number of operations required reduces to O(kN) in
2-dimensions: practically, it is about an order of magnitude faster than a full-matrix method.
This formulation is known as Narrow Band Level Set method, and is accurately described
in Sethian (1999); Peng et al. (1999). Without going into the formulation’s details, the main
steps performed by the algorithm used to integrate the G-equation are:
1. Two concentric tubes, of radii 0< β < γ respectively, are constructed around the G= 0
interface.
2. A G-field is constructed in the largest tube as a signed distance function (negative inside
the flame and positive outside) from the flame front. This is not a strict requirement,
but is a convenient way to keep the G-field smooth. In practice, this goal is achieved
by solving the Eikonal equation
|∇G|= 1 (1.13)
with a High-order Constrained Reinitialisation (HCR) scheme, as described in Hart-
mann et al. (2010). This is done every time step, since G-dependent quantities, such as
the local surface’s normal vector and curvature, enter in the evolution equation and
need to be accurately described.
3. A time step integration is performed. The integrated equation is:
∂G
∂ t
+ c(G)
(
u− sL ∇G|∇G|
)
·∇G = 0, (1.14)
1An explicit formulation assumes that the flame front is a single-valued function in a certain reference
system. This hypothesis may break down for simple flame configurations when a pinch-off is formed, and is
certainly not true for complex flame shapes.
1.3 Flame modelling 11
−1 −0.5 0 0.5 1
0.5
1
1.5
2
2.5
3
r
x
G < 0
G > 0
G = 0
Fig. 1.3 Narrow Band Level Set method: the G-equation is solved only for points that lie
in the red (boundary conditions), yellow (outer annulus), or light blue (inner tube) regions,
which are close to the flame surface (black line). For illustration purposes, the grid resolution
is reduced by a factor of 2 in both directions.
which is a modified version of Eq. (1.12), where
c(G)≡

1 if |G| ≤ β
(|G|− γ)2(2|G|+ γ−3β )/(γ−β )3 if β < |G| ≤ γ
0 if |G|> γ
. (1.15)
This means, equation (1.12) is solved in the smaller tube of radius β , no equation is
solved outside the largest tube of radius γ , and in the annular region in between them
an equation that smoothly connects the two regimes is integrated (see Fig. 1.3). After
this step, an iteration is completed, quantities such as the total flame surface area and
heat release are calculated and the process restarts from point 1.
The numerical code used to integrate the G-equation in this thesis is LSGEN. It was
originally developed by Hemchandra (2009) to investigate the nonlinear heat release response
of 2D flames subject to forced acoustic and equivalence ratio perturbations with a finite
amplitude. Kashinath et al. (2014) coupled the flame code with Fourier modes describing the
acoustics, enabling for the analysis of self-excited thermoacoustic oscillations in a straight
12 Introduction
pipe. Waugh (2013) extensively modified LSGEN to account for axisymmetric surfaces
and more general boundary conditions, so that effects like flashback and side-bulging could
be described. For the work presented in this thesis, LSGEN has been modify to account
for: (i) the effect of local curvature corrections on the flame speed and heat release; (ii) the
coupling between the flame dynamics and a generic acoustic state-space model – rather than
a simplistic Fourier series – to allow for the investigation of more complex acoustic networks;
(iii) non-uniform mean flows, allowing for the investigation of more complex flame shapes;
(iv) a travelling wave perturbation model in a non-uniform mean flow; (v) the advection of a
non-uniform equivalence ratio distribution in the flame domain and its interaction with the
flame speed and heat release.
Using G-equation based formulations, transfer functions of conical and V shaped flames
to harmonic velocity and heat release fluctuations have been evaluated analytically for a
variety of underlying flow fields in the linear limit (Schuller et al., 2003; Preetham et al.,
2008). In particular, a travelling wave model for flow perturbations has been shown to
accurately reproduce the flow-flame interaction. Also, transfer functions of heat release
fluctuations to harmonic velocity disturbances for these flame shapes have been proven to
compare favourably against experimental results (Durox et al., 2009; Karimi et al., 2009).
In the fully nonlinear case, G-equation models were shown to capture some of the complex
dynamical features of thermoacoustic systems, such as period-doubling, secondary Hopf
bifurcations, and routes to chaos (Kashinath et al., 2014; Waugh et al., 2014). The G-equation
model will be considered throughout this thesis as a low-order model for the heat release
response, with the exception of Chapter 5, in which an electrical heater is consider as a
source of unsteady heat release instead of a flame.
1.4 Thesis scope
The general aim of this thesis is to introduce novel methods for the analysis of thermoacoustic
networks in order to either reduce the cost currently required to investigate their linear and
nonlinear regimes, or to improve the current state-of-the-art of modelling and analysis of
thermoacoustic oscillations. This has been accomplished by achieving the following goals.
First, a framework in which the G-equation model for the flame dynamics is coupled with
an arbitrarily complex acoustic network has been developed. The method chosen, state-space
realisations, allows for naturally including the effects of mean flow, temperature rise across a
flame, and frequency dependent reflection coefficients in the acoustic model.
Second, the above framework was used to investigate the linear stability of thermoacoustic
systems with premixed flames. Adjoint sensitivity methods in thermoacoustic systems with
1.5 Thesis structure 13
premixed flames were introduced as a tool to quickly understand with respect to what
parameters the system’s stability is most susceptible, and how to change them in order to
make the system more stable2. Nonlinear calculations using state-of-the-art methods such
as time marching, FDF analysis, and numerical continuation have been performed. The
nonlinear calculations aim to (i) compare the numerical costs of the different nonlinear
methods, (ii) understand the strengths and weaknesses of the various available methods
and (ii) highlight the differences in the thermoacoustic system behaviour that arise when
a detailed description of the acoustic network is included, in comparison to the case in
which a straight duct with no mean flow or temperature variations is chosen as the acoustic
configuration.
Third, novel frameworks for the investigation of nonlinear effects in thermoacoustics
have been developed to go beyond the FDF paradigm, which can only predict harmonic
oscillations. Two methods have been developed, and their capabilities outlined by using them
to predict the amplitude and type of oscillations in thermoacoustic systems: (i) the Flame
Double Input Describing Function (FDIDF), which naturally extends the FDF concept to
oscillations containing more than one frequency, and highlights some weaknesses of the FDF
method that may arise when assessing the stability of limit cycle oscillations; (ii) weakly
nonlinear expansion of the thermoacoustic equations, which provides a very efficient way
to calculate the evolution of the amplitude of periodic but non-harmonic oscillations close
to Hopf bifurcations, via a so-called Stuart–Landau equation. Both methods give more
information about the nonlinear behaviour of a thermoacoustic system than the FDF analysis,
and represent a significant advance in the state-of-the-art.
Lastly, low-order models for thermoacoustic oscillations in complex combustors with
non-conical premixed flames using the G-equation have been developed, which may be
of interest in industrial applications as a quick tool to estimate the dangerous frequencies
and amplitudes of thermoacoustic instabilities. Focus has been put on generalising the
perturbation flow field models that have been originally developed for conical flames to more
elaborate flame shapes, and including in the flame model a non-uniform distribution of the
fuel/air mixture in the flame domain.
1.5 Thesis structure
All chapters of the thesis have been published in journal or conference proceeding articles3,
which are referred to at the beginning of each chapter, together with the relevant contributions
2An overview on adjoint methods is provided in Chapter 2, where they are introduced for the first time.
3Chapter 5 is currently under consideration for publication in the Journal of Fluid Mechanics.
14 Introduction
Linear Nonlinear
Conical Flame Chapter 2 Chapters 3 (FDF) and 4 (FDIDF)
Electric heater Chapter 5 (Weakly nonlinear)
Bluff-body stabilised flame Chapters 6 (perfectly premixed)
and 7 (imperfectly premixed)
Table 1.1 Schematic overview of the thesis structure.
from co-authors. This justifies the use of the subject “we”, which refers to all co-authors
of the publication associated with the various chapters. In the Introduction of each chapter,
a detailed literature review on the subject covered is outlined. The thesis is summarised in
Table 1.1, and outlined in details in the following.
In Chapter 2, the state-space representation of the acoustic response to heat release
fluctuations is presented in detail. It is then used to investigate the linear stability of a classic
thermoacoustic configuration, a flame driven Rijke tube, taking particular care in the acoustic
modelling. Using the G-equation, an analytical expression is derived for the conical FTF,
which corrects a previous expression found in the literature. A fully-coupled method for
the investigation of the thermoacoustic system stability is adopted rather than evaluating
separately the FTF. This is more versatile and allows for the use of adjoint sensitivity analysis
which, with one calculation, provides information about the system stability’s sensitivity
with respect to all (flame, flow and acoustic) parameters.
In Chapter 3, the same thermoacoustic system is investigated from a nonlinear point of
view. An FDF is evaluated and the harmonic balance is used to predict the amplitude and sta-
bility of harmonic limit cycle oscillations. These results are compared against those obtained
with time-marching simulations and a numerical continuation algorithm. All together, these
methods represent the current state-of-the-art for predictions about the nonlinear behaviour
of thermoacoustic systems. The strength and weaknesses of each method, together with their
practical cost, are highlighted. Super- and subcritical Hopf and Neimark–Sacker bifurca-
tions are identified, together with the Ruelle–Takens–Newhouse route to chaos. Results are
compared with experimental data and previous numerical analyses, which used a simpler
acoustic model.
In Chapter 4, the FDIDF analysis for a dynamic nonlinear flame model is presented.
This method generalises the idea of the FDF to non-harmonic perturbations. The theory is
developed, and criteria for the prediction of frequencies, amplitudes, and stability of periodic
and non-periodic thermoacoustic oscillations are derived. It is shown how the information
provided by the FDF in some cases is not sufficient to determine the stability of a limit cycle
1.5 Thesis structure 15
solution. This is because the FDF neglects the interaction between thermoacoustic modes,
which can destabilise some solutions. The FDIDF, instead, contains this information, and
can also predict the amplitude of non-periodic oscillations, but is a more expensive object to
obtain from experiments or simulations than the FDF. To overcome this issue, a methodology
that improves the FDF predictions exploiting only some of the FDIDF features is outlined,
which can be obtained roughly at the cost of two FDFs.
In Chapter 5, a high order weakly nonlinear analysis of a thermoacoustic system is
presented. For consistency with previous analysis in the literature, an electrical heater model,
consistent with fully nonlinear simulations of the heat released by an electrical wire, is
used as a heat source rather than a flame. The weakly nonlinear expansion presented here
greatly improves analogous analysis which can be found in the literature, because it considers
the acoustics expressed in state-space form (thus, an entirely different acoustic system can
be investigated using the very same equations), and takes into account the contribution of
multiple acoustic modes on the thermoacoustic system dynamics. Because of the latter
feature, adjoint methods are required to obtain the Stuart–Landau equation that governs
the evolution of the oscillations amplitude. The method is shown to accurately predict the
amplitude of thermoacoustic oscillations, and the frequency shift of the saturated oscillations
with respect to the frequency predicted by linear stability analysis. It also accounts for the
contribution of the harmonic components in the oscillations, which the FDF method ignores.
In Chapter 6, a more elaborate thermoacoustic configuration, consisting of a bluff-body
stabilised flame, is considered. A G-equation based model is developed accounting for a
realistic description of the flow field, obtained from URANS simulations. The travelling
wave perturbation model, originally developed for conical and V flames, is extended to
this configuration. Results show that the flow/flame model captures well the flame shape
dynamics as compared to experimental and URANS results. The heat release rate, however,
tends to overestimate the experimentally determined one. This is probably because neither
turbulent model nor shear layer effects on combustion have been included in the G-equation
model. Including these effects may improve the model, but would greatly increase the cost
of the simulations, probably making it not interesting as a low-order tool.
In Chapter 7, the same thermoacoustic configuration as in the previous chapter is consid-
ered, but focus is put on the interaction between the flame and equivalence ratio fluctuations
rather than flame and flow field interaction. Because equivalence ratio disturbances have a
much stronger impact on the heat release dynamics than velocity perturbations, a detailed
modelling of the flow field is superfluous. The G-equation based low-order model is shown
to qualitatively compare well with experimental results, and improves previous modelling
performed on the same system. FDFs are calculated and discussed, and are approximated
16 Introduction
with sum of time delays models. It is shown how the latter provide an interesting alternative
point of view to the FDF information, which can be used to understand what are the relevant
physical time and length scales at which the flame responds. An acoustic network for the
experimental geometry is constructed and coupled with the FDF, showing that the frequencies
observed in self-excited oscillations are captured by the low-order model.
In the Conclusions, the results achieved are summarised and discussed, and suggestions
for extensions of the methods developed and future work are presented.
Chapter 2
Linear stability and adjoint sensitivity
analysis of thermoacoustic networks with
premixed flames
This chapter contains a review of the linear stability theory and adjoint sensitivity analysis of
thermoacoustic systems with premixed flames. A detailed derivation of the acoustic equations
in an acoustic networks is presented, which is then used throughout the thesis. Furthermore,
a methodology that enables to cast the network frequency response into state-space form is
outlined, which was developed in collaboration with Dr. Simon Illingworth and published
in Orchini et al. (2015). The state-space model is coupled with a linearised, discretized
G-equation model, resulting in a versatile linear framework on which adjoint sensitivity
methods are applied.
The content of this chapter was published in Orchini & Juniper (2016b).
2.1 Introduction
The stability of a thermoacoustic system is usually analysed by first calculating the Flame
Transfer Function (FTF). This is the flame’s heat release response to velocity, pressure, or
equivalence ratio perturbations. The FTF is then combined with an acoustic network model.
There are several ways to formulate a linear acoustic model for a thermoacoustic analysis.
Three common methods are i) a Fourier-Galerkin discretization (Culick, 1976a,b; Zinn &
Lores, 1971); ii) a Green’s function approach (Heckl & Howe, 2007), which may subsequently
be used in a modal expansion (Schuermans, 2003); and iii) a wave-based approach (Lang
et al., 1987; Heckl, 1988). In any acoustic modelling method, it is useful to write the resulting
18 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
model in state-space form (to be defined in §2.2.2) because this allows powerful techniques
from dynamics and control to be used. These include, for example, stability analysis of
the coupled thermoacoustic system, analysis of its transient growth characteristics, and the
design of feedback control to eliminate oscillations. For the Fourier-Galerkin discretization
and the Green’s function approach, methods to describe the resulting systems in state-space
form have been developed. For the first case, a state-space description follows quite naturally
from the Fourier modes, and for a Green’s function approach, a state-space description can
be generated by performing a modal expansion (Schuermans, 2003). For the wave-based
approach, however, it is less straightforward to describe the resulting system in state-space
form. This is largely due to time delay terms, which make the system infinite-dimensional,
and which are not amenable to a state-space description in a straightforward way. A possibility
is to approximate time delays with, e.g., Padé approximations, as in Bothien et al. (2007),
or to make use of a state vector with memory, i.e., a state vector that includes the acoustic
states at previous times, as in Mangesius & Polifke (2011). Here, we develop an alternative
approach that allows us to cast the frequency response of the system in state-space form.
From the combustion point of view, we will focus on the response of laminar, coni-
cal flames, modelled with the linearised G-equation. Previous studies have shown that a
kinematic description of the flame front, using a front-tracking version of the G-equation
with a suitable velocity model, can capture the main features of the heat release response
of conical premixed flames to inlet velocity fluctuations. Birbaud et al. (2006) have shown
experimentally that acoustic perturbations are responsible for the formation of velocity per-
turbations that are advected along the flame at a characteristic speed, the convection speed,
which in general is a function of the amplitude and frequency of the forcing oscillation.
By measuring the velocity field inside the flame, they were able to identify two regions in
which the velocity field behaved differently: close to the burner the velocity field oscillates
with the acoustic field, whereas further downstream a travelling-wave behaviour of the flow
field was observed. A pure travelling wave model of axial velocity perturbations, which
uses the acoustic field as a boundary condition at the burner, qualitatively reproduces the
flow and flame dynamics (Baillot et al., 1992), and radial velocity fluctuations are found by
choosing a divergence free flow field. In some studies in the literature, the radial component
was neglected, because it was shown to be less important than the axial travelling wave
on the flame response (Preetham et al., 2008). The G-equation moving into a travelling
wave velocity field has been shown to capture the most relevant features of conical flames
dynamics - such as the formation of wrinkles on the flame surface - and the consequent heat
release response.
2.1 Introduction 19
This model has been developed and compared with experiments in several studies: Schuller
et al. (2002) considered an axial dependence of the mean flow field; Schuller et al. (2003)
compared the responses of conical and V -shaped flames; Karimi et al. (2009) compared
FTFs from experiment with analytical results from Schuller et al. (2003); Preetham et al.
(2008); Cuquel et al. (2011, 2013b) investigated the effect of confinement on conical flames
FTFs and compared with G-equation based analytical models; Kashinath et al. (2013b)
extracted a frequency-dependent convection speed from DNS and used it into a G-equation
low-order model. For a complete review of premixed combustion and acoustic waves cou-
pling see Lieuwen (2003). In this analysis we will assume that perturbations travel with a
constant speed, which in general is different from the mean flow velocity. We also allow
the flame speed to vary linearly with the local flame curvature, which avoids the formation
of unphysical cusps on the flame surface. This has already been considered for V -shaped
flames modelled with the G-equation (Wang et al., 2009; Preetham et al., 2010), but not for
conical flames, which is a typical experimental configuration (Noiray et al., 2008; Kabiraj &
Sujith, 2012). The flame model we derive only captures the most relevant characteristics of
conical flames dynamics. We do not model other effects which may be important in some
cases, such as gas expansion (Creta & Matalon, 2011) and flame base motion (Cuquel et al.,
2013a) to keep the low-order flame model simple.
The analysis of FTFs helps to explain the linear dynamics of flames, such as a conical
flame’s low-pass characteristics. Analytical results are usually not available, however,
meaning that simulations or experiments over a large range of frequencies are required,
which can be expensive. This becomes even more demanding when one wants to investigate
the effect of several parameters on the stability of a thermoacoustic system, because a new
FTF has to be evaluated for every set of parameters. On the other hand, if a relation between
the flame’s heat release response and acoustic velocity or pressure fluctuations is known, one
can apply classic linear stability techniques to the fully-coupled system, avoiding the explicit
evaluation of flame and acoustic transfer functions (Bloxsidge et al., 1988).
The aim of this chapter is to apply linear stability and adjoint sensitivity techniques to
thermoacoustic networks with premixed flames, in which the flame and heat release dynamics
are modelled by the kinematic G-equation. With this approach, the problem of identifying
thermoacoustic instabilities is reduced to a classic eigenvalue problem of the coupled system.
The resulting eigenvalue problem is small and all the eigenvalues can be calculated as the
parameters of the model are changed. If the model were larger, a similar approach could
be used, but only the eigenvalues with the largest growth rates would be calculated, using
iterative methods.
20 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
Having obtained a description of the linear coupled thermoacoustic system, adjoint
sensitivity analysis can be applied (Sipp et al., 2010; Luchini & Bottaro, 2014; Magri &
Juniper, 2013b). The premixed flame response is modelled with the linearised front-track
G-equation model1. Sensitivity results can be used, for example, to calculate how to change
the system in order to reduce the growth rate of each unstable eigenvalue. This change could
be in the shape of the combustion chamber, the shape of the flame, or the acoustic boundary
conditions. In this chapter, we demonstrate the usefulness of adjoint methods by calculating
how the convection speed affects the most unstable eigenvalues.
The chapter is structured as follows: in §2.2 the acoustic network modelling that is used
throughout the thesis is outlined, and we describe the state-space formulation used to couple
it with the heat release dynamics. In §2.3 the linear flame model is derived. In §2.4 we solve
the equations for the case where the flame speed is uniform and we present analytical results
that correct the FTF expression contained in Preetham et al. (2008). In §2.5 we extend the
model to the more general case of curvature-dependent flame speeds. In §2.6 we express the
fully-coupled self-excited thermoacoustic equations as an eigenvalue problem. In §2.7 we
present the parameters and geometrical details of the thermoacoustic network considered. In
§2.7.2 we calculate its eigenvalues while varying two parameters, build a stability map and
discuss the results. Finally, in §2.7.3, we perform a sensitivity analysis on the convection
speed on the entire stability map calculated with stability analysis, and provide physical
insights based on these results. In §2.8 we summarise our study and discuss the potential
applications of these methods to larger problems.
2.2 Linear acoustics: wave-based approach and state-space
representation
In this section the wave-based acoustic model is described, and the methodology used to
generate a state-space acoustic model from it outlined. We start with a simple case: a flame
in a one-dimensional duct open at both ends. This shows the important features of the
wave-based method and highlights the difficulties in formulating its state-space description.
1Note that it would be troublesome to use this approach on the entire G-field, because the G-field itself has
no physical meaning away from the G=0 level set, and care must be taken in calculating the sensitivities. This
problem does not appear with a linearised formulation, because only the flame front position is tracked, which
is a physical quantity.
2.2 Linear acoustics: wave-based approach and state-space representation 21
Fig. 2.1 Wave approach in a simple duct geometry. g1 and f2 are the waves radiated from the
heat source towards the boundaries in the upstream and downstream regions respectively. f1
and g2 are the waves reflected by the boundaries.
2.2.1 Wave-based modelling approach
Figure 2.1 shows a prototype thermoacoustic network: an acoustically compact flame inside
an open-ended duct. The mean flow, if non-zero, is from left to right. There is a temperature
jump across the flame, which causes the sound speed to increase from c1 to c2. A perturbation
in the heat release, q′(t), generates outward-travelling waves that propagate both upstream
(g1) and downstream ( f2). These waves partially reflect at the upstream/downstream ends
of the duct. The reflection coefficients are R1 and R2, and the reflected waves are f1 and g2.
The flame is at x = 0, so x ∈ [−l1,0) is upstream of the flame and x ∈ (0, l2] is downstream
of the flame.
We solve the wave equations (1.8) in each of the two regions shown in Figure 2.1.
Upstream of the flame, the acoustic pressure and velocity perturbations can be written in
terms of the upstream- and downstream-travelling waves, g1 and f1:
p′1(x, t) = g1
(
t+ xc1−u¯1
)
+ f1
(
t− xc1+u¯1
)
(2.1a)
u′1(x, t) =
1
ρ¯1c1
[
−g1
(
t+ xc1−u¯1
)
+ f1
(
t− xc1+u¯1
)]
. (2.1b)
We can write similar expressions for the downstream region (Evesque et al., 2003). Note
that, to simplify the notation, we have suppressed the subscript x for the axial velocity. These
equations are supplemented by boundary conditions at each end of the duct, which provide a
relationship between the outward-travelling waves, g1(t) and f2(t), and the inward-travelling
waves, f1(t) and g2(t):
f1
(
t+ l1c1+u¯1
)
= R1g1
(
t− l1c1−u¯1
)
(2.2a)
g2
(
t+ l2c2−u¯2
)
= R2 f2
(
t− l2c2+u¯2
)
. (2.2b)
22 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
We need to relate the upstream and downstream flow to the heat release rate. This is achieved
by considering the mass, momentum, and energy balances across the flame, through the
Rankine-Hugoniot jump conditions (Dowling, 1995):
A−ρ−u− = A+ρ+u+ (2.3a)
A+p−+A−ρ−u2− = A+p++A+ρ+u
2
+ (2.3b)
A−ρ−u−
(
γ
γ−1
p−
ρ−
+
1
2
u2−
)
= A+ρ+u+
(
γ
γ−1
p+
ρ+
+
1
2
u2+
)
, (2.3c)
where the subscripts − and + denote the flow variables just upstream and downstream the
flame, respectively.
No entropy waves are found in the upstream duct (cold flow), but they are generated at
the flame location. These waves travel downstream along the duct, and could be converted
into acoustic waves if the flow accelerates, for example at a chocked outlet (Marble & Candel,
1977; Duran & Moreau, 2013). The amplitude of waves so generated, referred to as indirect
noise, is typically small compared to that of direct noise, and is not modelled here. Thus,
although entropy waves are present in our model, we are not interested in calculating them,
as they are simply convected out of the domain.
We therefore solve only for the acoustic waves, given by the momentum and energy
equations into which the expression for the downstream density – calculated from the
mass flux equation – has been substituted. By considering the linearised disturbances
of equations (2.3), and substituting in the travelling-wave solutions (2.1) with boundary
conditions (2.2), we obtain the equations governing the time evolution of the outward-
travelling waves:
X
[
g1(t)
f2(t)
]
= Y
[
R1g1(t− τ1)
R2 f2(t− τ2)
]
+
[
0
1
]
q′(t)
c1
, (2.4)
where τ1 ≡ 2l1c1/(c21− u¯21), τ2 ≡ 2l2c2/(c22− u¯22), and
X ≡
[ −1+M1(2− u¯2u¯1 )−M21(1− u¯2u¯1 ) 1+M2
1−γM1
γ−1 +M
2
1 − 12M21(1−M1)(( u¯2u¯1 )2−1)
c2
c1
(
1+γM2
γ−1 +M
2
2
) ] (2.5a)
Y ≡
[
1+M1(2− u¯2u¯1 )+M21(1−
u¯2
u¯1
) −1+M2
1+γM1
γ−1 +M
2
1 − 12M21(1+M1)(( u¯2u¯1 )2−1)
c2
c1
(
1−γM2
γ−1 +M
2
2
) ] (2.5b)
2.2 Linear acoustics: wave-based approach and state-space representation 23
are 2×2 matrices and are functions of the mean flow properties only. These matrices are a
more general form of the matrices contained in Dowling (1997), as they are valid for any
(even frequency dependent) reflection coefficient.
In order to demonstrate the wave-based approach, we consider the zero mean flow case.
This simplifies the analysis significantly, whilst still retaining the most important features.
Taking the Laplace transform of equation (2.4) for no mean flow, and assuming that the
reflection coefficients are frequency-independent, we find[
1+R1e−sτ1 −1−R2e−sτ2
1−R1e−sτ1 c2c1 (1−R2e−sτ2)
][
g1(s)
f2(s)
]
=
[
0
γ−1
c1
]
q′(s), (2.6)
where s≡ σ + iω is the Laplace variable and σ the growth rate. The modes are now given
by those values of s for which the determinant Ω(s) of the matrix in (2.6) vanishes. This
gives us the relation:
Ω(s)≡
(
1+
c2
c1
)(
1−R1R2e−s(τ1+τ2)
)
+
(
1− c2
c1
)(
R2e−sτ2 −R1e−sτ1
)
= 0, (2.7)
which makes clear the influence of the ratio of the speeds of sound, c2/c1. When c2/c1 = 1
(i.e., no temperature change across the flame) the modes are equispaced and given simply by:
R1R2e−s(τ1+τ2) = 1. (2.8)
When c2/c1 ̸= 1 there is an extra term in equation (2.7) and the modes are no longer
equispaced.
Solving equations (2.6) for the waves g1(s) and f2(s), and substituting these into the
expressions (2.1) for the pressure and velocity, we find for the upstream region that
p′1(x1,s)
q′(s)
=
γ−1
c1
[1+R2e−sτ2]
[
1+R1e
−s
(
τ1−2x1c1
)]
Ω(s)
e−s
x1
c1 (2.9a)
u′1(x1,s)
q′(s)
=
γ−1
2ρ¯1c21
[1+R2e−sτ2]
[
−1+R1e
−s
(
τ1−2x1c1
)]
Ω(s)
e−s
x1
c1 . (2.9b)
Similar relations can be derived for the pressure and velocity in the downstream region.
Note that the transfer functions defined in equations (2.9) differ from those in the rest of
the thesis, which will include mean flow and area variations. The important point here is
24 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
that the relations for the pressure and velocity include time delay terms of the form e−sτ ,
and it is therefore not possible to cast them directly in state-space form. Time delays can
be approximated by Padé approximations in order to make them amenable to state-space
descriptions (Bothien et al., 2007). Alternatively, state vectors that includes the acoustic
states at previous times can be used (Mangesius & Polifke, 2011). However, we have opted
for a different approach to the state-space formulation, which is presented in the next section.
2.2.2 Finding a state-space model
Having outlined the most important features of the wave-based model, we now obtain its state-
space description using knowledge of its acoustic modes and frequency response. Rather
than discretize the system using an expansion such as Fourier modes, we discretize the
system by finding its eigenvalues directly. This involves finding the roots, λk ≡ σk + iωk, of
Ω(s) introduced in equation (2.7). This is achieved using Newton–Raphson iteration in the
complex plane. We also require the frequency response function, which is found by setting
s = iω in equations (2.9). We then approximate the frequency response with a state-space
model, as described in the next section.
Calculating the state-space matrices
We want to write the acoustic model described in §2.2 in state-space form:
x˙(t) = Ax(t)+B q′(t) (2.10a)
y′(t) =C x(t), (2.10b)
where x is the state vector, q′ is an input, and y′ is some output of interest. A is the state-space
matrix, and its eigenvalues determine the acoustic system stability. B and C are a column
and row vector respectively; they are matrices for Multi-Input Multi-Output systems. In our
case the system has a Single-Input – the fluctuating heat release – and a single output – either
velocity or pressure fluctuations. Taking Laplace transforms of (2.10) and rearranging, we
arrive at the transfer function
H(s)≡ y
′(s)
q′(s)
=C(sI −A)−1B, (2.11)
where I is the identity matrix.
A state-space realisation is not unique. There are many state-space realisations that give
the same transfer function. The realisation that is convenient for our purposes is a modal
2.3 Linear flame dynamics 25
form, where the A matrix is written as a diagonal matrix with its eigenvalues on its diagonal.
For this realisation, the term (sI −A)−1 can be written simply as
(sI −A)−1 =
s−λ1 · · · 0... . . . ...
0 · · · s−λn

−1
=

1
s−λ1 · · · 0... . . . ...
0 · · · 1s−λn
 , (2.12)
and the transfer function, H(s), becomes:
H(s) =
n
∑
k=1
ckbk
s−λk =
n
∑
k=1
θk
s−λk . (2.13)
Here we have defined θk ≡ ckbk. This is for the single-input-single-output case, for which B
is a column vector and C is a row vector, but is easily extended to the multi-input-multi-output
case. If we evaluate the frequency response of H(s)|s=iω at the frequencies ω1,ω2, . . . ,ωp,
then H(iω1)...
H(iωp)
=
(iω1−λ1)
−1 · · · (iω1−λn)−1
... . . .
...
(iωp−λ1)−1 · · · (iωp−λn)−1

θ1...
θn
 , (2.14)
which we can solve for
[
θ1 . . .θn
]T
. We need the frequency response at p = n frequencies
to make the matrix in (2.14) invertible. In practice, however, it is better to evaluate the
frequency response at many more than n frequencies, p≫ n, and then solve equations (2.14)
in the least squares sense. Note that we are free to choose any ck, bk, provided they satisfy
ckbk = θk.
A validation of the agreement between the original frequency response and the approxi-
mated state-space form has been given by Illingworth & Juniper (2014): for an approximation
order n = 23, the agreement is excellent at low-frequencies, and slightly deteriorates at high
frequencies. This is expected because the state-space model cuts off the contribution of
modes with high frequencies.
2.3 Linear flame dynamics
We describe the premixed flame’s dynamics with the kinematic G-equation model, assuming
that there is no density jump across the flame. This assumption precludes the Darrieus-Landau
instability in the flame. This instability can cause the formation of small-scale wrinkles
leading to turbulence (Darrieus, 1938; Landau, 1944a; Creta & Matalon, 2011). This, and
26 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
other physical phenomenon such as reaction mechanisms and turbulence effects, can be taken
into account in an LES simulation with a G-equation formulation, see for example Moureau
et al. (2009). However, for our purposes we want to keep the model low-order, and we
consider a laminar flame, assuming that the flame is an infinitely thin interface separating
reactant and products and neglecting temperature variations across the flame. Under this
assumption the G-equation model reads:
∂ G˜
∂ t˜
+ u˜ · ∇˜G˜ = s˜0L(1−Lκ˜)|∇˜G˜|, (2.15)
where u˜ is a prescribed flow field, s˜0L is the propagation speed of a laminar flat flame, L is the
Markstein length, and k˜ is the local flame curvature. The flame front is identified by the G˜= 0
level set. We describe axisymmetric flames in the laboratory framework, indicating with r˜ and
x˜ the radial and axial directions respectively. We denote mean quantities with overlines and
perturbations with primes. Dimensional quantities are indicated with a tilde. We also assume
that the mean flow is uniform in the axial direction, and that the axial flow perturbations
do not depend on the radial component. Radial velocity fluctuations are found by solving
the continuity equation, assuming that the flow is incompressible. This is a well-established
model that has been shown to accurately reproduce experimentally determined conical FTFs
when coupled with the G-equation dynamics. Comparisons between FTFs determined from
experiments and G-equation models that use this type of flow field can be found in Schuller
et al. (2002); Cuquel et al. (2011); Kashinath et al. (2013b). Therefore, we can write the two
components of u˜ as:
u˜x = u(1+ εu′x(x, t)), u˜r =−
1
2
εur˜
∂u′x
∂ x˜
, (2.16)
where ε ≪ 1 is the perturbation parameter, and u′x ∼O(1) is the axial velocity fluctuation,
which can be forced or self-excited. We will consider forced fluctuations in order to examine
how FTFs are affected by changes in flame speed due to curvature. However, we will not use
these FTF results in the self-excited configuration. Instead, we will rewrite the equations in
the frequency domain so that linear stability and adjoint methods can be applied without an
explicit knowledge of the FTF.
Because we study small perturbations and are interested in the linear limit, it is correct to
assume that the flame front is single-valued in a well-chosen reference system. The linearised
conical flame front is always single valued in the laboratory framework with respect to the
2.3 Linear flame dynamics 27
radial coordinate, with r˜ spanning the range [0,R] at any instant2. Thus the G˜ = 0 level set is
expressed as:
G˜(x˜, r˜, t˜) = x˜− F˜(r˜)− ε f˜ (r˜, t˜) = 0, (2.17)
where F˜ and f˜ are the explicit functions that define the shape of the mean flame and its
perturbation, respectively.
For an axisymmetric surface x˜ = S(r˜), the mean curvature is expressed by:
κ˜(S) =
d2S
dr˜2(
1+
(dS
dr˜
)2)3/2 + dSdr 1r√1+ (dSdr )2 . (2.18)
By substituting equations (2.16), (2.17) into (2.15) we obtain:
−ε ∂ f˜
∂ t˜
+u(1+ εu′x)+
1
2
ur˜ε
∂u′x
∂ x˜
(
dF˜
dr˜
+ ε
∂ f˜
∂ r˜
)
=
= s˜0L
√√√√1+(dF˜
dr˜
+ ε
∂ f˜
∂ r˜
)2(
1−Lκ˜
(
F˜ + ε f˜
))
,
(2.19)
where the velocity field has to be evaluated at the flame position x˜ = F˜ + ε f˜ .
Geometrical inspection shows that
√
1+(dF˜/dr˜)2 = 1/sinα(r˜), where α(r˜) is the
internal angle that the mean flame front forms locally with the vertical axis (see Figs. 2.2, 2.4).
By expanding the r.h.s. of equation (2.19) in ε , we obtain the zero and first order equations
for the flame dynamics:
1 =
s˜0L
u
[
1
sinα
−L
(
sin2α
d2F˜
dr˜2
+
dF˜
dr˜
r˜
)]
(2.20a)
1
u
∂ f˜
∂ t˜
−u′x−
1
2
r˜
dF˜
dr˜
∂u′x
∂ x˜
=− s˜
0
L
u
[
sinα
dF˜
dr˜
∂ f˜
∂ r˜
. . .
−L
(
sin2α
∂ 2 f˜
∂ r˜2
−2sin4α dF˜
dr˜
d2F˜
dr˜2
∂ f˜
∂ r˜
+
∂ f˜
∂ r˜
r˜
)]
.
(2.20b)
The flame is anchored at the burner at (r˜, x˜) = (R,0) and is axisymmetric and smooth with
respect to the centerline r˜ = 0. Therefore equations (2.20) are subject to the boundary
2The conical flame is single valued also in the axial direction. However, to work with a function which has
x˜ as an independent variable is an unfortunate choice, because the flame tip moves along this direction and
the domain of existence of the flame front becomes time dependent, [0, x˜end(t˜)], unnecessarily complicating the
formulation (Humphrey et al., 2014).
28 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
1 0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
x = F
x = F + ǫfα
β
1
r
x
u′n
s0L
Fig. 2.2 Schematic of the front-tracking model for a flame with constant flame speed. The
steady solution is a cone on top of which the perturbation oscillates. The transfer functions
are normalised by the normal velocity at the flame base.
conditions:
F˜(R) = 0,
dF˜
dr˜
∣∣∣∣∣
r˜=0
= 0, f˜ (R, t˜) = 0,
∂ f˜
∂ r˜
∣∣∣∣
r˜=0
= 0. (2.21)
We also need to evaluate the total heat released by the flame (Lieuwen, 2012):
Q≡ 2πρhr s˜0L
∫ R
0
(1−Lk˜)
√√√√1+(dF˜
dr˜
+ ε
∂ f˜
∂ r˜
)2
r˜dr˜. (2.22)
Expanding the integrand in ε we can evaluate the steady heat release Q and the fluctuations
Q′. The ratio q′ = Q′/Q is the non-dimensional quantity we are interested in:
q′ =
∫ R
0
[
sinα
dF˜
dr˜
∂ f˜
∂ r˜
−L
(
sin2α
∂ 2 f˜
∂ r˜2
−2sin4α dF˜
dr˜
d2F˜
dr˜2
∂ f˜
∂ r˜
+
∂ f˜
∂ r˜
r˜
)]
r˜dr˜
∫ R
0
[
1
sinα
−L
(
sin2α
d2F˜
dr˜2
+
dF˜
dr
r˜
)]
r˜dr˜
(2.23)
2.4 Forced response to harmonic fluctuations
We first consider the following case: we impose forced, harmonic fluctuations of angular
frequency ω at the flame’s inlet, we assume that they are convected axially with a velocity
2.4 Forced response to harmonic fluctuations 29
uc ̸= u (Birbaud et al., 2006), and we compute radial fluctuations assuming that the flow is
incompressible (Preetham et al., 2008; Kashinath et al., 2013b). Therefore:
u˜x = u(1+ ε cos(ω t˜− kx˜)), u˜r = ε ukr˜2 sin(ω t˜− kx˜), (2.24)
where kc is the wavenumber of the harmonic waves. For now, we neglect the contribution
of the curvature on the flame speed, by setting L = 0. This problem has already been
investigated by Preetham et al. (2008). The results we derive, however, differ slightly from
the ones they obtained.
To simplify a comparison between our results and those of Preetham et al. (2008), we
apply the same non-dimensional scheme to both. In particular, we scale axial lengths by the
mean flame height L f , radial lengths by the burner width R, speeds by the mean velocity u,
and time by the characteristic time L f /u. The complete set of non-dimensional variables,
which do not have a tilde, is presented in Appendix A. The parameters relevant for this
section are the ratio between the mean velocity and the convection speed K, the Strouhal
number St≡ ωL f /u and the flame aspect ratio β ≡ L f /R = [(u2− s˜02L )/s˜0
2
L ]
1/2 = cotα (see
Figure 2.2).
Note that the explicit non-dimensional form of the radial velocity component is:
ur =
u˜r
u
= ε
kr˜
2
sin(ω t˜− kx˜) = ε StKr
2β
sin[St(t−Kx)]. (2.25)
This is different from equation (13) in Preetham et al. (2008), because they missed the β
factor in the denominator. This missing factor affects only the results obtained by Preetham
et al. (2008) using Model B. We will now show how the transfer functions of heat release to
harmonic velocity fluctuations changes with this correction.
When L= 0, equation (2.20a) admits an analytical solution, which in non-dimensional
units is F = 1− r. The first order perturbation equation, (2.20b), simplifies to:
∂ f
∂ t
− β
2
1+β 2
∂ f
∂ r
= cos[St(t−K(1− r))]− 1
2
KStr sin[St(t−K(1− r))]. (2.26)
The solution of this linear PDE can be obtained, e.g., by performing a Laplace transform.
The heat release fluctuations simplify to:
q′(t) =− 2β
2
1+β 2
∫ 1
0
∂ f
∂ r
rdr =
2β 2
1+β 2
∫ 1
0
f (r, t)dr. (2.27)
30 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
St2
|F
T
F
|
0 5 10 15 20
0
5
10
15
20
25
St2
F
T
F
6
η = 0.1 - Current work
η = 0.1 - Preetham et al. (2008)
η = 1.1 - Current work
η = 1.1 - Preetham et al. (2008)
η = 2.0 - Current work
η = 2.0 - Preetham et al. (2008)
Fig. 2.3 Gain and phase of axisymmetric, conical, FTFs for different values of η . Lines with
empty markers correspond to our results (equation (2.29)). Lines with filled markers are the
results of Model B in Preetham et al. (2008). Differences can be observed mainly in the gain
response in the low- or high-frequency limits, depending on the value of η .
We define the Flame Transfer Function (FTF) as in Preetham et al. (2008), by:
FTF(St)≡ Qˆ
′/Q
ˆ˜u′n(x = 0,r = 1)/u
√
1+β 2
, (2.28)
where ˆ˜u′n(x = 0,r = 1) is the Fourier component with angular frequency St of the velocity
normal to the flame at the flame base. Introducing the parameters η ≡ Kβ 2/(1+β 2) and
St2 ≡ St(1+β 2)/β 2, the analytical expression we obtain for the transfer function is:
FTF(St2) =
2i
(
eiηSt2 − (η−1)2+η (η−2)eiSt2
)
+2St2η (η−1)
(
1+η
(
eiSt2 −1))
η(η−1)2 (2i+ηSt2)St22
.
(2.29)
This result can be recovered from the result of Preetham et al. (2008) Model B by setting
the explicit β terms to 1 in their equation (35), without changing the definitions of St2 and
η , which implicitly contain β . In Figure 2.3 we compare our FTF with that of Preetham
et al. (2008). Differences can be observed mainly in the gain response, especially in
the high-frequency region if η < 1, and in the low-frequency limit otherwise. No major
differences are observed in the phase response. Also, in Cuquel et al. (2011) an analytical
expression for the FTF was derived using the same flow field as in equation (2.24), but
assuming that perturbations convect with the same speed as the mean flow, which is not
necessarily the case (Birbaud et al., 2006; Kashinath et al., 2013b). They cannot be directly
2.5 Curvature effects 31
−1 −0.5 0 0.5 1
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x = F
x = F (M = 0)
α
β
1
r
x
u
′
n
sL
α
0 0.2 0.4 0.6 0.8 1
−6
−4
−2
0
2
4
6
r
 
 
1
sin α
 
dF
dr
d
2
F
dr2
Fig. 2.4 The steady flame shape for β = 6 and M= 0.02. Curvature corrections to the flame
speed round off the cusp at the flame tip, and the angle α now varies with respect to the
radial position of the mean flame front.
compared because different scalings for the FTFs were chosen. However, we have verified
that normalising the FTF with respect to axial (rather than normal to the flame) velocity
fluctuations, the result we obtain in the special case K = 1 collapses onto the one reported
by Cuquel et al. (2011).
In the low-frequency limit the transfer function (2.29) can be expanded as:
FTF(St2) = 1+
i
6
(2+3η)St2+O
(
St22
)
, (2.30)
and one can see that the particular case of uniformly perturbed flames, in which η = 0, yields
FTF(St2) ≈ 1+ i3St2, in agreement with the results reported by Schuller et al. (2003) in
the long flame limit, for which β → ∞ and radial fluctuations are negligible according to
equation (2.25). We conclude this section by highlighting that, although we have presented
here the correct expression for the FTF, the discussion and main results of the study conducted
by Preetham et al. (2008) are unaffected.
2.5 Curvature effects
In the previous section we derived analytical results for the special case L = 0. We now
allow the flame speed sL to vary linearly with curvature, and use the same velocity model and
non-dimensional scheme as in §2.4 to calculate FTFs. A linear curvature correction to the
flame speed was originally proposed by Markstein (1964) to explain the existence of stable
flame fronts, which were predicted to be unconditionally unstable due to gas expansion by
32 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
0 0.5 1 1.5 2
−20
0
20
t
u
′ n
/
si
n
α
r
=
0
−1
0
1
q′
0
0.5
1
x
T T +
1
9
T T +
2
9
T
0
0.5x
T +
3
9
T T +
4
9
T T +
5
9
T
0 0.5
0
0.5
r
x
T +
6
9
T
0 0.5
r
T +
7
9
T
−1 0 1
r
T +
8
9
T
Fig. 2.5 Time integration for β = 6, M= 0.01, η = 1.1, and St = 20. Top: normal velocity
at the flame base and heat release fluctuations during the transient (thin line) and steady-state
solution (thick line). Bottom: evolution over a steady cycle of the perturbed flame front
around the mean flame.
the constant flame speed analysis of Darrieus (1938); Landau (1944a). In Creta & Matalon
(2011) it was shown that analogous results are obtained by accounting also for flame speed
corrections due to hydrodynamic strain effects. Although we are not modelling gas expansion,
a curvature dependence on the flame speed avoids the formation of unphysically sharp cusps,
and it rounds the flame shape at the centerline guaranteeing continuity of the flame normals.
We do not account for hydrodynamic strain effects because the flow field model we adopt is
too simplistic, and only reproduces the travelling wave form of velocity disturbances.
We introduce the non-dimensional curvature κ ≡ L f κ˜ , and the Markstein number M≡
L/L f . Note that the characteristic length L f here is not the actual flame height, but is the
height that the flame would have without any curvature correction to the flame speed (see
Figure 2.4). This is because we can find no analytical expression for the mean flame shape
when L ̸= 0, and we cannot know the mean flame height a priori. The non-dimensional form
2.5 Curvature effects 33
of equations (2.20) is:
1 =
1√
1+β 2
[
1
sinα
−Mβ 2
(
sin2α
d2F
dr2
+
dF
dr
r
)]
(2.31a)
∂ f
∂ t
− cos[St(t−KF)]−StK r
2
dF
dr
sin
[
St(t−KF)]= . . .
− β
2√
1+β 2
[
sinα
dF
dr
∂ f
∂ r
−M
(
sin2α
∂ 2 f
∂ r2
−2β 2 sin4α dF
dr
d2F
dr2
∂ f
∂ r
+
∂ f
∂ r
r
)]
.
(2.31b)
We require numerical integration techniques to evaluate FTFs when L ̸= 0. The steady
equation (2.31a) can be efficiently solved numerically with a Newton–Raphson method. The
mean quantities dF/dr, d2F/dr2, sinα (Figure 2.4), and the perturbation f are discretized
in space using a second-order finite difference scheme. The linear PDE (2.31b) is then
marched forward in time using an explicit third-order Runge-Kutta method (Gottlieb & Shu,
1998). The final time is sufficiently large for the transient behaviour to disappear, and for a
few forced cycles to fully travel along the mean flame (Figure 2.5). The steady-state heat
release fluctuations q′ are evaluated according to equation (2.23) at every time step. The FTF
definition generalises equation (2.28):
FTFM(St) =
qˆ′
uˆ′n(x = 0,r = 1)
√
1+β 2
(
dF
dr
)2
r=1
, (2.32)
where
uˆ′n(x = 0,r = 1) =
uˆ′x−β dFdr uˆ′r√
1+β 2
(
dF
dr
)2
∣∣∣∣∣∣∣∣
x=0,r=1
(2.33)
is the velocity component normal to the mean flame at the flame base.
The numerical code has been benchmarked against theoretically known perturbed flame
evolutions such as equation (2.26), and transfer functions such as (2.29), showing that the
numerical results for gain and phase converge to the correct values in proportion to ∆r2, where
∆r is the grid spacing used in the discretization scheme. Figure 2.5 shows the integrated
heat release and normal velocity time-traces during the transient and steady-state solutions
of the forced system. The gain and phase can be calculated by taking the ratio between the
steady-state amplitudes and the (normalised) distance between the peaks of the time series
respectively. In the bottom panels the evolution of the perturbed flame front around the mean
one is shown over a forced cycle.
34 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
0 5 10 15 20
0
5
10
15
20
St2
F
T
F
M
Mincreases
6
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Mincreases
St2
|F
T
F
M
|
M = 0.0
M = 0.005
M = 0.01
M = 0.02
Fig. 2.6 Transfer functions of conical flames with curvature-dependent flame speed. Gain
and phase for β = 6, η = 1.1, and different M. Increasing the Markstein number decreases
the flame’s mean height. As a consequence the time delay of perturbations lowers (shifts in
phase), and in the low-frequency limit increases the effect of small perturbation on the heat
release.
The higher the frequency of the forcing fluctuations, the smaller the wavelengths of the
perturbations that are generated on the flame. This has a consequence for the numerical cost
required to evaluate the transfer functions: at high-frequencies, the perturbation wavelengths
are shorter and the grid spacing ∆r has to be smaller. The time step decreases accordingly
and the time-marching integration can become expensive. This phenomenon partly justifies
the technique developed in §2.6.
Figure 2.6 shows the heat release transfer function for β = 6 and η = 1.1 while increasing
the Markstein length. At small frequencies the perturbed flame is not highly wrinkled and
first order curvature effects are small. At the same time, when the Markstein length is larger,
the flame slope dF/dr at the base is smaller (see Figure 2.4). Therefore, the projection
of the radial velocity onto the flame’s normal reduces, the denominator in equation (2.32)
reduces, and the FTF gain increases. At higher frequencies the flame is highly wrinkled
and curvature corrections play a crucial role in reducing the size of these wrinkles, which
lowers the gain. These arguments are consistent with the gain’s dependence on M shown
in Figure 2.6. Furthermore, the fact that the FTF phase shifts towards zero with M has
a clear physical cause: when the Markstein number is larger, the flame is shorter and the
perturbations take less time to travel along the flame.
As shown in Figure 2.6, curvature corrections to the FTF are small, and comparisons
between the G-equation model without curvature effects and experiments can be found in
2.5 Curvature effects 35
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
St2
|F
T
F
|
 
 
G−equation model Experiments from Schuller et al. (2005)
0 5 10 15 20
0
5
10
15
20
St2
F
T
F
6
Fig. 2.7 Comparison between our low-order model and an experimentally deter-
mined (Schuller et al., 2002) FTF. The FTFs here are scaled with respect to axial velocity
fluctuations. Parameters for the simulation were chosen to match experimental values; the
gain responses compare reasonably well, especially at low frequencies, whereas we cannot
capture the phase inflection because we fix the convection speed value.
the literature (Schuller et al., 2002; Karimi et al., 2009; Cuquel et al., 2011; Kashinath
et al., 2013b). Nonetheless, in Figure 2.7 we compare the FTF derived from our model
(with curvature correction) with that determined from experiments, extracted from Schuller
et al. (2002). The flame aspect ratio was set to β = 2.64, and was derived by imposing a
mean velocity u = 1.1 m/s and a flame speed s0L = 0.39 m/s, as suggested in Schuller et al.
(2002). Because we have no information on the other parameters, we set the convection speed
ratio and the Markstein length to the reasonable values K = 1.2 and M= 0.01 respectively
(Kashinath et al., 2013b). Note that, in order to compare the results, in Figure 2.7 we have
scaled the FTF with respect to axial velocity fluctuations, and not to normal velocity at the
flame base as in the rest of this study. The gains compare reasonably well, especially at low
frequencies, which is a known result (Schuller et al., 2003; Cuquel et al., 2011). The phase
response difference may be adjusted at low frequencies by choosing a suitable value for K.
However, our current model cannot capture the inflection that occurs at St2 ≈ 7.5, because
we are fixing the value of K. To capture this phenomenon, a possibility would be to use a
frequency dependent convection speed extracted from experiments or DNS as in Kashinath
et al. (2013b). However, we also note that the inflection of the phase response is not always
observed in experiments. For example in Karimi et al. (2009); Cuquel et al. (2011) the phase
is shown to increase linearly and to saturate at high frequencies. In this study we do not aim
to accurately reproduce a specific experiment, and in the following we will retain K constant.
36 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
As a final remark, we note that Shin & Lieuwen (2013) showed that turbulence effects on
the flame dynamics – modelled with a low-intense stochastic component in the velocity field
– are analogous to curvatures effects on the flame speed (and consequent dynamics) of the
ensemble-averaged flame front. Using their result, we can thus expect an analogy between
the curvature corrections to FTFs discussed above and variations in the FTFs of flames with
small turbulent fluctuations.
2.6 Self-excited problem: eigenvalue problem formulation
So far, we have investigated the forced response of conical premixed flames to harmonic
fluctuations. Although the knowledge of FTFs has some importance, usually one is interested
in the analysis of a closed flame-flow-acoustics feedback loop. In this context the flame
dynamics is not forced, but is coupled with the acoustic equations and the flow field dynamics.
Perturbations will grow unboundedly in time if and only if at least one thermoacoustic mode
has a positive growth rate.
If a model that couples fluid and combustion fluctuations is prescribed, then explicit
knowledge of the FTF is not needed in order to assess the system’s stability. Indeed,
one can simply construct a larger eigenvalue problem, whose state vector contains all the
variables of the fully-coupled thermoacoustic system, and use Arnoldi methods to calculate
the eigenvalues with the largest growth rate. As we will show in §2.7.3, this formulation
also allows for straightforward use of adjoint methods to calculate eigenvalues’ sensitivities
with respect to any system’s parameter. The latter is precluded when a numerically (or
experimentally) measured FTF is used, because the explicit dependence of the linear operator
on the base state and system’s parameters is unknown. In the following, we will describe
the method using the same G-equation model as presented in §2.3, but it could as well be
used in a network with a higher order model for the premixed flame, as the ones developed
by Hemchandra (2012); Blanchard et al. (2015).
To keep the model low-order, we divide the thermoacoustic problem conceptually into
three components, which are connected in a feedback loop: (i) the flame front f and
heat release fluctuations q′ dynamics are governed by the kinematic, linear G-equation
(equations (2.20b) and (2.23)); (ii) the heat release acts as a point source in the (linear)
acoustic equations. Unsteady heat dilates the surrounding air and is a monopole source of
acoustic waves, which in turn provoke velocity fluctuations at the flame base (Dowling, 1999;
Kashinath et al., 2013a); (iii) axial velocity fluctuations are convected downstream in the
flame domain, and radial fluctuations are calculated to guarantee mass conservation (Birbaud
et al., 2006; Preetham et al., 2008).
2.6 Self-excited problem: eigenvalue problem formulation 37
The acoustic equations can conveniently be written in state-space form:
dx
dt
= Ax+Bq′(t) (2.34a)
u′x =Cux, p
′ =C px, (2.34b)
as described in §2.2.2. Equations (2.34) are scaled in non-dimensional units consistently
with (A.1). Note that, so far, no particular acoustic configuration has been chosen, and the
matrix A and vectors B, Cu/p refer to any possible acoustic configuration. Typical non-
dimensional numbers that appear in the matrices are the mean flow Mach number M ≡ u/c,
where c is the speed of sound, and the ratio between the flame and acoustic characteristic
lengths. In the Laplace space, the acoustic transfer function can be expressed as:
Hu(s) =
u′x(s)
q′(s)
=Cu (sI −A)−1 B, (2.35)
where s is the Laplace variable. Note that it is unimportant how the state-space model
has been derived from the acoustic equations. One can use a Galerkin decomposition in
Fourier modes as in Culick (2006); Kashinath et al. (2013a), a state vector with memory as
in Mangesius & Polifke (2011) or Padé approximations as in Bothien et al. (2007); Orchini
et al. (2015).
Finally, the convective model for the velocity field reads:
∂u′x
∂ t
+
1
K
∂u′x
∂x
= 0, u′x(x = 0, t) =Cux(t), (2.36)
where the second equation corresponds to the boundary condition that the axial velocity at
the flame base is given by the acoustic state-space model.
Because we are interested in finding the global modes of the coupled system, we look for
solutions of the type f (r, t) = fˆ (r)eλ t , x(t) = xˆeλ t , u′x(x, t) = uˆ′x(x)eλ t . We then discretize F ,
fˆ , and uˆ′x in space, as well as the derivative and integral operators, and combine the discrete
versions of equations (2.20b), (2.23), (2.34), (2.36) to obtain the eigenvalue problem:
Mz = λ z, (2.37)
38 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
where λ ≡ σ + iSt are the complex eigenvalues sought, z defines the thermoacoustic state as
z ≡ [ fˆ 1, . . . , fˆ Nr , xˆ1, . . . , xˆNs, uˆ′1x , . . . , uˆ′Nxx ]T , and the matrix M can be written as
M =
 M f f M f s M f uMs f Mss Msu
Mu f Mus Muu
 , (2.38)
where the submatrix M f f represents the coupling between the flame evolution and the flame
state, M f s the coupling between the flame evolution and the acoustic state, and so on. Their
explicit expressions are given in Appendix B.
We observe that equation (2.36) could be analytically solved, yielding:
uˆ′x(x) =Cuxˆe
−λKx =Cu (λ I −A)−1 Be−λKxqˆ, (2.39)
where we have used equation (2.35) in the second equality. Note that the eigenvalue λ
appears both in the exponential and in the matrix inversion operation. Because of this, if we
were to substitute the velocity field expression into the flame dynamics we would obtain a
smaller eigenvalue problem – the state would be fully determined by the flame shape fˆ – but
the problem would be nonlinear. Because of the small size of the eigenvalue problems we
are solving, we have decided to solve the larger but linear eigenvalue problem (2.37).
By solving the eigenvalue problem, we can determine whether, for a given acoustic
configuration and set of flame/flow parameters, there exist modes with a positive growth rate.
We can determine their frequencies without having explicit knowledge of the FTF. This is
useful because we do not need to evaluate a new FTF if we change a flame parameter, and
we can investigate a large set of parameters at a lower computational cost. Numerical cost for
the evaluation of FTFs becomes large at high-frequencies, where the wavelengths are short
and the CFL condition for the time step gives:
∆t < CFL
∆r
Umax
≈ CFL1+β
2
β 2
∆r2
∆r+M 2+β 2√
1+β 2
, (2.40)
where the advection velocity Umax can be estimated from equation (B.1a). For non-small
curvature effects, the time step scales as ∆r2, and evaluating transfer functions at high
frequencies becomes numerically expensive. Also, the dimension of the eigenvalue matrix
M increases at high-frequencies because the spatial discretization has to be very dense in
order to capture small wavelengths fluctuations. We are not interested in determining all the
eigenvalues of the matrix, however, but only those with the most positive real parts. Thus
2.7 Linear analysis of a thermoacoustic network 39
f2
g2g1
f1 qR1 R2
x = 0 x = xf x = 1
p¯1, ρ¯1, T¯1, M1, d p¯2, ρ¯2, T¯2, M2, D
Fig. 2.8 Acoustic network considered in this study. It is based on the experimental setup
described by Kabiraj & Sujith (2012). The flame position x f is the bifurcation parameter we
will investigate. The flame, assumed to be acoustically compact, induces a mean temperature
change and consequent mean flow properties variations. An area change models the blockage
introduced by the burner holding the flame.
one can use numerical algorithms such as Arnoldi methods to compute the few eigenvalues
of interest, greatly reducing the cost of the problem.
2.7 Linear analysis of a thermoacoustic network
We now apply the theory outlined so far to the case of a laminar conical flame confined in a
duct.
The acoustic geometry we consider, sketched in Figure 2.8, consists of a straight pipe
with a non-zero mean flow. The pipe radius abruptly increases at the flame location, to
emulate the blockage of the burner holding the flame. The acoustic mean properties suddenly
vary across the flame and across area variations to satisfy mass, momentum and energy
balances. We impose a closed, perfectly reflecting acoustic boundary condition at the inlet,
and an open frequency dependent reflection coefficient at the outlet.
2.7.1 Model parameters
The thermoacoustic network parameters were inspired by the experimental setup of Kabiraj
& Sujith (2012): a laminar conical flame is attached to a burner (diameter D = 11 mm),
and confined in an acoustically open-closed pipe of length L = 860 mm and diameter
DL = 25.6 mm. The area increase is treated as a discontinuity, and we use jump conditions
dictated by mass, momentum and energy conservation to solve for the flow after the area
change (Stow & Dowling, 2001). The temperature ratio is T2/T1 = 2. The value for T2
has to represent the mean temperature in the downstream part of the duct. This number
should be a function of the flame position in the duct: the more downstream is the flame, the
40 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
higher is the mean temperature in the last section, because less heat is dissipated through the
walls. Nonetheless, we decide to keep this parameter independent from the flame location
for simplicity. The mean flow speed is u = 1.55 m/s. A lean equivalence ratio regime is
analysed, with φ = 0.51. The Markstein length is fixed to L= 0.6 mm, and the laminar flame
speed is varied to change the flame aspect ratio. When s0L = 0.25 m/s, the steady flame has an
aspect ratio β ≡ L f /R≈ 6. Damping effects are included by means of losses at boundaries;
the closed inlet reflection coefficient is set to R1 = 1.0, whereas for the outlet reflection
coefficient we use the expression derived analytically, using the Wiener-Hopf technique, of
an unflanged cylindrical open end in the low Mach number limit (Munt, 1977; Cargill, 1982;
Eldredge & Dowling, 2003; Peters et al., 1993):
R2 =−(1+0.9M)
(
1− 1
2
ω2R2
c2
)
. (2.41)
(2.41) has been validated against experiments in Peters et al. (1993). Lastly, we fix the
convection speed to K = 1.2. Fixing K has a strong influence on the flame time lag response.
Experiments and DNS have shown that the convection speed ratio K is in general a function
of the perturbation amplitude and frequency (Birbaud et al., 2006; Kashinath et al., 2013b).
For simplicity we decided to fix K to a reasonable value that fits their findings. Similar values
have been used in other numerical analysis (Kashinath et al., 2013a; Waugh et al., 2014;
Orchini et al., 2015). The non-dimensional flame position is x f ≡ l1/L, where L≡ l1+ l2 is
the length of the pipe. The acoustic time scale is defined by the acoustic Strouhal number
St2 ≡ ωL/c1.
Although the acoustic configuration is simple, the presence of multiple jump conditions
(flame and area increase) makes it harder to find analytically an expression for the acoustic
transfer function to heat release fluctuations. Therefore, we rely on numerical methods. We
use LOTAN, which is a low-order thermoacoustic network model developed by Dowling &
Stow (2003) for the simulation of longitudinal and annular combustion systems. The low-
order modelling approach is based on the fact that the main nonlinearity is in the combustion
response to flow perturbations rather than in the acoustics (Chu, 1963; Culick, 1971). LOTAN
has been verified experimentally against both a sector rig (Stow & Dowling, 2001) and an
atmospheric test rig (Stow & Dowling, 2004). The combustion system is modelled as a series
of interconnected modules. Longitudinal ducts, annular ducts, combustion zones and area
changes are amongst the module types that can be modelled. The model decomposes the flow
into a steady mean axial component and small perturbations. The perturbations throughout
the combustor are related via wave propagation, in which acoustic, entropy, and vorticity
waves are all included. The flow conservation equations are used to track the evolution of
2.7 Linear analysis of a thermoacoustic network 41
102 103
−60
−40
−20
0
20
40
|H
x
f
=
0
.3
4|
[d
B
]
 
 
Frequency response
State space fit
102 103
−360
−270
−180
−90
0
Frequency    [Hz]
6
H
x
f
=
0
.3
4
[d
eg
]
Fig. 2.9 Comparison between the frequency response calculated with LOTAN (solid black)
and the state-space approximation (dashed red) at x f = 0.34. The approximation works well
over a wide range of frequencies.
these waves. The connecting modules are modelled as acoustically compact, meaning that
their axial length is short in comparison to the acoustic wavelengths of interest. The acoustic
boundary conditions at the inlet and outlet of the combustor are assumed to be known, and
these can be prescribed as frequency-dependent functions. The flame is assumed to combust
at one axial location, i.e., to be acoustically compact.
Although LOTAN is able to model quite complex acoustic geometries, the range of flame
models currently available in LOTAN is relatively limited. We therefore use LOTAN in
an unconventional way. We take full advantage of LOTAN’s advanced acoustic modelling
capabilities by extracting the acoustic eigenfrequencies and the network frequency response.
Then we cast it in state-space form as in §2.2.2. This is necessary to extend the frequency
response – calculated at s = iω – in the full Laplace space, in which the growth rate σ
can be non-zero. Figure 2.9 shows that the state-space approximation fits the frequency
response evaluated with LOTAN well over a wide range of frequencies. We find that an
approximation order n = 20 is always sufficient to properly describe the frequency response
with a state-space model. Finally, rather than coupling this with a flame model directly in
LOTAN, we couple the extracted acoustic model with the G-equation model described in
§3.2. By doing so, we are able to model the coupled dynamics of complex acoustics with a
42 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
0 0.25 0.5 0.75 1
0
0.02
0.04
0.06
0.08
xf
σ
∗
 
 
St
∗
0
1
2
3
4
5
6
Fig. 2.10 Growth rates of the eigenvalues λ with a non-negative real part. Line colours
indicates the frequency of the mode. Thick lines and circles at σ∗ = 0 correspond to brute-
force time marching results, as described in 3. Full and empty circles correspond to super-
and subcritical Hopf bifurcations respectively.
flame model which is also complex. This can be used to study the thermoacoustic system
both in the frequency and the time domain.
2.7.2 Linear stability of a ducted flame
The first bifurcation parameter we investigate is the position of the flame in the duct x f ∈ [0,1],
fixing the flame aspect ratio to β = 6. The corresponding FTF is analogous to the one shown
in Figure 2.6 for M= 0.02. We can take advantage of the fact that (i) its gain never exceeds
unity, and (ii) it acts as a low-pass filter, to deduce that if a thermoacoustic mode has a
positive growth rate, then (i) it is associated with one of the acoustic resonant frequencies,
for which the gain is much larger than one and (ii) it is a low-frequency mode. This will be
shown to be true in Chapter 3 also in the nonlinear limit, by means of the harmonic balance
technique. Only modes associated with the first two acoustic resonant frequencies were found
to be unstable. Therefore, we can reduce the size of the eigenvalue problem by using a small
state-space model that characterizes only the first few acoustic modes (Ns = 4), and by using
a relatively large grid spacing ∆r, because we do not need to describe short wavelengths
(Nr = Nx = 399). Limiting the analysis to 4 acoustic modes is a good approximation for
the case considered here, because only low-frequency modes can be unstable, and their
projection on high-frequency modes is negligible. A higher order approximation needs to be
considered if modes with higher frequencies can be unstable.
2.7 Linear analysis of a thermoacoustic network 43
fˆ
0 0.25 0.5 0.75 1
r
uˆ
′ x
1 2 3 4
k
xˆ
k
Fig. 2.11 Real parts of the eigenstate of the unstable mode for x f = 0.27. Top left: perturbed
flame shape state. Bottom left: axial velocity fluctuations. Right: components of the acoustic
state-space vector.
Figure 2.10 shows the results of the eigenvalue problem. Only the growth rates σ∗ ≡
σ/(2π) and frequencies St∗ ≡ St/(2π) of the eigenvalues with a non-negative real part are
shown. We clearly identify two modes, whose frequencies can be shown to be very close
to the first and second acoustic resonant frequencies. We have verified with convergence
tests that these modes are physical modes, and not spurious modes that arise because of the
discretization of the equations. We observe that for some flame positions, x f , both modes
may have a positive growth rate. We have indicated with squares the Hopf bifurcations of
the high-frequency mode, which lie inside the region of instability of the low-frequency
mode. When both modes have a positive real part, the growth rate of the of the low-frequency
mode dominates over the growth rate of the high-frequency one, which is consistent with the
system’s low-pass characteristics. All these results agree with the fully nonlinear analysis
carried out by Orchini et al. (2015). We have plotted in Figure 2.10 with thick black lines
and circles the linearly stable regions identified by brute-force time marching of the fully
nonlinear equations. According to these results, filled and empty circles locate super- and
subcritical Hopf bifurcations respectively.
Figure 2.11 shows the eigenvector of the unstable mode at x f = 0.27. Moving from the
flame anchoring point r = 1 to the centreline r = 0, the top-left frame shows that the flame
perturbation grows in amplitude spatially. On the other hand, velocity fluctuations, shown
in the bottom panel, decrease in amplitude while moving in the same direction. This is
physically correct because we are using a convective model with speed 1/K, so the velocity
amplitude at r = 0 at time t is the amplitude that the velocity perturbation had at r = 1 at
time t−1/K and, because λ has a positive growth rate, the earlier perturbation had a smaller
amplitude than the current one. We showed in equation (2.39) the analytical solution for the
convective model along the flame. The analytical result is plotted with circles in Figure 2.11,
44 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
and is in perfect agreement with the numerically evaluated eigenstate. This comparison
can be used as a test to assess the convergence of the eigenvalue problem. For example,
high-frequency modes will not converge with the current grid-space size but we know that
they will be stable because the flame acts as a low-pass filter. Finally, the right panel shows
the intensity of the acoustic eigenstate. The fact that only the first mode has a large intensity
is an indicator of the fact that this thermoacoustic mode is closely linked to the first acoustic
resonant mode.
Given the relatively small size of the eigenvalue problem, we can quickly calculate the
stability of the thermoacoustic system for a wide range of parameters. We start by focusing
on two parameters: the position of the flame in the duct and the flame aspect ratio β , which
can also be interpreted as the ratio between the flame speed and the mean flow velocity (see
equations (A.1)). We vary x f along the full length of the duct in steps of ∆x f = 0.01, and
β in the range [0.5,10] in steps of ∆β = 0.1. For each couple of parameters, we solve the
eigenvalue problem (2.37), and we investigate the two eigenmodes with the largest growth
rate. We will refer to the low- and high-frequency modes as mode 1 and mode 2 respectively
because they are closely linked to the acoustic fundamental and second hamonic resonant
modes. Note that we are not tracking the eigenvalues, but we are just looking at the two
eigenvalues with the largest growth rate. We find that, in our system, some of the eigenvalues
may cross each other in the complex plane while varying the bifurcation parameters. Because
they cross, the growth rate map we evaluate is continuous, but its derivative is not, as is
discussed in §2.7.3. We have verified that the eigenvalue crossing always occurs when the
growth rates of the crossing modes are negative (dashed lines in Figure 2.12), and we are not
missing any region of instability in the following analysis.
The growth rate maps of the two modes are shown in Figure 2.12. We have highlighted
with black lines the zero growth rate contours, which are the neutral curves along which
Hopf bifurcations are located. The thin straight lines (and circles) at β = 6 on the two maps
correspond to the growth rates (and Hopf bifurcations) shown in Figure 2.10. Note that we
have used two different colour maps for the stability diagrams of the two modes to highlight
that the growth rate of mode 1 is higher than the growth rate of mode 2, which agrees with
the fact that the flame acts as a low-pass filter.
Looking at the size of the regions with positive (or negative) growth rates, we observe
that mode 1 shows very large zones in which the sign of the growth rate does not change,
and they approximately extend over the whole length of the duct. On the other hand, mode 2
shows several smaller regions with a positive growth rate, and they extend approximately
over 1/3 of the duct.
2.7 Linear analysis of a thermoacoustic network 45
xf
β
(M
o
d
e
1)
 
 
σ∗
0 0.2 0.4 0.6 0.8 1
0.5
2
4
6
8
10
−0.15 −0.1 −0.05 0 0.05 0.1
xf
β
(M
o
d
e
2)
 
 
σ∗
0 0.2 0.4 0.6 0.8 1
0.5
2
4
6
8
10
−0.06 −0.04 −0.02 0 0.02
Fig. 2.12 Growth rate maps of two thermoacoustic modes. Dark regions are associated with
a negative growth rate (the mode is stable), and light regions with positive growth rates (the
mode is unstable). Two different colour map scales are used to highlight the growth rate
gradients. Neutral lines (zero growth rate) are highlighted in black. The straight thin line and
the circles correspond to the results and the Hopf bifurcations shown in Figure 2.10. In mode
1 we observe an eigenvalue crossing phenomenon: the curves along which the eigenvalues
cross are highlighted with dashed white lines.
0 0.25 0.5 0.75 1
M
od
e 
1
xf
0 0.25 0.5 0.75 1
xf
M
od
e 
2
 
 
uˆ′ pˆ
′ pˆ′uˆ′
Fig. 2.13 Fundamental (mode 1) and second hamonic (mode 2) modes in a closed-open pipe,
which roughly approximates the acoustic network under consideration. The location of nodes
and maxima/minima closely resembles those observed in Figure 2.10. Thus, we find that
the system is more prone to thermoacoustic instabilities where the Rayleigh index (2.42) is
largest, as expected.
46 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
This can be explained through the Rayleigh criterion, which states that a necessary
condition for thermoacoustic oscillations to occur is that pressure and heat release fluctuations
need to be sufficiently in phase (Rayleigh, 1878). More quantitatively, the Rayleigh index,
which quantifies the rate of energy produced over a thermoacoustic cycle and is defined by
Ry≡ γ−1
γ p
∫ T
0
p′(x f , t)q′(x f , t)dt, (2.42)
has to be larger than the sum of the acoustic energy losses (Ferreira & Carvalho Jr., 1997;
Nicoud & Poinsot, 2005). Assuming that the heat release fluctuations are proportional to
(delayed) velocity fluctuations, the Ry index becomes
Ry ∝ pˆ′(x f )uˆ(x f )
∫ T
0
cos(ωt)cos(ω(t− τ))dt, (2.43)
where pˆ′(x f ) and uˆ′(x f ) are the values that the acoustic modeshapes of eigenfrequency ω
assume at the flame location. τ is the time delay between acoustic velocity and heat release
fluctuations, which is connected to the convection speed and flame length, as discussed later.
Because the acoustic network we are considering has a temperature jump across the flame
and a cross sectional area variation, the shape of the modes is different for every value of x f .
However, our network can be roughly approximated by a closed-open pipe with no area or
temperature variations, as shown in Figure 2.13. In mode 1, the interaction between pressure
and velocity acoustic waves has nodes ( pˆ′(x f ) = 0 or uˆ′(x f ) = 0) at the boundaries and a
maxima (or minima) is found at the centre of the duct. For the second mode, the acoustics
waves have nodes at x f = 1/3 and 2/3, and maxima or minima are found at x f = 1/4, 2/4,
3/4. These same patterns can be observed (with some modifications due to temperature and
area variations of the acoustic network we are considering) in the growth rate maps of the
two thermoacoustic modes. Varying the delay τ causes a time shift between velocity and
pressure fluctuations, which may alter the sign of the Rayleigh index, as discussed later.
2.7.3 Adjoint-based sensitivity
Having evaluated steady solutions and the stability of small perturbations around them over
a fairly large parameter space, a natural question that arises is: how does the stability map
change by making a tiny change to one of the parameters in the governing equations? This
information is contained in the so-called base state sensitivity map, which measures the drift
of the eigenvalues δλ with respect to a given perturbation in a parameter δK (Giannetti &
Luchini, 2007; Sipp et al., 2010; Luchini & Bottaro, 2014). Adjoint methods for eigenvalue
sensitivity were introduced in thermoacoustics by Magri & Juniper (2013b) on a Rijke tube
2.7 Linear analysis of a thermoacoustic network 47
system, and extended to the analysis of enclosed diffusion flames with a Fourier-Galerkin
based acoustics by Magri & Juniper (2014). Here we extend the use of adjoint-based
sensitivity methods to wave-based acoustic networks with premixed flame models. This is
indispensable information for gradient-based optimization algorithms, in which one changes
the flame or cavity shape to minimize the growth rate of a particular mode.
To perform a sensitivity analysis, we need to define the adjoint operator of the direct
equations (2.37). We can either derive the continuous adjoint equations from the continuous
governing equations, or we can obtain a discrete version of the adjoint equations directly
from the discretized version of the direct equations (Chandler et al., 2012; Magri & Juniper,
2013b). Here we will use the second approach, i.e., we will define the adjoint matrix M†
to be M† = MH , where the superscript H stands for Hermitian conjugation. Under this
assumption, the drift in the eigenvalue is given by the known relation (Sipp et al., 2010;
Luchini & Bottaro, 2014):
δλi =
〈
z†i ,δMzi
〉
〈
z†i ,zi
〉 , (2.44)
where zi is a right eigenvector of the direct matrix, Mzi = λizi, and z†i is the corresponding
right eigenvector of the adjoint matrix, MHz†i = λ
∗
i z
†
i , where the superscript
∗ stands for
complex conjugation. The symbol ⟨·, ·⟩ denotes an inner product. To define the adjoint
discrete operator as the transpose conjugate of the direct discrete operator, is equivalent to
choosing the scalar product as an inner product, i.e.:
⟨z1,z2⟩= zH1 · z2 =
Nr+Ns+Nx
∑
i=1
zi
∗
1 z
i
2. (2.45)
Note that here we are only interested in evaluating sensitivities of the eigenvalues. The drift
in the eigenvalue is a well-defined physical quantity, and can be proven to be independent of
the choice of the inner product (Luchini & Bottaro, 2014). This choice allows a very simple
connection between the adjoint right eigenvectors z† and the left eigenvectors zL of the direct
matrix M . In particular we have that z† = zHL , meaning that, by computing both the right and
left eigenvectors of M , we have all the information required to evaluate sensitivities. Finally,
recall that M is the discrete operator of the governing equations linearised around a steady
solution. Indicating the base state with z, the eigenvalue drift with respect to parameter p is:
δλ =
zL · δM(z, p)z
zL · z =
(
zL · ∂M∂ p z
zL · z +
zL · ∂M∂ z ∂ z∂ pz
zL · z
)
δ p, (2.46)
48 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
0 0.25 0.5 0.75 1
−0.4
−0.2
0
0.2
0.4
0.6
xf
 
 
d
σ
∗
/
d
K
St∗
Mode 1
Mode 2
0 2 4 6
Fig. 2.14 Growth rate sensitivity at β = 6 of the two modes with the largest growth rate to
convection speed perturbations. Lines correspond to sensitivity calculations with adjoint
methods; the line’s colour refers to the frequency of the unperturbed modes. Markers refer to
sensitivity calculations with a finite difference approach, and validate the adjoint calculations.
where we have used the chain rule to account for the fact that the steady-state solution
is in general a function of the parameter p as well. The quantity ∂ z∂ p cannot be easily
evaluated in general, and another adjoint problem can be set up to calculate it. The left and
right eigenvectors of M have already been evaluated when solving the eigenproblem (2.37).
Therefore, with simple matrix multiplication operations, we can compute the sensitivity of
the system with respect to any parameter.
Here, we will focus on the role of the convection speed, K, on the stability of the system.
This parameter does not affect the steady solution so the sensitivity is simply given by the
first term in the r.h.s. of equation (2.46). The convection speed is of crucial importance in
modelling premixed flames and it has been investigated by Preetham et al. (2008); Kashinath
et al. (2013b). In Orchini & Juniper (2015) it is shown that it severely affects the gain
response of non-conical Flame Describing Functions. The matrix ∂M∂K can be evaluated
explicitly from equations (B.5), and only the derivatives of the blocks (B.5h) and (B.5i) are
non-zero.
It is known that the adjoint eigenvectors of the finite-difference discretized equations
can be noisy close to non-Dirichlet boundary conditions (Hartmann, 2007; Chandler et al.,
2012). This is because the discrete forms of the boundary conditions have been chosen to
2.7 Linear analysis of a thermoacoustic network 49
xf
β
(M
o
d
e
1
)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
7
8
9
10
xf
β
(M
o
d
e
2
)
0 0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
7
8
9
10
 
 
dσ∗
dK
−1 −0.5 0 0.5 1
Fig. 2.15 Growth rate sensitivity to convection speed variations in the parameter space x f ×β .
Thick black lines are the neutral lines of the growth rate map. Thin light lines are the zero
contours of the sensitivity maps, i.e., where the system is insensitive to small changes in the
parameter K. The red lines highlight results shown in Figure 2.14: solid and dashed red lines
have a negative and positive sensitivity, respectively.
work well with the right eigenvectors, but there is no guarantee that they will work with the
same accuracy on the left ones. Indeed, we observe some noise in the left eigenvectors of M
close to the Neumann boundary condition relative to the flame tip, so we first validate the
adjoint-based sensitivity results with a brute-force finite difference method.
Using the same parameters as in §2.7.2, and fixing β = 6, we show in Figure 2.14 the
growth rate sensitivity to variations in the convection speed evaluated with adjoint-based
(solid lines) and finite-difference (markers) methods for thermoacoustic modes 1 and 2.
Although we observe some noise in the adjoint eigenvectors close to the flame Neumann
boundary condition (not shown), the results obtained with the two methods agree perfectly.
From now on we will always refer to the adjoint-based sensitivity results. At a given x f
location in Figure 2.14, if the sensitivity of a mode is positive, then a small decrease of the
parameter K will make the growth rate smaller and vice versa. Note that a change in K could
make one mode grow and another mode decay. This is typically a disadvantageous result,
because a change that tends to stabilise one mode may destabilise another one. Also, at this
stage, no clear pattern can be observed for the behaviour of the two modes.
To gain more insight into the physical mechanisms that trigger the instabilities, we extend
the growth rate sensitivity analysis to the parameter space x f ×β as in §2.7.2. Figure 2.15
shows the sensitivity of the growth rate for mode 1 (top panel) and mode 2 (bottom panel).
50 Linear stability and adjoint sensitivity analysis of thermoacoustic networks
Thick black lines correspond to the zero growth rate level sets of Figure 2.12, and thin
light lines to the zero contour lines of the sensitivity maps, i.e., the lines along which the
system is insensitive to small changes in the convection speed. Every horizontal slice of
Figure 2.15 produces a graph analogous to Figure 2.14, which corresponds to the horizontal
lines highlighted in red at β = 6. Note that the sensitivity of mode 1 is actually discontinuous
across the lines where one eigenvalue takes over from another, the dashed lines of Figure 2.12.
This is because, as discussed in §2.7.2, the dominant eigenvalue switches along these lines.
Note how in mode 2’s sensitivity map, the zero growth-rate isolines cross the zero
sensitivity isolines at the acoustic nodes x f ≈ 1/3 and 2/3. This is because the location of the
nodes is a purely acoustic property, and it is unaffected by a change in the convection speed,
which only alters the flow and flame dynamics. Also, note how all the zero sensitivity isolines
cross the zero growth rate isolines always in the same fashion in both mode’s maps: the lower
parts of the neutral curves always lie in regions with a positive sensitivity (light colours),
and the upper parts in regions with a negative sensitivity (dark colours). As a consequence,
increasing the convection parameter K, shifts the edges of the instability towards a smaller β
value, i.e., towards shorter flames. This has a very simple physical explanation: according to
the Rayleigh criterion (2.42), a thermoacoustic oscillation can occur only when unsteady heat
release and acoustic pressure fluctuations are sufficiently in phase. Changing the parameter
K does not affect the acoustics, but it does change the phase of heat release fluctuations.
In particular, we can relate the time delay of heat release fluctuations with the time that a
perturbation needs to travel from the base to the tip, given by:
τ ∝ L f
1
uc
= βR
K
u
. (2.47)
The onset of thermoacoustic instabilities is found at specific values of time delays τ . Thus, if
we increase the convection parameter K and we wish to find where the new zero growth rate
contours lies by keeping the time delay constant, then we need to reduce β , in agreement
with the sensitivity analysis. This also means that the sensitivities with respect to K and β
must have the same sign, because increasing K or β causes an increase in the time delay.
This can be verified by qualitatively evaluating dσ∗/dβ along vertical lines in the growth
rate maps of Figure 2.12.
All the results obtained with adjoint sensitivity are in line with an intuitive physical
argument based on the time delay induced by the convection speed model, and can be used
to understand changes in the stability of thermoacoustic system’s with premixed flames.
This example shows how sensitivity analysis has great potential for the design and passive
2.8 Conclusions 51
control of thermoacoustic systems, one of its main advantages being that the process can be
embedded within a gradient-based optimization algorithm.
2.8 Conclusions
In this chapter we have outlined the wave-based acoustic network modelling adopted in this
thesis. A methodology to cast the acoustic response in state-space has been presented. We
have evaluated Flame Transfer Functions (FTF) for laminar, conical flames excited by an
incompressible, travelling wave flow model. We obtain analytical results for the case of
uniform flame speeds (which contains a correction to a previously published result), and
numerical results when we account for curvature’s corrections on the flame speed. We show
how in the latter case the numerical evaluation can become unnecessarily expensive even
for low-order models. Therefore we tackle the thermoacoustic problem from a different
perspective, by avoiding the explicit evaluation of FTFs and directly evaluating the stability
of the fully-coupled thermoacoustic system. In this way the stability problem is cast in a
classic eigenvalue problem and one can efficiently solve only for the eigenvalues of interest,
typically the ones with the largest growth rates. With minimal computational resources we
can calculate the stability of the system over a wide range of parameters, and we identify
and discuss the location of stability lines (Hopf bifurcations) in the parameter space. Finally,
we show how adjoint sensitivity methods can be applied to this model without any further
expensive calculation. Sensitivity results help to extract even more information out of the
eigenvalue problem. We explicitly evaluate the sensitivity with respect to the convection
speed parameter, but any other parameter of the system could be chosen. We observe that the
system’s stability is strongly connected to the time delay L f /uc induced by the convective
model, as expected. Future analysis can extend them to systems with premixed flames using
more elaborated models for the flame and heat release dynamics and to the more sophisticated
question of optimizing a burner shape.

Chapter 3
Frequency domain and time domain
nonlinear analysis of thermoacoustic
oscillations with wave-based acoustics
In this chapter a thorough nonlinear analysis of the thermoacoustic system described in §2.7
is performed. Current state-of-the art methods in the frequency and time domain are used
to predict the nonlinear behaviour of this system. The aim is to compare the strengths and
weaknesses of the various methods, together with their cost. This information will be used
in the next chapters of the thesis as a guidance to develop novel techniques that fill the
deficiencies of the current methods, or reduce their cost.
The content of this chapter was published in Orchini et al. (2015).
3.1 Introduction
The calculation of thermoacoustic oscillations’ amplitude in systems with confined premixed
flames involves the interaction of linear acoustics and nonlinear combustion. In order to
model these oscillations in a numerically tractable way, it is usually necessary to use a
low-order model for the flame. This reduced order model allows the system’s stability to be
investigated either in the frequency or time domains, and gives insight into the underlying
mechanisms without resorting to full CFD (Dowling, 1997, 1999; Culick, 2006; Noiray et al.,
2008; Lieuwen, 2012).
A widely-used frequency domain approach is to measure or calculate the Flame De-
scribing Function (FDF), which approximates the nonlinear unsteady heat release response
to finite amplitude harmonic velocity or equivalence ratio fluctuations. In the linear limit,
54 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
the FDF reduces to the Flame Transfer Function (FTF), for which analytical expressions
can be derived for some flames (Schuller et al., 2003; Preetham et al., 2008; Shreekrishna
et al., 2010). In a nonlinear framework, the FDF can be coupled with a model for the
acoustics, and can predict the stability of a thermoacoustic system, as well as the frequency
and amplitude of steady-state harmonic oscillations (Dowling, 1997; Noiray et al., 2008).
However, it is restrictive to assume that the velocity fluctuations are harmonic. This precludes
non-harmonic periodic oscillations, as well as more complex dynamical behaviour such as
period-2, quasiperiodic, and chaotic oscillations, which are common in thermoacoustics
(Subramanian et al., 2010; Gotoda et al., 2011; Kabiraj & Sujith, 2012; Kabiraj et al., 2012a;
Gotoda et al., 2014).
On the other hand, time domain simulations do not make any a priori assumptions about
the form of the acoustic fluctuations. These have been used successfully by Kashinath
et al. (2014) and Waugh et al. (2014) to simulate the nonlinear behaviour of thermoacoustic
systems by time integration and numerical continuation of limit cycles. These simulations
qualitatively reproduce the elaborate dynamical behaviour found experimentally. However,
so far only a simplistic acoustic configuration has been considered in such studies: a duct
with uniform cross-sectional area and no temperature variation across the flame. In this
analysis we present a general method to evaluate the acoustic response of an axisymmetric
resonator containing, for example, area changes and temperature variations, and how to
couple it with a kinematic flame model based on the G-equation both in the frequency and
time domains. This allows the methods of Kashinath et al. (2014) and Waugh et al. (2014) to
be generalised to practical thermoacoustic systems.
The chapter is structured as follows: in §3.2 we present the fully nonlinear combustion
and flow models. Details on the methods used to numerically integrate it and on the acoustic
model have already been presented in the thesis. In §3.3 the acoustic model improvements
with respect to previous analogous analyses are outlined. In §3.4 a numerically evaluated
FDF is shown, and we perform a nonlinear stability analysis of our thermoacoustic system
by means of the harmonic balance technique. The same thermoacoustic configuration is
investigated in §3.5.1 in the time domain with two different techniques. In §3.6 bifurcation
diagrams are shown, and we discuss the analogies and differences between the results
predicted by the various methods. Finally, in §3.7 we discuss the nonlinear dynamical
behaviour that we have observed, and we compare it with other analogous studies.
3.2 Nonlinear flame modelling 55
3.2 Nonlinear flame modelling
In this section we describe the nonlinear model for the flame dynamics and unsteady heat
release fluctuations. It is based on the well-known G-equation model (Markstein, 1964;
Williams, 1985), accounting for curvature effects on the flame speed. We also introduce the
flow field model, which is based on previous studies (Birbaud et al., 2006; Kashinath et al.,
2013b).
The G-equation model provides a low-cost method that enables premixed flame dynamics
and heat release to be described qualitatively. In contrast to Chapter 2, here we consider the
fully nonlinear dynamics of the flame front (the G = 0 level set), which is governed by the
transport equation:
∂G
∂ t
+u ·∇G = sL |∇G| . (3.1)
The local flame speed sL is in general a function of the equivalence ratio φ , the type of fuel,
the flame curvature, and, for turbulent flames, the turbulent intensity. In this analysis, we
consider conical, laminar, fully premixed flames. Therefore, the equivalence ratio is constant
and uniform, and the flame speed depends only on local curvature effects (Markstein, 1964;
Creta & Matalon, 2011):
sL = s0L (1−Lκ) . (3.2)
Here s0L is the speed of a laminar, flat flame sheet, κ ≡ ∇ · nˆ is the local flame curvature, and
L is the Markstein length.
We consider the same axisymmetric thermoacoustic configuration as described in §2.7,
using a cylindrical coordinate system
(
rˆ, θˆ , xˆ
)
. The velocity field u = (ur,uθ = 0,ux) is
decomposed into mean (u) and fluctuating (u′) components. The mean flow is assumed to
be uniform and flow in the axial direction xˆ. For the perturbation (Birbaud et al., 2006), the
axial velocity (forced or self-excited) is imposed at the burner (xb) every instant and advected
downstream, and the radial velocity is found by continuity:
∂u′
∂ t
+Uc
∂u′
∂x
= 0 u′(x = xb) = u′ac(t),
1
r
∂ (ru′r)
∂ r
+
∂u′
∂x
= 0 u′r(r = 0) = 0.
(3.3)
DNS calculations show that these perturbations advect axially at a frequency-dependent
speed that is slower than the mean flow (Kashinath et al., 2013b). We do not know the
acoustic frequencies a priori, so we fix this speed at Uc = 0.833ux, which is the largest value
observed by Kashinath et al. (2013b).
56 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
Substituting the flow field and flame speed expression (3.2) into (3.1), and introducing
the non-dimensional parameters
x
L f
→ x r
R
→ r ux
ux
→ ux urux → ur
t ux
L f
→ t, (3.4)
we obtain
∂G
∂ t
+βu′r
∂G
∂ r
+
(
1+u′x
) ∂G
∂x
=
s0L
ux
(1−Mκ)
√(
β
∂G
∂ r
)2
+
(
∂G
∂x
)2
, (3.5)
where L f is the flame height, R is the burner radius, β ≡ L f /R is the flame aspect ratio, and
M is the ratio between the Markstein length and the flame height.
The nonlinear PDE (3.5) summarises the nonlinear combustion model adopted. It is
integrated numerically using LSGEN, as discussed in §1.3.
3.3 Effect of the temperature jump on the acoustic response
The acoustic network and the corresponding state-space model chosen for the analysis in this
chapter are the same as those described in §2.7. Because our wave-based acoustic approach
includes a mean flow, a temperature jump, and frequency dependent reflection coefficients,
it significantly improves previous low-order time domain modelling of analogous thermoa-
coustic networks, as discussed below. Furthermore, the state-space formulation allows us
to perform calculations both in the frequency and time domain, with no approximations
between the two domains from an acoustic point of view. Therefore, differences in the
nonlinear results obtained with the various methods that will be applied can only be due to
approximations introduced in the nonlinear flame response.
Previous studies in the time domain (Kashinath et al., 2014; Waugh et al., 2014) have
examined straight pipes with uniform acoustic mean properties and zero mean flow. The low-
order thermoacoustic framework proposed here, however, includes a mean flow, temperature
and cross-sectional area variations, and can therefore analyse more general configurations.
Varying the flame position in the acoustic network, we evaluate transfer functions Hx f (s)
and fit them to state-space models as described in §2.2.2. When the reflection coefficients are
equal to ±1, the acoustic response (evaluated at, say, the flame position) can be thought of as
a Galerkin discretization onto a basis set formed by the orthogonal acoustic eigenmodes ψack :
u′(x f , t)≈
N
∑
k=1
uˆ′k(t)ψ
ac
k (x f ). (3.6)
3.3 Effect of the temperature jump on the acoustic response 57
−1
0
1
ψ
a
c
1
[a
.
u
.
]
T2/T1=1
ψ
a
c
2
[a
.
u
.
]
ψ
a
c
3
[a
.
u
.
]
0 0.25 0.5 0.75 1−1
0
1
x
ψ
a
c
1
[a
.
u
.
]
0 0.25 0.5 0.75 1
x
T2/T1=2
ψ
a
c
2
[a
.
u
.
]
0 0.25 0.5 0.75 1
x
ψ
a
c
3
[a
.
u
.
]
Fig. 3.1 Acoustic velocity modeshapes evaluated with a Fourier-Galerkin basis (top panels),
analogous to the limit T2/T1 → 1, and wave-based acoustics (bottom panels), with T2/T1 =
2. In the former case the modes are continuous and correspond to the natural acoustic
eigenmodes of the open-closed pipe, in the latter a discontinuity is located at the flame
position because of a sudden change in the mean flow temperature.
100 101 102
−150
−100
−50
0
50
|H
(i
ω
)|
[d
B
]
 
 
Fourier−Galerkin, T2/T1 = 1
LOTAN, T2/T1 = 2
100 101 102
−2880
−2160
−1440
−720
0
720
6
H
(i
ω
)
[d
eg
]
St2 [rad]
Fig. 3.2 Transfer functions of velocity fluctuations with respect to heat release fluctuations
for a straight open-closed duct with a flame positioned at x f = 0.213. Fourier-Galerkin (solid
line) state-space, as in Kashinath et al. (2013b): pole’s locations are equispaced and remain
at the natural frequencies of the open-closed pipe. LOTAN (dot-dashed line) state-space,
with T2/T1 = 2: the position of the poles is shifted because the mean acoustic properties vary
with T2/T1, and they are no longer equispaced.
58 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
Indeed, one can choose the state-space vector C =
[
ψac1 (x f ), . . . ,ψ
ac
N (x f )
]
in equation (2.10b).
Note that this analogy is not true in general, because the acoustic eigenvectors are non-
orthogonal for non-trivial reflection coefficients (Nicoud et al., 2007). Figure 3.1 shows a
comparison between the first three eigenmodes of an open-closed pipe with R1 = 1.0 and
R2 =−1 with and without a temperature jump. In the latter case, the Galerkin basis functions
correspond to a Fourier expansion, i.e., ψack = e
ikx, which we find in the limit T2/T1 → 1. In
both cases, the thermoacoustic modes contain a discontinuity at the flame location, which
is typically modelled with a Dirac delta in systems with no mean temperature effects, as
was seen for example in Magri & Juniper (2013b); Waugh et al. (2014). In networks with a
temperature jump across the flame, a discontinuity is naturally present in all the acoustic basis
functions when a wave-based approach is adopted, which relaxes the Gibbs phenomenon.
Furthermore, thermoacoustic oscillation frequencies are often found to be close to the
acoustic natural frequencies (Dowling & Stow, 2003; Noiray et al., 2008). This will be the
case in our configuration, as we discuss in the next section. These acoustic eigenfrequencies
change significantly when the mean temperature is non-uniform, as is shown in Figure 3.2,
and are no longer multiples of the fundamental frequency (see equation (2.7)). This reduces
coupling between the modes, which will be discussed later.
3.4 Nonlinear analysis in the frequency domain
In this section we analyse thermoacoustic oscillations in the frequency domain. This is ac-
complished by numerically evaluating a Flame Describing Function (FDF) to inlet harmonic
velocity fluctuations. The FDF is coupled in a feedback loop with acoustic transfer functions.
3.4.1 Flame Describing Function
In most thermoacoustic systems, the flame is the main source of nonlinearity (Chu, 1963;
Culick, 1971). The FDF method exploits this characteristic; it approximates the flame’s
nonlinear response to an imposed input signal. We denote with FDF(St,A) the FDF of a
sinusoidal-input with angular frequency 2πSt and amplitude A. When we close the FDF
in a feedback loop with an acoustic transfer function H(St), we can predict the frequency,
amplitude, and stability of limit cycles via the harmonic balance (or Describing Function
method) (Khalil, 2001).
We numerically construct an FDF by harmonically forcing the axial flow at the flame
inlet at various frequencies f = ω/(2π) and amplitudes. Solving continuity and advection
3.4 Nonlinear analysis in the frequency domain 59
(a)
r
x
 
 
0.3 0.2 0.1 0 0.1 0.2 0.3
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
(b)
Fig. 3.3 Flame wrinkling due to vortex formation and roll up. (a) Experimental forced conical
flame image, reprinted from Karimi et al. (2009), with permission from Elsevier. (b) G-field
obtained from a numerical simulation of a forced conical flame. In both cases, the forcing
is harmonic, with a frequency St = 20/(2π) and an amplitude A = 0.3. The flame contour
G = 0 is highlighted and it qualitatively reproduces experimental results.
equations, the perturbation flow reads
u′ = Acos [2πSt(Kx− t)] u′r = A
K
β
r
2
2πStsin [2πSt(Kx− t)] , (3.7)
where St≡ f L f /u is the flame Strouhal number, and K = 1.2 is the ratio between the axial
mean flow speed and the convection speed, as in Kashinath et al. (2013b). This perturbation
model has been proven to reproduce the characteristic vortex formation at the burner lip
and roll up along the flame. It causes flame wrinkling, which strongly influences flame
surface area and consequent heat release fluctuations (Preetham et al., 2008). A qualitative
comparison with experimental results is shown in Figure 3.3.
The heat released by the flame is evaluated through:
Q(A, t)≡ 2πρs0LhR
∫ ∫
D
(1−Lκ) |∇G|δ (G)r dr dx. (3.8)
Heat release fluctuations q′(A, t) = Q(A, t)−Q are decomposed into Fourier modes
q′(A, t) =
∞
∑
k=1
q′k(A)cos(2π kSt t+ϕk(A)). (3.9)
60 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
0 1 2 3 4 5 6
10−2
10−1
100
St
|F
D
F
(S
t,
A
)|
 
 
Α = 0.02
Α = 0.10
Α = 0.20
Α = 0.30
Α = 0.40
0 1 2 3 4 5 6
−10pi
−8pi
−6pi
−4pi
−2pi
0
St
F
D
F
(S
t,
A
)
 
 
∠
Α = 0.02
Α = 0.10
Α = 0.20
Α = 0.30
Α = 0.40
Fig. 3.4 FDF gain (left) and phase (right) of the heat release rate of a laminar, conical flame
to velocity fluctuations. The flame aspect ratio is β = 6 and curvature effects on the flame
speed are accounted for, with a non-dimensional Markstein length M= 0.02. The top and
bottom panels show the same data.
3.4 Nonlinear analysis in the frequency domain 61
0 0.1 0.2 0.3 0.40.5
1
1.5
A
G
ai
n
St = 1.91
0 0.1 0.2 0.3 0.40.5
1
1.5
A
G
ai
n
St = 4.97
Fig. 3.5 Amplitude dependence of the closed-loop Describing Function gain |N0.5(St,A)| for
two St values. Left panel: at small frequencies, a monotonical decay of the gain suggests that
the system can be either linearly stable (if the gain at A= 0 is less than one) or supercritically
unstable and saturate to a limit cycle. Right panel: at higher frequencies, the gain does not
monotonically decay and subcritical instabilities may exist, as well as multistability. Filled
and empty circles indicate hypothetical stable and unstable limit cycles (fixed points at zero
amplitude) respectively. For these limit cycles to be solutions, a phase condition has to be
met, which is not considered here.
Since we are looking for harmonic cycles, we consider only the first harmonic contribution
in the Fourier transform of heat release fluctuations:
qˆ′(St,A)≈ q′1(A)eiϕ1(St,A). (3.10)
The FDF is defined by
FDF(St,A)≡ qˆ
′(St,A)/Q
uˆ′(A)/u
, (3.11)
and is shown in Figure 3.4. It contains the features that are characteristic of describing
functions of conical flames: unitary gain and a linear behaviour at small frequencies, the
presence of amplitude-dependent zeros due to destructive wave interference, and a general
monotonic decrease of the phase, typical of time delayed systems. They are described in
detail in experimental and numerical studies (Schuller et al., 2003; Karimi et al., 2009).
3.4.2 Harmonic balance
We can now study the stability of the thermoacoustic system. The FDF is closed in a positive
feedback loop with the acoustics, as shown in Figure 1.2. Figure 3.5 shows the amplitude
dependence of the gain of the closed-loop system Nx f (St,A)≡ FDF(St,A)Hx f (St) at two
frequencies. It gives insight into the possible types of bifurcations of the thermoacoustic
62 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
Growth Rate
A
 
 
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
σLf/ux
0
0.02
0.04
0.06
xf
A
 
 
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0
0.02
0.04
0.06
(a)
Frequency
A
 
 
0 0.25 0.5 0.75 1
0
0.1
0.2
0.3
0.4
St
0
2
4
6
xf
A
 
 
0 0.25 0.5 0.75 1
0
0.1
0.2
0.3
0.4
0
2
4
6
(b)
Fig. 3.6 Harmonic balance stability analysis of the first (top) and second (bottom) thermoa-
coustic modes. (a) Positive growth rates of the modes as a function of the velocity fluctuations
amplitudes. Moving from low to high amplitudes, if the growth rate passes from positive
(negative) to negative (positive), a stable (unstable) limit cycle is found. (b) Frequency maps
of regions with positive growth rates.
system. In particular, we can predict that supercritical bifurcations are expected at low
oscillation frequencies, whereas a subcritical response can be found at higher frequencies.
However, the condition described in Figure 3.5 is only a necessary condition for the
existence of limit cycles. In order to correctly predict limit cycles amplitudes and frequencies,
we have to consider the closed-loop plant phase dependence as well. On saturated limit
cycles the growth rate σ of thermoacoustic modes is zero, and harmonic oscillations are
found for the pairs (A,St) that satisfy the dispersion relation (harmonic balance):
Nx f (St,A)≡ FDF(St,A)Hx f (St) = 1. (3.12)
We determine the stability of limit cycles by looking at the change in sign of the growth rate
around the saturated amplitudes (Schmid et al., 2013). Growth rate and frequency maps away
from limit cycles are determined by fitting the FDF to a state-space model for each amplitude
section, and replacing St with the Laplace variable s in the dispersion relation (3.12) (Schmid
et al., 2013).
We investigate the stability of modes with a maximum frequency Stmax = 40/(2π)≈ 6.37,
which is the maximum frequency at which we forced the system when evaluating the FDF.
3.4 Nonlinear analysis in the frequency domain 63
Fig. 3.7 Harmonic balance bifurcation diagram. The amplitudes and frequencies of the
thermoacoustic modes with non-negative growth rate are shown while varying the position
of the flame in the duct. Super- and subcritical bifurcations of two thermoacoustic modes
are observed. Stable and unstable limit cycles are indicated with solid and dashed lines,
respectively. In some regions, both modes have a positive growth rates.
Note that, according to the harmonic balance condition (3.12), an oscillation can arise only if
the product between the FDF and the acoustic transfer function’s gain exceeds unity. The
gain of the laminar, conical FDF shown in Figure 3.4 does not exceed unity, so oscillation
frequencies for this flame are always expected to be close to an acoustic eigenfrequency; this
is verified in §3.6, Figure 3.10. Therefore, we label thermoacoustic modes by the acoustic
mode with the closest frequency.
Two thermoacoustic modes can have a positive growth rate in the range of parameters
considered. The first mode (Figure 3.6, top panels) has a positive growth rate in the region
0.04 ≤ x f ≤ 0.56. Within this region, fixed point solutions are unstable and small pertur-
bations cause oscillations with a frequency St ∈ (2.1,2.3) to grow in amplitude, until they
saturate on stable limit cycles indicated with solid lines. Two supercritical Hopf bifurcations
are located at the edges of this region. The second mode (Figure 3.6, bottom panels) exhibits
several super- and subcritical Hopf bifurcations, fold points, and multistable regions. Stable
and unstable limit cycles are identified, and oscillate at a frequency in the range St ∈ (4.5,6.3).
In some cases, the saturation amplitude of the oscillations exceeds the maximum forcing
amplitude we have considered when obtaining the FDF, and cannot be predicted from our
data.
Figure 3.7 superposes the frequency maps of the positive growth rate regions of the two
modes. In some regions, both thermoacoustic modes have positive growth rates. This raises
questions about the physical interpretation of results obtained with the harmonic balance
64 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
analysis. The main assumption, that the limit cycles are harmonic, becomes particularly
weak if more than one mode is linearly unstable. If both modes grow, how does the flame
behave? If the two modes have incommensurate frequencies, is the final attractor of the
thermoacoustic system a limit cycle? If the oscillations are non-harmonic, how different is
their amplitude compared with those predicted with the FDF? These questions cannot be
tackled in a single-mode frequency domain framework, and different approaches need to be
used.
3.5 Nonlinear dynamics in the time domain
The general behaviour of thermoacoustic oscillations is not limited to harmonic limit cycles.
Several experimental and numerical studies report that small variations in a single parameter
in the thermoacoustic state – equivalence ratio, position of the flame in a duct, intensity of
the mean flow – lead to profound changes in the oscillations dynamics, which can exhibit
for example quasiperiodic and chaotic behaviour (Gotoda et al., 2011; Kabiraj & Sujith,
2012; Kashinath et al., 2014). In this section we use two different time domain techniques to
analyse the thermoacoustic system that was investigated with the harmonic balance technique
in the previous section. We interpret the results with nonlinear time series analysis and
Floquet theory.
3.5.1 Time-marching
Analysis in the time domain does not require the restrictive assumptions required for analysis
in the frequency domain, particularly that the acoustic oscillations are harmonic. We couple
the acoustic state-space models with the G-equation dynamics (3.5) and we time integrate the
thermoacoustic system. The acoustics induce axial velocity perturbations at the base of the
flame. Continuity and advection equations are integrated numerically in the flame domain.
The G-field is updated. The unsteady heat release is evaluated through equation (3.8). This
acts as a source in the acoustic equations, closing the loop. The analysis is performed by
integrating the thermoacoustic dynamics from fixed points for 350 non-dimensional time
units in the range 0≤ x f ≤ 1, with steps of ∆x f = 0.01, and a higher resolution in regions of
interest. For 0.65≤ x f ≤ 0.78 we have also integrated the thermoacoustic response starting
from a highly perturbed initial condition, in order to analyse the subcritical response of the
system. The time series of acoustic velocity fluctuations at the flame position are stored and
used for post-processing analysis. The resulting bifurcation diagram (Figure 3.11) is shown
in §3.6.
3.5 Nonlinear dynamics in the time domain 65
  0°   90°   180°   270°   360°   450°   540°   630°   720°
xf = 0.43
xf = 0.95
xf = 0.51
Fig. 3.8 Instantaneous self-excited flame images at different flame locations over two cycles
of the PSD dominant frequency. At x f = 0.51 the system exhibits periodic, non-harmonic
oscillations, with a dominant frequency associated with the first thermoacoustic mode. At
x f = 0.95 the oscillations are still periodic, but they are associated with the second ther-
moacoustic mode, with a higher frequency; the characteristic flame perturbation wavelength
varies accordingly. At x f = 0.43 the system has undergone a Neimark–Sacker bifurcation:
two incommensurable frequencies describe the motion, which is aperiodic and the flame
shape is never exactly the same.
Figure 3.8 shows instantaneous flame shapes at different flame locations over two cy-
cles of the dominant oscillation frequency. We observe that for some flame locations,
x f = 0.51,0.95, the flame shape evolution repeats itself after one cycle; i.e., the system ex-
hibits non-harmonic limit cycle oscillations. On the other hand, at x f = 0.43 the flame shape
does not repeat itself, meaning that the motion is aperiodic; at least two incommensurable
frequencies are governing the dynamics, which cannot be described by a limit cycle.
Nonlinear time series analysis
To further characterise the system, we use methods from nonlinear time series analysis
(Gotoda et al., 2011; Kabiraj & Sujith, 2012). Figure 3.9 left panels show the time series of
velocity fluctuations at the flame position for different flame locations. Moving from top to
bottom, they identify limit cycle, frequency-locked, quasiperiodic, and chaotic oscillations
respectively; the region 260 < t < 265 has been enlarged to highlight the shape of the
oscillations. In limit cycles the time trace exactly repeats itself after a period T . Frequency-
66 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
20
0
22
0
24
0
26
0
26
2.
5
26
5
28
0
30
0
32
0
−
0.
10
0.
1
t
u
′
20
0
22
0
24
0
26
0
26
2.
5
26
5
28
0
30
0
32
0
−
0.
10
0.
1
t
u
′
20
0
22
0
24
0
26
0
26
2.
5
26
5
28
0
30
0
32
0
−
0.
20
0.
2
t
u
′
20
0
22
0
24
0
26
0
26
2.
5
26
5
28
0
30
0
32
0
−
0.
20
0.
2
t
u
′
0
5
10
15
−
10
0
−
500
S
t
PSD
0
5
10
15
−
10
0
−
500
S
t
PSD
0
5
10
15
−
10
0
−
500
S
t
PSD
0
5
10
15
−
10
0
−
500
S
t
PSD
f 1
2f
1
3f
1
f 1
f 2
f 2−
2f
1
f 1+
f 2
2f
1+
f 2
f 1
f 2
f 2−
f 1
f 1+
2f
2
f 1
f 2
f 3
−
0.
2
0
0.
2
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
−
τ
)
u
′
(t
)
u
′
(t−2τ)
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
)
u
′
(t−τ)
−
0.
2
0
0.
2
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
−
τ
)
u
′
(t
)
u
′
(t−2τ)
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
)
u
′
(t−τ)
−
0.
2
0
0.
2
−
0.
20
0.
2
−
0.
20
0.
2
u
′
(t
−
τ
)
u
′
(t
)
u
′
(t−2τ)
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
)
u
′
(t−τ)
−
0.
2
0 0
.2
−
0.
20
0.
2
−
0.
20
0.
2
u
′
(t
−
τ
)
u
′
(t
)
u
′
(t−2τ)
−
0.
2
0
0.
2
−
0.
20
0.
2
u
′
(t
)
u
′
(t−τ)
Fig. 3.9 Characterisation of the dynamical behaviour observed in our thermoacoustic model.
From top to bottom: limit cycle, frequency-locked, quasiperiodic, and chaotic oscillations.
From left to right, saturated time series, Power Spectral Density, phase space portraits, and
Poincaré sections. For illustration purposes, the time series have been stretched between
260≤ t ≤ 265.
3.5 Nonlinear dynamics in the time domain 67
locked oscillations exhibits several peaks and troughs, and the time series exactly repeats
itself after a certain number of oscillations, which are characterised by two well-defined time
scales. Also in the quasiperiodic time trace we can identify two time scales, but on close
inspection these oscillations never repeat themselves. Finally, in chaotic solutions it becomes
harder to distinguish precise time scales and the time series is certainly aperiodic.
It is difficult to distinguish between the various types of oscillations just by looking at
the temporal evolution of the velocity fluctuations. We use Power Spectral Density (PSD),
phase portraits, and Poincaré sections to characterise the dynamical behaviour of each
thermoacoustic oscillation (Guckenheimer & Holmes, 1983; Thompson & Stewart, 2001;
Kantz & Schreiber, 2004). Figure 3.9 shows the results of nonlinear time series analysis for
each type of dynamics we observe. All these methods play a role in a detailed discussion
of the bifurcations and the nonlinear thermoacoustic regime, which is postponed until §3.6
and §3.7.
3.5.2 Numerical continuation of limit cycles
Time-marching is the most accurate technique to study a nonlinear system. The main
shortcoming of time-marching is its computational cost. The transient between an unstable
thermoacoustic state and the final stable attractor can pass through several intermediate
states and can take a long time (Kashinath et al., 2014). Therefore, there is the need for a
tool that gives more information than the FDF technique, but that has lower computational
cost than time integration. Numerical continuation of limit cycles achieves this. Assuming
that a periodic (but not necessarily harmonic) solution is known for a set of thermoacoustic
parameters, we can vary one parameter slightly and iteratively solve a linear problem to
find a thermoacoustic cycle with the new set of parameters. The use of this technique on
low-order models is particularly efficient because the number of relevant degrees of freedom
of a thermoacoustic state x is fairly small, O(103). Matrix-free methods can be used to
decrease even further the memory and time required to solve the set of linear equations.
Matrix-free method for limit cycles have been introduced to thermoacoustics by Waugh
et al. (2013) and have been successfully used in analysing a ducted premixed flame with
Fourier-Galerkin acoustics (Waugh et al., 2014).
Let us indicate with
[
x0(0,x f ),T 0
]
an initial guess for a starting state and period of a
thermoacoustic cycle. We integrate the system forward in time over a cycle and compute the
residual r0 ≡ x0(0,x f )− x0(T 0,x f ). Then we iteratively solve the linear problem J0∆x0 =
68 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
−r0, with the Jacobian matrix J0 defined by a shooting iteration
J0i j ≡
∂
(
x0i (0,x f )− x0i (T 0,x f )
)
∂x0j(0,x f )
, (3.13)
to find a correction
[
∆x0,∆T 0
]
to be added to the initial guess and obtain a new starting state[
x1(0,x f ),T 1
]
=
[
x0(0,x f ),T 0
]
+
[
∆x0(0,x f ),∆T 0
]
, (3.14)
which is closer to a limit cycle solution. The process is iterated until the residual falls below
a threshold value, set to 5×10−4.
Matrix-free methods are used to solve the linear problems Jn∆xn = −rn (n indicates
the n+ 1 shooting iteration) by means of an implementation of the Generalised Minimal
Residual (GMRES) algorithm (Saad & Schultz, 1986). Solution approximations ∆xn ≈ ∆xnk
are evaluated on a k-dimensional Krylov subspace by minimising the GMRES residual
β nk ≡ ||−rn− Jn∆xnk || . (3.15)
The algorithm is stopped when the residual (3.15) falls below a predefined threshold,
β nk ≤ 5×10−2. One period of time-marching integration is required whenever we add a new
dimension to the Krylov subspace. For our system, convergence is achieved within k = 30
iterations. Considering that the thermoacoustic state is formed from around 600 variables,
the matrix-free method is 20 times faster than the computation of the entire Jacobian. Further
information on the method can be found in Waugh et al. (2013).
Once two (or more) limit cycles have been found for different values of the bifurcation
parameter, we can improve the initial guess of the next cycle by extrapolating the values of the
variables in the thermoacoustic state, allowing for a further speed-up in the calculations. The
bifurcation diagram obtained with numerical continuation is shown in Figure 3.12. In some
regions no solution is shown because limit cycles do not exist, or because the continuation
algorithm does not converge. This happens when the flame shape becomes very complicated,
i.e., when the flame is oscillating at high frequencies. This is a technical problem with our
implementation, not the technique in general. Nevertheless, we have been able to track limit
cycles in most of the parameter regions we have considered.
Floquet analysis of limit-cycles
We can straightforwardly analyse the stability of cycles found with continuation by means of
Floquet theory (Guckenheimer & Holmes, 1983). Floquet multipliers are complex numbers
3.6 Comparison between frequency and time domain results 69
associated with the linear response of limit cycle oscillations to infinitesimal perturbations.
If one of them has a magnitude larger than one, the cycle is unstable. Thus, bifurcation
locations and types are found by studying when and where Floquet multipliers cross the unit
circle. We evaluate numerically the four Floquet multipliers with the largest magnitude. If
a multiplier crosses the circle at +1, a Limit Point of Cycle (LPC) is found, meaning that
the limit cycle branch changes its stability; this corresponds to cyclic fold or saddle-node
bifurcations. Period-doubling (flip) bifurcations happen when a Floquet multiplier crosses
the circle at −1. Finally, Neimark–Sacker (secondary Hopf) bifurcations are found when
a pair of complex conjugate multipliers crosses the unit circle. The next section contains a
discussion on the location of the bifurcations predicted by Floquet theory.
3.6 Comparison between frequency and time domain re-
sults
In this section we compare the results obtained with the three types of nonlinear analysis.
We start by comparing the oscillation frequencies on stable attractors. Then we compare the
results obtained with the harmonic balance against those obtained with time-marching. In
particular we examine the location of bifurcation points, the nature of the oscillatory states
and the amplitudes of oscillations.
3.6.1 Quasi-linear regime: frequency of oscillations
Figure 3.10 top panel shows the frequencies of stable limit cycles found with the harmonic
balance method. It shows that the predicted limit cycle frequencies are always very close
to the acoustic natural frequencies, as we had anticipated when discussing the FDF. In this
analysis, the oscillation frequencies are well-defined because the cycles are harmonic. On
the other hand, oscillations in the time domain are not necessarily harmonic. For time-
marching simulations, we compute the PSDs of steady-state oscillations and normalise them
with respect to the intensity of the highest peak. We assign to the PSD’s intensity at each
frequency a colour: the higher the intensity, the darker the colour. For each vertical slice
of Figure 3.10 central panel, the black region indicates the frequency at which we have
the highest peak, and grey regions are secondary peaks. For numerically continued cycles,
which are plotted in the bottom panel, we plot only the fundamental oscillation frequency.
Solid lines indicate stable cycles and dashed lines indicate unstable cycles, which have a
complex pair of Floquet multipliers with magnitude larger than one, i.e. cycles that have
undergone a Neimark–Sacker bifurcation. For these cycles, Floquet theory also predicts
70 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
the frequency of the mode that is linearly unstable around the cycle. These frequencies are
plotted in Figure 3.10 bottom panel with dot-dashed lines. They are close to the frequencies
in Figure 3.10 central panel. They are not exactly the same because the frequency shifts as
these unstable cycles evolve towards the stable attractor.
The frequencies predicted by the three methods compare favourably. Referring to the
time domain simulations, note that the first mode dominates in the region x f < 0.6, shown by
the fact that its PSD intensity is dominant and the second mode never oscillates alone. This
is in agreement with the harmonic balance predictions (see Figure 3.7): the fixed point of the
first mode is mainly unstable in this region, whereas the fixed point of the second mode is
mainly stable. Finite amplitude oscillations are therefore needed to trigger limit cycles with
high-frequencies. For x f > 0.6, the second mode is dominant. However, a major difference
between the methods is observed here: the harmonic balance does not predict oscillations
of the first mode in this region, but its frequency appears in the time-marching time series.
This is probably due to nonlinear effects, which couple the response of the modes and may
induce oscillations that cannot be observed when examining the modes one by one. This
suggests that, if the thermoacoustic feedback enhances higher harmonics of the nonlinearity,
a frequency domain analysis based on a single mode does not accurately predict the system’s
nonlinear behaviour.
3.6.2 Fully nonlinear regime: limits of the FDF method
Figure 3.11 superposes the bifurcation diagram obtained with the harmonic balance (lines, as
in Figure 3.7) and that obtained with the time domain (dots). Dots correspond to peaks and
troughs of the acoustic velocity time series. Solid lines at u′ = 0 indicate linearly stable fixed
points.
Let us compare results for the region x f > 0.6 in detail. The harmonic balance predicts a
supercritical Hopf bifurcation at x f = 0.98, which ends in a fold at x f = 0.61. The subcritical
Hopf associated with the fold point is located at x f = 0.72. A second set of stable limit
cycles with higher amplitudes is also found within the same range of parameters, and it has a
triggering amplitude u′ ≥ 0.2.
Time-marching simulations qualitatively reproduce this behaviour: the system has a
supercritical Hopf bifurcation at x f = 0.96 and the resulting oscillations persist as the flame
moves upstream. Figures 3.11 and 3.12 show that there are two stable solutions in the region
0.65 ≤ x f ≤ 0.78: a stable fixed point and an oscillating solution. Time-marching finds
a subcritical Hopf bifurcation at x f = 0.78. Thus, time-marching simulations confirm the
bistability predicted by the harmonic balance in this region.
3.6 Comparison between frequency and time domain results 71
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
S
t
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
S
t
 
 
PSD
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
xf
S
t
Fig. 3.10 Comparison between predicted frequencies. Top panel: circles indicate the first two
natural acoustic frequencies of the geometry considered. Lines refer to Strouhal numbers of
stable limit cycles found with the harmonic balance technique. Central panel: intensity of
the normalised PSDs of converged time series. Bottom panel: dominant frequency of cycles
obtained with numerical continuation (lines). Dot-dashed lines correspond to the frequency of
the incipient unstable mode predicted by Floquet analysis when a Neimark–Sacker bifurcation
occurs, as indicated by the arrows.
However, we observe quantitative differences between the two analyses: one of the main
problems is that the positions of Hopf and fold bifurcations obtained with the two methods
do not exactly match. One reason for this is that the heat release response to forced harmonic
oscillations can be very non-harmonic, and it is a large approximation to consider only the
contribution of the first harmonic. This is particularly true for high-amplitude oscillations,
because the heat release response is highly nonlinear and can affect the position of fold points.
However, it cannot explain a shift of the Hopf bifurcations. The latter is probably due to
the fact that the frequency domain analysis relies on interpolation and extrapolation in the
Laplace variable and amplitude of the FDF data when solving the dispersion relation (3.12).
If the FDF is not well-resolved around the most important frequency regions (the frequencies
of the thermoacoustic modes), it causes inaccuracies in the harmonic balance predictions.
If the interpolation underestimates the FDF gain, then we underestimate also the overall
gain of Hx f in (3.12), and the Hopf bifurcations are shifted. Increasing the FDF frequency
72 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
0 0.2 0.4 0.6 0.8 1
−0.5
−0.25
0
0.25
0.5
xf
u
′
Fig. 3.11 Time integration bifurcation diagram (dots) and FDF predicted stable and unstable
limit cycles (lines). Light and dark lines refer to the first and second thermoacoustic modes,
respectively. Time series peaks and troughs of the final stable thermoacoustic state are shown
as a function of the flame position in the duct. Solid lines at u′ = 0 indicate linearly stable
fixed points.
resolution improves the agreement between the location of the bifurcations, as demonstrated
in Chapter 4, but cannot address the second major difference between the two analysis, which
is one of the main results of this chapter: steady-state solutions found from time-marching
are usually not limit cycles. Looking at the corresponding PSDs while moving the flame
from x f = 1 to x f = 0, we observe the appearance of a new frequency incommensurate with
respect to the first one at x f = 0.94, meaning that the oscillations are quasiperiodic. When this
happens, the absolute maximum and minimum velocity often greatly exceed those predicted
by the harmonic balance method (see Figure 3.11). For some cases (x f = 0.83,0.86,0.88),
we find that another quasiperiodic stable attractor exists with an even higher maximum
amplitude of oscillation, u′ > 0.5. It is thus possible that the thermoacoustic system exhibits
a triggering mechanism between two quasiperiodic attractors, although this phenomenon is
not investigated in this analysis.
Similar features are found in the region x f ≤ 0.6, where the harmonic balance predicts
stable and unstable limit cycles of two modes. As mentioned in §3.4, there exist parameter
regions where the fixed points of two thermoacoustic modes are simultaneously unstable:
because of the nonlinearity in the governing equations, a superposition of the mode responses
does not represent the correct evolution of the system. One cannot infer a criterion from the
3.6 Comparison between frequency and time domain results 73
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
xf
u
′
Time integration
Continuation − Stable cycles
Continuation − Unstable cycles: quasiperiodicity
Continuation − Unstable cycles: saddle node
Fig. 3.12 Thermoacoustic bifurcation diagrams obtained with time integration (dots, see
Figure 3.11) and numerical continuation (lines). The bifurcation parameter x f is the di-
mensionless flame’s position in the combustion chamber. The amplitude of self-excited
acoustic velocity fluctuations is shown. Lines connect the absolute maximum and minimum
amplitudes of cycles found with numerical continuation: solid lines indicate stable limit
cycles, dashed lines two different types of limit cycle instability mechanisms: saddle-node
(dot-dashed) and Neimark–Sacker (dashed).
FDF analysis to determine whether the system will oscillate with one of the two frequencies
or if the oscillations will be aperiodic. Thus, we conclude that the FDF is not suitable for
studying the correct nonlinear dynamics. In theory, it would be possible to obtain more
accurate solutions by evaluating a multi-input FDF, FDFN (iω1, . . . , iωN ,A1, . . . ,AN), and
solving a set of dispersion relations as described by Moeck & Paschereit (2012) for a static
nonlinear flame model.
3.6.3 Numerical continuation results
Cycles found with numerical continuation and the harmonic balance fundamentally differ. In
numerical continuation, we impose no further constraints on the cycles dynamics other than
those included in the governing equations. Therefore limit cycles evaluated with numerical
continuation are solutions of the thermoacoustic model, not harmonic approximations. In
addition, numerical continuation methods reveal the stability of cycles by means of Floquet
74 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
theory. This is particularly useful because we can predict the location of Neimark–Sacker
bifurcations, which cause transitions from periodic to quasiperiodic dynamics.
Results from numerical continuation and time-marching are compared in Figure 3.12.
The locations of Hopf and Neimark–Sacker bifurcations predicted by Floquet theory compare
extremely well with those found from time-marching analysis. For stable limit cycles, the
oscillation amplitudes of the two methods compare well. This is, of course, not the case for
unstable limit cycles. Floquet analysis shows the parameter values at which a quasiperiodic
solution exists but cannot predict its amplitude. It can be very different from that of the
unstable limit cycle, as shown in Figure 3.12.
Although it would be theoretically possible to perform a Floquet analysis in the frequency
domain (Basso et al., 1997), this requires a multi-input FDF, which is expensive to evaluate.
The results in this section show that numerical continuation combined with Floquet analysis
is a useful tool for the analysis of thermoacoustic models. Its main weakness is that, if any
parameter is changed, the entire analysis has to be performed again. This is impractical when
one wants to study the effect of several parameters on the nonlinear behaviour of the system.
3.7 Nonlinear dynamical behaviour
We conclude by looking in detail at the nonlinear dynamical behaviour we observe, and by
discussing analogies and differences between the nonlinear regimes of our system and those
found in similar experimental and numerical studies.
Our bifurcation diagram (Figure 3.12) can be compared with the one determined in the
experiments performed by Kabiraj & Sujith (2012); Kabiraj et al. (2012b), on which we
have based our geometry. Major differences are found between the two diagrams: as an
example, Kabiraj & Sujith (2012) find that the thermoacoustic system is linearly stable for
1≤ x f ≤ 0.343 (however, in the experiments the response of the system to finite perturbations
was not investigated), whereas for the same flame position range we observe a wide region
of oscillations.
We were, however, not expecting any quantitative agreement between the two results,
because we have introduced several approximations in our thermoacoustic system. Our flame
model is very simplistic and a more detailed model might be needed to accurately reproduce
the correct flame dynamics and heat release response. Also, our lack of knowledge about the
acoustic reflection coefficients of the experimental setup might be important: although we
have chosen reasonable models for the geometry considered, the real reflection coefficients
may differ. For example, if the damping of high-frequency modes were stronger than the one
we considered, the instability of the second mode observed in our analysis at the end of the
3.7 Nonlinear dynamical behaviour 75
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.47
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.46
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.45
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.43
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.42
−0.3 0 0.3
−0.3
0
0.3
u
′(t)
u
′
(t
−
τ
)
xf = 0.41
Fig. 3.13 Ruelle–Takens–Newhouse route to chaos. Poincaré sections show the creation of a
strange attractor from a quasiperiodic solution. The torus of a quasiperiodic solution first
folds, then develops corrugations on its surface and eventually disintegrates into a strange
attractor.
duct would be suppressed. Including experimentally measured reflection coefficients may
improve the agreement with experiments.
Nevertheless, we observe some qualitative common features between the two studies.
For example, experimental oscillations after the first Hopf bifurcation have a period of about
5.4 ms, which correspond to a Strouhal number St ≡ L f f/u≈ 2.4 (assuming a flame height
of 4 cm). This is consistent with the frequency of the low-frequency mode observed in
our numerical analysis (Figure 3.10), which is 2.1. Also, the types of bifurcations found
by Kabiraj & Sujith (2012) match the ones we observe: a Hopf bifurcation, followed by a
Neimark–Sacker bifurcation, chaos, and flame blowout. In the experiments, before chaos
an intermittent behaviour associated with the flame lifting off and returning to the burner
was observed; we cannot describe this effect because our flame can slide on the burner, but
cannot lift off. For the same reason, our low-order simulations cannot model flame blowout,
but we reproduce the same route to chaos. By analysing the phase portraits and PSDs of the
thermoacoustic time series in the whole range of parameters considered, we identify one type
of route to chaos, which is summarised in Figure 3.13. It starts from a quasiperiodic solution,
76 Frequency and time domain nonlinear analysis of thermoacoustic oscillations
whose Poincaré section (a torus) folds, develops corrugations and finally breaks down. The
Poincaré sections of the first and last frames of Figure 3.13 correspond to the time series in
the last two rows of Figure 3.9, respectively. By looking at the PSDs, we see that during
the torus breakdown process a new frequency appears in the spectrum, a sign that a Hopf
bifurcation has occurred. Thus, we can identify three subsequent Hopf bifurcations which
lead to chaos: at x f = 0.59 a first Hopf bifurcation creates limit cycle oscillations from fixed
points; at x f = 0.5, a secondary Hopf transforms the dynamics into quasiperiodic oscillations;
finally at x f = 0.42 a third Hopf occurs, and the torus starts to break down, leading to chaotic
oscillations. This is the signature of the Ruelle–Takens–Newhouse route to chaos.
The same route to chaos, together with the period-doubling route to chaos, is observed
by Kashinath et al. (2014). However, they considered as acoustic resonators straight pipes
with no area changes, and assumed zero temperature jump across the flame. The same
acoustic configuration is considered by Waugh et al. (2014), where fold, period-doubling,
and Neimark–Sacker bifurcations are identified. As mentioned in §3.3, in those simple
acoustic configurations the natural acoustic eigenfrequencies are all integer multiples of
the fundamental duct frequency. This enhances the interaction between the modes: if the
fundamental frequency (or a subharmonic) is excited, its harmonics respond as well due to
nonlinear effects and, because these harmonics are the resonant frequencies of the higher
modes, a coupling between the modes is induced. Thus, one may argue that the highly
nonlinear dynamics observed is influenced by the too simplistic geometry considered. In
this chapter we have shown that this is not the case. Accounting for temperature and area
variations, the frequencies of the acoustic modes are no longer equispaced, and the interaction
between the modes is reduced. Nevertheless, we still observe very rich nonlinear dynamics.
3.8 Conclusions
In this chapter we have developed a nonlinear low-order thermoacoustic model for premixed
flames. Particular care has been taken to describe the acoustics by adapting an existing
tool, LOTAN, and casting wave-based frequency response functions into suitable state-space
models. This allows us to consider complex acoustic geometries accounting for area and
temperature variations, and to easily analyse the stability of a thermoacoustic system both in
the frequency and time domains.
An FDF has been evaluated numerically and the stability of this thermoacoustic network
has been investigated via the harmonic balance technique. Stable and unstable limit cycles of
the first two thermoacoustic modes have been calculated while varying the flame position
in the duct. By comparing these results with time-marching simulations, we observe that
3.8 Conclusions 77
the harmonic balance technique predicts the onset of instability reasonably accurately and
captures the oscillation frequencies well. However, the harmonic balance technique does
not, in general, predict well the amplitudes observed in time-marching results. This is
because the combustion model we use, which is based on the kinematic G-equation, is highly
nonlinear at high amplitudes of oscillations. Consequently, accounting for only the first
harmonic component of the heat release fluctuations induces large approximations in the
Flame Describing Function, which is a crucial ingredient in the description of the system
in the frequency domain. Secondly, we observe that neglecting the interaction between
the modes can lead to very different results in the dynamical behaviour of the system. If
the harmonic balance predicts simultaneous growth of two thermoacoustic modes with
incommensurate frequencies, limit cycle oscillations cannot be always expected. We observe
this in the time domain simulations, in which we find aperiodic solutions. We determine
their dynamical nature by means of nonlinear time series analysis techniques. We identify
the position of secondary Hopf bifurcations which lead to quasiperiodic oscillations through
Neimark–Sacker bifurcations. We also identify the Ruelle–Takens–Newhouse route to chaos.
Finally, we apply a numerical continuation algorithm to the system. The method is
able to track non-harmonic limit cycles in the parameter space, and does not contain the
approximations required when analysing the system in the frequency domain. As a result,
the frequencies and amplitudes of cycles obtained with this method approximate results from
time-marching simulations very well. Importantly, we can straightforwardly apply Floquet
theory on cycles found with numerical continuation and determine their stability. This is
particularly important because we can predict the location of Neimark–Sacker bifurcations:
when they occur, oscillations cease to be periodic and we need to rely on time-marching
methods to have a full characterisation of the nonlinear dynamics.
This chapter shows that time domain techniques, such as numerical continuation analysis,
can be applied to reduced order models containing a G-equation flame and complex acoustics.
Furthermore, it shows that these provide more accurate information on the stability and
nature of thermoacoustic oscillations than the harmonic FDF method. On the other hand,
a new set of calculations based on numerical continuation has to be performed when a
parameter of the system is changed, whereas the same FDF can be used in several acoustic
configurations. This is a promising new development for the analysis and understanding of
realistic thermoacoustic systems.

Chapter 4
Flame Double Input Describing Function
analysis
In this chapter a novel method for the prediction of non-periodic thermoacoustic oscillations
is developed. The method relies on an extension of the FDF, called the Flame Double Input
Describing Function (FDIDF), which quantifies the nonlinear heat release response when a
flame is subject to two, simultaneous harmonic forcings. Starting from the FDF assumptions,
the theory that couples linear acoustics with a generic non-static nonlinear flame model ap-
proximated by the FDIDF is developed. The dispersion relations that identify the frequencies
and amplitudes of non-periodic oscillations are derived, together with analytical criteria to
assess their stability. It is shown how the FDIDF (i) improves the FDF assessment on the
stability of periodic oscillations; (ii) can predict the onset of Neimark–Sacker bifurcations,
together with the frequency of the oscillations that arise on top of unstable limit cycles; (iii)
can calculate the amplitude and stability of quasiperiodic solutions.
The content of this chapter was published in Orchini & Juniper (2016a).
4.1 Introduction
In Chapter 3, we have seen how the Flame Describing function (FDF), together with the
harmonic balance method, can be used to calculate the amplitude and stability of periodic
thermoacoustic oscillations. Its solutions are, however, only harmonic approximations
of the actual response of the system. Furthermore, if the growth rate of more than one
thermoacoustic mode is positive, then the oscillations are non-periodic. Because the FDF is
calculated by forcing the flame harmonically, it cannot be used to predict the amplitude of
non-periodic oscillations. In particular, one cannot linearly superpose two periodic solutions
80 Flame Double Input Describing Function analysis
that are found from the harmonic balance at a given operating point. This is simply because
the flame’s behaviour when forced by two finite amplitude signals is not a linear superposition
of its behaviour when forced by each finite amplitude signal independently. When using
the FDF, therefore, one cannot rule out the possibility that the long time behaviour is non-
periodic.
A detailed investigation of the interplay between two oscillating modes has been per-
formed experimentally only for simple configurations (Kabiraj et al., 2012a; Balachandran
et al., 2008), due to its high cost. Nonetheless, the presence of multiple, incommensurate
frequencies in the spectrum of thermoacoustic oscillations has been reported in several
experimental studies (Gutmark et al., 1992; Dunstan et al., 2001; Lamaroui et al., 2011).
The study of the nonlinear interaction between the modes may be relevant for the analysis of
these systems. It has also been observed in experiments that, although a single eigenmode
is found to be linearly unstable, nonlinear effects may actually stabilise the oscillations at
this frequency and trigger oscillations at a different frequency (Noiray et al., 2008). The
FDF can predict the existence of oscillations of the two modes independently, but will fail
in predicting their stability, as the latter is connected to the nonlinear coupling between the
two modes. This phenomenon is usually called mode-switching, and was observed also
by Moeck & Paschereit (2012), and in gas turbines experiments by Cazalens et al. (2008);
Anisimov et al. (2015). In Moeck & Paschereit (2012) it was shown that mode-switching
can be attributed to the existence of an unstable quasiperiodic attractor in the phase-space of
thermoacoustic trajectories, which the FDF framework cannot calculate.
In order to predict the amplitude of at least some classes of non-periodic oscillations, a
different approximation of the nonlinear flame model has to be calculated. This is known as
the Double Input Describing Function (DIDF), and is created by forcing the flame with a
signal composed of two harmonic components with independent amplitudes and incommen-
surate frequencies (Gelb & Velde, 1968). The calculated Flame DIDF (FDIDF) can then be
fed into an acoustic network in a similar manner to that in Figure 1.2. The harmonic balance
procedure yields two coupled dispersion relations which have to be solved simultaneously,
as was first shown by Moeck & Paschereit (2012) for a thermoacoustic system.
The aim of this chapter is to present a numerical analysis that exploits frequency domain
calculations of a non-static (or dynamic1) nonlinearity based on a low-order model for the
flame dynamics. This is the major difference between our analysis and that of Moeck &
Paschereit (2012), where a static model for the flame was considered. For static nonlinearities,
a Wiener-Hammerstein model can be adopted, which decouples the nonlinear amplitude
saturation process from the linear dynamic response. This is not possible for dynamic
1 A nonlinearity is non-static if it depends on time derivatives of the input state.
4.2 Review of the FDF analysis approximations 81
nonlinearities, and the FDIDF we calculate is a nonlinear object that couples the input
amplitudes and frequencies. We also obtain an analytical criterion for the stability of
quasiperiodic oscillations, which is different from the one discussed in Moeck & Paschereit
(2012). A different attempt to extend the concept of the FDF was proposed by Selimefendigil
& Polifke (2011); Selimefendigil et al. (2012), where higher order transfer functions that
account for modal coupling were derived using Volterra series expansions. However, the
dependence of the higher order transfer functions upon the relative amplitude of the input
modes was not considered in these studies. The non-static model we adopt for the flame is the
kinematic nonlinear G-equation, which is known to lead to quasiperiodic oscillations when
coupled with an acoustic network (Kashinath et al., 2014; Waugh et al., 2014; Orchini et al.,
2015). The FDIDF method is able to predict the location of Neimark–Sacker bifurcations,
the frequency of unstable oscillations around limit cycles, and also the saturation amplitude
and the stability of quasiperiodic oscillations.
The chapter is structured as follows: in §4.2 we present FDF results and calculate
harmonic limit cycles amplitudes and frequencies, together with their stability, highlighting
strong points and weaknesses of the method; in §4.3 and §4.4 the FDIDF is presented and
tested against the FDF in the limit of a small forcing amplitude; the dispersion relations that
couple it with the acoustic response are derived and solved; the frequencies and amplitudes
of periodic and non-periodic solutions are calculated with the harmonic balance method
based on the FDIDF; criteria for the stability of these solutions are outlined; results are
compared with the FDF method analysis and with time domain simulations of the same
nonlinear system. Finally in §4.5 the chapter is summarised and the benefits and problems of
the methods are discussed.
4.2 Review of the FDF analysis approximations
The thermoacoustic model we will investigate in this chapter, sketched in Figure 4.1, is the
same as that considered in Chapter 3, to allow a comparison between the FDF and the FDIDF
methods. The acoustic network is described in §2.7, and the nonlinear flame model based on
the G-equation, as discussed in §3.2.
By FDF, we refer to the frequency domain approximation of the nonlinear flame response
to harmonic velocity perturbations (see Figure 4.2). We therefore set u′ac = Asin(ωt), time
march equations (3.1)-(3.3), and calculate the heat release according to equation (3.8). Given
that the velocity perturbation is harmonic with angular frequency ω , it is reasonable to
82 Flame Double Input Describing Function analysis
R1 R2
f1
g1
f2
g2Lf
x˜f0 L x
Q
M1 p1 T1
M2 p2 T2
Fig. 4.1 Sketch of the thermoacoustic network. A compact flame connects two ducts with
different diameters and mean temperatures. The linearised Euler equations are solved on
top of a uniform mean flow solution. Jump conditions at the flame and prescribed reflection
coefficients at the inlet/outlet provide closure for the model.
assume that the heat release response can be expanded in a Fourier series as:
Q =
∞
∑
k=1
qˆk sin(kωt+φk). (4.1)
This assumes that the heat release is periodic, with the same period as the forcing. For
laminar flames, this is supported by experimental evidence (Ducruix et al., 2000; Schuller
et al., 2002; Durox et al., 2009; Karimi et al., 2009). This model cannot capture a possible
response of the nonlinearity at subharmonics. Also, for laminar flames that oscillate in
the absence of forcing at an intrinsic frequency (Juniper et al., 2008), it cannot capture the
response that may appear at non-integer multiples of the forcing frequency.
The FDF that is fed into the harmonic balance dispersion relation, defined by (1.4), is
then defined as:
FDF(A, iω)≡ qˆ1e
iφ1
uˆ
u
Q
, (4.2)
where uˆ is the Fourier component of the input velocity signal at the burner.
Rather than performing the FDF calculations over all possible frequencies, in the follow-
ing we provide an argument that allows us to limit the calculations only over certain sets of
dangerous frequencies. We first recall that the dispersion relation (1.4) is derived from the
harmonic balance method (Gelb & Velde, 1968). Its solutions, which for a fixed value of the
amplitude can be interpreted as the poles of the closed-loop thermoacoustic system, are those
for which the loop-gain, |FDF||H|, is equal to 1 and the total (wrapped) phase is equal to 0.
To find limit cycle oscillations, we impose the additional condition that the growth rate is
equal to zero. From the loop-gain condition, one can infer that a necessary (but not sufficient)
condition for a thermoacoustic oscillation to exist is that either the acoustic transfer function
4.2 Review of the FDF analysis approximations 83
0 5 10 15
−100
−50
0
St
PS
D
f1
2f1
3f1
0 5 10 15
−100
−50
0
St
PS
D
f1
f2
f2−f1
f1+2f2
−0.2
0
0.2 −0.2
0
0.2
−0.2
0
0.2
u
′(t − τ)u
′(t)
u
′
(t
−
2
τ
)
−0.2 0 0.2
−0.2
0
0.2
u
′(t)
u
′
(t
−
τ
)
−0.2
0
0.2 −0.2
0
0.2
−0.2
0
0.2
u
′(t − τ)u
′(t)
u
′
(t
−
2
τ
)
−0.2 0 0.2
−0.2
0
0.2
u
′(t)
u
′
(t
−
τ
)
Fig. 4.2 Velocity Power Spectral Density (PSD), phase plane and Poincaré sections of self-
excited thermoacoustic oscillations. Top row: periodic oscillations. The system responds
also at the harmonics, which are neglected in the FDF framework. Bottom row: quasiperiodic
oscillations. The system responds also at the harmonics and linear combination of the
fundamental frequencies, which are neglected by the FDIDF.
H or the FDF must have a gain larger than 1. For the acoustic transfer function, this happens
close to the acoustic eigenfrequencies, whereas the FDF may or may not have regions in
which the gain is larger than 1. If the FDF gain is larger than 1 over some frequency range,
the frequency of thermoacoustic oscillations may lie in this region, and can be far from
the acoustic eigenfrequencies (Tay-Wo-Chong et al., 2012). It has been shown that these
oscillations may persist even in the extreme case in which anechoic boundary conditions for
the acoustic network are imposed, and therefore no purely acoustic mode exists (Hoeijmakers
et al., 2014b; Emmert et al., 2014; Hoeijmakers et al., 2014a; Courtine et al., 2015). For this
reason, these thermoacoustic modes have been labelled as intrinsic thermoacoustic modes.
When the unconfined laminar conical flame model we are considering is forced har-
monically, its gain |FDF| never exceeds 1. Within the G-equation framework, this can be
proven analytically in the low forcing amplitude limit when curvature corrections on the
flame speed are neglected (Preetham et al., 2008; Cuquel et al., 2011; Orchini & Juniper,
2015). Numerical and experimental studies have shown that this holds true even in the fully
nonlinear case (Schuller et al., 2003; Karimi et al., 2009; Kashinath et al., 2013a; Orchini
et al., 2015). For this reason, no intrinsic thermoacoustic instabilities can be observed in
our system, and we can deduce that thermoacoustic oscillations are possible only in certain
frequency bands, given by the regions in which the acoustic gain |H| is larger than one. For
84 Flame Double Input Describing Function analysis
St
A
 
 
2 3 4 5 6 7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.05
0.1
0.15
0.2
0.25
0.3
St
A
 
 
2 3 4 5 6 7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
−20
−15
−10
−5
0
Fig. 4.3 Gain (a) and phase (b) of a laminar, conical flame’s FDF. The FDF is evaluated
numerically only around frequencies that may give rise to thermoacoustic oscillations.
(a) (b)
example, at x f = 0.34 one can see from Figure 2.9 that oscillations can be expected only in
the [118, 197] Hz and [303, 371] Hz band regions.
This is useful information because we can reduce the cost of the FDF calculations by
evaluating the FDF only over these frequency regions2. We identify these regions while
varying the bifurcation parameter x f over the entire range [0, 1]. Let us define the Strouhal
number St≡ L f f/u, where L f and u are the characteristic flame length and mean flow
speed respectively. For the thermoacoustic system under consideration in this chapter,
oscillations are possible only in the frequency ranges St ∈ [1.273, 3.024], associated with the
fundamental acoustic eigenfrequency, and St ∈ [4.138, 7.162], associated with the second
acoustic eigenfrequency. Note that this range is obtained by considering all possible values
of x f and is therefore different from the one discussed in the previous paragraph, because
the latter was considering only a specific position of the flame. We carry out a detailed
evaluation of the FDF in these frequency ranges, varying the amplitude of the oscillation
between 0 and 0.4. The FDF gain is shown in Figure 4.3, and contains the usual features
of conical, premixed flames: the gain is larger at low frequencies and overall it tends to
decrease with the amplitude, a signature of the nonlinearity saturation effect. This holds true
at low frequencies, whereas at high frequencies the gain can also increase with the amplitude,
meaning that subcritical bifurcations and triggering may be observed.
2A broad knowledge of the FDF is needed to ensure that a flame’s gain never exceeds unity. For our model,
we already have this information from Chapter 3. Practically, one could first evaluate the FDF on a coarse set of
frequencies, and them refine the measurements around those frequencies at which the gain is high.
4.2 Review of the FDF analysis approximations 85
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
xf
S
t
 
 
PSD0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
xf
u
′
 
 
NS bifurcation
St
2
2.5
3
3.5
4
4.5
5
5.5
6
Fig. 4.4 Bottom frame: bifurcation diagrams calculated with time domain simulations (dots)
and the FDF method (lines), using the flame position, x f , as the bifurcation parameter. Thick
and thin lines correspond to stable and unstable limit cycles, respectively, and their colour to
the limit cycle oscillation frequency. Dots represent peaks of time domain simulations, as
described in 3. The FDF method predicts Hopf bifurcations and periodic oscillations well
(e.g. between x f = 0.50 and 0.60) but cannot predict quasiperiodic oscillations (e.g. between
x f = 0.34 and 0.50). The top graph shows the PSD of time domain simulations at every
flame location.
Having calculated both the acoustic transfer function H and the FDF, we can close the
thermoacoustic feedback loop (as shown in Figure 1.2) and calculate the thermoacoustic
eigenfrequencies according to the dispersion relation (1.4). We recall that (1.4), deriving from
the harmonic balance method, works well when the so-called filtering hypothesis is satisfied,
meaning that the closed-loop system does not respond greatly at the harmonics of the input
frequency. Experimental and numerical studies have shown that laminar, conical flames act
as low-pass filters (Schuller et al., 2002, 2003; Preetham et al., 2008), and Figure 4.3 shows
that our model contains this feature. Also the acoustics tends to damp high frequencies more,
although gain peaks are found at the resonance frequencies. For these reasons, we shall
assume that the filtering hypothesis is satisfied.
86 Flame Double Input Describing Function analysis
Limit cycles are found when the dispersion relation (1.4) is satisfied, with the additional
constraint that the growth rate of the oscillations is equal to zero. At this stage, no further
approximation has been introduced, because we have knowledge of the FDF at harmonic
oscillations. However, to assess the stability of the cycles we need to perturb the saturation
amplitude and calculate the shift in frequency and growth rate that it causes, i.e., we want to
find the ∆s = ∆σ + i∆ω that satisfies:
FDF(ALC +∆A,sLC +∆s)H (sLC +∆s)−1 = 0, (4.3)
where ALC and sLC = iωLC are a limit cycle solution of (1.4), and ∆A is an imposed infinitesi-
mal perturbation. If ∆σ/∆A is positive, the limit cycle is unstable, and if it is negative then
the cycle is stable. To solve (4.3) the FDF needs to be extended into the complex plane C.
Following Schmid et al. (2013), we have tried two different techniques: (i) extrusion, by
assuming that the FDF does not vary with σ , and (ii) analytic continuation, by fitting every
amplitude slice of the FDF onto a state-space. Both methods give the same results.
The bifurcation diagram we obtain by varying the flame position is shown in Figure 4.4.
Thick and thin lines indicate stable and unstable limit cycles as predicted by the harmonic
balance method. Results are compared with time domain simulations of the same system (Or-
chini et al., 2015). Solid lines at A= 0 indicate regions in which the time domain simulations
are linearly stable, and Hopf bifurcations are marked with circles. The dots represent peaks
of velocity fluctuations in the time domain. At x f locations where multiple dots are plotted,
the oscillations therefore are non-periodic. For example, at x f = 0.5 time domain oscil-
lations cease to be periodic, and quasiperiodic solutions arise through a Neimark–Sacker
bifurcation, marked with an arrow in Figure 4.4. The two methods give similar locations of
Hopf bifurcations and amplitudes of periodic oscillations. However, the FDF method fails
to predict the amplitude of non-periodic oscillations. Further, although many stable limit
cycles are predicted by the FDF method, time domain simulations rarely converge to these
solutions. This is because they are not, in fact, stable. The FDF criterion for stability misses
this because it only considers growth or decay of the mode that is already oscillating. It
cannot consider growth or decay of another mode on top of the oscillating limit cycle, which
is considered with the FDIDF in the next section.
Note that, a similar comparison between the time and frequency domain methods was
presented in Chapter 3. However, a mismatch between the locations of the Hopf bifurcations
predicted by the FDF and those found by time integration was observed there. In this chapter
we have resolved the FDF more accurately around the frequencies at which thermoacoustic
oscillations are expected, which has led to a better match between the two methods.
4.3 FDIDF assumptions and calculation 87
4.3 FDIDF assumptions and calculation
By FDIDF, we refer to the frequency domain approximation of the nonlinear flame response
to a quasiperiodic velocity perturbation of the form:
u′ac = A1 sin(ω1t)+A2 sin(ω2t), (4.4)
where ω1 and ω2 are incommensurate frequencies. This choice guarantees that the phase be-
tween the two signals does not affect the resulting dynamics. In the following subsections we
discuss in details the approximations and assumptions we make concerning the nonlinearity.
4.3.1 FDIDF definition
First, as in the FDF case, we assume that the nonlinearity does not excite the subharmonics
of the forcing frequencies, and that no intrinsic dynamical instabilities exist. Because the
heat release is a nonlinear function of the forcing signal (4.4), we expect that its response
will contain all the possible combinations of the input frequencies. By using a double Fourier
series expansion (Gelb & Velde, 1968), we can write:
q′ =∑
m
∑
n
qˆmn sin [(mω1+nω2) t+φmn], (4.5)
where the heat release amplitude coefficients qˆmn and the phases φmn are functions of the
input velocity frequencies and amplitudes. The integers m, n ∈ Z are varied over all the
possible combinations giving a non-negative value of the angular frequency mω1+nω2.
In order to proceed with the harmonic balance analysis, we need to assume that the heat
release response is dominated by the frequency components at the two input frequencies (see
Figure 4.2), so that it can be approximated by:
q′ ≈ qˆ10 sin(ω1t+φ10)+ qˆ01 sin(ω2t+φ01) . (4.6)
This assumption is less well-justified than the filtering hypothesis of the previous section,
because the latter only requires that high-frequency oscillations are damped by the system.
For the FDIDF, the coupling between the frequencies can also lead to low-frequency oscilla-
tions (e.g., at an angular frequency of |ω2−ω1|) for which the filtering hypothesis does not
necessarily hold. Therefore, we are implicitly assuming that the nonlinearity’s response at
these frequencies is either filtered by the system or is weak. This holds true at small forcing
amplitude, for which nonlinear effects are small, but it has to be tested at larger amplitudes.
88 Flame Double Input Describing Function analysis
5 10 15
−0.01
0
0.01
t
q
′
5 10 15
−0.1
0
0.1
t
q
′
0 2 4 6 8 10
0
1
2
3 x 10
−3
St
|F
F
T
|
(a)
St1
St2
St1 + St2 2St2
0 2 4 6 8 10
0
0.01
0.02
0.03
0.04
St
|F
F
T
|
(b)
St1
St2 St1 + St2
2St2
2St1 − St2
Fig. 4.5 FFT of the heat released by the flame when the forcing is quasiperiodic with the
form (4.4). Top frames: (a) At low forcing amplitudes, A1 = 0.01, A2 = 0.05, the forcing
frequencies dominate the heat release spectrum. (b) At large forcing amplitudes, A1 = 0.2
and A2 = 0.25, peaks at other frequencies become relevant. Bottom frames: nonlinear heat
release fluctuations (thin black) and heat release reconstructed using only the peaks at the
forcing frequencies (thick red).
Figure 4.5 shows examples on the quality of the FDIDF approximations: at low forcing
amplitude nonlinear effects are weak and the heat release approximated by (4.6) compares
well with the fully nonlinear output. At larger input amplitudes the quality of the approxima-
tion deteriorates. This is because the nonlinearity couples the modes, and high peaks can
be observed in the heat release FFT at frequencies which are simple combinations of the
input ones. For example, in Figure 4.5b one can see that the peak at the very low frequency
2St1−St2 has a large amplitude, meaning that the heat release exhibits large fluctuations over
long time scales. The FDIDF approximation cannot see these long time scale fluctuations, as
shown in the bottom frame. This is because it ignores all the FFT contributions which are
not at St1 and St2. For this reason, we cannot expect the FDIDF method to work well at large
amplitudes. Therefore, we limit the FDIDF calculations in amplitude so that both A1 and A2
are smaller than 0.4, and their sum is less than 0.5.
The FDIDF is defined as:
FDIDF≡ [F10,F01]≡ u
Q
[
qˆ10eiφ10
uˆ10
,
qˆ01eiφ01
uˆ01
]
, (4.7)
4.3 FDIDF assumptions and calculation 89
where uˆ′10 and uˆ
′
01 are the Fourier components of the input velocity at ω1 and ω2 respectively.
F10 (F01) contains information on how the amplitude and phase of heat release fluctuations
at ω1 (ω2) vary when the flame is forced quasiperiodically. The total (non-dimensional) heat
release fluctuations are then approximated by:
qˆ′ ≈ FDIDF · [uˆ10, uˆ01]T = F10uˆ10+F01uˆ01. (4.8)
Note that the heat release in (4.8) is not a simple linear superposition of two FDFs. This
is because the FDIDF’s gains and phases are functions of all the four input variables
(A1,ω1,A2,ω2). Finally, notice that the FDIDF is a symmetric object with respect to the
input pairs (A1,ω1) and (A2,ω2) so that:
F10(A1,ω1,A2,ω2) = F01(A2,ω2,A1,ω1). (4.9)
4.3.2 FDIDF amplitude saturation
In §4.2, using knowledge of the flame’s gain response from the literature, we performed
calculations only for frequencies close to the first two acoustic modes. No information is
available about the gain response of conical flames when they are forced with quasiperiodic
signals. However, it is reasonable to assume that, when fixing the amplitude A1 and increasing
the amplitude A2 (or viceversa), the gains of the FDIDF will decrease. This is because we
expect the flame nonlinear responses F10 and F01 to saturate, at least on average, with respect
to the amplitudes A1 and A2 of both forcing modes. This is proven to be correct for a simple
cubic nonlinearity in Moeck & Paschereit (2012), where also some experimental evidence of
this fact is provided.
We therefore assume that the FDIDF gains are less than 1. As a consequence, we expect
that self-excited thermoacoustic oscillations can only be found at frequencies for which
the acoustic gain is larger than one (see the FDIDF dispersion relations (4.11)). These
frequency ranges are the same as in the FDF case, because we have not modified the acoustic
system. Because we expect two modes to be unstable, it is reasonable to guess that one of
the mode’s frequencies will be close to the fundamental acoustic frequency, and the other
one will be close to the acoustic second acoustic eigenfrequency. Note that, if two modes
with similar frequencies were to oscillate simultaneously, beating phenomena could occur,
and one should also investigate the coupling between these close frequencies. However, this
does not happen for the system we are considering, as was also shown via the time domain
analysis carried out in Orchini et al. (2015) on the same thermoacoustic system. Given
this, and the symmetry condition (4.9), we will limit the calculations to the cases in which
90 Flame Double Input Describing Function analysis
0.2
0.2
0.2 0.2
0.
24
0.24
0.24
0.24
0.
27
0.27
0.
3
0.3
0.33
0.3
3
0.3
6
A1
A
2
|F10|
0.1 0.2 0.3 0.4
0.01
0.1
0.2
0.3
0.4
0.01
0.03
0.03 0.03
0.03
0.03
0.03
0.
05
0.05
0.1
0.1
0.3
A1
 
 
|F01|
0.01 0.1 0.2 0.3 0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Fig. 4.6 Amplitudes dependence of the FDIDF gains. The forcing frequencies have been
fixed at the arbitrary values St1 = 1.513 and St2 = 5.153. The region A1+A2 > 0.5 has not
been investigated.
the non-dimensional frequencies Stn ≡ L f fn/u lie in the ranges St1 ∈ [1.273, 3.024] and
St2 ∈ [4.138, 7.162] respectively. In the following, we will refer to mode 1 and mode 2 when
referring to oscillations with a frequency in the range spanned by St1 and St2 respectively.
4.3.3 FDIDF calculation and validation
Figure 4.6 shows an example of the FDIDF gains as a function of the two forcing amplitudes.
The forcing frequencies are fixed at arbitrary values. We observe that the gain of F10 (low
frequencies) is generally larger than the gain of F01 (high frequencies); this is in line with
the low-pass filter characteristics of the conical flame we are investigating. Also, for F10
we see that the gain tends to decrease with respect to both amplitudes, as was discussed in
the previous section. This is not always true for the F01. It is not surprising because F01
contains the heat release response at frequencies spanned by St2. Even in the FDF analysis
we observed that, in this frequency range, the gain does not decrease monotonically with the
amplitude, meaning that subcritical Hopf bifurcations and regions with multi-stable solutions
may be observed.
In rare cases, we observe that the gain ofF01 is larger than one. This always happens when
the amplitude of A1 is large (between 0.3 and 0.4), and the amplitude of A2 is at its minimum,
0.01. This is due to the fact that, although we numerically ensure that the two forcing
frequencies are incommensurate, their ratio can be close to a simple fraction. For example, in
some cases the frequency St2 is close to a harmonic of St1. If the velocity amplitude at St1 is
large, the heat release responds significantly also at its harmonics. Because we perform FFTs
4.3 FDIDF assumptions and calculation 91
2 3 4 5 6 7 0.1
0.2
0.3
0.4
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
A
St
|F
D
F
|
Fig. 4.7 In the limit in which one of the two amplitudes vanishes, the FDIDF tends to
the FDF. FDIDF limits are plotted as surfaces and the FDF results as dots. In the region
St∈ [1.273, 3.024] the limit of |F10| is plotted fixing St2 = 5.153 and A2 = 0.01 in (4.10); in
the region St ∈ [4.138, 7.162] the limit of |F01| is plotted fixing St1 = 1.513 and A1 = 0.01.
The results compare well over the entire set of parameters investigated.
on signals of finite length, the FDIDF component at St2 will contain part of the harmonic
contribution of St1, artificially increasing the gain of the second mode (see Figure 4.6). This
is a source of error which increases when the thermoacoustic eigenfrequencies are close to
multiples of each other. It could be reduced by integrating the governing equations over
a longer time period, in order to have a better frequency resolution in Fourier space and
distinguish the various peak contributions. However, this would lead to an extra numerical
cost, which is undesirable.
A good test to assess the accuracy of the FDIDF calculations is to look at the limit in
which the amplitude of one of the two modes goes to zero. From the definitions of the FDIDF
and FDF one can verify that:
lim
A2→0
F10(A1,ω1,A2,ω2) = FDF(A1,ω1) ∀ω2
lim
A1→0
F01(A1,ω1,A2,ω2) = FDF(A2,ω2) ∀ω1
, (4.10)
92 Flame Double Input Describing Function analysis
meaning that F10 tends to the FDF results in the region covered by St1 when A2 vanishes
and, by exploiting the symmetry condition (4.9), F01 tends to the FDF results in the region
covered by St2 when A1 vanishes.
By assuming that the FDIDF is a continuous function, we use the calculations at the
smallest amplitudes we have investigated (0.01) as limits. Therefore, the horizontal slice
of |F10| at A2 = 0.01 and the vertical slice of |F01| at A1 = 0.01 in Figure 4.6 need to
match the FDF gain at St = 1.513 and St = 5.153 respectively (vertical slices of Figure 4.3).
Figure 4.7 shows this comparison over the entire range of frequencies and amplitudes we
have investigated. The limits agree well with the FDF results, with the largest difference
between the FDF and the FDIDF limit being about 10−3.
4.4 FDIDF analysis
We now couple the FDIDF with the acoustic response in a similar fashion as in Figure 1.2
and find the dispersion relations that need to be satisfied for quasiperiodic oscillations to
exist. The coupling between the acoustic network and the FDIDF is sketched in Figure 4.8.
Note that, although qˆ10 is explicitly proportional only to uˆ10 through F10, the latter is an
implicit nonlinear function of both uˆ10 and uˆ01. Therefore, the dispersion relations we obtain
when imposing the harmonic balance condition are coupled, and need to be simultaneously
satisfied:
F10(A1,s1,A2,s2)Hx f (s1)−1 = 0
F01(A1,s1,A2,s2)Hx f (s2)−1 = 0.
(4.11)
Quasiperiodic oscillations of the form (4.4) exist when the growth rates of the Laplace
variables sn = σn + iωn are both equal to zero, which is the condition under which the
FDIDF was calculated. However, to investigate the stability of the FDIDF solutions, we
want to calculate the rate of change of the growth rates when the calculated amplitudes are
perturbed. This will yield solutions of (4.11) with non-zero growth rates. Because we are
working with a non-static nonlinearity, the FDIDF is a function of two complex variables,
and it is not straightforward to extend it to the complex C2 space. Thus, we decide to use
the extrusion method of Schmid et al. (2013), by assuming that FDIDF(A1,s1,A2,s2) =
FDIDF(A1, iω1,A2, iω2), which is a zero-order approximation of the FDIDF around the
steady-state solutions. This complication is not present in the study of Moeck & Paschereit
(2012), because static nonlinearities were used. In that case, the FDIDF is a simpler object
and depends only on the forcing amplitudes, not on the frequencies.
4.4 FDIDF analysis 93
Fig. 4.8 Sketch of the FDIDF feedback loop with the acoustics. The FDIDF (dashed block)
is a two-input, two-output nonlinear object. The output is formed with the superposition of
two, coupled, nonlinear elements which respond at different frequencies. The two harmonic
components of the quasiperiodic signal are indicated with subscripts 10 and 01 respectively.
The implicit dependence of F10, F01 with respect to both uˆ10 and uˆ01 has been highlighted
to emphasise that the dispersion relations (4.11) are coupled.
4.4.1 Linear stability of limit cycles: Neimark–Sacker bifurcations
A first set of solutions of the FDIDF are those for which the amplitude of one of the two
modes is equal to zero. These are the FDF harmonic solutions. For example, if A2 = 0 then
we look for periodic solutions (with zero growth rate) of mode 1. From (4.11) we have:
FDF(A1, iω1)Hx f (iω1)−1 = 0
F01(A1, iω1,0,s2)Hx f (s2)−1 = 0.
(4.12)
The first equation derives from the limit (4.10) and converges to the FDF dispersion
relation (1.4). It is now decoupled from the second equation. We have already calculated
its solutions, shown as yellow lines in Figure 4.4. The second dispersion relation, however,
contains information that the FDF cannot provide. It has to be solved for the frequency
ω2 and the growth rate σ2 by fixing the frequency and amplitude of the other mode at the
FDF solution. If the growth rate σ2 is positive, then oscillations at frequency ω2 are linearly
unstable around the limit cycle with amplitude A1 and frequency ω1. The onset of these
instabilities is known as a secondary Hopf or Neimark–Sacker bifurcation.
Figure 4.9 shows the bifurcation diagram of periodic solution when their stability is
assessed with the FDIDF method. Most of the limit cycles that were found to be stable
with the FDF method are now predicted to be unstable because, according to the solution
94 Flame Double Input Describing Function analysis
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
xf
u
′
 
 
NS bifurcation
St
2
2.5
3
3.5
4
4.5
5
5.5
6
Fig. 4.9 Comparison between time domain (as in Figure 4.4) and FDIDF period bifurcation
diagrams. The flame position x f is used as a control parameter. Thin and thick lines are used
to plot unstable and stable limit cycles respectively. Neimark–Sacker (NS) bifurcations are
found at the edges of the stable solution with a non-zero amplitude.
of (4.12), oscillations at a different frequency will grow around them. This is consistent with
the time integration results, in which we rarely observe periodic oscillations. Time domain
and FDIDF results cannot compare perfectly throughout the entire bifurcation map, because
the latter neglects contributions away from the input frequencies, which may be important at
large amplitudes. However, the FDIDF correctly captures some of the system’s bifurcations.
For example, analysing Figure 4.9 from x f = 0.60 backwards, time marching results show
a supercritical Hopf bifurcation at x f = 0.59, a Neimark–Sacker bifurcation at x f = 0.50,
and an inverse Neimark–Sacker at x f = 0.11. The FDIDF method locates correctly the first
Hopf bifurcation for mode 1, and predicts Neimark–Sacker bifurcations at x f = 0.53 and
x f = 0.045.
With the FDIDF we can also calculate the frequency of oscillations that grow around limit
cycles after Neimark–Sacker bifurcations. At x f = 0.53 the FDIDF predicts that oscillations
with a non-dimensional frequency St2 = 5.0136 are linearly unstable (σ2 = 2.86×10−5)
around the limit cycle with A1 = 0.1305 and St1 = 2.1132. This prediction can be compared
with self-excited time domain results. Figure 4.10 shows the FFT of the velocity signal just
before and after the Neimark–Sacker bifurcation in the time domain (see Figure 4.9). In
the former case, the oscillation is dominated by a component at frequency St1 = 2.148 with
4.4 FDIDF analysis 95
0 2 4 6 8 10
10−6
10−4
10−2
100
St
|F
F
T
|
St1 = 2.148
A1 = 0.1399
2St1
3St1 4St1
St2
0 2 4 6 8 10
10−6
10−4
10−2
100
St
|F
F
T
|
St1 = 2.185
A1 = 0.0935
St2 = 5.005
A2 = 0.0744
Fig. 4.10 FFT of velocity fluctuations of time domain simulations as described in 3. (a):
x f = 0.51, the solution is periodic, only a dominant peak at St1 = 2.148 is found; its
harmonics are present but negligible. (b) x f = 0.49, a second intense peak is found at
St2 = 5.005; the system has undergone a subcritical Neimark–Sacker bifurcation.
(a) (b)
intensity A1 = 0.1399. Just after the bifurcation a second high peak appears at St2 = 5.005.
All these results are consistent with the FDIDF predictions.
Discussion on cost and practical implementation
The FDIDF is a function of four independent input parameters. As a consequence, the
numerical cost of building such an object increases quickly when wide ranges of parameters
are investigated. By using the arguments in §§4.3.1-4.3.2, we limit as much as possible
the width of these ranges. However, it is non-trivial to determine how to discretize these
regions to appropriately estimate the FDIDF response. Because our model is low-order,
we can afford to carry out a very detailed calculation of the FDIDF, and then investigate
its dependence on the number of points used. We use about 60 discretization points for
each frequency range. We vary the amplitudes in the range [0.01,0.4] in 9 steps, with the
additional constraint that their sum does not exceed the threshold value of 0.5. With these
limits, the total number of simulations we run to build the FDIDF is about 200 000. About 70
000 CPU hours were required to perform the analysis, which is approximately 5 times more
expensive than the continuation method used by Waugh et al. (2014) to calculate limit cycle
bifurcations on a similar thermoacoustic system with a continuation algorithm. We then use
a four-dimensional cubic spline method to interpolate the real and imaginary parts of the
FDIDF between calculated points onto a much finer grid, as functions of the input amplitudes
96 Flame Double Input Describing Function analysis
103 104 105
0
0.2
0.4
0.6
0.8
Np
||F
D
ID
F
2
.1
·
1
0
5
−
F
D
ID
F
N
p
||
||F
D
ID
F
2
.1
·
1
0
5
||
Fig. 4.11 Interpolated FDIDF relative error dependence with respect to the number of
discretization points Np used. Choosing fewer than 30 000 points leads to deviations from
the actual response larger than 10%.
and frequencies. The interpolation is performed using either the full set of simulations or
partial information only, to assess the effect of the discretization on the system dynamics.
Figure 4.11 shows the relative error of the interpolated FDIDF as a function of the
number of points (always uniformly spaced) used for the interpolation. The results at the
finest discretization, which are those used for the analysis in the rest of this chapter, are used
as reference. By halving the number of discretization points used for St1 and St2, the cost
of the FDIDF is reduced by a factor of 4 and the percentage error is about 5%. However,
further reduction in the number of points used for the interpolation lead to larger errors, and
significant deviation from the actual dynamic response should be expected. This shows that
a large number of calculations is required to accurately estimate the FDIDF. This makes it
currently non-affordable for, say, compressible LES studies, in which many CPU hours are
already required to calculate the FDF only (Han et al., 2015).
Part of the high cost of the current FDIDF analysis is due to the fact that all possible
flame positions are investigated. Because a temperature jump follows the flame, the eigenfre-
quencies vary significantly when x f spans from 0 to 1, and wide a range of frequencies needs
to be investigated. In practical situations, this is probably not the case, and the frequency
bands of interest may be narrower, thus reducing the number of calculations required for the
FDIDF. Also, we emphasise that, to calculate the stability of limit cycles found with the FDF
(as discussed in §4.4.1), we need only a part of the FDIDF calculation. This is because we
examine cases in which one of the two amplitudes is small. The only parameter that has to be
varied is the frequency of the small amplitude mode. In this framework, the FDIDF method
is much cheaper (it approximately reduces to the cost of two FDFs), and is comparable in
cost with the continuation method described by Waugh et al. (2014). The latter remains
4.4 FDIDF analysis 97
more accurate, because it studies the stability of periodic solutions (i.e., the spectrum of the
oscillations may contain peaks at the harmonics of the fundamental frequency), whereas the
FDIDF is limited to harmonic solutions (i.e., the spectrum of the oscillations contains only
one peak at the fundamental frequency). The advantage of the FDIDF is that it can be reused
in different acoustic networks to calculate the stability of several thermoacoustic systems,
whereas a new set of calculations has to be performed with numerical continuation for every
acoustic network.
The use of the FDIDF to assess the stability of periodic solutions could also be exploited
in experiments at approximately the cost of two FDFs by means of the following procedure:
(i) measure an FDF; (ii) obtain harmonic solutions and their stability – with respect to a
single mode – with the harmonic balance; (iii) for solutions that are predicted to be stable
by the FDF method, perform another set of experiments to assess again their stability with
respect to other forcing frequencies. This is accomplished by forcing the flame with a signal
of the form (4.4), by fixing the amplitude and frequency of a mode at the FDF solution and
the amplitude of the other mode at a small value. The only parameter left is the frequency of
the second mode. It has to be varied over a range of dangerous frequencies, which can be
obtained by the FDF results and the acoustic response. The stability of the FDF solutions
with respect to other frequencies can then be calculated following the procedure described in
§4.4.1.
4.4.2 Prediction and stability of quasiperiodic oscillations
Once limit cycles have become unstable, thermoacoustic oscillations converge towards
another stable solution. This can be another periodic solution, with a different frequency and
amplitude, a quasiperiodic attractor, or even a strange attractor. The FDIDF can approximate
the location and stability of periodic and quasiperiodic solutions, but cannot predict the
existence of other types of attractors, which were shown to exist in this type of thermoacoustic
system by Kashinath et al. (2014); Orchini et al. (2015).
When looking for quasiperiodic attractors, the dispersion relations (4.11) need to be
solved by fixing the growth rates σ1 and σ2 at zero, and looking for solutions with finite
amplitudes for both modes. We rely on numerical techniques to find the roots of (4.11)
that satisfy this conditions starting from a good initial guess. Because five parameters (two
amplitudes, two frequencies and the bifurcation parameter) can be varied, a large number of
initial guesses can be chosen, which is numerically inefficient. To reduce the numerical cost,
we first locate isolated solutions by starting from a coarse grid of initial guesses that covers
the parameter space. Then, we extend the solutions to continuous lines with an arclength
continuation method by slowly varying the bifurcation parameter.
98 Flame Double Input Describing Function analysis
Stability criterion
We find several sets of quasiperiodic solutions and require a criterion to assess their stability.
From a dynamical system viewpoint, the coupled evolution of the oscillations’ amplitudes
can be written in terms of a linear operator L and a nonlinear operator N(A) as
dA j
dt
= L j jA j +N j(A)≡ σ j(A)A j, (4.13)
where A is the amplitudes vector. Although the explicit expressions for the linear and
nonlinear operators are not known, σ j represents a nonlinear growth rate, in the sense that
its intensity varies with the amplitudes of the oscillations. A steady-state solution is found
when (4.13) equals zero. A solution for which at least one A j ̸= 0 is a non-trivial solution
of the dynamical system (i.e., not a fixed point). The amplitude of each mode varies with
respect to the value of its growth rate only, which is implicitly a function of all the amplitudes.
For our system, which contains only two modes, the dynamical system (4.13) reduces to:
A˙1 = σ1(A1,A2)A1
A˙2 = σ2(A1,A2)A2.
(4.14)
Equations (4.14) will slowly vary the oscillations’ amplitudes, which in turn will change the
growth rates and frequencies according to the solution of (4.11) at the current amplitudes.
Equations (4.14) were also discussed in Moeck & Paschereit (2012), where their interpretation
in terms of an averaging procedure was also provided.
Let us now indicate a solution of (4.14) with overlines. These solutions are fixed point if
both amplitudes are equal to zero, limit cycles if only one amplitude is zero, or quasiperiodic
if both amplitudes are non-zero. By linearising equation (4.14) around a solution, the
evolution of small perturbations, indicated with ∆, is given by:
d
dt
[
∆A1
∆A2
]
=
[
∂σ1
∂A1
A1+σ1 ∂σ1∂A2 A1
∂σ2
∂A1
A2
∂σ2
∂A2
A2+σ2
][
∆A1
∆A2
]
≡ J∆A, (4.15)
where the partial derivatives are evaluated at the solution. If the eigenvalues of the Jacobian
J have negative real parts, the solution under consideration is stable.
It is worth discussing the forms that the Jacobian assumes for the different types of
solutions. For a fixed point, both amplitudes vanish and J simply contains the growth rates
σ1 and σ2 on the main diagonal, retrieving the classic linear stability result. For a limit cycle
4.4 FDIDF analysis 99
solution (say of mode 1), the Jacobian takes the form
JLC =
[
∂σ1
∂A1
A1
∂σ1
∂A2
A1
0 σ2
]
(4.16)
and has eigenvalues ∂σ1/∂A1A1 and σ2. Because A1 is positive, the stability is determined
by the signs of ∂σ1/∂A1 (the FDF condition) and σ2. This is a rigorous derivation of the
stability condition that was intuitively discussed in the previous section. Furthermore, the
eigenvector corresponding to the limit cycle eigenvalue ∂σ1/∂A1A1 is orientated along the A1
direction. The second eigenvector, however, has a non-trivial direction and can be calculated
only having the FDIDF. We will shortly return to the significance of these eigenvectors in the
FDIDF analysis.
Lastly, for quasiperiodic solutions we obtain that the stability is determined by the
eigenvalues of the Jacobian
JQP =
[
∂σ1
∂A1
A1
∂σ1
∂A2
A1
∂σ2
∂A1
A2
∂σ2
∂A2
A2
]
(4.17)
This is not exactly the condition that was suggested by Moeck & Paschereit (2012), whose
Jacobian does not depend on the solution amplitudes. Nonetheless, condition (4.17) derives
from the linearisation of the amplitudes’ evolution around a solution. Given that we retrieve
correct physical conditions for the stability of fixed point and limit cycles, we shall expect it
to hold even for quasiperiodic oscillations.
Two methods can be used to calculate the partial derivatives of the growth rates with
respect to the amplitudes. By brute force, in analogy with equation (4.3), one can fix one
amplitude at its solution’s value, slightly perturb the other amplitude, and determine the
variations in frequency and growth rate of both modes by solving (4.11) with an iterative
method. Alternatively, the implicit function theorem may be used, as suggested by Moeck &
Paschereit (2012). The latter is quicker and more reliable because no iterative methods need
to be used. Details on the implicit function theorem method are given in C. Both methods
have been tested and yield the same results.
FDIDF bifurcation analysis
Figure 4.12 contains the FDIDF solutions when the bifurcation parameter is varied between
0.40 ≤ x f ≤ 0.60. Limit cycle solutions lie on A1 = 0 and A2 = 0, whereas solutions for
which both amplitudes are non-zero are quasiperiodic. We have plotted with black filled
circles attractors (solutions for which both eigenvalues have a negative real part), with red
100 Flame Double Input Describing Function analysis
0
0.1
0.2
0
0.1
0.2
0.3
0.4
0.45
0.5
0.55
0.6
A2
A1
x
f
0
0.1
0.2
0
0.1
0.2
0.3
0.4
0.45
0.5
0.55
0.6
A1
A2
x
f
Fig. 4.12 FDIDF bifurcation diagram in the region 0.40≤ x f ≤ 0.60. The FDF solutions of
mode 1 and 2 lie on the A2 = 0 and A1 = 0 planes, respectively. The fixed point solutions
lie on the line A1 = A2 = 0. Super- and subcritical quasiperiodic oscillations are found. The
stability of all solutions is assessed with the FDIDF conditions. Stable attractors are indicated
with filled black circles, repellors with empty red circles and saddle-nodes with empty red
squares. Two views of the same bifurcation diagram are shown.
empty circles repellors (both eigenvalues have a positive real part), and with red squares
saddle-nodes (one eigenvalue has a positive real part, and the other a negative real part). The
latter are particularly interesting because thermoacoustic oscillations can be first attracted
towards them along their stable manifold, and only later diverge along the unstable manifold
towards an attractor. If the growth rate of the unstable mode is small, the oscillations may
persist for a long time around the saddle-node state. This can be problematic for time domain
simulations or experiments, because the system has to be observed for a long time before
being sure that the final attractor has been reached. Saddle-nodes in thermoacoustic systems
were also discussed in Kashinath et al. (2014), where they were referred to as “unstable
attractors”.
A convenient way of representing the FDIDF results is through phase-planes. A phase-
plane contains the trajectories that the amplitudes will follow before converging to an
attractor. Starting from different initial conditions can lead thermoacoustic oscillations
towards different attractors. The set of initial conditions that converge towards an attractor
is known as the basin of attraction of the attractor. On a theoretical basis, it should be
possible to identify the basins of attraction boundaries by investigating the growth rates of
the thermoacoustic modes while varying the oscillations’ amplitudes. However, we find
4.4 FDIDF analysis 101
that when we are not close to steady-state solutions of our system, the growth rates that
satisfy equations (4.11) quickly become large. The FDIDF was not evaluated under these
conditions, therefore it cannot be used to build the phase-planes because the extrusion method
we adopted is no longer valid. Note that, for a static nonlinearity as the one considered
by Moeck & Paschereit (2012), this problem does not arise because the FDIDF is a function
of the amplitudes only.
For the non-static nonlinearity we are considering in this chapter, the FDIDF can still be
used to estimate the phase-planes. This is accomplished by calculating the eigenvectors of
the Jacobian (4.15). By means of the Centre Manifold Theorem (Guckenheimer & Holmes,
1983), the eigenspaces spanned by the eigenvectors associated with the stable and unstable
eigenvalues are locally tangent to the stable and unstable manifolds respectively. A sketch of
the phase-planes of our system across the Neimark–Sacker bifurcation at x f = 0.53 is shown
in Figure 4.13. Stable and unstable solutions are plotted with the same shape and colour
scheme of Figure 4.12, together with vectors pointing in the direction of their eigenvectors.
For saddle-nodes, these vectors are locally tangent to the stable and unstable manifolds.
For attractors and repellors, the eigenvalues and eigenvectors of J can be complex-valued.
In this case, trajectories will spiral inwards/outwards the solution. We have also sketched
with dashed lines possible heteroclinic orbits. A heteroclinic orbit is a path that connects an
unstable solution to a stable one. Note that some solutions may be missing from our maps,
because they can lie in a region we have not investigated (large amplitudes or amplitudes
smaller than 0.01), or they can be strange attractors that we cannot locate.
Although we have only partial information about phase-planes, they help to identify
possible routes that thermoacoustic oscillations undertake before converging to an attractor.
For example, let us consider Figure 4.13a, which corresponds to the x f location just before
the Neimark–Sacker bifurcation marked in Figure 4.9. Starting from the quiescent state
A1 = A2 = 0, the oscillations will be attracted towards the stable limit cycle solution along
A2 = 0. However, starting from an excited state with A2 ̸= 0 or by triggering the system,
the oscillations may converge to a different attractor. Here, the other possible attractors are
quasiperiodic. We have added a non-calculated solution (marked with a question mark) in
order to sketch some heteroclinic paths. The position of this solution however is not entirely
arbitrary; by slowly varying the bifurcation parameter, the limit cycle solution at A2 = 0 loses
its stability. A possible scenario is that, at the bifurcation point, a quasiperiodic repellor (or
another type of oscillation) collapses onto the stable limit cycle solution. After the bifurcation
(Figure 4.13b), the limit cycle on A1 turns into a saddle-node, changing the topology of
the phase-plane. Now, starting from a quiescent state thermoacoustic oscillations are first
attracted towards the limit cycle solution along the A2 = 0 axis (which is the limit cycle
102 Flame Double Input Describing Function analysis
Fig. 4.13 Sketch of phase-planes before (a) and after (b) the Neimark–Sacker bifurcation at
x f = 0.53. The arrows pointing inwards and outwards the solutions indicate the directions
of the Jacobian eigenvectors with a negative and positive growth rate, respectively. Some
hypothetical heteroclinic orbits are sketched with dashed lines. Across the bifurcation, the
amplitude A1 (A2) of the attractor reached starting from a quiescent state suddenly decreases
(increases).
(a) (b)
stable manifold), and only later are repelled from it along the unstable manifold towards the
stable quasiperiodic oscillation. This is exactly what is observed in time domain simulations,
although not shown here. Analogous time domain results can be find in Kashinath et al.
(2014).
Lastly, Figure 4.13 also shows that the amplitudes A1 and A2 suddenly vary across the
bifurcation. This is possible across a Neimark–Sacker bifurcation, as solutions are suddenly
attracted towards a different attractor. Time domain results of the same bifurcation shown in
Figure 4.10 are in line with this FDIDF prediction. Indeed, The FFT of the time signal before
and after the bifurcation shows that the amplitudes A1 (A2) suddenly decreases (increases)
across the bifurcation. This feature of Neimark–Sacker bifurcations is also seen in the time
domain results shown in Figure 4.9. At x f = 0.50 the maximum amplitude of the oscillations
suddenly deviates from the limit cycle amplitude before the bifurcation. A fair comparison
between the oscillation amplitudes predicted by the FDIDF and time marching is seen by
looking at the position of the stable quasiperiodic attractor in Figure 4.13 and the intensity of
the peaks in Figure 4.10. Note that, however, these figures contain information at slightly
different values of x f , because the location of the Neimark–Sacker bifurcation predicted by
the two methods is slightly different, due to the FDIDF approximations.
4.5 Summary and conclusions 103
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
xf
u
′
Fig. 4.14 Overview of the FDIDF bifurcation diagram. PDF of stable quasiperiodic solutions’
peaks (shaded regions) are plotted on top of stable limit cycle amplitudes (lines). The
PDF intensity is higher in darker regions. The location of Neimark–Sacker bifurcations is
highlighted with dots.
We conclude the chapter by showing in Figure 4.14 the bifurcation diagram calculated
with the FDIDF in the entire range 0≤ x f ≤ 1. We have plotted only the peaks of stable os-
cillations, which are those observable in self-excited experiments or time domain simulations.
For limit cycle solutions, these peaks are shown as lines corresponding at the oscillations’
amplitude. For quasiperiodic solutions, we calculate the Probability Density Functions
(PDF) of their peaks. These are shown as shaded regions in Figure 4.14. The locations of
Neimark–Sacker bifurcations have been highlighted with black dots. One can see that there is
a nice match between their location and the onset of stable quasiperiodic solutions. This does
not happen if we apply the stability criterion for quasiperiodic solutions contained in Moeck
& Paschereit (2012). In the region 0.60 ≤ x f ≤ 1, quasiperiodic oscillations tend to have
a large amplitude, which exceeds the A = 0.5 threshold we have set when calculating the
FDIDF. This is partly consistent with time domain results, in which very large oscillations,
e.g. at x f = 0.83, are occasionally observed. In some regions, multiple stable solutions are
found. With time marching methods, a thorough investigation of the initial condition is
required to find these solutions.
4.5 Summary and conclusions
We have presented a numerical approach for the investigation of non-periodic thermoacoustic
oscillations. A Flame Double Input Describing Function (FDIDF) of a non-static nonlin-
104 Flame Double Input Describing Function analysis
ear flame model based on the G-equation has been calculated by forcing the flame with a
quasiperiodic signal composed of two harmonic components with independent amplitudes
and incommensurate frequencies. The FDIDF assumptions and limitations have been out-
lined, and the FDIDF has been tested against the Flame Describing Function (FDF) in the
limit in which the amplitude of a mode is small. The FDIDF has been embedded into a
thermoacoustic network and, through the harmonic balance method, stable and unstable
thermoacoustic oscillations have been calculated. Furthermore, a criterion to assess their
stability has been derived.
The FDIDF contains a far more accurate approximation of the nonlinear flame response
than the FDF. Exploiting all its information, one can predict the amplitude and stability of
quasiperiodic solutions. Also, via the Centre Manifold Theorem, it can be used to sketch
phase-planes to understand the path that thermoacoustic oscillations trajectories will take.
Quantitative comparisons between the FDIDF and time marching results have been presented.
We have shown that the FDIDF is capable of predicting the location of Neimark–Sacker
bifurcations, the frequency of the unstable modes and the amplitude of the final quasiperiodic
oscillations. We have discussed in detail the change in behaviour of a system at a Neimark–
Sacker bifurcation, across which a new mode becomes unstable and the amplitude of the
oscillations varies abruptly. This can lead to quasiperiodic oscillations or mode-switching to
another stable periodic oscillation at a different frequency. Neither type of behaviour can be
predicted by linear stability analysis nor by the FDF framework.
Although the FDIDF is an expensive object to calculate, for simple dynamical flame
models, such as the G-equation, this is affordable. Also, we have shown how its cost can
be greatly reduced if one is interested in calculating only the stability of limit cycles. This
accounts for the nonlinear interaction between modes, which the FDF ignores, and provides
the location of Neimark–Sacker bifurcations. Only the information at which one of the
amplitudes is fixed at a very small value is needed for this, and the cost of the FDIDF reduces
to the cost of a second FDF, making it affordable for experimental purposes too. We find that,
for our system, most of the limit cycles that are predicted to be stable by the FDF method,
are predicted to be unstable by the FDIDF method. This is consistent with self-excited time
marching results of the same thermoacoustic system. Within this framework, the FDIDF
is capable of predicting the frequency of oscillations that will grow in time around limit
cycles. Knowing these frequencies, Helmholtz resonators can be tuned and retro-fitted to the
thermoacoustic system in order to make it less prone to oscillations.
Chapter 5
Weakly nonlinear analysis of
thermoacoustic bifurcations
In this chapter, a weakly nonlinear formulation of the equations governing thermoacoustic
oscillations in the Rijke tube is presented. In contrast to previous expansions available in
the literature, which approximate the acoustic response with a single Fourier mode, here the
contribution of multiple modes is retained. Because of this, adjoint methods are required to
derive the Stuart–Landau equations, which govern the oscillations’ saturation mechanism.
Although the weakly nonlinear expansion is strictly valid only close to the expansion point,
it is shown how it provides an accurate and inexpensive method for the investigation of
nonlinear effects.
This work was the result of a collaboration with Dr. Georgios Rigas. Part of the content
of this chapter is under consideration for publication in Orchini et al. (2016).
5.1 Introduction
Weakly nonlinear analysis is a nonlinear method capable of tracking the evolution of the
oscillations’ amplitude. It is based on an asymptotic expansion of the governing equations in
the vicinity of a bifurcation point. This approach provides an analytical description of the
perturbation dynamics, which is exact up to the order of the truncated expansion. In general,
a weakly nonlinear solution is calculated as a superposition of one or more spatial modes
with a time dependent amplitude. The temporal evolution of the amplitude is reduced to an
ordinary differential equation (ODE) - appearing as a Stuart–Landau equation - for every
linearly unstable mode. Solving the latter equation is much faster than time marching the full
106 Weakly nonlinear analysis of thermoacoustic bifurcations
nonlinear system, and also provides physical insight into the nonlinear interactions between
the modes.
Weakly nonlinear analysis has been widely used in hydrodynamics to study the transition
of globally unstable flows and derive low-order models of the Navier–Stokes equations
around bifurcation points (Chomaz, 2005). In the simple case of the cylinder flow, which
undergoes a supercritical Hopf bifurcation at a diameter-based Reynolds number Re≈ 46,
the Stuart–Landau equation accurately captures the amplitude of the most unstable global
shedding mode close to the threshold of bifurcation (Landau, 1944b; Provansal et al., 1987).
Sipp & Lebedev (2007) showed how the Stuart–Landau equation can be derived from the
Navier–Stokes equations using global stability analysis and a multiple timescale expansion.
As a consequence of the global character of the analysis, adjoint methods were required to
identify the Landau coefficients of the model.
Weakly nonlinear tools have been applied also to thermoacoustic systems. Culick (2006)
used the method of averaging to derive the amplitude evolution for thermoacoustic models
with one or two oscillating modes. In the same framework, Juniper (2012) described how
the averaged quantities can be connected to the Flame Describing Function methodology.
Ghirardo et al. (2015) applied the method of averaging to azimuthal thermoacoustic in-
stabilities, in which two counter-rotating azimuthal modes with the same frequency are
simultaneously unstable. Subramanian et al. (2013) used the method of multiple scales to
derive a Stuart–Landau equation at third order describing the evolution of the oscillation
amplitude in a Rijke tube. The Landau coefficients showed that the Hopf bifurcation was
subcritical, and the low-amplitude limit cycles arising close to the Hopf point were unstable,
in agreement with experimental studies.
The unstable limit cycles arising from subcritical bifurcations in a Rijke tube may undergo
a fold bifurcation to become stable, and create a region of bi-stability (Ananthkrishnan et al.,
2005; Juniper, 2011; Jegadeesan & Sujith, 2013). In the weakly nonlinear analysis performed
by Subramanian et al. (2013), however, the expansion was truncated at third order. Therefore,
the fold point, the amplitude of stable limit cycles, and the bistable region could not be
predicted by their weakly nonlinear methods, because this would require expansion to at
least fifth order. In their case, however, even an expansion at higher order would not have
captured the fold point for reasons explained in section §5.2.2. It is also worth noting that in
the weakly nonlinear studies on the Rijke tube (Juniper, 2012; Subramanian et al., 2013) only
one Galerkin mode was used to describe the dynamics. This can be a rough approximation
for some thermoacoustic networks, because considering only one acoustic mode may alter
the nature and amplitude of the oscillations, as was discussed by Jahnke & Culick (1994);
Ananthkrishnan et al. (2005).
5.2 Thermoacoustic modelling 107
R1 R2
f1
g1
f2
g2
α2
xh0 L x
Q
u1 p1 T1
u2 p2 T2
Fig. 5.1 Horizontal Rijke tube model. Subscripts 1 and 2 indicate flow and wave properties in
the upstream and downstream ducts respectively. The intensity of the heat release fluctuations
will be used as a control parameter.
In this chapter, we perform a high order weakly nonlinear analysis of thermoacoustic
oscillations in a Rijke tube close to a subcritical Hopf bifurcation. The high order expansion
allows us to obtain analytical expressions for the location of the bistable region and the
amplitude of both unstable and stable limit cycles. In our analysis, this is achieved by using a
wave-based approach when solving the linear acoustic equations. It provides a more general
description of a thermoacoustic network, and enables us to include temperature and area
variations in the analysis. We also retain the contribution of multiple acoustic modes to the
dynamics of the thermoacoustic system. This corresponds to approaching the problem in
a global framework, and adjoint methods are required to calculate the Landau coefficients
through solvability conditions (Sipp & Lebedev, 2007).
The chapter is organised as follows. In §5.2 we present the Rijke tube thermoacoustic
set-up and the wave-based governing equations; the nonlinear heat release model adopted
for this study is discussed; the damping mechanisms adopted to stabilise high-frequency
modes are presented. In §5.3 we perform a linear stability analysis of the system and we
identify the location of Hopf bifurcations, when the heat release power is used as a control
parameter. In §5.4 we present the theoretical framework for the weakly nonlinear analysis,
deriving in detail the equations for the amplitude evolution of the dominant mode up to fifth
order. In §5.5 we validate our weakly nonlinear results against the exact solutions of the
fully-nonlinear equations, obtained with a continuation algorithm method and time domain
simulations. For our system, we find that, depending on the width of the subcritical region, an
expansion up to seventh order is needed to obtain a good agreement with the exact solutions.
In §5.6 we discuss how the method could be applied to more complex systems. Finally, in
§5.7 we summarise our findings and discuss possible future applications.
108 Weakly nonlinear analysis of thermoacoustic bifurcations
5.2 Thermoacoustic modelling
The configuration considered in this chapter is a horizontal Rijke tube, as shown in Fig-
ure 5.1. It has been extensively considered by many authors (Matveev, 2003; Juniper, 2011;
Subramanian et al., 2013; Magri & Juniper, 2013b; Mariappan et al., 2015) for the analysis
of thermoacoustic phenomena. However, weakly nonlinear analyses of thermoacoustic
oscillations presented in these studies have been approximated by considering a single pair
of Galerkin modes for the acoustic response. It was shown by Jahnke & Culick (1994);
Ananthkrishnan et al. (2005); Kashinath et al. (2014) that considering only one acoustic
mode may alter the amplitude and type of thermoacoustic oscillations. Also, mean flow and
temperature jump effects are often neglected, although their presence affects the thermoa-
coustic eigenmodes and the stability of the system. Although our model remains low-order,
we consider a wave-based approach which naturally yields a more realistic description of the
acoustic response of the system (Dowling, 1995; Orchini et al., 2015), and is easily scalable
to more complex acoustic networks, which is important when considering, say, gas turbines.
5.2.1 Acoustic model
The acoustic network we consider is a duct of length L with an inlet of area A1. We prescribe
mean flow u1, mean pressure p1, and inlet temperature T1. The heater is located at a distance
xh downstream. Across the heater we impose a temperature jump ∆≡ (T2/T1)1/2 = c2/c1,
determined by the heater mean heat release, and an area change Θ≡ A2/A1. We decompose
the acoustic velocity, pressure and density fluctuations into downstream ( f ) and upstream
(g) travelling acoustic waves and an entropy (α) wave. Entropy waves play a minor role
in this study, but their contribution would be more relevant if the system were to have a
choked exit. Mass, momentum and energy fluxes are conserved through the heater via the
Rankine-Hugoniot jump conditions (Dowling, 1995; Stow & Dowling, 2001).
The reflection coefficients provide the relations f1 = R1g1e−sτ1 and g2 = R2 f2e−sτ2 ,
where τ1 ≡ 2xhc1/(c21−u21) and τ2 ≡ 2(L− xh)c2/(c22−u22). The inlet reflection coefficient
is chosen to be R1 =−0.9. For the outlet, we use either R2 =−0.9 or a frequency dependent
reflection coefficient, depending on the damping model adopted, as discussed in §5.2.3.
By following a procedure analogous to that described in Dowling (1995); Orchini et al.
(2015), we calculate the linear acoustic response to heat release fluctuations q′, which is
5.2 Thermoacoustic modelling 109
given by the equations:
M
 g1f2
α2
=
 M11 M12 M13M21 M22 M23
M31 M32 M33

 g1f2
α2
=
 00
q′/(A1c1)
 , (5.1)
which represent mass, momentum and energy conservation across the heater element.
The coefficients in the matrix M are reported in Appendix D. By setting the determinant
of M equal to zero (i.e., by solving a nonlinear eigenvalue problem) we find the acoustic
eigenfrequencies. Solving for the wave amplitudes as a function of q′ provides the acoustic
transfer functions of pressure, velocity and density fluctuations with respect to heat release
oscillations at any point in the duct. We are interested in the velocity response u′ just upstream
the flame for the coupling with the heater. We cast this response in state-space form:
dx
dt
= Ax+Bq′ (5.2a)
u′ =Cx, (5.2b)
by following the procedure described in Chapter 2. If an expression for the heat release is
provided, the above equation describes the dynamics of thermoacoustic oscillations.
5.2.2 Heat release model
King’s law (King, 1914) expresses the heat transferred from a hot-wire to the flow under
steady flow conditions. With a quasi-steady argument, (i.e., assuming that the instantaneous
heat transfer is determined by the instantaneous velocity), an unsteady, nonlinear model for
the heat release fluctuations is obtained:
Q = k(Tw−T1)Lw
(
1+
√
2πcpdw
k
|u+u′(t− τ)|
)
, (5.3)
where k, Tw and Lw are the air thermal conductivity, wire temperature and total length
respectively, cp the specific heat per unit volume, dw the wire diameter, and τ a time delay.
The heat release time delayed response with respect to acoustic velocity fluctuations models
the low-frequency response of the heater, as found by Lighthill (1954). The non-dimensional
fluctuations are given by:
q′ = K
(√
|1+u′(t− τ)|−1
)
, (5.4)
110 Weakly nonlinear analysis of thermoacoustic bifurcations
with K ≡√2πcpdwu/k. Note that the gain of (5.4) does not decrease at high frequencies, be-
cause it is a static heat release model (i.e., q′ does not depend on du′/dt). This is problematic
because, if not damped, high-frequency modes are as likely to be unstable as low-frequency
modes, which is unphysical. To overcome this issue, two damping mechanism are used, to
highlight some features of the weakly nonlinear expansions, and are discussed in §5.2.3.
In Subramanian et al. (2013) a Taylor expansion of the heat release fluctuations (5.4)
around u′ = 0 was considered for a weakly nonlinear analysis. However, due to the presence
of an absolute value in equation (5.4), such a Taylor series converges to the original heat
release expression only for u′ >−1, as shown in Figure 5.2a. This is problematic because
the fact that q′ increases when u′ <−1 decreases is the only saturation mechanism, which
we want to model in this analysis. The weakness of the Taylor expansion for this model
can be quantitatively shown by calculating the Describing Function (DF) of King’s model.
Imposing harmonic velocity oscillations u′ = Acos(ωt), the DF is defined as (Gelb & Velde,
1968):
DF(q′)≡ K
Aπ
∫ 2π
0
(√
|1+Acos(θ −ωτ)|−1
)
eiθ dθ . (5.5)
Its gain is shown in Figure 5.2b together with the DF gain of the Taylor expansion, which
does not saturate when the amplitude increases. Note that the unsteady King’s law DF gain
first increases up to an amplitude of A = 1 and then decreases. The initial increase in gain is
probably a non-physical feature of the model. It arises upon the introduction of a time-delayed
unsteady velocity dependence in King’s law, which was originally derived under steady-flow
assumptions. Although time-delayed unsteady flow effects provide the correct connection
between King’s law at small amplitudes with the linear dynamical theory of Lighthill (1954)
at low-frequencies, it is questionable if these effects correctly capture the nonlinear dynamics
of the heater at high amplitudes. Indeed, CFD simulations performed by Selimefendigil et al.
(2012) on the fully nonlinear unsteady equations of a pulsating flow around a hot-wire have
shown that the gain of heat release describing function monotonically decreases when the
forcing amplitude increases. Furthermore, Witte & Polifke (2015) reported that the Reynolds
number Re based on the wire diameter has a great influence on the linear response of the
heater, which, for low values of Re, is very different from the one predicted by Lighthill
(1954).
Because a smooth, nonlinear, dynamical model for a heater response is not available in
the literature, we perform a least square fit of equation (5.4) onto a fifth order polynomial of
the form:
q′(t) = K
5
∑
n=1
αnu′
n
(t− τ). (5.6)
5.2 Thermoacoustic modelling 111
−2 −1 0 1 2
−1
−0.5
0
0.5
1.0
u′(t − τ)
q′
(t
)
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
A
|D
F
(q
′ )
|
(a) (b)
 
 
Quasi−steady King’s law model
Taylor expansion
Least−square fit
CFD (Selimefendigil et al., 2012)
Fig. 5.2 Nonlinear heat release models. (a): comparison between unsteady King’s law, fifth
order Taylor expansion and polynomial fit approximations. K is fixed to 1. (b): the DF’s gain
of King’s model and its approximations. The saturation mechanism starts at A = 1 and is
due to the abrupt change of sign of the heat release derivative. The Taylor expansion around
u′ = 0 cannot capture this mechanism. CFD results from Selimefendigil et al. (2012) (scaled
to match the linear gain) are in qualitative agreement with the least-square fit model that we
use in this chapter.
In order to ensure that the linear stability of the thermoacoustic problem is not affected
by the fitting, we constrain the linear coefficient to be α1 = 1/2, which is the first order
Taylor coefficient of King’s law (5.4). The α coefficients depend on the range chosen for
the fit. We choose the range u′ ∈ [−2,2], which is wide enough to capture the saturation
mechanism up to amplitudes which are much larger than those we will consider in this
study. The coefficients obtained with the fit are α2 =−0.108, α3 =−0.044, α4 =+0.059,
α5 =−0.012. From Figure 5.2, one can see that the least square fit model has the same linear
behaviour as the unsteady King’s law and a qualitatively similar nonlinear behaviour, but with
a smooth saturation mechanism. The saturation starts at values smaller than u′ = 1, which
is consistent with the experimental observations of Heckl (1990). Also, the gain decreases
monotonically with the amplitude forcing, which is consistent with the nonlinear unsteady
calculations of Selimefendigil et al. (2012). Note that a monotonic decrease in the gain does
not preclude the possibility of having a subcritical bifurcation, as this can be caused by the
variations of the phase response with the amplitude, as is the case for our system. In the
following, we will find that oscillations saturate with amplitudes |u′|< 1, for which our fit
is in good agreement with CFD results. If an appropriate model for the nonlinear dynamic
response of the heater is provided, the α coefficients can be obtained via a Taylor expansion
up to the desired order.
112 Weakly nonlinear analysis of thermoacoustic bifurcations
In the following, variables are presented in a non-dimensional form. We scale lengths with
the duct length L, velocities with the mean flow velocity u1 and time with the characteristic
timescale L/c1.
5.2.3 Damping models
Two damping models that reduce the gain of the system at high frequency are discussed next.
They rely on different physical damping mechanisms, and, although lead to analogous results,
exhibit quantitative differences in the weakly nonlinear analysis, as will be discussed in §5.5.
Model I: boundary and viscous-thermal losses
As a first damping model, we model acoustic losses at the boundaries via a frequency
dependent reflection coefficient for the outlet, of the form (Cargill, 1982; Hughes & Dowling,
1990):
R2 =−(1+0.9M2)
(
1− 1
2
ζ
(
ωrd
c2
)2)
, (5.7)
where M2 is the Mach number after the heater, rd is the duct radius, and ζ is a constant
(ζ ≈ 3) that accounts for the fact that hot air meets ambient temperature at the outlet. As a
drawback, equation (5.7) determines a threshold frequency at which the reflection coefficient
changes sign. This is non-physical, so we cannot investigate modes above this threshold
frequency. This can be problematic because, for the weakly nonlinear expansion to converge,
a large number of acoustic modes need to be included in the state-space model. However, the
number of modes that we can consider is limited by the threshold frequency at which the
reflection coefficient changes sign with this model.
Additional damping due to viscous-thermal boundary layer losses is accounted for via a
wavenumber correction (Peters et al., 1993):
k± =
ω
c±u
(
1+
a√
ω
+
b
ω
)
(5.8a)
a≡ (1− i)
√
ν√
2rd
(
1+
γ−1√
Pr
)
b≡− iν
r2d
(
1+
γ−1√
Pr
− γ(γ−1)
2Pr
)
, (5.8b)
where c is the speed of sound, Pr is the Prandtl number, ν the kinematic viscosity and γ the
specific heat ratio.
5.2 Thermoacoustic modelling 113
0 10 20 30
0
0.2
0.4
0.6
0.8
1
ωdw/u
G
ai
n
 
 
a)                                                                        b)Lighthill theory
Extrapolation
Linear acoustic
response
Lighthill linear
gain reduction
Nonlinear heater
model with
time delay
State−space model
Fig. 5.3 a): Lighthill (1954) gain saturation with frequency of the linear hot-wire response.
The theoretical predictions (dashed lines) are extrapolated at all frequencies (solid line). b):
schematic of the feedback mechanism with damping model II. High frequencies are saturated
with Lighthill’s theory that, being linear, can be combined with the acoustic response to
construct a single state-space approximation.
Model II: Lighthill’s dynamic response
In Lighthill (1954), the linear dynamic response of a hot-wire was derived analytically,
showing that its behaviour is analogous to that of a low-pass filter. Lighthill’s results for the
gain saturation with the frequency are reported in Figure 5.3a. The low-pass filter cutoff
frequency is a function of the wire diameter dw and mean flow intensity u. A gain reduction
with the frequency stabilises the high-frequency modes, which is what we are seeking.
The total response of the heater is therefore given by the product between a linear dynamic
response and a nonlinear amplitude saturation, which is a Wiener-Hammerstein model. Note
that here we are retaining the heater phase response due to the time delay in the nonlinear
element. Furthermore, because Lighthill’s theory is linear, it can be combined with the linear
acoustic response, as shown in Figure 5.3b.
In this way, their combined linear response can be approximated with a single state-space
model. Thus, the thermoacoustic system is modelled as a Lur’e system (a linear component
in feedback loop with a nonlinear component), as was the case for damping model I, and the
theory developed in the next can be applied using both models with no changes.
114 Weakly nonlinear analysis of thermoacoustic bifurcations
5.3 Linear stability analysis
By coupling the acoustic response (5.2) with the heat release model (5.6), we obtain a non-
linear, delayed dynamical system that describes the evolution of thermoacoustic oscillations.
The dynamics of the state variables x is coupled upon the introduction of the heat element
through the acoustic velocity u′ =Cx:
x˙(t) = Ax(t)+KB
5
∑
n=1
αn(Cx(t− τ))n, (5.9)
where K is the control parameter, which determines the intensity of the heat release fluctua-
tions. The matrix A and the column and row vectors B and C are functions of the state-space
model order of approximation and of the flame location xh. The latter is fixed to xh = 0.17
and 0.13 for damping mechanisms I and II respectively, which is the location at which the
thermoacoustic network is most prone to thermoacoustic oscillations at its lowest eigen-
frequency. This was obtained by looking for the maximum value of the Rayleigh index
Ry ≡ ∫ T0 p′xh(t)q′(t)dt, assuming harmonic oscillations and using equation (5.6) to relate
heat release to velocity fluctuations.
Fixing all the other parameters, there are specific values of K, called critical points, at
which the system is marginally stable, meaning that all the eigenvalues of the linearised
spectrum have a negative growth rate except for a complex conjugate pair, which has a zero
growth rate. At these critical points, Hopf bifurcations occur, and the linear stability of the
system changes across them. In order to perform a weakly nonlinear analysis, we need to
first locate critical points and then expand the governing equations around them. This yields
the amplitude and frequency of limit cycle oscillations that occur after the bifurcation.
The fixed point x = 0 is a solution of the dynamical system (5.9) for any value of K. Its
stability is determined by the eigenvalues of the linearised system:
x˙(t) = Ax(t)+α1KBCx(t− τ). (5.10)
Taking the Laplace transform we have:(
sI −A−α1KBCe−sτ
)
xˆ = 0, (5.11)
where s≡ σ + iω is the non-dimensional Laplace variable, and I the identity matrix. Note
that, because B and C are column and row vectors respectively, their (outer) product results
in a matrix. The values of s for which the determinant of the linear operator vanishes are the
5.3 Linear stability analysis 115
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
 
 
ω
c(a) (b)
K
c
τ
3.5
4
4.5
5
−1 −0.5 0
−30
−20
−10
0
10
20
30
Re(s)
Im
(s
)
Fig. 5.4 (a): Neutral line in the K− τ plane. Along it, the eigenmode with the smallest
eigenfrequency has zero growth rate. Colours refer to the frequency of the marginally stable
mode. Black lines indicate that mode(s) with a higher frequency have a positive growth rate.
(b): Spectrum of the acoustic (circles) and thermoacoustic (crosses) system for the set of
parameters chosen for the weakly nonlinear analysis, marked with a circle in the left plot.
The paths of the eigenvalues from their acoustic values (K = 0, dots) to their thermoacoustic
values (K = Kc, crosses) are shown as dotted lines. These graphs have been calculated using
damping model I; analogous results are obtained with model II.
thermoacoustic eigenvalues. The nonlinear eigenvalue problem (5.11) is solved iteratively
while varying K until a marginally stable solution is found.
We first perform a parametric study in the K−τ plane to identify a set of Hopf bifurcations
of the thermoacoustic mode with the smallest eigenfrequency (see Figure 5.4a). This is
achieved using the open-source package DDE-BIFTOOL (Engelborghs et al., 2002; Sieber
et al., 2015), tracking the critical values of the heater power Kc at which the growth rate
associated with the smallest eigenfrequency is zero. However, we find that, for some of these
solutions, mode(s) with a higher frequency have a positive growth rate along the neutral
lines of the first mode. This is physically possible when the values of Kc or τ are large. We
show these solutions with black lines in Figure 5.4a. At these locations the system is not
marginally stable, and the theory presented in the following cannot be applied. A typical
value for the time delay can be estimated from Lighthill (1954)’s theory, τ = 0.2dw/u. In the
following, two values of τ are chosen depending on the damping model: for model I, we will
fix τ = 0.02, for which the critical power is Kc = 5.04; for model II, we will fix τ = 0.04,
for which the critical power is Kc = 1.422. Figure 5.4b shows the spectrum of the acoustic
and thermoacoustic systems at this Hopf bifurcation for damping model I.
Because, for damping model I, the value of Kc is fairly large, the thermoacoustic eigen-
modes are significantly shifted from the pure acoustic ones. The order of approximation of
116 Weakly nonlinear analysis of thermoacoustic bifurcations
1 2 3 4 5 6 7
5
10
15
Number of modes
K
c
4.5
5
5.5
6
6.5
7
ω
c
Damping model I
2 4 6 8 10 12
1.2
1.3
1.4
1.5
1.6
Number of modes
2 4 6 8 10 12
4.15
4.16
4.17
4.18
4.19
Damping model II
Fig. 5.5 Convergence of the marginally stable angular frequency ωc (top) and critical power
Kc (bottom) with respect to the number of modes considered in the state-space approximation.
A single-mode approximation does not accurately capture the correct response of the system.
For damping model I, we cannot consider modes above the seventh, because their frequency
exceeds the threshold frequency imposed by the choice of the reflection coefficient.
the state-space - which is twice the number of modes considered - has to be large enough
in order to capture the thermoacoustic response correctly, as was shown by Ananthkrishnan
et al. (2005); Kashinath et al. (2014). A single-mode approximation of the acoustic response
(as was considered by Juniper (2011); Subramanian et al. (2013) for weakly nonlinear
expansions) is not able to capture the system response accurately in this case. Figure 5.5
shows that the marginally stable eigenfrequency and heat power obtained with a single-mode
approximation are significantly different from the saturated ones obtained with more modes.
For damping model I, modes above the seventh have a frequency which is above the threshold
frequency of the reflection coefficient (5.7). They give rise to non-physical high-frequency
oscillations that are not damped, and are not included in the analysis. In this case, we will
retain all 7 modes (Nmodes) when performing the weakly nonlinear analysis. For damping
model II we can account for more modes, and we will retain 12 modes to analyse this case.
Our thermoacoustic system is composed of 2×Nmodes coupled differential equations. As
a consequence, the use of adjoint methods is required to obtain solvability conditions of the
weakly nonlinear equations, as discussed in §5.4.3.
5.4 Weakly nonlinear analysis 117
5.4 Weakly nonlinear analysis
We now perturb the bifurcation parameter from the critical point, K = Kc + ∆K, with
|∆K| ≪ 1. After the Hopf bifurcation, the system is linearly unstable and oscillations will
grow and can saturate to limit cycle oscillations due to nonlinear effects. In the case in which
the Hopf bifurcation is subcritical, a bistable region exists before the Hopf point in which, by
triggering the system, stable solutions with a finite amplitude can be found (Ananthkrishnan
et al., 2005; Juniper, 2011). To calculate the amplitudes and frequencies of these oscillations,
nonlinear methods are required. We accomplish this with a weakly nonlinear analysis, by
expanding the evolution of the dynamical system (5.9) around the Hopf location. We denote
with 0 < ε ≪ 1 a small quantity that quantifies the amplitude of the oscillations close to the
Hopf point. We then seek for solutions x expressed as power series of ε:
x = εx1+ ε2x2+ ε3x3+ ε4x4+ ε5x5+O(ε6). (5.12)
For a subcritical bifurcation, an expansion at third order yields unstable limit cycle solutions.
However, such oscillations are typically not observable in self-excited experiments, although
they can be studied experimentally by forcing the system close to the unstable solutions (Je-
gadeesan & Sujith, 2013). For subcritical phenomena, however, one is also interested in
calculating the amplitude of stable solutions, and in identifying the width of the bistable
region. This requires terms of at least order 5 in a weakly nonlinear expansion.
We choose to work with the method of multiple scales. With this method, one assumes
that several, independent timescales act on the system. One is the fast timescale t0, at which
the oscillations of the marginally stable frequency respond. The slow timescales t˜2 and t˜4 are
associated with long time saturation or growth processes. The total time derivative therefore
reads:
d
dt
=
∂
∂ t0
+
∂
∂ t˜2
+
∂
∂ t˜4
+ . . . . (5.13)
By Taylor expanding the dynamical system (5.9) around the fixed point solution x = 0 at
the critical point K = Kc we obtain:
∂x
∂ t0
+
∂x
∂ t˜2
+
∂x
∂ t˜4
= Ax+
5
∑
n=1
αnKcB(Cx(t− τ))n . . .
+
5
∑
n=1
αn∆KB(Cx(t− τ))n.
(5.14)
It is important to note that the orders of magnitude of O(x) = ε and those of O(∆K),
O(t˜2), and O(t˜4) are not independent. Upon the expansion of the equations, one can show
118 Weakly nonlinear analysis of thermoacoustic bifurcations
that, at odd orders larger than 1, secular terms (i.e., forcings at resonant frequencies) arise due
to nonlinear interactions. Solvability conditions need to be imposed on these forcings, which
have to be balanced by contributions arising from slow timescales and control parameter
terms (Rosales, 2004; Strogatz, 2015). This reasoning leads to balances between the order of
magnitudes of the various terms, which read:
O(x∆K) = ε3, O
(
∂x
∂ t˜2
)
= ε3, O
(
∂x
∂ t˜4
)
= ε5. (5.15)
We shall then rewrite all the quantities in terms of ε by defining ∆K ≡ ε2δ2, t2 ≡ ε2t˜2 and
t4 ≡ ε4t˜4. The parameter δ2 can take the values ±1 depending on the side of the Hopf point
we are investigating. From the definition of ∆K, we also obtain a measure of the expansion
parameter in terms of the distance from the Hopf location:
ε =
√
|K−Kc|. (5.16)
Lastly, the time delay contained in our system acts at all the timescales we are con-
sidering (Das & Chatterjee, 2002). Considering the ε scalings just discussed for the slow
timescales, delayed variables are therefore functions of t0− τ , t2− ε2τ , and t4− ε4τ . These
terms are then expanded in series of ε . For ease of notation, in the following we will adopt
the short notation x(t) for x(t0, t2, t4), and x(t− τ) for x(t0− τ, t2, t4).
We now substitute the relations (5.12), (5.15) into equation (5.14). The complete list of
terms we obtain is given in Appendix E. By matching these terms by their ε order, we obtain
a set of linear, inhomogeneous differential equations which have to be solved in ascending
order. Although we perform the weakly nonlinear expansion up to O(ε7) for the case with
damping model I, in the following subsections we will solve and discuss the equations in
detail up to order ε5 for brevity, and will justify the need of the seventh order expansion
in §5.5.
5.4.1 O(ε): eigenvalue problem
At order ε we retrieve the homogeneous linear equations (5.17) for the evolution of x1:
∂x1
∂ t0
−Ax1−α1KcB(Cx1(t− τ)) = 0. (5.17)
5.4 Weakly nonlinear analysis 119
Because the left hand structure of the equations will be the same at all ε orders, it is
convenient to define the spectral operator:
Ms ≡
(
sI −A−α1KcBCe−sτ
)
, (5.18)
where s≡ σ + iω denotes the Laplace variable, so that the nonlinear eigenvalue problem in
the frequency domain can be rewritten as Msx1 = 0. This is the same eigenvalue problem
of equation (5.11). Also, because here we have fixed K = Kc, we know that the system is
marginally stable (its spectrum is shown in Figure 5.4b with crosses).
We can simplify the evolution of the dynamical system to the evolution of only the
marginally stable thermoacoustic eigenmode, i.e., we ignore the contribution of the eigen-
modes with a negative growth rate, because they will quickly be damped (Sipp & Lebedev,
2007). Close to the Hopf bifurcation, we expect the dynamical system to saturate to limit
cycle oscillations at the slow timescales, therefore we can write:
x1 ≈W (t2, t4)xW1 eiω
ct0 + c.c., (5.19)
where ωc is the angular frequency of the marginally stable eigenmode, xW1 the correspond-
ing right eigenvector and W (t2, t4) a complex valued variable which depends on the slow
timescales only. W contains information on the amplitude saturation and frequency shift
effects caused by nonlinear effects. At the next odd orders, we will explicitly find the
dependence of W with respect to the slow timescales.
5.4.2 O(ε2): mean shift and second harmonic
At this order we obtain the equations for the evolution of x2. From this order on, forcing
terms will appear in the r.h.s. of the equations. In general, the forcing terms at order εN are
due to the nonlinear interactions between the solutions xk at orders k < N, which are known.
The only forcing term at this order is α2KcB(Cx1(t− τ))2, which is due to the interaction of
x1 with itself. By using the expression (5.19) for x1, we obtain the inhomogeneous linear
equations:
∂x2
∂ t0
−Ax2−α1KcB(Cx2(t− τ)) = |W |2F |W |
2
2 +
(
W 2FW
2
2 e
2iωct0 + c.c.
)
, (5.20)
120 Weakly nonlinear analysis of thermoacoustic bifurcations
where
FW
2
2 ≡ α2KcB(CxW1 )2e−2iω
cτ
F |W |
2
2 ≡ 2α2KcB|CxW1 |2.
(5.21)
The superscripts are used to classify the forcing terms by their dependence on the complex
amplitudes W . This forcing is composed of two components: a steady forcing with zero
frequency (due to the interaction between the eigenmode xW1 and its complex conjugate),
and second harmonic contributions at frequency 2ωc (due to the interaction between the
eigenmode xW1 and itself). These are not resonant terms, because the spectrum of the linear
operator does not contain 2ωc or 0 as eigenvalues (see Figure 5.4b), and equations (5.20) can
be readily solved.
We look for a steady-state solution x2 which has the same shape as the forcing, by using
the ansatz
x2 = |W |2x|W |
2
2 +
(
W 2xW
2
2 e
2iωct0 + c.c.
)
. (5.22)
By substituting the latter into equation (5.20), taking the Laplace transform (with respect to
the fast timescale t0), and matching the terms according to their amplitude dependence, we
obtain the sets of linear equations:
M2iωcxW
2
2 = F
W 2
2 (5.23a)
M0x
|W |2
2 = F
|W |2
2 . (5.23b)
The matrices M2iωc and M0 are non-singular and can be inverted, yielding the solutions at
the various amplitude levels of x2.
In particular, xW
2
2 (and its c.c.) describes second harmonic oscillations, whereas x
|W |2
2 ,
having zero frequency, will cause a shift in the mean acoustic level. This is a well-known
effect in hydrodynamics, where zero frequency corrections are due to quadratic terms
arising from the nonlinear convective term of the Navier–Stokes equations. A weakly
nonlinear expansion allows distinction between the base flow (solution of the steady Navier–
Stokes equations) and the mean flow (time averaged solution of the unsteady Navier–Stokes
equations) (Sipp & Lebedev, 2007; Meliga et al., 2009). These effects are not found if
the nonlinearity expansion contains only odd terms. This is often the case for low-order
thermoacoustic modelling (Noiray et al., 2011; Noiray & Schuermans, 2013b; Ghirardo
et al., 2015), although some experimental evidence of acoustic level mean shifts can be found
in the literature (Flandro et al., 2007).
5.4 Weakly nonlinear analysis 121
5.4.3 O(ε3): third harmonic and saturation
At this order we obtain the equations for the evolution of x3:
∂x3
∂ t0
−Ax3−α1KcB(Cx3(t− τ)) =−∂x1∂ t2 − τBα1K
cC
∂x1
∂ t2
(t− τ) . . .
+
(
WFW3 e
iωct0 + |W |2WF |W |2W3 eiω
ct0 +W 3FW
3
3 e
3iωct0 + c.c.
)
.
(5.24)
The explicit expressions of the forcing terms are reported in Appendix F.1. The slow timescale
derivative may be rewritten as:
∂x1
∂ t2
(t) =
∂W
∂ t2
xW1 e
iωct + c.c.. (5.25)
Resonant forcings with angular frequencyωc arise at this order, which act on two different
amplitude levels, W and |W |2W . A solvability condition, known as the Fredholm alternative
(Oden & Demkowicz, 2010), needs to be satisfied for a solution to exist. This condition
requires the sum of the resonant forcing terms to be orthogonal to the kernel (nullspace)
of the (singular) adjoint operator M†iωc (Sipp & Lebedev, 2007; Meliga et al., 2009). This
generalises the idea of cancelling the secular terms used for weakly nonlinear analysis of
scalar problems.
The adjoint matrix Miωc is defined through the scalar product:〈
y,Miωcx
〉
=
〈
M†iωcy,x
〉
, (5.26)
and corresponds to the Hermitian of the direct matrix Miωc . The latter is singular, because
s = iωc is an eigenvalue of (5.11). Thus, Miωc is also singular, and its kernel is spanned by
the adjoint eigenvector x†1 only. This can be calculated as the Hermitian of the left nullspace
of the operator Miωc . The solvability condition therefore requires:〈
x†1,−
∂W
∂ t2
PxW1 +WF
W
3 + |W |2WF |W |
2W
3
〉
= 0, (5.27)
where the right terms in the bracket are all the resonant forcings, and we have defined the
matrix P ≡ (I + τα1KcBCe−iωcτ). By rearranging equation (5.27), we obtain:
∂W
∂ t2
= λ3W +ν3|W |2W, (5.28)
122 Weakly nonlinear analysis of thermoacoustic bifurcations
where the complex values λ3, ν3, known as the Landau coefficients, are defined by:
λ3 ≡
〈
x†1,F
W
3
〉
〈
x†1,Px
W
1
〉 , ν3 ≡
〈
x†1,F
|W |2W
3
〉
〈
x†1,Px
W
1
〉 . (5.29)
For damping model I, the values we found for the Landau coefficients when K < Kc are
λ3 =−0.0291−0.2181i and ν3 = 0.0157−0.0203i. For damping model II, the dependence
of the Landau coefficients on the number of modes retained in the acoustic state-space model
is shown in Figure 5.6a. Note that, in this case, we find that the sign of Re(ν3) with one
mode is different from the sign of the saturated value, containing multiple modes, whereas
the sign of Re(λ3) does not change. This shows that the nature of the bifurcation predicted
with a single mode approximation may be different from the actual response of the system.
For the case shown in Figure 5.6a, the single mode approximation predicts a supercritical
bifurcation, whereas the saturated values show that the bifurcation is actually subcritical.
We then seek a solution x3 via the ansatz:
x3 =WxW3 e
iωct0 + |W |2Wx|W |2W3 eiω
ct0 +W 3xW
3
3 e
3iωct0 + c.c.. (5.30)
By using the relation (5.28), we match the solution and forcing terms of (5.24) by their
dependence on the amplitude W , and obtain the sets of linear equations:
MiωcxW3 = F
W
3 −λ3PxW1 (5.31a)
Miωcx
|W |2W
3 = F
|W |2W
3 −ν3PxW1 (5.31b)
M3iωcxW
3
3 = F
W 3
3 . (5.31c)
Although the matrix Miωc is singular, the values of the Landau coefficients guarantee that
solutions for xW3 and x
|W |2W
3 exist. They can be calculated, e.g., by using the pseudo-inverse
matrix of Miωc . These solutions provide a nonlinear correction to the shape of the linearly
unstable mode. Equation (5.31c), instead, can be readily solved by inverting the matrix
M3iωc , which is non-singular. xW
3
3 accounts for third harmonic contributions to the oscillatory
solution.
Stuart–Landau equation: O(ε3)
Equation (5.28) is known as the Stuart–Landau equation. Its roots yield the amplitude of
limit cycle solutions and the frequency shift of the nonlinear oscillation with respect to
the marginally stable eigenfrequency. By using the polar representation W = reiθ , and by
5.4 Weakly nonlinear analysis 123
 
 
1 2 3 4 5 6 7 8 9 10 11 12
−10
−5
0
5
x 10−3
Number of modes
1 2 3 4 5 6 7 8 9 10 11 12
0
0.1
0.2
La
nd
au
 C
oe
ffi
cie
nt
s
Re(λ3) Im(λ3) Re(ν3) Im(ν3)
1.41 1.42 1.43 1.44
0
0.2
0.4
0.6
0.8
1
K
|u
′ iω
|
K
c
Third order subcritical
bifurcation diagram
Fig. 5.6 (a): Dependence of the Landau coefficients λ3, ν3 with respect to the number of
modes. (b): Subcritical Hopf bifurcation diagram. Solid and dashed lines represent stable
and unstable solutions, respectively. The oscillation amplitude at the fundamental frequency
with corrections up to ε3 is shown. Results are shown for damping model II; analogous
graphs are found for damping model I.
splitting the real and the imaginary parts, we have:
∂ r
∂ t2
= Re(λ3)r+Re(ν3)r3 (5.32a)
∂θ
∂ t2
= Im(λ3)+ Im(ν3)r2. (5.32b)
The equation for the phase θ is valid only for solutions with non-zero amplitude. Steady-state
solutions are reached when the amplitude r does not vary in time. The amplitude levels at
which this happens are:
r1 = 0, r2 =
√
−Re(λ3)
Re(ν3)
. (5.33)
From the definitions (5.29), (F.2a), one can see that λ3, being proportional to δ2 =±1,
changes sign across the critical point Kc, whereas ν3 does not vary when we change the
bifurcation parameter. Therefore, the solution r2 is real only on one side of the Hopf location.
The stability of the solutions is connected to the sign of the eigenvalue of the Jacobian
J ≡ Re(λ3)+3Re(ν3)r2 evaluated at the two solutions:
J(r1) = Re(λ3), J(r2) =−2Re(λ3). (5.34)
124 Weakly nonlinear analysis of thermoacoustic bifurcations
Note that, in the region where two solutions coexist, their stability is different. This dis-
tinguishes between super- and subcritical Hopf bifurcations. For the set of parameters we
have chosen, we find that the bifurcation is subcritical: before the Hopf bifurcation, stable
fixed points and unstable limit cycle solutions exist, and after it only unstable fixed point
solutions are found, as shown in Figure 5.6b. The present analysis could be also applied to
configurations in which the bifurcation is supercritical.
The solution of the phase equation (5.32b) on the unstable limit cycle solution reads:
θ = ε2
(
Im(λ3)− Im(ν3)Re(λ3)Re(ν3)
)
t0 ≡ ∆ωt0, (5.35)
where we have used the scaling t2 = ε2t0 between the fast and slow timescales. ∆ω represents
the frequency shift between the fundamental oscillation frequency of limit cycles and the
marginally stable frequency ωc.
Combining the power expansion (5.12), the weakly nonlinear solutions (5.19), (5.22),
(5.30) and the solution of the Stuart–Landau equation, we obtain an analytical expression for
the time evolution of the thermoacoustic states up to third order, which reads:
x = ε2r2x|W |
2
2 +
[
εrxW1 + ε
3rxW3 + ε
3r3x|W |
2W
3
]
ei(ω
c+∆ω)t0 . . .
+ε2r2xW
2
2 e
2i(ωc+∆ω)t0 + ε3r3xW
3
3 e
3i(ωc+∆ω)t0 + c.c.+O(ε4).
(5.36)
Figure 5.6b shows the subcritical bifurcation diagram we obtain at this order. It contains
the amplitude level of the velocity fluctuations u′ =Cx calculated from equation (5.36) at the
fundamental frequency. We shall postpone the discussion of the response at other frequencies
to section §5.5. Because these limit cycle oscillations are unstable, the amplitudes that
the Stuart–Landau equation predicts at this order correspond approximately to the level of
triggering which is required to excite finite amplitude oscillations. In order to predict the
amplitude of stable limit cycles, we need to extend the weakly nonlinear expansion to higher
orders.
5.4.4 O(ε4): mean shift and fourth harmonic
At this order we obtain the equations for the evolution of x4:
∂x4
∂ t0
−Ax4−α1KB(Cx4(t− τ)) = |W |4F |W |
4
4 + |W |2F |W |
2
4 . . .
+
(
W 2FW
2
4 e
2iωct0 + |W |2W 2F |W |2W 24 e2iω
ct0 +W 4FW
4
4 e
4iωct0 + c.c
)
.
(5.37)
5.4 Weakly nonlinear analysis 125
None of the forcing terms resonates. Their expressions are provided in Appendix F.2. Using
the ansatz:
x4 = |W |2x|W |
2
4 + |W |4x|W |
4
4 . . .
+
(
W 2xW
2
4 e
2iωct0 + |W |2W 2x|W |2W 24 e2iω
ct0 +W 4xW
4
4 e
4iωct0 + c.c.
)
,
(5.38)
we can readily calculate the solutions
x|W |
2
4 =M
−1
0 F
|W |2
4 (5.39a)
x|W |
4
4 =M
−1
0 F
|W |4
4 (5.39b)
xW
2
4 =M
−1
2iωcF
W 2
4 (5.39c)
x|W |
2W 2
4 =M
−1
2iωcF
|W |2W 2
4 (5.39d)
xW
4
4 =M
−1
4iωcF
W 4
4 , (5.39e)
which provide contributions to the mean acoustic level shift, and second and fourth harmonic
oscillations.
5.4.5 O(ε5): fifth harmonic and saturation
At this order we obtain the equations for the evolution of x5:
∂x5
∂ t0
−Ax5−α1KcB(Cx5) =−P ∂x1∂ t4 . . .
+
(
WFW5 + |W |2WF |W |
2W
5 + |W |4WF |W |
4W
5
)
eiω
c(t0−τ) . . .
+
(
|W |2W 3F |W |2W 35 +W 3FW
3
5
)
e3iω
c(t0−τ)+W 5FW
5
5 e
5iωc(t0−τ)+ c.c..
(5.40)
In this section, we will not explicitly calculate the solution x5, but we will only derive the
dependence of the amplitude W with respect to the slow timescale t4. This is achieved by
applying the Fredholm alternative solvability condition on the resonant forcings at frequency
ωc which appear on the r.h.s. of equation (5.40).
By imposing that the resonant terms are orthogonal to the kernel of the adjoint operator
M†iωc , we obtain:〈
x†1,−
∂W
∂ t4
PxW1 +WF
W
5 + |W |2WF |W |
2W
5 + |W |4WF |W |
4W
5
〉
= 0. (5.41)
126 Weakly nonlinear analysis of thermoacoustic bifurcations
The resonant forcing expressions can be found in Appendix F.3. Equation (5.41) can be
simplified into the Stuart–Landau equation:
∂W
∂ t4
= λ5W +ν5|W |2W +µ5|W |4W, (5.42)
where the Landau coefficients are defined by:
λ5 ≡
〈
x†1,F
W
5
〉
〈
x†1,Px
W
1
〉 , ν5 ≡
〈
x†1,F
|W |2W
5
〉
〈
x†1,Px
W
1
〉 , µ5 ≡
〈
x†1,F
|W |4W
5
〉
〈
x†1,Px
W
1
〉 . (5.43)
For damping model I, the values we obtain for these coefficients when K < Kc are λ5 =
0.0019− 0.0077i, ν5 = −0.0235+ 0.004i, µ5 = −0.0147− 0.0136i. The coefficient ν5
changes sign across the Hopf location.
Stuart–Landau equation: O(ε5)
The overall slow timescale evolution of the amplitude W is obtained by combining the results
at the two timescales t2 and t4 (Fujimura, 1991; Gambino et al., 2012). By using the scaling
t4 = t2ε2, we obtain:
dW
dt2
=
∂W
∂ t2
+
∂W
∂ t4
∂ t4
∂ t2
=
(
λ3+ ε2λ5
)
W +
(
ν3+ ε2ν5
) |W |2W + ε2µ5|W |4W. (5.44)
By using the polar representation W = reiθ , this decouples into:
dr
dt2
= Re(λ )r+Re(ν)r3+Re(µ)r5 (5.45a)
dθ
dt2
= Im(λ )+ Im(ν)r2+ Im(µ)r4, (5.45b)
where λ = λ3+ ε2λ5, and similarly for ν and µ . The fixed points of the amplitude equation
are:
r1 = 0, r2,3 =
√
−Re(ν)±
√
(Re(ν))2−4Re(λ )Re(µ)
2Re(µ)
. (5.46)
The existence of real solutions for r2,3 depends only on the sign of the terms under the
square roots. Because the bifurcation is subcritical, we have two solutions after the Hopf
location (an unstable fixed point and a stable limit cycle), a bistable region between the Hopf
and the fold points with three solutions (a stable fixed point and two limit cycles, one unstable
and one stable), and a region with only one stable fixed point before the fold. This is shown
5.5 Results validation 127
5.05 5.1 5.15 5.2 5.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
K
|u
′ iω
|
K
c
K
F
 
 
O(ǫ3)
O(ǫ5)
Fig. 5.7 Comparison between third order (black) and fifth order (red) bifurcation with damp-
ing model I. Analogous results are found with damping model II. The analytically calculated
amplitude of the acoustic velocity at the fundamental frequency is shown, according to (5.36).
At fifth order the limit cycle saturates, so the limit cycle amplitude and the location of the
fold point KF can be calculated.
in Figure 5.7, where the path followed by the oscillations when the bifurcation parameter is
varied across the bistable region is shown with arrows. As expected, close to the Hopf point
the fifth order expansion correctly matches the third order one.
The stability of the solutions is determined by the sign of the Jacobian J = Re(λ )+
3Re(ν)r2+5Re(µ)r4 evaluated at the solutions. These values are:
J(r1) = Re(λ ) (5.47a)
J(r2,3) =−4Re(λ )+ (Re(ν))
2
Re(µ)
∓Re(ν)
√
(Re(ν))2−4Re(λ )Re(µ). (5.47b)
The frequency shift ∆ω on the limit cycles can be readily calculated from equation (5.45b)
by using the scaling t2 = ε2t0:
∆ω2,3 ≡ ε2
(
Im(λ )+ Im(ν)r22,3+ Im(µ)r
4
2,3
)
. (5.48)
5.5 Results validation
We now compare the weakly nonlinear analysis discussed in §5.4 with the fully-nonlinear
results obtained by solving the nonlinear dynamical system (5.9) with no approximations.
This is achieved with two methods: time marching the governing equations, and numerical
128 Weakly nonlinear analysis of thermoacoustic bifurcations
0.000 0.500 1.000 1.500 1.890 2.289 2.290
−1
−0.5
0
0.5
1
t
u
′
0 5000 10000
10−4
10−3
10−2
σ =2.068e-04
×10
4
0 5 1000 2000 3000 4000 5000 6000
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
t
u
′
3000 3500 4000
10−4
10−2
100
σ =-3.785e-03
Fig. 5.8 Time marching simulations with damping model I. a): at K = Kc +∆K the fixed
point solution becomes unstable and oscillations grow to a limit cycle. b): at K = 5.03 no
limit cycle solutions exist, and the system converges to fixed point solutions even when
initialised to a highly perturbed state. The insets show the exponential growth/decay rates of
the oscillations in log-scale.
continuation of limit cycles. Time domain simulations are performed using a delay differential
equations solver, and the numerical continuation algorithm is based on the DDE-BIFTOOL
package (Engelborghs et al., 2002). Although the latter can only predict periodic oscillations
and their stability, we find that, for the set of parameters we have investigated, the system
always converges towards fixed points or periodic oscillations. Therefore, the two methods
yield the same results, as was verified.
For time domain simulations we proceed as follows: we initialise the integration to a
state which is slightly perturbed from the fixed point solution. Because the equations are time
delayed, the initial state covers the history of the system for a time−τ ≤ t ≤ 0. We start from
a value of K < Kc, and then increase it in steps of ∆K. Until K ≤ Kc, the initial perturbations
are damped and the system converges to fixed points solutions. Focusing on the numerical
results of model I, at Kc = 5.17, the oscillations start growing and converge towards a limit
cycle attractor with a large amplitude (see Figure 5.8a). This solution is used to initialise
the subsequent integrations, for which the control parameter K is varied in both directions
in steps of ±∆K = ±0.01. The amplitude of the oscillations gets larger as K increases.
The largest value of the control parameter that we consider is Kmax = 5.25, for which the
parameter expansion of the weakly nonlinear analysis is εmax =
√
Kmax−Kc = 0.29. On
the other hand, the amplitude of the velocity fluctuations decreases smoothly for K < Kc,
until we reach the fold location at KF = 5.03. At this location, the highly perturbed initial
state decays to the fixed point solution, as shown in Figure 5.8b.
To validate the linear analysis, the growth/decay rates close to the Hopf/fold locations
are extracted from the time series by using a linear regression on the logarithm of the
5.5 Results validation 129
5 5.1 5.2 5.3
0
0.2
0.4
0.6
0.8
1
K
|u
′ iω
|
K
c
K
F
 
 
O(ǫ7)
1.421 1.422 1.423 1.424
0
0.1
0.2
0.3
0.4
0.5
K
|u
′ iω
|
K
c
K
F
 
 
O(ǫ3) O(ǫ5)
Numerical
Continuation
Fig. 5.9 Comparison between the weakly nonlinear analysis at various orders (lines) and
numerical continuation results (circles). The bistable region is highlighted in grey. Solid
and dashed lines indicate stable and unstable solutions, respectively. Left and right panels
contain the results obtained with damping models I and II respectively.
fluctuations amplitude. The latter is obtained using the Hilbert transform on portions of
the time series. The growth and decay rates, reported in Figure 5.8, can be compared with
those we obtain when solving the eigenvalue problem (5.11) at the same locations. These
values are σ = 2.097×10−4 and σ =−3.82×10−3 respectively, and are in good agreement
with the time marching results. To validate the weakly nonlinear analysis, we compare the
oscillation amplitudes at the various frequency components with those predicted by numerical
continuation of limit cycles. This is because the latter also yields information about the
unstable limit cycles, which are more difficult to investigate with time marching simulations.
The bifurcation diagram in Figure 5.9 shows the amplitude of the oscillations at the
fundamental frequency as predicted from the weakly nonlinear analysis at fifth order (red
line) and limit cycle continuation (black circles). For damping model II (right inset), the
prediction between the amplitude of the limit cycles calculated with numerical continuation
compares well with the fifth order weakly nonlinear expansion, in red. However, this is
not true for damping model I. In this case, although the fifth order analysis improves the
predictions of the third order one (it predicts the existence of stable limit cycles of the bistable
region), these results do not compare well with the exact solutions. This is because, in this
case, the width of the bistable region is much larger than in the other case. The value of
the expansion parameter ε at the fold point for the two damping models is εI = 0.35 and
εII = 0.045, where εI ≡
√
KcI −KFI . The weakly nonlinear expansion is strictly valid only
130 Weakly nonlinear analysis of thermoacoustic bifurcations
for ε ≪ 1. This condition is well satisfied at the fold point for model II, i.e., εII ≪ 1, but it is
not for model I. Thus, to improve the quality of the weakly nonlinear analysis, expansion to
a higher order is required in the latter case.
This is achieved by introducing a third slow timescale t6 = ε6t0 and expanding equa-
tions (5.14) up to order ε7. The expansion terms at these orders are lengthy and are not
reported here. We can solve straightforwardly the equations at O(ε6), which yields even
harmonic contributions, because they contain no resonant terms. At order ε7 we look only at
the response at the fundamental frequency, which contains resonant terms. By imposing the
solvability condition we obtain the Stuart–Landau equation:
∂W
∂ t6
= λ7W +ν7|W |2W +µ7|W |4W +ξ7|W |6W. (5.49)
The Landau coefficient definitions are analogous to those in equation (5.43). The values
we obtain for these coefficients when K < Kc are λ7 = (9.63+5.59i)×10−5, ν7 = 0.012−
0.013i, µ7 = 0.0168+ 0.003i, and ξ7 = 0.003+ 0.002i. λ7 and µ7 change sign across
the bifurcation. The overall evolution of the complex-valued amplitude W is obtained by
summing the contributions at all orders:
dW
dt2
=
3
∑
n=1
ε2n−2
∂W
∂ t2n
= λW +ν |W |2W +µ|W |4W +ξ |W |6W, (5.50)
with λ = λ3+ ε2λ5+ ε4λ7 and similarly for the other coefficients. Amplitude roots and the
corresponding stability and phase shifts are then calculated numerically.
The seventh order weakly nonlinear bifurcation diagram is shown in Figure 5.9a (green
line). No significant changes are observed for the unstable solutions close to the Hopf
location, but the fold location and the amplitude of oscillation predicted greatly improves at
this order.
Further information can be extracted from the weakly nonlinear analysis, which is shown
in Figure 5.10 for damping model I only, at the seventh order expansion. This is the difference
between the frequency of limit cycle oscillations compared with that of the marginally stable
eigenvalue, and the amplitude of the harmonics in the spectrum of the oscillatory solution.
The zeroth harmonic corresponds to the mean shift in the acoustic level previously discussed,
which is indeed observed in time domain simulations, due to the fact that the nonlinearity
is not odd. All these results show a good agreement with those obtained from the exact
solution.
5.6 Practical implementation 131
5.05 5.1 5.15 5.2 5.25
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
K
∆
ω
5 5.05 5.1 5.15 5.2 5.25
0
0.02
0.04
0.06
0.08
0.1
0.12
K
|u
′ n
iω
|
n = 3
n = 0
n = 2
Fig. 5.10 Left: frequency shift ∆ω of the oscillations with respect to the marginally stable
frequency at the Hopf bifurcation. Right: magnitude of the harmonic contributions in the
spectrum of the stable oscillatory solutions. Results from weakly nonlinear analysis (lines)
and amplitude of FFT of time domain simulations (circles) are compared.
5.6 Practical implementation
One of the attractive characteristics of the weakly nonlinear analysis is that it is computa-
tionally very cheap. The expansion of the equations can be tedious at high orders, but the
process can be automatised with symbolic solvers. The information needed for the analysis
is (i) a marginally stable solution, (ii) the direct and adjoint eigenvectors corresponding to
the marginally stable solution, and (iii) the Taylor expansion of the nonlinear heat release
fluctuations. Once these components have been found, the calculation of a bifurcation dia-
gram reduces to a series of matrix inversions and multiplications. This is affordable even
for systems with a large number of degrees of freedom, as was shown by Sipp & Lebedev
(2007); Meliga et al. (2009) for the analysis of flow bifurcations. An alternative technique,
numerical continuation (Waugh et al., 2014), requires time-marching for several cycles in
order to converge to a limit cycle. Another technique, time marching, requires even more
cycles and can converge only to stable oscillations. For our low-order model, we find that the
weakly nonlinear analysis is 20 times faster than numerical continuation, and three orders of
magnitude faster than time marching.
It can be challenging, however, to obtain an analytical model for the nonlinear heat
release when the combustion system is complex, e.g. turbulent. A possible solution is to
measure the Flame Transfer Function (FTF) for small amplitude oscillations and to combine
this with an acoustic network to obtain a marginally stable solution, which will be at a
132 Weakly nonlinear analysis of thermoacoustic bifurcations
given frequency. The direct and adjoint eigenvectors of this solution can be calculated
easily. The Taylor expansion coefficients α for the nonlinear heat release dynamics can
be obtained through experiments or DNS, by forcing the system at various amplitudes at
the marginally stable frequency. One can then calculate the Taylor coefficients by finite
difference methods, although this may be prone to inaccuracies at large amplitudes due to
error propagation. Alternatively, as in this study, the nonlinear response can be fitted, in a
least squared sense, onto a polynomial over the amplitude range of interest. This expansion
can also be generalised to non-static nonlinearities, meaning that the heat release fluctuation
q′ is a function of the velocity fluctuation u′ and its time derivative u˙′.
With this procedure, all the information required for the weakly nonlinear analysis is
known and the theory can be applied. Assuming that the measured nonlinear behaviour
changes slowly with frequency, the entire bifurcation diagram can be constructed, which
would require many more measurements if determined experimentally or numerically. More-
over, the dynamic evolution of the oscillations associated with the global mode can be
obtained for various values of the bifurcation parameter, by direct integration of the Stuart–
Landau equation.
5.7 Conclusions
In this chapter we have investigated nonlinear thermoacoustic oscillations in a Rijke tube
using a high order weakly nonlinear expansion of the governing equations. The framework
we analysed, composed of a wave-based approach for the linear acoustic equations and a
nonlinear model for the heat release response, is general and can be extended to more complex
networks easily. We have shown how a high order expansion close to the Hopf bifurcations
can be used to predict the amplitude of stable and unstable limit cycles, the frequency shift
with respect to the marginally stable solution, and the region of bistability for subcritical
bifurcations. We have compared our results with the exact solutions of the fully-nonlinear
equations and shown that the method yields accurate results when the expansion is truncated
at fifth/seventh order. Using this type of analysis, the numerical/experimental effort needed
to construct a bifurcation diagram can be significantly reduced.
The weakly nonlinear analysis we have performed is valid for a deterministic system,
close to a Hopf bifurcation, and assumes that only one mode is marginally stable and all the
other thermoacoustic modes are damped. The influence of turbulence and combustion noise
can be modelled phenomenologically as random forcing (Noiray & Schuermans, 2013b;
Rigas et al., 2015); their effects can be captured by including stochastic terms in the Stuart–
Landau equation. Other possible extensions of the theory are the following: varying the
5.7 Conclusions 133
bifurcation parameter, a second mode may become marginally stable in the vicinity of the
first Hopf location, resulting in a sudden change in the system dynamics, as discussed by
(Sipp, 2012). In this case, the theory could be extended by considering a codimension two
bifurcation, as described in Meliga et al. (2009). It can also be extended to investigate the
response of the system to external forcing. In this case, the forcing terms appear explicitly
in the amplitude equations, as shown by Sipp (2012), and the weakly nonlinear interaction
between the global mode and the forcing is taken explicitly into account. Moreover, the
unknown Landau coefficients of the model can be identified from self-excited limit cycles
oscillations of the system, as described in Noiray & Schuermans (2013a), or from transient
and steady-state measurements when the system is subjected to external forcing (Rigas et al.,
2016). This method could provide a systematic way for devising open-loop control strategies
for the regulation of thermoacoustic oscillations (Paschereit & Gutmark, 2008; C´osic´ et al.,
2012).

Chapter 6
Heat release response to forced flow
oscillations of a low-order modelled
laboratory scale dump combustor
In this chapter, a low-order model for the dynamics of a bluff-body stabilised flame is
developed. The model, based on the kinematic G-equation, aims to obtain a numerically
cheap but qualitatively reasonable approximation for the heat release dynamics. This could
be coupled in an acoustic network to estimate the frequencies at which thermoacoustic
oscillations can occur. The low-order model developed mimics the flame dynamics over
forced cycles reasonably well. This is because a rather accurate description of the flame-
flow interaction is considered here, using an Unsteady Reynolds Averaged Navier–Stokes
(URANS) meanflow, provided by Dr. Carol Armitage. However, it is found that the heat
release response tends to be overestimated by the model, which is also very sensitive to
parameters such as the convection speed.
The content of this chapter was published in Orchini & Juniper (2015).
6.1 Introduction
Low-order modelling in thermoacoustics relies on radical (but physically-based) approxima-
tions in the flame-flow-acoustic coupling, and allows for a great reduction of the number of
degrees of freedom needed to describe the thermoacoustic interaction. As the computational
cost required to integrate the system is highly reduced, low-order models can be used to
investigate a large variety of parameters and gain insight into the physical mechanisms that
cause thermoacoustic oscillations to arise. For premixed flames, the kinematic G-equation
136 Heat release response to forced flow oscillations of a low-order modelled combustor
is a common model used to determine the position and evolution of a thin flame front, as
well as the amount of heat released by the flame. In the linear limit, transfer functions of
conical and V shaped flames to harmonic velocity and heat release fluctuations have been
evaluated analytically for a variety of underlying flow fields (Schuller et al., 2003; Preetham
et al., 2008). In particular, a travelling wave model for flow perturbations has been shown
to accurately reproduce the flow-flame interaction. Also, transfer functions of heat release
fluctuations to harmonic velocity disturbances for these flame shapes have been proven to
match experimental results (Durox et al., 2009; Karimi et al., 2009). In the nonlinear regime,
2-dimensional tent and axisymmetric conical flames have been investigated with a variety of
tools both in the frequency and time domain. Frequency domain analyses are based on the
Flame Describing Function (Dowling, 1999; Noiray et al., 2008), and allow for the prediction
of frequencies and amplitudes of oscillations when the oscillations are associated with a
single thermoacoustic mode. Time domain analyses allow for a more extensive investigation
of thermoacoustic systems, and phenomena linked to the nonlinear dynamical nature of the
system, such as period-doubling, secondary Hopf bifurcations, and routes to chaos, can be
observed (Kashinath et al., 2014; Waugh et al., 2014; Orchini et al., 2015), as was shown in
Chapter 3.
In this chapter, we extend the models derived for conical flames to a more realistic
combustor configuration. The combustor we investigate is a dump combustor which has been
developed by Balachandran et al. (2005), and is shown in Figure 6.1. As the experiments
performed on this setup represent one of the first set of measurements of forced turbulent
flames in the nonlinear regime, low-order and CFD simulations of this rig have already been
performed. In Graham (2012) a G-equation based low-order model has been developed.
However, in that study the flow field was treated as a potential flow generated by a pulsating
spherical source upstream of the bluff-body. This does not include the characteristic formation
of vortices at the slot and their roll up. Also, each side of the flame was treated explicitly as a
single-valued function in one of the coordinates; this condition is not necessarily satisfied
if the flame shape becomes very wrinkled. More accurate numerical simulations based on
unsteady Reynolds-averaged Navier–Stokes (URANS) models were performed by Armitage
et al. (2006); Ruan et al. (2013).
Our goal is to develop a low-order model based on the fully implicit G-equation that
describes the flame-flow interaction in this configuration. The implicit formulation naturally
includes the possibility for the flame to assume configurations that are not necessarily single-
valued in any simple coordinate system. The flame is stabilised by a bluff-body, and, although
it resembles the structure of a V -flame, it differs from it in the fact that the reactants are
enclosed between inner and outer flame surfaces, which are assumed to be axisymmetric.
6.1 Introduction 137
Fig. 6.1 Schematic of the bluff-body combustor rig developed by Balachandran et al. (2005).
Air and fuel are premixed far upstream of the dump plane. A schematic of the flow field in
the combustion zone is shown.
The flame mean shape and evolution are strongly affected by the interaction of the flow with
the walls and the presence of recirculation zones in the combustion chamber (see Figure 6.1).
Here, we propose a flow model which is based on experimental and numerical observations
of the acoustically forced flow field. Then, we perform a qualitative comparison between
the forced response of flame and heat release dynamics of our model and results obtained
in previous studies. Although qualitatively the flame shape mimics the experimental flame
dynamics, major quantitative differences are observed in the heat release response. This is
because the model is low-order, and we are neglecting, e.g., turbulence effects and heat losses
in the outer recirculation zone. We conclude discussing our results and commenting on how
our low-order model could be improved in a future study. Once a satisfactory agreement with
the realistic forced response is achieved, the flame-flow model can be coupled with a linear
acoustic network as in Orchini et al. (2015), to obtain a flame-acoustic feedback loop and
have a complete low-order thermoacoustic model able to predict frequencies and amplitudes
of oscillations both in the frequency and time domain with a small computational cost.
The chapter is structured as follows: in §6.2 the flame and flow field modelling for
the configuration under consideration are developed, considering experimental results and
extending models available in the literature. In §6.3 the model FDF is evaluated, by forcing
the flame acoustically at various frequencies and amplitudes. Results are compared against
138 Heat release response to forced flow oscillations of a low-order modelled combustor
experiments and URANS simulations. In §6.4 the role of the convection speed on the system
response is discussed. Finally, in §6.5 the analysis results are summarised.
6.2 Modelling
In this section we describe in detail the low-order model and our assumptions. The main
concept is to first obtain a mean flow representative of the geometry under consideration and
freeze it, thus solving only for the linearised acoustic equations which induce fluctuations in
the mean flow. As a consequence, the full Navier–Stokes equations need to be solved only
once, for the mean flow, greatly reducing the computational cost needed. The geometry we
consider is axisymmetric, and we will therefore use cylindrical polar coordinates.
6.2.1 Flow field
We decompose the flow field u into a mean component u and a fluctuation u′. The latter is
not necessarily small, and we will perform a fully nonlinear analysis. In the literature, most
low-order analyses involving a kinematic description of the flame assume the mean flow to
be uniform in the axial direction, and zero in the radial and azimuthal components. However,
for the combustor we are modelling, we cannot make this assumption: the presence of a
conical bluff-body just upstream the combustion zone introduces a strong radial component
of the flow at the inlet, creating a side recirculation zone at the dump plane. Also, the wake
induced by the bluff-body forms a large central recirculation zone. These recirculation zones
guarantee the flame stabilization, as they promote flame anchoring at the dump plane, and
also greatly affect the flame’s shape and dynamics.
On account of this, we want to use a realistic mean flow, i.e., a mean flow that accurately
describes the location of the recirculation zones. This can be obtained solving the Reynolds-
averaged Navier–Stokes (RANS) equations in the enclosed domain of Figure 6.1. The mean
flow we use in the following analysis has been obtained by time-averaging unsteady RANS
simulations performed by Armitage et al. (2006). The radial (ur) and axial (uz) components
of the averaged velocity field are shown in Figure 6.2, and the azimuthal component is zero,
because we are considering a perfectly axisymmetric configuration.
On top of the mean flow, fluctuations develop when the system is acoustically forced
by a loudspeaker, or when a thermoacoustic oscillation arises. Experimental cold flow
smoke visualizations have shown that acoustic fluctuations cause the formation of a pair
of counter rotating vortices in the inner and outer sides of the flame (one in each recircu-
lation zone) (Balachandran et al., 2005). The vortices are transported downstream with a
6.2 Modelling 139
r [mm]
z
[m
m
]
ur
 
 
0 10 20 30
0
10
20
30
40
50
60
70
[m/s]
−3
−2
−1
0
1
2
3
r [mm]
uz
 
 
0 10 20 30
[m/s]
−6
−4
−2
0
2
4
6
8
Fig. 6.2 Averaged URANS flow field components in the combustion zone. Solid thick lines
indicate walls. The dashed lines indicate the centerline (left boundaries) and the streamline
r = ρs(z) that passes through the slot midpoint (only in the right frame).
characteristic velocity, the convection speed, causing flame wrinkling and consequent heat
release fluctuations. The scenario is similar to the one observed for conical flames. In that
case, oscillations are axisymmetric with respect to the centerline, and it has been shown that
an axial developing travelling wave velocity model, with radial fluctuations computed by
means of mass conservation, allows for a realistic description of the flame-flow coupling
mechanism (Schuller et al., 2003; Preetham et al., 2008; Durox et al., 2009).
Mimicking this description, we want to generate a fluctuating flow field which is con-
vected axially with a certain convection speed and generates vorticity in the inner and outer
sides of the flame. For the axial component, we impose acoustic velocity fluctuations at the
inlet, and we convect them downstream. For the convection, we use as a reference velocity
the axial velocity of the streamline r = ρs(z) that passes through the centre of the slot (see
Figure 6.2). This assumes that axial fluctuations do not vary in the radial direction. Finally,
we solve for mass conservation to obtain radial fluctuations. We want these oscillations
to have opposite sign on the two sides of the flame, in order to emulate the formation of
counter rotating vortices. This is achieved by imposing radial fluctuations to be zero along
the streamline r = ρs(z). Summarising, we solve
∂u′z
∂ t
+uz (ρs (z) ,z)
1
K
∂u′z
∂ z
= 0, u′z (z, t)|z=0 = u′ac(t) (6.1)
140 Heat release response to forced flow oscillations of a low-order modelled combustor
for the convection, and
1
r
∂ (r u′r)
∂ r
+
∂u′z
∂ z
= 0, u′r (r,z, t)|r=ρs(z) = 0 (6.2)
for mass conservation. The parameter K introduced in equation (6.1) is the ratio between
the mean and convection speeds, which in general may not be equal to 1 (Preetham et al.,
2008; Kashinath et al., 2013b). In the following, we fix K = 1.5; this particular choice for K
is discussed in §6.4.
As we have assumed that axial fluctuations are not a function of the radial direction,
equation (6.2) can be solved yielding an expression for radial fluctuations, which reads:
u′r(r,z, t) =−
1
2
∂u′z
∂ z
r
(
1− ρ
2
s (z)
r2
)
. (6.3)
Note that, with this model, radial fluctuations diverge when approaching the centerline r = 0.
However, because we use a kinematic equation for the dynamics of the flame (see next
section), we need to evaluate the velocity field only around the flame front, which is always
distant from the centerline, and the use of equation (6.3) is justified.
Flame model
We use the kinematic G-equation to describe the flame and heat release dynamics. We
define a field G whose level set G = 0 identifies the flame surface, which separates reac-
tants (G < 0) from products (G > 0). The flame front evolves according to the transport
equation (Markstein, 1964):
∂G
∂ t
+u ·∇G = sL |∇G| . (6.4)
Aside for its sign, the G-field away from the zero contour line has no physical meaning, and
we define it as a signed distance function. We define the zero level set line implicitly, i.e., as
G(r,z, t) = 0, without assuming that the flame is single-valued in one of the two coordinates
as is done in Preetham et al. (2008); Graham (2012). This allows us to naturally describe
topological changes in the solution, which occur when pocket of fuels detach from the flame’s
main body, and to be able to describe very wrinkled flame shapes, which in general cannot be
found with an explicit treatment of the flame front. Topological changes and highly wrinkled
structure are the main features of this flame, as can be observed in Figures 6.5 and 6.6, and
need to be accounted for. In equation (6.4) u = u+u′ is the underlying flow field and sL the
flame speed. The flame speed is in general a function of the equivalence ratio, turbulence
intensity, and local stretch effects. In this analysis, we consider fully premixed, laminar
6.2 Modelling 141
flames. The equivalence ratio φ is used to define the flame speed of the laminar flat flame
sheet, which is given by the empirical relation:
s0L(φ) = k1φ
k
2 e
−k3(φ−k4)2. (6.5)
The fuel we consider is an air-ethylene gaseous mixture, for which the coefficients in equa-
tion (6.5) are given by k1 = 1.32176, k2 = 3.11023, k3 = 1.72307, k4 = 0.36196 (Armitage
et al., 2006). We consider a uniform equivalence ratio φ = 0.65, yielding s0L = 0.3 m/s.
Corrections to the unstretched flame speed are due to curvature effects (Markstein, 1964;
Tien & Matalon, 1991), and are typically accounted for by:
sL = s0L (1−Lκ) , (6.6)
where L is the Markstein length and κ = ∇ · nˆ =−∇ · ∇G|∇G| the flame curvature. We fix
L= 1.2×10−3 m; this value is consistent with the Markstein numbers which have been used
in similar studies on the conical flame modelled with the G-equation (Waugh et al., 2014;
Orchini et al., 2015). However, equation (6.6) is a linear correction to the flame speed with
respect to the local curvature. Looking at the flame shapes that were observed experimentally
and numerically, we expect to find very wrinkled flame structures and the formation of
pinch-offs. When a pinch-off forms, the flame becomes cuspy and the local curvature can be
very high: higher-order stretch corrections are needed. Other studies (Ronney & Sivashinsky,
1989; Kelley & Law, 2009) have addressed the question of developing nonlinear models for
the influence of highly stretched flame sheets on the flame speed. We shall adopt one of the
nonlinear models they have proposed, which reads:
sL(κ) :

(
sL
s0L
)2
log sL
s0L
=−Lκ κ ≤ 12Le
sL = s0Le
− 12 κ > 12Le
, (6.7)
and is plotted in Figure 6.3. In equation 6.7 the flame speed sL is defined implicitly as a
function of the curvature κ .
Expanding equation (6.7) around sL = s0L (i.e., small curvature) yields:
κ(sL)≈− 1L
[
κ|sL=s0L +
dκ
dsL
∣∣∣∣
sL=s0L
(
sL− s0L
)]
=
=− 1L
sL− s0L
s0L
, (6.8)
142 Heat release response to forced flow oscillations of a low-order modelled combustor
−6 −3 0 3 6
0
0.5
1
1.5
2
Lκ
s
L
[m
/
s]
 
 
Linear model
Nonlinear model
Fig. 6.3 Nonlinear model adopted for the flame speed curvature dependence for s0L = 0.3.
High-order terms decrease the curvature’s influence on the flame speed.
showing that in this limit we recover the linear model (6.6). The saturation in equation (6.7)
for κ > 1/(2Le) is necessary to keep the flame speed single-valued with respect to the
curvature. Nevertheless, it is unlikely that curvature values will lie within this range because
premixed flames propagate normal to themselves and cusps form pointing towards the
products (Lieuwen, 2012). Large positive curvature values would indicate that a cusp
has formed pointing towards the reactants, which is unlikely to arise and can be checked
afterwards. Adopting the nonlinear model (6.7), the influence that the curvature has on the
flame speed is reduced for very wrinkled flame sheets.
6.2.2 Numerical implementation and mean flame shape
To numerically integrate equation (6.4) we define the following set of non-dimensional
variables:
t∗ = t
ure f
H
u∗ =
u
ure f
s∗L =
sL
ure f
z∗ =
z
H
r∗ = β
r
H
M= L
H
, (6.9)
where β ≡ H/D, H and D are the enclosure height and the diameter of the bluff-body
respectively (see Figure 6.1), and ure f is a reference speed chosen to be the one at the slot
midpoint. In contrast to the G-equation notation widespread in the literature, in which the
stretch parameter β represents the flame’s aspect ratio (Preetham et al., 2008; Kashinath
et al., 2014), here β does not have a particular physical meaning and is just a geometrical
parameter. This is because the mean flow is non-uniform and therefore we cannot derive an
analytical expression that relates the flame’s aspect ratio to the flame speed.
6.3 Forced acoustic response 143
r
∗
β
z
∗
0 0.5 1 1.4
0
0.5
1
1.5
2
2.5
3
Fig. 6.4 Mean flame shape contour G = 0 (red line) and mean flow vector plot for φ = 0.65,
M= 0.015.
Setting H = 80 mm, D = 25 mm, and ure f = 10 m/s, we fully determine the non-
dimensional variables in (6.9). The numerical method we use to integrate equation (6.4) was
discussed in §1.3.
Figure 6.4 shows the G-field zero level set obtained by setting the acoustic fluctuations
u′ac = 0 in equation (6.1), thus solving the stationary problem:
βur∗
∂G
∂ r∗
+uz∗
∂G
∂ z∗
= s∗L(κ)
√(
β
∂G
∂ r∗
)2
+
(
∂G
∂ z∗
)2
. (6.10)
The steady flame shape compares favourably with the shapes that have been computed
numerically by URANS simulations (Armitage et al., 2006; Ruan et al., 2013). This steady
solution is used as an initial condition in the following analysis.
6.3 Forced acoustic response
For the enclosure length we have considered (H = 80 mm), the system is thermoacoustically
stable. Consequently we investigate the forced (rather than self-excited) response of the
system to axial flow oscillations. This will allow us to compare the unsteady flame struc-
tures and heat release response predicted by the low-order model with the ones observed
experiments and URANS simulations.
We force the system by imposing velocity fluctuations with amplitudes in the range
|u′| ∈ [0.1 , 3.0] m/s and frequencies in the range f ∈ [40 , 500] Hz at the inlet of the domain,
144 Heat release response to forced flow oscillations of a low-order modelled combustor
(a) ϕcycle = π/6 (b) ϕcycle = 3π/6 (c) ϕcycle = 5π/6
Fig. 6.5 Comparison between (left) URANS and (right) low-order model flame shapes and
flow fields over a forcing cycle with St = 1.28, A = 0.1.
(a) ϕcycle = π/6 (b) ϕcycle = 3π/6 (c) ϕcycle = 5π/6
Fig. 6.6 Comparison between (left) URANS and (right) low-order model flame shapes and
flow fields over a forcing cycle with St = 1.28, A = 0.25.
i.e., by setting the boundary condition of equation (6.1) to
u′ac
∗
(t∗) = Asin(2π St t∗) , (6.11)
where we have defined the non-dimensional amplitude of forcing A = u′/ure f , and the
Strouhal number St = f H/ure f .
We numerically integrate the system forward in time for a time sufficient to reach the
steady-state response and compute a few limit cycles on it.
6.3 Forced acoustic response 145
6.3.1 Qualitative comparison
As our model is low-order, we do not expect a quantitative agreement with the experimental
results. We are nevertheless interested in a qualitative comparison, in order to understand
whether the model is capturing the main physical features of the system.
We focus the discussion on the two cases forced at a frequency St = 1.28 ( f = 160 Hz)
and amplitudes A = 0.1,0.25. In Figures 6.5 and 6.6, we compare our instantaneous flame
fronts and velocity vector fields with those of Armitage et al. (2006)’s URANS simulations1
at three phase angles over a forced cycle. A comparison with the simulations is easier,
because we can compare the flame front G = 0 from our simulations with the contour lines
of URANS snapshots corresponding to the c = 0.5 isocontour of the progress variable c,
which is a good indicator of the flame front position. Experimental results only provide OH
chemiluminescence fields, which are difficult to compare with the G-field, because the latter
has no physical meaning away from the flame surface. An indirect comparison with the
experiments can be found in Armitage et al. (2006).
At small forcing amplitudes (Figure 6.5), the intensity of the vortices is fairly small, and
the flame tends to be long, undergoing a severe stretching. This eventually creates a large
pinch-off which detaches from the flame far downstream and is then advected while burning;
this large pocket of reactants does not burn completely before exiting the computational
domain, meaning that part of the heat release is not accounted for. However, the latter
problem is found also in DNS simulations and in the experiments, where the domain of
computation of heat release is limited by the camera window for the OH chemiluminescence
measurements.
At higher amplitudes (Figure 6.6), the flame structure over a cycle is very different. As
the strength of the vortices is higher, the flame pinches off at a shorter axial distance, and
as a consequence the flame is shorter on average. Also, the roll up of the vortices causes
the formation of a characteristic mushroom-shape at the top of the flame, which we capture
well. On the other hand, there are a few features that we cannot describe, such as the flame’s
impingement at the wall, which is responsible for some differences in the flame front shapes.
Also, we note that the flames evaluated with the low-order model tend to be shorter than
URANS flames. This is a common feature of all our simulations, and is probably due to the
fact that we are not modelling gas expansion across the flame. The flame average length
could be adjusted by varying the unstretched flame speed s0L or the equivalence ratio as in
equation (6.5). Also, we recall that the flow field model has to be valid only close to the
flame surface, and no comparison should be sought far away from the G = 0 lines.
1Reprinted from Armitage et al. (2006), with permission from Elsevier.
146 Heat release response to forced flow oscillations of a low-order modelled combustor
0 pi/2 pi 3pi/2 2pi
−0.2
−0.1
0
0.1
0.2
ϕ
 
 
1)
2)
3)
u’ *
ac
q’
1) 2) 3)
Fig. 6.7 Heat release response over a forced cycle with St = 1.28 and A = 0.1. Flame front
snapshots corresponding to key moments in the heat release response.
Although improvements are possible, we find that the low-order flame-flow interaction
model gives a reasonable description of the main kinematic features of the flame front, and
it represents a significant improvement in the description of this type of flames respect to
previous models involving the G-equations, such as the one presented in Graham (2012).
6.3.2 Heat release response
The heat release response is computed by calculating the amount of fuel burned at every
instant in the domain, by
Q = 2πhr
∫ R
0
∫ H
0
ρ(r,z)sL(κ)
√(
∂G
∂ r
)2
+
(
∂G
∂ z
)2
δ (G)r dr dz, (6.12)
where δ is the Dirac delta function. The non-dimensional heat release fluctuations are defined
by q′ = (Q−Q)/Q, where Q is the heat release averaged over a forcing cycle.
The integrated heat release over a forced cycle with St = 1.28 and A = 0.1 is shown in
Figure 6.7. The heat release signal is strongly non-harmonic, meaning that it is a strongly
nonlinear function of the imposed velocity. The amplitude of heat release oscillations is
6.3 Forced acoustic response 147
greater than the amplitude of the forcing, meaning that the gain of the system exceeds 1, and
the two oscillations are about π/2 out of phase.
A few key points of the heat release response have been highlighted in Figure 6.7. Point
1) corresponds to the absolute maximum of the heat release response, and is one of the cusps
in the signal. To gain insight into the physical causes of the appearance of sharp peaks, we
have plotted in the corresponding bottom panel of Figure 6.7 the flame front at the same
instant. The main body of the flame is quite stretched, and there is a pocket of fuel, which
had been released during the previous cycle, that is exiting the domain. On closer inspection,
one can see that, at the instant under consideration, this pocket of fuel is breaking into two
parts, i.e., the flame has just pinched off. A pinch-off creates a discontinuity in the flame
shape, and the local flame curvature close to the pinch-off location suddenly becomes highly
negative. Thus, according to equation (6.7), the local flame speed becomes higher. This has
two effects:
(i) from equation (6.12) we see that the flame speed directly enters into the evaluation of
heat release fluctuations. Therefore, it is expected that the moment at which the flame
pinches off corresponds to an instantaneously high heat release response;
(ii) at the same time, a higher flame speed increases the rate of kinematic restoration
(kinematic restoration is the process that smooths out the flame’s wrinkles, and it
is due to the fact that the flame propagates normal to itself (Lieuwen, 2012)). As a
consequence, the cusps formed on the flame front are smoothed out quickly, decreasing
the magnitude of curvature corrections and destroying flame surface area. Because
both these effects have a negative impact on the total heat released by the flame, a rapid
decrease of q′ is expected after a pinch-off.
This is consistent with our observations. The same features just discussed are observed
also in the second peak of the heat release time series, at point 2), where the main body
of the flame undergoes a pinch-off. Finally, point 3) is the absolute minimum of q′, as it
corresponds to a moment in which the flame is short, and cusps on the flame surface have
been smoothed out by kinematic restoration. From this instant on, the flame will elongate,
increasing flame surface and thus heat release, until point 1) is reached again and the cycle
restarts.
These arguments suggest that, for this particular flame structure, the formation of pinch-
offs has an important role in the heat release response, because it induces high-amplitude
fluctuations and the formation of cusps in the time-signal, which enhance nonlinear effects.
Note that this effect cannot be observed in the simpler configuration of a conical flame (Waugh
et al., 2014; Orchini et al., 2015). For conical flames, pinch-offs occur at the centerline
148 Heat release response to forced flow oscillations of a low-order modelled combustor
0 pi/2 pi 3pi/2 2pi
−0.1
−0.05
0
0.05
0.1
0.15
(a)
ϕ
 
 
u’ *
ac q’ q1
0 250 500 750 1000
10−15
10−10
10−5
100
(b)
f [Hz]
 
 
PSD u’ *
ac PSD q’
Fig. 6.8 (a): heat release response over a forced cycle with St = 1.28 and A = 0.01. The
contribution of the first harmonic is highlighted. (b): PSD of velocity and heat release
fluctuations.
r = 0: because the flame is axisymmetric, the contribution of a flame element to the heat
release is proportional to the radial distance (see equation (6.12)), and even if the local flame
structure is highly wrinkled on the pinch-offs, its weight on the evaluation of the heat release
is negligible. On the other hand, for the flame under consideration in this chapter, pinch-offs
occur at a large radial distances, and provide large contributions to the integration of total
heat release.
Flame Describing Function
Having integrated the forced response of heat release to velocity fluctuations, we can evaluate
the Flame Describing Function (FDF) for this flame-flow configuration. The FDF is evaluated
by extracting the heat released at the forcing frequency q1, neglecting higher harmonics, and
is calculated by:
FDF(St,A) = F1 (St,A)eiφ1(St,A), (6.13)
where i is the imaginary unit, F1 (St,A)≡
∣∣∣ qˆ1(St,A)A ∣∣∣ and φ1 (St,A) are respectively the gain
and phase of the heat release first harmonic component to inlet velocity fluctuations.
Figure 6.8 (a) shows the heat release signal (and its first harmonic component) when
the system is forced at 160 Hz with the smallest amplitude we have considered, A = 0.01:
although the forcing amplitude is small, it is sufficient to induce the formation of pinch-
offs, which is a nonlinear effect. This causes heat release fluctuations to have a very large
amplitude, and the system’s gain exceeds 1 by a large amount. Note that, in experiments and
URANS simulations, the system response is linear for this forcing amplitude; this suggests
6.3 Forced acoustic response 149
0 0.1 0.2
0
2
4
6
8
A
F
1
(S
t,
A
)
f = 60 Hz
f = 160 Hz
f = 320 Hz
fexp = 40 Hz
fexp = 160 Hz
fexp = 310 Hz
0 0.1 0.2
−4pi
−3pi
−2pi
−pi
0
A
φ
1
(S
t,
A
)
Fig. 6.9 FDF gain and phase as a function of the forcing amplitude at the frequencies
St = 0.48, 1.28, 2.56. Thick lines correspond to the low-order model results, circles to
experimental measurements.
that the low-order model is too nonlinear, and corrections should be made to prevent the
formation of pinch-offs at small forcing amplitudes. The presence of cusps in the heat release
signals causes higher harmonics to have large contributions. The PSD plot of heat release
fluctuations in Figure 6.8 (b) shows how strong the contribution of higher harmonics is
even at this small forcing amplitude. Thus, if we want to use this FDF in a thermoacoustic
feedback loop, we must assume that higher harmonics in the heat release are filtered out by
the acoustic transfer function, which has to act as a strong low-pass filter for the feedback
loop analysis to work (this assumption is not required if one uses time-domain techniques
such as continuation analysis (Waugh et al., 2014; Orchini et al., 2015)).
Figure 6.9 shows the FDF amplitude dependence for three forcing frequencies. For all
the frequencies we have considered, the gain shows an overall monotonical decay with the
amplitude. At small amplitudes the gain exceeds the unitary value for all the frequencies
shown. This is expected (to some extent) because of the similarity between this flame shape
and V shaped flames (Schuller et al., 2003; Durox et al., 2009). In the linear limit, we find
that the gain increases with the forcing frequency, reaches a maximum value at a forcing
of 160 Hz, and then decreases for higher frequencies; the same feature was observed in
experimental and URANS investigations (Balachandran, 2005; Balachandran et al., 2005;
Armitage et al., 2006; Ruan et al., 2013). By linear limit we mean the smallest forcing
amplitude we have considered, A= 0.01. Nevertheless, as we have shown that pinch-offs can
form at this amplitude, inducing nonlinear effects, the behaviour of the system in the linear
limit is not properly characterised, and simulations with even smaller forcing amplitudes
need to be performed, or the contribution of pinch-offs on the heat release has to be reduced.
150 Heat release response to forced flow oscillations of a low-order modelled combustor
We note that the gain values we find are much higher than those of the real system,
especially in the low-amplitude limit. The gain reaches a maximum value of about 8, against
the maximum value of 3 found in experiments at the same frequency. Excluding the influence
of pinch-offs on the heat release that we have already discussed, a second possible explanation
for such a strong overestimation of the gain can be found in the lack of a turbulence model
in the underlying flow field in our equations. Indeed, in Hemchandra et al. (2011); Shin &
Lieuwen (2013) it has been shown that, because of nonlinearities in the governing equations,
turbulent fluctuations modelled as stochastic fluctuations couple with the forced oscillations
and flame dynamics, affecting the kinematic restoration process, hence the formation of
cusps, the rate at which flames wrinkles are smoothed out, and the evolution of the heat
release. Lastly, we are also neglecting heat losses through the combustor wall in the outer
recirculation zone. Because the temperature here is lower, the combustion process is less
efficient, as was observed experimentally by Balachandran et al. (2005), via OH Planar Laser
Induced Fluorescence (PLIF) (see Figure fig:Comparebalach).
6.4 Convection speed
In order to have a qualitatively reasonable comparison between URANS and low-order
simulations in Figures 6.5 and 6.6, we have only looked at the flame’s structures. As it
turns out, this results in a disagreement between the velocity phases of the images we have
compared. For the URANS simulations, the starting point of the limit cycle corresponds to
a minimum in the velocity fluctuations at the slot; we denote this condition as ϕ0min. For the
low-order simulations, we find that the phase that allows for a good comparison of flame
fronts does not correspond to ϕ0min, but it has a shift ∆ϕ ≈ π/4 rad.
This shift is likely to be caused by our choice for the convection speed of velocity
fluctuations. Indeed, several studies have shown that this parameter plays a key role in the
correct description of the dynamics (Preetham et al., 2008; Kashinath et al., 2013b). In
the advection equation (6.1) we have chosen the ratio between the mean and convection
velocities to be K = 1.5. This was based on DNS simulations performed by Kashinath et al.
(2013b), which predicted a value of K in the range [1.1 , 2.0]; values in this range have
already been used in other studies (Waugh et al., 2014; Kashinath et al., 2014; Orchini et al.,
2015). However, the latter results were obtained for a laminar, conical flame configuration,
and it is not surprising that the value of the convection speed may significantly vary for a
bluff-body stabilised, turbulent flame. We nevertheless used laminar flame results because of
the lack of information on the convection speed for the dump combustor under investigation
when we started this analysis. More recently, new experiments have been performed on the
6.4 Convection speed 151
0.5 1 1.5
0
0.5
1
1.5
K
G
ai
n
0.5 1 1.5
0.5
1
1.5
2
K
Ph
as
e 
  [r
ad
]
Fig. 6.10 Effect of the convection speed on the heat release gain and phase response. Forcing
amplitude and frequency have been fixed to A = 0.25 and St = 1.28 respectively.
same combustor by Kypraiou et al. (2015). Applying Proper Orthogonal Decomposition
(POD) analysis on phase-averaged OH∗ chemiluminescence images and computing the Power
Spectral Density (PSD) of the POD time coefficients on self-excited thermoacoustic cycles,
they have estimated the ratio between the mean flow and the convection speeds to be in the
range K = [0.59 , 0.72]. Unfortunately this information was not available to us when we
performed the numerical simulations for the calculation of the FDF. A lower value of K
means that fluctuations are advected downstream with a velocity that on average is higher
than that of the mean flow, whereas in our calculations we have assumed them to move more
slowly. A smaller convection speed corresponds to a larger wavelength between two vortices
on the same side of the flame. This has a direct impact on the phase between the shedding
of the vortices and the flame structure over time, and thus on the formation of pinch-offs
and the heat release phase with respect to velocity fluctuations. It is worth mentioning that
the convection speed is in general a function of the frequency (as found by Kashinath et al.
(2013b)), and the analysis performed by Kypraiou et al. (2015) is not exhaustive, as it is
performed on self-excited oscillations, which tend to have a frequency of about 350 Hz, close
to the first resonant frequency of the burner.
Having gained knowledge of experimentally determined values for the convection speed,
we have performed some numerical simulations fixing the amplitude and frequency of inlet
oscillations, and varying the parameter K in the range [0.4,1.5] in steps of ∆K = 0.1. To
quantify the impact of K on the dynamics of the flame, we show in Figure 6.10 the gain and
phase dependence on K of the FDF (defined as in equation (6.13)) evaluated at A = 0.25 and
f = 160 Hz. The phase follows a clear path: it reaches a maximum when the convection
speed is K = 1 and monotonically decreases for other values of K. Looking at the phase
152 Heat release response to forced flow oscillations of a low-order modelled combustor
values for K = 1.5 (our simulations), and K = 0.6 (experimentally determined), we observe
that we can correct the phase of heat release fluctuations of about π/6 rad. On the other hand,
the gain has an oscillatory dependence with respect to the convection speed, and the system’s
amplitude response is not so sensitive to variations in K. This means that convection speed
variations are not enough to explain the strong gain overestimation we have observed.
6.5 Conclusions
In this chapter we have developed a low-order model to investigate the forced response of a
bluff-body stabilised dump combustor, with a view to perform low-order analysis on self-
excited oscillations in this combustor in the future. The flame front has been described by the
kinematic G-equation evolving in a non-uniform flow field, which has been computed with
URANS simulations. We have attempted to extend the classical travelling wave model that
describes the interaction between conical flames and flows to this configuration. The model
has been developed by analysing experimentally determined cold flow smoke visualizations
of the forced system.
The resulting forced dynamics has been investigated by examining the system’s response
over a cycle: although a quantitative comparison cannot be performed because of fundamental
differences between the systems, an acceptable qualitative agreement between URANS and
low-order flame shapes is found, meaning that our model captures the main kinematic
features of the flame surface dynamics, such as the formation of distorted mushroom-shaped
fronts and pinch-offs. We have evaluated the Flame Describing Function (FDF) to inlet
velocity fluctuations, and shown its amplitude dependence for three forcing frequencies. The
dependence of the gain on the amplitude and frequency qualitatively compares favourably
with experimental results, but we overestimate the system’s gain by a large amount. This can
have two causes. First, we find that pinch-offs form even at fairly small forcing amplitudes
for this flame’s structure, and they have a major role in the description of the flame and
heat release dynamics. In particular, because they create cuspy flame fronts, they lead to
large maximum values in the heat release, followed by a rapid destruction of flame surface.
This occurs even if we have adopted a nonlinear model for the flame speed dependence
on the curvature, which reduces the importance of very wrinkled flame front portions on
the flame dynamics and heat release evaluation. As a consequence, the amplitude of heat
release fluctuations is enhanced, and peaks in its signal are created, which in turn augments
the heat release nonlinear response. This is a characteristic of this flame shape, and cannot
be observed on conical flames, where pinch-offs occur at the centerline and have a very
small influence on the heat release evaluation. The second cause of the gain overestimate is
6.5 Conclusions 153
probably due to the lack of a turbulent model for the flow field, which has a non-negligible
impact on the flame and heat release dynamics.
The level of reliability of the low-order model can be improved. The convection speed
of velocity perturbations can be varied according to the latest experimental results on the
experimental apparatus. It was shown that adjusting the convection speed to experimentally
determined values has a strong effect on the system response. In particular, it improves the
agreement between the phases of heat release and velocity fluctuations, but cannot address
the overestimation of the gain. The impact of pinch-offs on the heat release response can be
reduced by varying the value of the Markstein length. Heat losses in the outer recirculation
zone can be included using temperature profiles. Finally, a stochastic component can be
added to the flow field to emulate turbulence effects. The latter two changes should reduce the
heat release gain response to velocity fluctuations. The qualitative agreement obtained so far
is encouraging for further research on this model. If further modifications prove satisfactory,
the flame-flow model could then be used as a reliable, low-cost tool that can be coupled with
an acoustic solver in order to obtain a complete low-order thermoacoustic network.

Chapter 7
Modelling of equivalence ratio
fluctuations
In this chapter, an alternative model for the response of bluff-body stabilised flames is
considered. An imperfectly premixed flame is considered here. Rather than focusing
on the flame-flow interaction, we accurately model the convection of equivalence ratio
fluctuations, which dominate the heat release response. An FDF is evaluated and compared
with experiments, showing a reasonable quantitative agreement. The FDF calculated is
approximated with a sum of time delays. It is shown that the sum of time delays provides
an alternative point of view to the FDF results, which can help to understand the physical
mechanisms dominating the heat release response. The FDF is embedded in an acoustic
network and the harmonic balance method is applied to the system.
Part of the content of this chapter has been published in Semlitsch et al. (2016).
7.1 Introduction
In the previous chapter, we have seen how a realistic flow field model could be included in
the G-equation framework, as an attempt to describe the flame and heat release dynamics
in a bluff-body stabilised flame combustor with low-order models. Although it was found
that considering a realistic flow field yields reasonable results for the flame structure, the
heat release response largely overestimates the experimental one, because of the crude
assumption that the heat release is proportional to the flame surface area. Although this could
be improved, for example by considering turbulent flame speeds and shear layer effects on
the combustion processes, the model would become computationally more expensive, in
156 Modelling of equivalence ratio fluctuations
contrast with the idea lying behind the G-equation as a quick tool to estimate the heat release
response.
The latter is of interest to industry, as it would provide a tool able to quickly assess the
stability of a given combustor, and estimate the potentially dangerous frequencies and the
amplitude of the oscillations. This was one of the goals of Rolls-Royce’s AMEL (Advanced
Methods for prediction of Lean burn combustor unsteady phenomena) project, which is part
of European research Clean Sky. This motivates the modelling and analysis of this chapter.
Although I was not directly part of this project, I have developed the G-equation numerical
framework that was used into it, and have actively collaborated with Prof. Ann Dowling, Prof.
Matthew Juniper, and Dr. Bernhard Semlitsch (who ran some of the numerical simulations
presented here) to improve the model and couple it with LOTAN. I also acknowledge
discussions with Dr. Simon Stow, particularly for the suggestion of approximating an FDF
with a sum of time delays.
To obtain a low-order model that can predict thermoacoustic instabilities with bluff-body
stabilised flames, a simpler flame model than the one of the previous chapter is considered.
The main idea, which consists in modelling the flow field with a potential flow, is based on the
analysis performed by Dr. Owen Graham during his PhD (Graham, 2012). This model was
shown to predict heat release dynamics comparable with that measured in the experiments.
However, here we treat the G-equation in its fully nonlinear implicit form, whereas Graham
tracked the two sides of the flame separately as single-valued functions, which were merged
as a post-processing step. With such an approach, it is not straightforward to detect and
track pinch-offs, and curvature effects cannot be applied at the flame tip, where they have the
largest contribution. These issues are naturally solved by adopting an implicit formulation.
In this analysis, we will consider the effect of equivalence ratio fluctuations caused by
velocity fluctuations at the injection location. The stiff injector model discussed by Polifke &
Lawn (2007) is adopted. We perform calculations with and without curvature corrections,
and compare the resulting FDFs. A sum of time delays model is used to fit the measured
FDFs onto analytic functions. Insight into the physics captured by the model can be gained
from the intensity of the response at the various time delays. We show, using the harmonic
balance, how the resulting frequency associated with the most unstable mode matches the
one observed in the experiments.
7.1.1 Mean flow model
Following the modelling approach proposed by Graham (2012), we adopt a potential flow
field generated by a spherical source to model the mean flow around the bluff-body. This is a
crude model, as it does not account for the presence of recirculation zones or the interaction
7.1 Introduction 157
r [mm]
z
[m
m
]
−30 −20 −10 0 10 20 30
0
5
10
15
20
25
30
35
40
45
50
Fig. 7.1 Comparison between the instantaneous OH Planar Laser Induced Fluorescence
(PLIF) of an unforced flame with φ = 0.55 from Balachandran (2005) and the steady G-
equation model (red line). The parameters L and k0 are tuned to match the experimental
results at the flame base.
between the flame and the walls. However, the results obtained by Graham (2012) with this
flow field model showed a qualitative good comparison with the experimental results. The
velocity potential P for a spherical source in cylindrical coordinates (rˆ, θˆ , zˆ) is:
P =
−M˙√
r2+(z+L)2
, (7.1)
where M˙ is the source flow rate, and L is the axial distance between the source and the origin.
The mean velocity field is:
u = ∇P = (ur,uθ ,uz) =
(
M˙r
[r2+(z+L)2]3/2
,0,
M˙(L+ z)
[r2+(z+L)2]3/2
)
. (7.2)
The coefficient M˙ is chosen so that the absolute value of the velocity measured at a reference
point matches the experiments. The reference velocity (ure f = 10 m/s) is measured at the
centre of the burner slit, located at r = (rin+rout)2 , z = zb = 0, where rin = 12.5mm, rout =
17.5mm are the inner and outer radial distances of the burner slit, and zb is the burner axial
158 Modelling of equivalence ratio fluctuations
location (see Figure 7.1). Thus we have:
M˙ = ure f
[(
rin+ rout
2
)2
+L2
]
. (7.3)
The value of L is adjusted so that the angle that the steady flame forms with the inlet plane
matches the one observed experimentally (see Figure 7.1). A good comparison is observed
when L = 40mm, in agreement with with the analysis of Graham (2012).
7.1.2 Velocity and equivalence ratio perturbations
If the thermoacoustic system is unstable or acoustically forced, acoustic velocity and pressure
fluctuations will arise in the combustor. These oscillations will directly influence the flame
response, but will also cause modulation of the velocity at the fuel injection location. We
model the acoustic velocity perturbations as a pulsation of the spherical source, so that the
total velocity can be written as:
u ≡ u (1+u′ac(t)) , (7.4)
where u′ac(t) is the instantaneous, non-dimensional acoustic velocity. Given the naïve
modelling of the flow field for this analysis, no further modelling of the unsteady flow field
(e.g., travelling waves) is introduced. This is also because we will consider fluctuations in
the equivalence ratio field, which have a stronger influence on the flame dynamics than the
direct velocity perturbations (Lieuwen, 2003; Cho & Lieuwen, 2005).
For the combustor geometry we are considering here, the fuel injection location is close to
the combustion chamber inlet, zin j =−55mm. Thus, fuel and air cannot mix uniformly before
reaching the flame. To model the equivalence ratio fluctuations, we adopt the stiff-injector
model proposed by Polifke & Lawn (2007). Here, one assumes that acoustic oscillations affect
only the air mass flow rate, and not the fuel one. Therefore the instantaneous equivalence
ratio at the injection point is given by:
φ(zin j, t)≡ m˙
st
air
m˙stf uel
m˙ f uel
m˙air
=
m˙stair
m˙stf uel
m˙ f uel
m˙air + m˙′air
=
m˙stair
m˙stf uel
m˙ f uel
m˙air
1+ m˙
′
air
m˙air
=
φ
1+u′ac(t)
, (7.5)
where overlines and primes denotes mean and fluctuating quantities, and the superscript st
indicates stoichiometric conditions. In the last equality, we have exploited the fact that the
distance between the injection point and the flame (the speakers, if the system is forced) is
7.1 Introduction 159
acoustically compact, so that the acoustic fluctuations generated by the flame (by the speakers)
directly influence the equivalence ratio fluctuations at the injector. These fluctuations are then
convected in the upstream duct, from the injection point to the flame inlet. We shall assume
that, in this region, a uniform flow advects the equivalence ratio fluctuations. Imposing
mass flux conservation in the sections upstream of the combustor inlet, one can calculate
the time needed for the equivalence ratio perturbations to travel from the injector to the
flame inlet (Graham, 2012). The total time is τc = τ1 + τ2, where τ1 = 9× 10−3 s and
τ2 = 1.6×10−3 s are the times needed for the equivalence ratio perturbations to be transported
from the injection point to the base of the bluff-body, and from the latter to the combustion
chamber inlet, respectively.
For forced simulations, where u′ac(t) = Asin(ωt), the convection process can be treated
as a time delay, so that the equivalence ratio at the combustion chamber inlet can be explicitly
written as:
φ(zb, t) =
φ
1+Asin(ω(t− τc)) . (7.6)
In self excited cases instead, the history of the acoustic velocity perturbations from time
t− τc is stored, so that the nonlinear equivalence ratio perturbations at the burner can be
calculated as:
φ(zb, t) =
φ
1+u′ac(t− τc)
. (7.7)
7.1.3 Flame dynamics
We assume that the flame is anchored to the bluff-body, and its dynamics is modelled through
the kinematic G-equation:
∂G
∂ t
+u ·G = sT |∇G|, (7.8)
where the turbulent flame velocity is given by:
sT = s0T (φ)(1−Lκ). (7.9)
Here s0T contains the turbulent flame speed dependence on the local equivalence ratio, and
curvature corrections on the flame speed are proportional to the Markstein length L. An
analytical model for a laminar flame speed that fits experimental data was provided by Abu-
Orf (1996); Cho & Lieuwen (2005), and reads:
s0L = k1φ
k2e−k3(φ−k4)
2
, (7.10)
160 Modelling of equivalence ratio fluctuations
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
φ
s0 T
[m
/s
]
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
φ
h
r
[M
J
/k
g]
Fig. 7.2 Equivalence ratio dependence of the flame properties. Left: turbulent flame speed s0T .
Right: enthalpy of reaction hr.
with k1 = 1.322m/s, k2 = 3.110, k3 = 1.723 and k4 = 0.362. As in Graham (2012), we then
model the (higher) turbulent flame speed as s0T = k0s
0
L (see Figure 7.2), i.e., we assume that
it has the same equivalence ratio dependence as in the laminar case. The scaling factor
k0 = 1.513 is obtained by matching the length of the experimental steady flame shape with
the numerical one (see Figure 7.1).
Lastly, the total heat release rate in the domain D is then given by:
Q = 2πρ
∫
D
hr(φ)sT (φ)|∇G|δ (G)r dr dz. (7.11)
This expression accounts for the fact that the flame speed and enthalpy of reaction are
functions of the local equivalence ratio, which is not uniformly distributed in space. The
enthalpy of reaction, shown in Figure 7.2, is assumed to increase with the intensity of
the equivalence ratio and saturate at stoichiometric conditions, φ = 1, according to the
expression:
hr =
3.2×106min(φ ,1)
1+0.067φ
J/kg. (7.12)
7.2 FDF calculation and comparison with the experiments
In this section, the bluff-body stabilised flame model is forced harmonically over a wide
range of amplitudes and frequencies, and an FDF is extracted from the simulations results.
We perform the calculations for a flame with no curvature correction on the flame speed
(Markstein numberM≡L/H = 0, where H = 80 mm is a geometrical characteristic length),
and for a configuration with M= 0.005. Although the steady flame shape is not affected
7.2 FDF calculation and comparison with the experiments 161
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Gain
1
2
3
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Phase
−2
0
2
Fig. 7.3 Perfectly premixed FDF. The velocity model here is very simplistic and no relevant
saturation effects are observed even at high forcing amplitudes.
much by curvature corrections, the heat release response varies greatly at high frequencies.
This is because, as discussed in Chapter 6, curvature corrections on the flame speed enhance
the effect of kinematic restoration, whose contribution to the heat release dynamics is large
when the flame exhibits a pinch-off.
7.2.1 Perfectly premixed response
In the perfectly premixed case, no equivalence ratio fluctuations are considered, and the flame
oscillates only due to acoustically induced velocity perturbations according to equation (7.4),
where u′ac = Asin(ωt). The effect of the local flame’s curvature on the flame speed is
neglected here, by setting M= 0. Because of the simplistic velocity field model, the FDF
calculated here is not expected to compare well with the experimental results. However, the
perfectly premixed FDF is calculated in order to have a reference case against which the
imperfectly premixed case can be compared. This will help in understanding the role of
equivalence ratio fluctuations and flame speed curvature corrections on the FDF.
The perfectly premixed FDF is shown in Figure 7.3. One can see that the flame acts
as a low-pass filter as the gain tends to decrease with the frequency. The large gain at
low-frequencies is typical of V - and M-shaped flames (Durox et al., 2009). However, no
saturation mechanism is observed as the forcing amplitude is increased. This is because the
simplistic velocity field model adopted (a pulsating spherical source) causes the flame to
stretch and compress along the radial direction, without causing wrinkles on the flame surface.
Therefore, the relevant saturation mechanisms of the G-equation model, such as kinematic
162 Modelling of equivalence ratio fluctuations
restoration of flame front corrugations and pinch-offs, are not triggered, and the response
is quasi-linear even at large amplitudes. Lastly, one can observe that regions with high and
low gain (and similarly for the phase) alternate when the frequency is varied. The distance
between two peaks or troughs is about 80 Hz. This is due to constructive or destructive
interference between acoustic perturbations on the flame front and perturbations caused by
the anchoring boundary condition (Preetham et al., 2008). For simple flame configurations,
such as conical or V flames in uniform flow fields, the frequency gap between two consecutive
gain peak/troughs can be predicted with analytical results in the linear limit. This frequency
gap is related to the characteristic time that perturbations need to travel from the flame base
to the tip (Preetham et al., 2008; Blumenthal et al., 2013). However, the radial dependence
of the flow field makes it harder to find such a scaling analytically for the current model. In
the next subsection a method is discussed through which a physical understanding of the
frequency gap between peaks in the gain peaks can be obtained.
7.2.2 Sum of time delays approximation
As is common practice, the FDF was measured only for purely harmonic excitations, i.e.,
for values of the Laplace variable s≡ σ + iω having zero growth rate σ . The FDF is then
extended in the entire Laplace space via the analytic continuation technique1.
In Chapter 3, analytic continuation was achieved by fitting the measured FDF data onto
state-space models. This was used to estimate the growth rate of thermoacoustic oscillations
far from limit cycle or fixed point solutions, and build stability maps. Although this method
is correct, it provides little insight into the heat release dynamics mechanisms, which are
hidden in the values of the fitted parameters. An alternative approach that provides more
physical understanding of the flame dynamics is provided by approximating the FDF with a
sum of time delays (Macquisten et al., 2014; Subramanian et al., 2015).
A transfer function (here a slice of the FDF at a given amplitude level A), can be calculated
as the Laplace transform of the impulse response of the system, hA(t):
FDF(A,s) =
+∞∫
−∞
hA(t)e−st dt ≈
N−1
∑
n=0
hA(n∆τ)e−sn∆τ∆τ, (7.13)
where the integral has been approximated with a Riemann sum, and we have exploited the
fact that the system does not respond at times t < 0. ∆τ needs to be sufficiently small for the
approximation to work well, and the largest time considered in the sum, tmax = (N−1)∆τ ,
1This assumes that the FDF is an analytic function, i.e., that it is differentiable with respect to the complex
variable s at all points of interest.
7.2 FDF calculation and comparison with the experiments 163
has to be large enough for the impulse response to have decayed. By defining the coefficients
k(n∆τ,A)≡ hA(n∆τ)∆τ, (7.14)
one can see that the FDF can be approximated as a superposition of time delay responses,
each with a different time delay, τ = n∆τ , and a specific coefficient, k:
FDF(A,s)≈
N−1
∑
n=0
k(n∆τ,A)e−sn∆τ . (7.15)
In order to obtain the coefficients k, we can make use of the information collected when
measuring the FDF at the purely imaginary values of the Laplace variable s = iω . Indeed,
the inverse Fourier transform of FDF(iω,A) yields the impulse response hA(t), so we have:
hA(n∆τ) =
1
2π
∞∫
−∞
FDF(iω,A)eiωn∆τ dω =
1
π
∞∫
0
Re[FDF(iω,A)eiωn∆τ ]dω. (7.16)
In the latter equality we have exploited the fact that, because the impulse response is a real
valued object, the FDF satisfies the property FDF(−iω,A) = FDF∗(iω,A), where ∗ denotes
complex conjugation.
Combining equations (7.15), (7.14) and (7.16), an analytic continuation of the FDF is
obtained, which can be exploited when its coupling with a linear acoustic network response
is considered.
Because (7.15) is a sum of time delayed responses, the amplitudes of the coefficients
k are determined by the influence of a perturbation on the heat release response after a
time τ . The intensity of the coefficients k(τ,A) is shown in Figure 7.4 as a function of the
forcing amplitude and the time delay. The values of the coefficients k are approximately
independent of the amplitude level, because no saturation mechanism is captured by this
model, as discussed earlier. The dependence of k upon the time delay, however, has a clear
physical explanation. Acoustic perturbations directly influence the flame dynamics at time
t = 0, therefore we have k(0,A) ≥ 0. These perturbation propagate along the flame front
from the base to the tip. Because the geometry we are considering is axisymmetric and the
flame has an M-shape, the effect of downstream perturbations on the heat release is larger,
as more flame surface area is present. This explains why the value of k increases with τ ,
until a critical time delay is reached. After that, the coefficients at larger times drop to zero.
This is because the flame tip has been reached, and after this critical time delay perturbations
generated at time t = 0 can no longer affect the heat release dynamics.
164 Modelling of equivalence ratio fluctuations
A
τ
[s
]
 
 
τf = 1.24 · 10
−2 s
0 0.2 0.4
0
0.01
0.02
0.03
0.04
k
0
1
2
3
x 10−4
Fig. 7.4 Weights of the sum of time delays approximation for the perfectly premixed flame.
The time delay at which the coefficients k have the largest magnitude corresponds to the time
τ f that an acoustically induced perturbation needs to travel from the flame base to the tip.
The time needed for an acoustic perturbation to travel along the flame front can be
roughly estimated as the convective time needed for a perturbation to be transported by the
mean flow from the centre of the burner slit (at a distance ρ1 = 42.72 mm, where ρ is a
the radial distance in spherical coordinates centred in the source) to the steady flame tip (at
a distance ρ2 ≈ 90.14 mm, from Figure 7.1). This is an approximation because it ignores
the flame speed contribution to the total speed at which perturbations are convected along
the flame front. However, for the configuration we are considering, the mean flame speed
is s0T (φ = 0.55) = 0.29 m/s, which is much smaller than the smallest value that the radial
mean velocity assumes, u(ρ2) = M˙/ρ22 = 2.25 m/s, and this approximation should yield a
reasonable result. The convective time along the flame is:
τ f ≈
∫ ρ2
ρ1
1
u(ρ)
dρ =
∫ ρ2
ρ1
ρ2
M˙
dρ =
1
3M˙
(
ρ32 −ρ31
)
= 0.0120 (7.17)
which compares well with the time delay at which the largest k coefficient is observed in the
sum of time delays approximation (τ f = τ(kmax) = 0.0124, see Figure 7.5).
Furthermore, perturbations proportional to sin(ωt) and to sin[(ω+∆ω)t], will generate
the same type of constructive/destructive interference if the perturbations at the flame base
7.2 FDF calculation and comparison with the experiments 165
0 200 400 600
0
1
2
3
4
f [Hz]
|F
D
F
|
20 Hz
10
0 H
z
18
0 H
z
26
0 H
z
0 0.02 0.04
0
1
2
3
x 10−4
τ [s]
k
τ (kmax) = 0.01239
Fig. 7.5 Slice of the FDF gain (left) and of the sum of time delays coefficients (right) at
A = 0.05. The frequency gap between the FDF peaks is about 80 Hz, which is consistent
with the reciprocal of the characteristic time τ(kmax) that acoustically induced perturbations
take to travel along the flame front.
and tip are the same at every instant t, and in phase. This is achieved when:
∆ωτ f = 2πn =⇒ ∆ f = nτ f , n ∈ N
+. (7.18)
Therefore we can expect the frequency gap between gain peaks to be ∆ f = 1/τ f = 80.7 Hz,
which compares well with the results from the measured FDF. Note that, when multiple
sources of heat release disturbances are included, multiple waves, which in general propagate
at different speeds, affect the heat release dynamics. Although the various contributions may
be separated linearly (Lieuwen, 2003), nonlinear terms may trigger non-trivial interactions
between the waves, and a simple scaling as the one discussed above cannot expected to be
found in general (C´osic´ et al., 2014).
The discussion in this section shows how an analytic continuation of the FDF based on a
sum of time delays yields physical understanding into the heat release dynamics, which a
brute force fit onto a state-space model fails to capture. In the following sections, equivalence
ratio perturbations and curvature corrections are included, and their effect on the FDF and on
the coefficients of the sum of time delays approximation is discussed.
166 Modelling of equivalence ratio fluctuations
z
[m
m
]
ϕ =0◦
r [mm]
0 10 20 30
0
10
20
30
40
ϕ =49◦
r [mm]
0 10 20 30
ϕ =98◦
r [mm]
0 10 20 30
ϕ =147◦
r [mm]
0 10 20 30
ϕ =196◦
r [mm]
0 10 20 30
ϕ =245◦
r [mm]
0 10 20 30
ϕ =294◦
r [mm]
0 10 20 30
ϕ =343◦
r [mm]
0 10 20 30
 
 
φ
0.6 0.62 0.64 0.66 0.68 0.7 0.72
Fig. 7.6 Imperfectly premixed flame shape (red line) on top of the equivalence ratio field
over a forced cycle u′ac(t) = Asin(St t) with φ = 0.65, A = 0.1, St = 17.093. Snapshots at
eight instants over a cycle are plotted as a function of the angle ϕ = 360(t− t0)/T , where T
is the oscillation period. Where the equivalence ratio is higher, the flame burns faster and
eventually forms a pinch-off.
7.2.3 Imperfectly premixed response
We now consider the case in which fuel is injected close to the combustion chamber, so that
air and fuel have no time to mix perfectly together. The equivalence ratio field is transported
in the flame domain as a passive scalar, and obeys the advection equation:
∂φ
∂ t
+u ·∇φ = 0, (7.19)
where u is the pulsating flow field (7.4), and the equivalence ratio at the burner inlet is given
by the stiff-injector model (7.6). Curvature corrections to the flame speed are neglected for
now, and the system is forced harmonically at various amplitudes and frequencies.
The FDF extracted from the imperfectly premixed heat release rate response is shown
in Figure 7.7. Equivalence ratio fluctuations strongly vary the flame dynamics: where the
equivalence ratio fluctuations are higher, the flame burns faster and tends to close, as shown
in Figure 7.6. When the amplitude of the perturbations is high enough, this causes the flame
to pinch off. The formation of pinch-offs has a strong influence on the heat release time
series analysis, as discussed in Chapter 6. However, because here pinch-offs are always
found in regions in which the equivalence ratio has low values, and therefore the flame speed
is smaller, the influence of pinch-offs on the total heat release here is less relevant than in the
previous chapter. The saturation mechanism induced by the equivalence ratio fluctuations
are seen in Figure 7.7 as a reduction in the FDF’s gain with respect to the forcing amplitude.
A pattern can be observed in the location of the FDF’s gain minima (and maxima) as
the amplitude increases. The gap between the gain’s minima can be predicted with an
7.2 FDF calculation and comparison with the experiments 167
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Gain
0
2
4
6
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Phase
−2
0
2
Fig. 7.7 Imperfectly premixed FDF with M= 0. Equivalence ratio fluctuations cause the
flame to pinch off at large amplitudes, and the gain to saturate with the amplitude. However,
at low amplitudes the gain remains very high at all frequencies. The location of the FDF gain
minima calculated using the sum of time delay approximation is shown with dashed lines.
A
τ
[s
]
 
 
τc = 1.06 · 10
−2 s
τf = 1.24 · 10
−2 s
0 0.2 0.4
0
0.01
0.02
0.03
0.04
k
−1
0
1
2
3
4
x 10−3
Fig. 7.8 Weights of the sum of time delays approximation for the imperfectly premixed flame
with no curvature corrections. Equivalence ratio perturbations generated at the injection
location need to travel for a time τc = 1.06 ·10−2 before influencing the flame dynamics.
168 Modelling of equivalence ratio fluctuations
argument similar to that discussed in the previous section. The map of the sum of time
delays coefficients for the imperfectly premixed flame is shown in Figure 7.8. Its structure
is very different from the perfectly premixed one. Two characteristic time scales can be
distinguished: τc, the time that equivalence ratio perturbations need to be convected from the
injection location to the flame base, which does not vary with the forcing amplitude level,
and τ f , the time the perturbations need to travel along the flame surface to to tip, which
varies with the amplitude level. At times 0 < τ < τc, only acoustic perturbations influence
the flame dynamics, as in the perfectly premixed scenario. However, at time τc equivalence
ratio perturbations reach the flame and start contributing to the heat release response, until
time τc + τ f , when they reach the flame tip. Their contribution to the flame dynamics is
stronger than the acoustic perturbations (the magnitude of the coefficients is larger), because
they significantly alter the flame speed, and hence the flame shape and rate of heat release.
Also note, in Figure 7.8, that the sign of the coefficients k suddenly changes across τc. This
is because the effect of equivalence ratio perturbations at first order is given by:
φ(zb, t) =
φ
1+u′ac(t− τc)
≈−φ u′ac(t− τc), (7.20)
i.e., equivalence ratio perturbations are linearly proportional to (delayed) velocity fluctuations
perturbations, but have opposite sign.
Lastly, we also observe that the time τ f that perturbations need to travel along the flame
decays with the amplitude. This is because equivalence ratio fluctuations cause the flame to
pinch off (see Figure 7.6). The larger is the perturbation amplitude, the lower is the axial
location at which the flame pinches off, thus reducing the effective mean length of the flame
over a cycle. In the small amplitude limit, we recover the value of τ f calculated in the
previous section, because the flame does not pinch off in this limit, and the mean flame length
corresponds to the steady flame length. From equation (7.17), the mean flame length can be
estimated as:
L f (A)≈ ρ2(A)−ρ1 = 3
√
3M˙τ f (A)−ρ31 −ρ1 (7.21)
This provides a physically relevant length scale for the flame dynamics (as a function of the
forcing amplitude), which could be used in a non-dimensional scheme to scale the data. We
can also make use of the characteristic time responses of this flame to predict the frequency
gaps between two peaks/troughs in the FDF gain. Because equivalence ratio fluctuations
largely dominate the dynamics, it is reasonable to assume that the most important time scale
is τ f (A). With the same argument provided in the previous section, we can predict that the
frequency gap between gain’s minima or maxima is given by ∆ f (A) = 1τ f (A) . The dashed
7.2 FDF calculation and comparison with the experiments 169
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Gain
0
2
4
6
A
f
[H
z]
 
 
0 0.2 0.4
0
200
400
600
Phase
−2
0
2
Fig. 7.9 Imperfectly premixed FDF with M= 0.005. The gain at low forcing amplitudes
and large frequencies is greatly reduced because of kinematic restoration effects.
lines in Figure 7.7 have been plotted using this equation, and match well the FDF regions in
which the gain reaches minimum values.
Curvature effects
One of the weaknesses of the model presented so far is that the flame response at low
amplitudes has a large gain at all frequencies. Indeed, from Figure 7.7 one can see that the
gain in the A→ 0 limit is almost always larger than one.
This is not consistent with the experimental results, in which a gain larger than one is
measured at low-frequencies, but quickly drops at values less than unity as the frequency
is increased. In order to account for frequency dissipation effects, we include curvature
corrections to the flame speed. These corrections are known to damp high-frequency oscilla-
tions in flames modelled with the G-equation, because they enhance the effect of kinematic
restoration, which quickly smooths highly wrinkled flame sheets, thus reducing the flame
surface area (Lieuwen, 2003). It was also shown by Shin & Lieuwen (2013) that their
effect is analogous to that of turbulent flow effects (modelled with white noise) on the flame
dynamics.
The FDF of the imperfectly premixed flame with curvature corrections on the flame
speed is shown in Figure 7.9. The structure of the FDF with respect to the case with no
curvature corrections (Figure 7.7) is mostly unchanged, but the gain at high frequencies
in the low-amplitude limit has greatly decreased. Also the structure of the sum of time
delays’ coefficients, shown in Figure 7.10, is mostly unchanged, and the physics underlying
the response at the various time delays is the same as in the case with no curvature. The
170 Modelling of equivalence ratio fluctuations
A
τ
[s
]
 
 
0 0.2 0.4
0
0.01
0.02
0.03
0.04
k
−1
−0.5
0
0.5
1
x 10−3
Fig. 7.10 Weights of the sum of time delays approximation for the imperfectly premixed
flame with curvature corrections. The structure of the coefficients is very similar to the case
with no curvature corrections on the flame speed (see Figure 7.8), but the intensity of the
coefficients is decreased, because the FDF gain is now lower.
reduction in gain can be seen here as a reduction of the intensity of the coefficients k at small
amplitudes.
This FDF contains all the physical features we are aiming to model here, and will be the
one used in the next section for the modelling of thermoacoustic oscillations. A comparison
between the FDF evaluated with the current model, the model of Graham (2012), and
the experimental results of Balachandran (2005) is shown in Figure 7.11. A qualitatively
good agreement between the model and the experiments is observed at all frequencies and
amplitudes. Our model improves the results of Graham (2012) in that it is able to capture
better the increase in gain at low-frequencies. The differences between our results and those
of Graham (2012) can be justified by recalling that we are solving the G-equation in its fully
implicit form, whereas Graham (2012) tracked independently the two sides of the flame front
and merged them. This causes curvature effects to be stronger in our model (particularly at
the flame tip), and therefore the flame and heat release dynamics to vary.
7.3 Harmonic balance analysis
Having obtained a reasonable model for the heat release rate response of the imperfectly
premixed flame, we need to develop an acoustic model of the combustor geometry for
7.3 Harmonic balance analysis 171
0 100 200 300 400 500 600
0
2
4
|F
D
F
|
A = 0.1
 
 
Current analysis
Graham (2012) model
Experiments, Balachandran (2005)
0 100 200 300 400 500 600
0
2
4
|F
D
F
|
A = 0.2
0 100 200 300 400 500 600
0
2
4
f [Hz]
|F
D
F
|
A = 0.3
Fig. 7.11 Comparison between the FDF gains extracted from the current analysis (solid line),
Graham (2012) model (dashed line), and experimental results Balachandran (2005) at three
forcing amplitudes. The model captures the increase in gain at low-frequencies and shown
a qualitative good agreement for all frequencies and amplitudes. The phase behaviour is
reported and commented in Semlitsch et al. (2016).
investigating thermoacoustic oscillations. A detailed sketch of the acoustic network under
consideration is shown in Figure 7.12. The geometry is axisymmetric, but we consider only
acoustic waves propagating in the axial direction z. The inlet mean flow is chosen so that the
velocity at the burner slit corresponds to the experimental reference velocity, ure f = 10 m/s.
A temperature jump T2/T1 = 4 is imposed at the inlet of the combustion chamber. The outlet
is modelled as acoustically open, whereas the reflection coefficient of the inlet is chosen to
be R1 = 0.8.
In the experiments performed by Balachandran (2005), the setup in which the FDF was
measured had an enclosure duct of length Lduct = 80 mm. This length was chosen to minimise
the acoustic interference between the acoustically forced waves used to calculate the FDF,
and the natural acoustic response of the combustor. In this non-resonant configuration, the
combustor did not show self-sustained thermoacoustic oscillations. However, by increasing
the length of the enclosure duct to Lduct = 350 mm, the resonance of the combustor was
exploited to investigate self-excited oscillations. The latter is the scenario we will start
considering here.
172 Modelling of equivalence ratio fluctuations
z
r
∅ 70
∅ 35
∅ 25
∅ 8
z i
n
j
45
◦
fuel injection
air supply inlet
fuel injection
open outlet
8
0
5
0
2
0
0
5
0
4
7
3
.5
8
4
5
L
d
u
c
t
=
8
0
∅ 35
∅ 100
1
Fig. 7.12 Schematic of the acoustic network modelled with LOTAN. Dimensions are in mm.
The burner and and plenum dimensions are based on the experimental setup of Balachandran
(2005). The fuel injection location for the imperfectly premixed case is highlighted. The
length of the enclosure duct, Lduct , is varied. Image credit to Dr. B. Semlitsch.
The acoustic geometry of Figure 7.12 is implemented in LOTAN, and the transfer function
H – the response of velocity fluctuations just upstream of the flame with respect to heat
release excitations – is calculated and fitted onto a state-space model:
u′ac = H(s)q
′ =C(sI −A)−1B q′. (7.22)
The harmonic balance is then used to couple the FDF (expressed in terms of the sum of time
delays) and acoustic response. The harmonic balance dispersion relation FDF(A,s)H(s) = 1
is solved. Poles with a non-negative growth rate are shown in Figure 7.13. Five thermoa-
coustic modes are found to be linearly unstable: a low-frequency mode, with a frequency
of 65 Hz, 3 modes with frequencies between 290 and 360 Hz, and a mode at 480 Hz. This
is roughly consistent with the acoustic characterisation of the experimental setup carried
out by Balachandran (2005), in which acoustic peaks in the velocity response (in cold flow
conditions) were measured at about 30, 290 and 390 Hz.
7.3 Harmonic balance analysis 173
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
0.5
f [Hz]
A
 
 
σ
0
10
20
30
40
50
Fig. 7.13 Stability map of thermoacoustic oscillations in the configuration in which Lduct =
350 mm. Only solutions of the harmonic balance dispersion relation with non-negative
growth rates σ (grey scale) are shown. The mode with the linearly largest growth rate has a
frequency of 323 Hz.
Because multiple modes are found to be linearly unstable, in the fully nonlinear regime it
is possible that one mode dominates over the others – resulting in periodic oscillations with a
frequency close to one of the modes detected here – or that multiple modes contribute to the
oscillations – resulting in non-periodic oscillations (Moeck & Paschereit, 2012; Orchini &
Juniper, 2016a). In the experiments, it was found that the spectrum of self-excited oscillations
in this configuration contained only a dominant peak, with frequency 348 Hz. To interpret
the FDF analysis results, we may argue that the nonlinear interaction between the modes
suppresses the oscillations at all frequencies but one. If this frequency is chosen to be the
one with the largest growth rate in the linear regime, we should expect oscillations at 325
Hz. This is, however, just a speculation, as the transient data from the experiments are not
available in the literature. Also, we do not have information about the nonlinear coupling
between the modes, such as an FDIDF, as discussed in Chapter 4.
To test the accuracy of the model, we verify that, by decreasing the length of the enclosure
duct, the system becomes more stable. The growth rate maps for different values of the
enclosure duct length are shown in Figure 7.14. At Lduct = 300 mm no significant differences
are observed with respect to the Lduct = 350 mm case. At Lduct = 250 mm the modes at 60
and 480 Hz stabilise, but new unstable modes are observed at frequency 580 and 780 Hz.
Because these modes have high frequencies, it is possible that the acoustic model under
consideration does not damp high-frequency oscillations enough. Nonetheless, the mode at
174 Modelling of equivalence ratio fluctuations
 
 
σ
0 10 20 30 40
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
A
Lduct =300 mm
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
Lduct =250 mm
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
f [Hz]
A
Lduct =150 mm
0 200 400 600 800 1000
0
0.1
0.2
0.3
0.4
f [Hz]
Lduct =80 mm
Fig. 7.14 Stability map of thermoacoustic oscillations for various values of the enclosure
duct length Lduct . Consistently with the experiments, the system becomes more stable when
Lduct is decreased.
325 Hz still has the largest growth rate. At Lduct = 150 mm most of the modes have stabilised.
The mode at 325 Hz remains the most unstable mode, but its growth rate has largely decreased.
Lastly, all the thermoacoustic modes are stabilised when Lduct = 80 mm (the growth rate of
the only unstable mode is only marginally positive). Thus, the thermoacoustic characteristics
of the modelled network are consistent with the experimental results.
An interesting result of this analysis is that it predicts a low-frequency mode to be
thermoacoustically unstable when the length of the enclosure duct is long enough. This
low-frequency thermoacoustic mode originates from the fact that the configuration under
investigation has a low-frequency acoustic resonant mode, f ≈ 50 Hz. This was reported
in the experimental characterisation of the rig conducted by Balachandran (2005), and is
captured by our acoustic network model. The amplitude of the pressure modeshape at this
frequency as calculated by LOTAN is shown in Figure 7.15. Furthermore, the imperfectly
premixed FDF gain at low-frequencies is high (see Figs. 7.9 and 7.11). This makes it likely
that the thermoacoustic modes close to this frequency will satisfy the harmonic balance
criterion and be unstable.
7.3 Harmonic balance analysis 175
 
a
m
pl
itu
de
 o
f s
ta
tic
 p
re
ss
ur
e 
pe
rtu
rb
at
io
n 
(ar
bit
rar
y s
ca
le)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 7.15 Amplitude of pressure modeshape for the acoustic eigenmode at frequency 49
Hz and growth rate σ = −28.83 s−1. Configuration with Lduct = 80 mm. The mode has a
maximum at the inlet (approximately a closed boundary), and a zero at the outlet (open
boundary).
Because this mode has such a low-frequency, its wavelength is therefore high, with an
order of magnitude of 5 m. For this reason, it is likely that such waves also interact with the
environment surrounding the combustor in the real configuration. The reflection of acoustic
waves on the walls surrounding the combustor may play a crucial role in determining the
stability of this mode. This could be modelled with complex-valued reflection coefficients,
whose imaginary part (which supposedly quickly decays with the frequency) represents the
time needed for waves to be reflected from the room’s walls and travel back to the combustor
inlet/outlet.
We also note that, in the analogous low-order modelling of this thermoacoustic net-
work performed by Graham (2012), no unstable thermoacoustic modes were found at low-
frequencies. This was surprising to us because, by coupling the FDF contained in Graham
(2012) in our acoustic network, modes with a positive growth rate and a frequency of about
50 Hz were still observed. A good agreement was instead observed between modes with
f ≈ 340 Hz. We realised that, in the work of Graham (2012), the diameter of the combustion
chamber (70 mm), was mistakenly doubled (a radius of 70 mm was used, see for exam-
ple Figure 6.6 in Graham (2012)). This appears to be the only difference in the acoustic
modelling between the two analysis. This shows how the system stability, particularly at
low frequencies where the FDF gain is very high, is very sensitive to the acoustic network
geometry and, probably, reflection coefficients.
176 Modelling of equivalence ratio fluctuations
7.4 Conclusions
In this chapter, a potential flow field generated by a pulsating spherical source was con-
sidered in order to develop a low-order model for a bluff-body stabilised flame combustor.
Equivalence ratio fluctuations have been included in the flame model, by assuming that
acoustic velocity fluctuations modulate the inlet air mass flow rate, whereas the fuel mass
flow rate remains constant. Including curvature effects on the flame speed, a kinematic model
based on the G-equations was shown to reproduce reasonably well the flame heat release
response measured in experiments. The agreement is due to the fact that equivalence ratio
perturbations, which are convected by the flow along the flame, dominate the heat release
response. Thus, a detailed modelling of the coupling between the flame and the acoustic field
is not required.
It was shown how the measured FDF can be approximated as a sum of time delays. The
latter provides an alternative point of view on the flame response, which can be useful to
understand what mechanisms dominate the heat release response. In particular, it was shown
how the coefficients of the sum of time delays approximation can be used to separate acoustic
and equivalence ratio fluctuations effects, confirming that the latter have a stronger impact
on the flame dynamics. Also, they provide estimates for the relevant time and length scales
of the system. This information gives a better understanding on the FDF results, and could
be used to develop strategies to change the system in order to make it less susceptible to
thermoacoustic oscillations.
Lastly, by coupling the calculated FDF with an accurate linear acoustic model, multiple
thermoacoustic modes are found to have a positive growth rate. Consistently with the
experiments, we verified that the system becomes more stable when the length of the
enclosure duct is reduced, and that the mode with the largest growth rate has a frequency of
about 350 Hz.
This model provides a tool that can be used to quickly estimate the frequencies at which
a thermoacoustic system may be prone to instabilities. It improves the work of Graham
(2012) because it relies an a fully implicit G-equation formulation, which avoids problems in
tracking pinch-offs or including curvature effects on the flame speed. It has been used in the
context of Rolls-Royce AMEL project to investigate the susceptibility of the thermoacoustic
system with respect to geometrical (e.g., fuel injection location) and operating condition
(e.g., mean equivalence ratio value) parameters. These results have been partly published
in Semlitsch et al. (2016).
Chapter 8
Conclusions and future work
In this thesis, methods for the analysis of thermoacoustic systems have been developed.
In Chapter 2, linear stability and adjoint methods on a thermoacoustic system with a
premixed flame have been demonstrated for the first time. In particular, adjoint methods
were shown to be a quick and reliable tool to understand what physical mechanisms drive the
instabilities, and what parameters in the system can be changed to make it less susceptible
to thermoacoustic oscillations. The analysis, however, was applied to a simplistic system.
Even if a more complex network were to be considered, approximations would be introduced
in the thermoacoustic model in order to make it analytically treatable. The effect of such
approximations on the eigenvalues’ sensitivity is non-trivial, and needs to be assessed. A
series of experiments was recently performed in Cambridge by Dr. George Rigas and
Nicholas Jamieson to compare the actual shifts of a thermoacoustic system’s eigenvalues (a
Rijke tube) with those obtained with adjoint methods on a low-order model. They showed that
a good comparison is generally observed for the growth/decay rate shifts, but the frequency
shift was not predicted so well. The sensitivity information gathered with the experiments
can be used to improve the accuracy of the model, which can then be reused in different
thermoacoustic configurations to assess their stability/eigenvalues sensitivity. Future analysis
can extend the adjoint analysis to systems with premixed flames using more elaborate models
for the flame and heat release dynamics and to the more sophisticated question of optimizing
the shape of a burner or combustion chamber. Note that, however, when addressing this
question for a real application, one has to consider many factors that the manufacturer faces in
the design process, such as the system efficiency, structural stability, and boundary conditions
due to the connection between the combustion chamber and the compressor/turbine stages. It
is a big challenge to accurately formulate an optimization problem that accounts for all these
constraints. This research subject is currently being considered by José Aguilar.
178 Conclusions and future work
In Chapter 3, nonlinear methods were introduced to investigate the amplitude and na-
ture of saturated thermoacoustic oscillations. Time marching, Flame Describing Function
(FDF) and numerical continuation methods have been used. This analysis improved previous
nonlinear analysis on similar networks in that it considered more realistic acoustic configu-
rations. The results obtained with the various methods have been compared, highlighting
their strengths and weaknesses. This information was crucial for understanding what novel
methods had to be developed to improve nonlinear modelling of thermoacoustic systems.
In Chapter 4, the FDF method, which can predict only harmonic oscillations, was
extended to consider cases in which two frequencies simultaneously excite the flame. A
similar analysis was performed by Prof. Jonas Moeck in Berlin, but only static (frequency
independent) nonlinearities were considered there, which makes the analysis simpler. Real
flames have a dynamic (frequency dependent) nonlinear response, and the theory developed
here for this case is one of the main novelties of this thesis. The approximation of the
nonlinearity calculated, called the Flame Double Input Describing Function (FDIDF), can be
embedded in an acoustic network in a similar manner to an FDF. The dispersion relations
that determine the thermoacoustic system behaviour were derived, together with conditions
to assess the stability of saturated states. These saturated states can be either fixed points,
limit cycles or quasiperiodic oscillations. It was shown how, for limit cycles, the FDIDF
corrects the predictions on the system stability obtained with the FDF in some cases, because
it accounts for the nonlinear interaction between the modes, which is neglected by the
FDF. Although more expensive to obtain from experiments or numerical simulations, the
FDIDF contains more accurate information about the flame nonlinear response. A natural
follow-up for this analysis would be to measure an FDIDF from experiments, and verify its
accuracy. To overcome the cost issue, a low-cost procedure to only assess the stability of
limit cycle solutions using the FDIDF was proposed in this thesis. It could be exploited in
experiments at a cost roughly equal to that of two FDFs. This information can be used to
predict more accurately the stability of limit cycle solutions, the frequency of modes that
are linearly unstable around limit cycles, and can explain phenomena like mode switching.
The latter can be important information when designing feedback loop controllers/tuning
Helmholtz resonators to stabilise thermoacoustic oscillations, which need to be tuned at
specific frequencies.
In Chapter 5, a rigorous weakly nonlinear expansion of the equations governing ther-
moacoustic oscillations in a Rijke tube around a Hopf point was performed. This extended
previous weakly nonlinear analysis formulations of thermoacoustic oscillations, as it con-
sidered wave-based acoustics, which can be easily scaled to larger systems, and was not
limited to a single Galerkin-mode approximation. The latter was the state-of-the-art for
179
weakly nonlinear expansions of thermoacoustic systems, and it was shown here how a single
mode approximation can lead to inaccurate results. The formulation was general in that the
acoustic response was embedded in a state-space model, thus not relying on a particular
configuration. The nonlinear element considered here was an electrical heater, for which a
Wiener-Hammerstein model based on King’s law and Lighthill’s theory was considered. As a
side result, it was discussed how the nonlinear, unsteady King’s law model, typically adopted
in thermoacoustics to describe the nonlinear dynamics of the heat released by a hot-wire,
appears to be non-physical and does not agree with fully nonlinear CFD calculations per-
formed in TU Munich. This calls for research in this area, as a reliable nonlinear dynamical
model for the unsteady heat released by a wire is needed if quantitative comparisons with
experiments are sought. Although the weakly nonlinear analysis is strictly valid only close to
the bifurcation point, it was shown how performing a high-order expansion of the governing
equations, one can obtain accurate results on the amplitude, stability and frequency of limit
cycle solutions around the Hopf points. This framework could be extended in multiple ways
in future analyses. For example, one can consider a realistic model for a premixed flame as a
nonlinear element rather than a model for a heater, or can use the weakly nonlinear frame-
work to investigate the nonlinear interaction between multiple unstable modes. Deriving
the weakly nonlinear equations can be tedious at high-orders or for complex nonlinearities,
but the process can be easily automatised with symbolic solvers. The main advantage of
the method is its numerical cost: only matrix multiplications and inversions need to be
performed to calculate the amplitude and stability of thermoacoustic oscillations, making it
much cheaper than any nonlinear numerical method currently available in thermoacoustics.
Finally, in Chapters 6 and 7 nonlinear low-order models based on the G-equation for
turbulent bluff-body stabilised flames were developed. The goal was to obtain a cheap but
qualitatively accurate estimate of the nonlinear response of non-conical flame shapes, which
can be used in industry as a tool to quickly assess the stability of a thermoacoustic network.
The idea underlying the first model was to describe the flame-flow interaction mechanism,
and the consequent formation of vortices on the flame structure and their roll-up. Although
the dynamic response of the flame front was shown to compare well against experimental
results, the heat release response overestimated the actual response of the system. This
model can be improved by taking into account more physical mechanisms in the heat release
dynamics, such as turbulent flame speeds and the role of shear layers in the combustion
process. However, this would make the model more expensive, in contrast with the original
low-order modelling requirement. The second model was based on the interaction between
the flame and equivalence ratio fluctuations in imperfectly premixed flames. Equivalence
ratio fluctuations largely dominate the heat release dynamics, and an accurate description of
180 Conclusions and future work
the flame-flow interaction was not needed here. Results showed that the low-order modelled
heat release response compares well with that measured from experiments. An FDF was
evaluated and approximated with a sum of time delays. The latter yields an alternative
point of view to the heat release response. It can be used to separate the roles of acoustic
and equivalence ratio fluctuations, and to estimate the relevant time and length scales of
the system. The FDF was embedded in an acoustic network and was able to predict the
frequency at which thermoacoustic oscillations grow in experiments. This model was used in
the context of Rolls-Royce’s AMEL project to investigate the sensitivity of a thermoacoustic
system’s stability to parameters like the mean equivalence ratio and fuel injection location.
Appendix A
Non-dimensional variables
The complete set of non-dimensional variables and parameters we define is:
x =
x˜
L f
β =
L f
R
r =
r˜
R
= β
r˜
L f
F =
F˜
L f
f =
f˜
L f
∇= L f ∇˜=
(
∂
∂x
, β
∂
∂ r
)
t = t˜
U
L f
St =
ωL f
U
K =
U
Uc
=
kL f
St
ux =
u˜x
U
ur =
u˜r
U
s˜0L
U
=
1√
1+β 2
κ = κ˜L f M= LL f sinα(r) =
[
1+β 2
(
dF
dr
)2]−1/2
.
(A.1)
Appendix B
Eigenvalue problem
The eigenvalue problem we want to solve reads:
− β
2√
1+β 2
(
h1(r)
d fˆ
dr
+h2(r)
d2 fˆ
dr2
)
+
(
uˆ′x+
r
2
dF
dr
duˆ′x
dx
)
x=F
= λ fˆ (B.1a)
Asˆ+B
β 2
2
√
1+β 2
∫ 1
0
(
h1(r)
d fˆ
dr
+h2(r)
d2 fˆ
dr2
r
)
dr = λ sˆ (B.1b)
− 1
K
duˆ′x
dx
= λ uˆ′x, (B.1c)
where h1 and h2 are functions of the mean flame shape only, obtained by inspection from the
relation:
h1(r)
d fˆ
dr
+h2(r)
d2 fˆ
dr2
=
(
sinα
dF
dr
+2Mβ 2 sin4α dF
dr
d2F
dr2
−M1
r
)
d fˆ
dr
−Msin2α d
2 fˆ
dr2
.
We discretize the radial and axial components into:
r → r = [r0 = 1,r1, . . . ,r j, . . . ,rNr+1 = 0]
fˆ (r)→ fˆ (r) = fˆ = [ fˆ 0, fˆ 1, . . . , fˆ j, . . . , fˆ Nr+1] , (B.2)
and
x→ x = [x0 = 0,x1, . . . ,x j, . . . ,xNx+1]
uˆ′x (x)→ uˆ′x (x) = uˆ′x =
[
uˆ′0x , uˆ′1x , . . . , uˆ
′ j
x , . . . , uˆ′Nx+1x
]
,
(B.3)
where Nr +2 is the number of discretization points for the flame, and Nx+2 the number of
discretization points for the velocity field, which has to be sufficiently high to store the value
183
of the axial velocity inside the flame. Also, the velocity has to be evaluated at the mean flame
positions: it is therefore natural to discretize the radial and axial directions with the same
number of points, i.e., Nr = Nx with x = F(r). The boundary conditions (2.21) determine the
flame position at r = 0 and r = 1, so we are left with Nr degrees of freedom. We indicate with
Dx, Dr and Dr,r the (Nr +2)× (Nr +2) first and second differentiation matrices in the axial
and radial directions respectively; Dx and Dr differ because, if M ̸= 0, the steady solution is
not a straight line, and the discretization in the axial direction is non-uniform. Finally, the
heat release integration is discretized as:
∫ 1
0
(
h1(r)
d fˆ
dr
+h2(r)
d2 fˆ
dr2
)
rdr ≈
Nr+1
∑
k=0
∆rµk rk
(
h1(rk)
d fˆ
dr
∣∣∣∣
k
+h2(rk)
d2 fˆ
dr2
∣∣∣∣
k
)
=
=
Nr+1
∑
k=0
Nr
∑
j=1
∆rµk rk
(
hk1D
k, j
r fˆ
j +hk2D
k, j
r,r fˆ
j
)
= (B.4)
=
Nr
∑
j=1
[
Nr+1
∑
k=0
∆rµk rk
(
hk1D
k, j
r +h
k
2D
k, j
r,r
)]
fˆ j ≡
Nr
∑
j=1
∆q j fˆ j,
where µk are the weights of the integral discretization chosen.
We can finally express the submatrices of the eigenvalue problem as (2.37), defining:
M i, jf f =−
β 2√
1+β 2
(
hi1D
i, j
r +h
i
2D
i, j
r,r
)
i = 1, . . . ,Nr j = 1, . . . ,Nr (B.5a)
M i, jf s =
1
2
ri
dF
dr
∣∣∣∣
i
ξi (δ1,i+δ2,i)C ju i = 1, . . . ,Nr j = 1, . . . ,Ns (B.5b)
M i, jf u = δi, j +
1
2
ri
dF
dr
∣∣∣∣
i
Di, jx i = 1, . . . ,Nr j = 1, . . . ,Nx (B.5c)
M i, js f =
β 2
2
√
1+β 2
Bi∆q j i = 1, . . . ,Ns j = 1, . . . ,Nr (B.5d)
M i, jss = A
i, j i = 1, . . . ,Ns j = 1, . . . ,Ns (B.5e)
M i, jsu = 0 i = 1, . . . ,Ns j = 1, . . . ,Nx (B.5f)
M i, ju f = 0 i = 1, . . . ,Nx j = 1, . . . ,Nr (B.5g)
M i, jus =−
1
K
ξi (δ1,i+δ2,i)C ju i = 1, . . . ,Nx j = 1, . . . ,Ns (B.5h)
M i, juu =−
1
K
Di, jx i = 1, . . . ,Nx j = 1, . . . ,Nx, (B.5i)
184 Eigenvalue problem
where δi, j is the Kronecker delta, and ξi are finite difference coefficients which apply the
boundary condition (2.36) for the velocity at the flame base; because we use a second order
scheme, only two of them are non-zero.
Appendix C
Growth rate variations by implicit
function theorem
For convenience, let us rewrite the dispersion relations (4.11) in a compact form by splitting
them into real and imaginary part as N(A,y) ≡ (NRe10 ,NIm10 ,NRe01 ,NIm01 ) = 0, where we have
defined
NRe10 ≡ Re [F10(A1,ω1,A2,ω2,)H(σ1+ iω1)−1]
NIm10 ≡ Im [F10(A1,ω1,A2,ω2,)H(σ1+ iω1)−1]
NRe01 ≡ Re [F01(A1,ω1,A2,ω2,)H(σ2+ iω2)−1]
NIm01 ≡ Im [F01(A1,ω1,A2,ω2,)H(σ2+ iω2)−1] ,
(C.1)
where A ≡ (A1,A2) is the vector of amplitudes, and y ≡ (σ1,ω1,σ2,ω2) is the vector of
growth rates and frequencies. This is a system of four equations through which the four
dependent variables (frequencies and growth rates) are implicit functions of the amplitude
levels, i.e., y = y(A). By implicit differentiation of the dispersion relations, one obtains
dN =
∂N
∂y
dy+
∂N
∂A
dA = 0, (C.2)
or, by rearranging
dy
dA
=

∂σ1
∂A1
∂σ1
∂A2
∂ω1
∂A1
∂ω1
∂A2
∂σ2
∂A1
∂σ2
∂A2
∂ω2
∂A1
∂ω2
∂A2
=−
(
∂N
∂y
)−1 ∂N
∂A
. (C.3)
The latter expression yields the growth rates and frequencies sensitivities with respect to
amplitudes variations. The right hand side terms can be evaluated by finite difference by
186 Growth rate variations by implicit function theorem
imposing small perturbations (one by one) in the dispersion relations (C.1). No iterative
methods are required when using the implicit function theorem, which makes the method
more reliable because it is not susceptible to convergence problems.
Appendix D
Acoustic matrix coefficients
The coefficients of the matrix in equation (5.1) are given by:
M11 = (1−M1)− (1+M1)R1e−sτ1 (D.1)
M12 =
(
(1+M2)+(M2−1)R2e−sτ2)
)Θ
∆
(D.2)
M13 = M2
Θ
∆
(D.3)
M21 = (2−M1)M1−Θ− ((2+M1)M1+Θ)R1e−sτ1 (D.4)
M22 =
(
(1+M2)2+(M2−1)2R2e−sτ2
)
Θ (D.5)
M23 = M22Θ (D.6)
M31 =
(1−M1)(2+(M1−2)M1(γ−1))− (1+M1)(2+M1(2+M1)(γ−1))R1e−sτ1
2(γ−1)
(D.7)
M32 =
(M2−1)(2+(M2−2)M2(γ−1))R2e−sτ2 +(1+M2)(2+M2(2+M2)(γ−1))
2(γ−1) ∆Θ
(D.8)
M33 =
1
2
M32∆Θ. (D.9)
When damping model I is adopted (see §5.2.3), the Laplace variable s implicitly accounts
for the wavenumber correction defined in equation (5.8). In case of damping model II, no
further correction to s is introduced.
Appendix E
Nonlinear expansion terms
In the following all the terms obtained by expanding equation (5.9) to fifth order in ε are
listed and classified by their physical origin.
1. Fast timescale ( ∂x∂ t0 ):
ε
∂x1
∂ t0
+ ε2
∂x2
∂ t0
+ ε3
∂x3
∂ t0
+ ε4
∂x4
∂ t0
+ ε5
∂x5
∂ t0
. (E.1)
2. First slow timescale ( ∂x∂ t2 ):
ε3
∂x1
∂ t2
+ ε4
∂x2
∂ t2
+ ε5
∂x3
∂ t2
. (E.2)
3. Second slow timescale ( ∂x∂ t4 ):
ε5
∂x1
∂ t4
. (E.3)
4. Linear acoustics (Ax):
εAx1+ ε2Ax2+ ε3Ax3+ ε4Ax4+ ε5Ax5. (E.4)
5. Linear heat release (α1KcB(Cx(t− τ))):
εα1KcB(Cx1(t− τ))+ ε2α1KcB(Cx2(t− τ))+ ε3α1KcB(Cx3(t− τ)) . . .
+ε4α1KcB(Cx4(t− τ))+ ε5α1KcB(Cx5(t− τ)).
(E.5)
189
6. Quadratic heat release (α2KcB(Cx(t− τ))2):
ε2α2KcB(Cx1(t− τ))2+ ε32α2KcB(Cx1(t− τ))(Cx2(t− τ)) . . .
+ε42α2KcB(Cx1(t− τ))(Cx3(t− τ))+ ε52α2KcB(Cx1(t− τ))(Cx4(t− τ)) . . .
+ε4α2KcB(Cx2(t− τ))2+ ε52α2KcB(Cx2(t− τ))(Cx3(t− τ)).
(E.6)
7. Linear heat release and ∆K coupling (α1δ2B(Cx(t− τ))):
ε3α1δ2B(Cx1(t− τ))+ ε4α1δ2B(Cx2(t− τ))+ ε5α1δ2B(Cx3(t− τ)). (E.7)
8. Cubic heat release (α3KcB(Cx(t− τ))3):
ε3α3KcB(Cx1(t− τ))3+ ε43α3KcB(Cx1(t− τ))2(Cx2(t− τ)) . . .
+ε53α3KcB(Cx1(t− τ))2(Cx3(t− τ))+ ε53α3KcB(Cx1(t− τ))(Cx2(t− τ))2.
(E.8)
9. Quadratic heat release and ∆K coupling (α2δ2B(Cx(t− τ))2):
ε4α2δ2B(Cx1(t− τ))2+ ε52α2δ2B(Cx1(t− τ))(Cx2(t− τ)). (E.9)
10. Quartic heat release (α4KcB(Cx(t− τ))4):
ε4α4KcB(Cx1(t− τ))4+ ε54α4KcB(Cx1(t− τ))3(Cx2(t− τ)). (E.10)
11. Cubic heat release and ∆K coupling (α3δ2B(Cx(t− τ))3):
ε5α3δ2B(Cx1(t− τ))3. (E.11)
12. Quintic heat release (α5KcB(Cx(t− τ))5):
ε5α5KcB(Cx1(t− τ))5. (E.12)
190 Nonlinear expansion terms
13. Time delay (proportional to τ , τ2)
−τBα1KcC ∂x1∂ t2 (t− τ)ε
3 . . .
+τB
(
−2α2KcCx1(t− τ)C ∂x1∂ t2 (t− τ)−α1K
cC
∂x2
∂ t2
(t− τ)
)
ε4 . . .
+τB
[
−α1KcC ∂x1∂ t4 (t− τ)−α1δ2C
∂x1
∂ t2
(t− τ) . . .
−3α3Kc(Cx1(t− τ))2C ∂x1∂ t2 (t− τ)−2α2K
cCx2(t− τ)C ∂x1∂ t2 (t− τ) . . .
−2α2KcCx1(t− τ)C ∂x2∂ t2 (t− τ)−α1K
cC
∂x3
∂ t2
(t− τ) . . .
+
1
2
τα1KcC
∂ 2x1
∂ t22
(t− τ)
]
ε5.
(E.13)
Appendix F
Forcing terms
F.1 O(ε3)
The list of forcing terms is:
2α2KcB(Cx1(t− τ))(Cx2(t− τ))+α3KcB(Cx1(t− τ))3+α1δ2B(Cx1(t− τ)).
(F.1)
Using the solutions (5.19),(5.22), we can expand these terms and classify them by their
amplitude dependence as:
FW3 ≡ α1Bδ2(CxW1 )e−iω
cτ , (F.2a)
F |W |
2W
3 ≡ BKc
[
3α3(CxW1 )|(CxW1 )|2 . . .
+2α2
(
(CxW
2
2 )(Cx
W
1 )
∗+(CxW1 )(Cx
|W |2
2 )
)]
e−iω
cτ ,
(F.2b)
FW
3
3 ≡ BKc(CxW1 )
(
α3(CxW1 )
2+2α2(CxW
2
2 )
)
e−3iω
cτ . (F.2c)
192 Forcing terms
F.2 O(ε4)
The list of forcing terms is:
−∂x2
∂ t2
+ τB
(
−2α2KcCx1(t− τ)C ∂x1∂ t2 (t− τ)−α1K
cC
∂x2
∂ t2
(t− τ)
)
. . .
+2α2KB(Cx1(t− τ))(Cx3(t− τ))+α2KB(Cx2(t− τ))2 . . .
+α1δ2B(Cx2(t− τ))+3α3KB(Cx1(t− τ))2(Cx2(t− τ)) . . .
+α2δ2B(Cx1(t− τ))2+α4KB(Cx1(t− τ))4.
(F.3)
The definitions of the forcings in eq (5.37) read:
FW
4
4 ≡ BK
(
α4(CxW1 )
4+3α3(CxW1 )
2(CxW
2
2 )+2α2(Cx
W
1 )(Cx
W 3
3 )+α2(Cx
W 2
2 )
2
)
, (F.4a)
F |W |
4
4 ≡ BKc
(
6α4|CxW1 |4+(CxW
2
2 )
∗
(
3α3(CxW1 )
2+2α2(CxW
2
2 )
)
. . .
+3α3(CxW
2
2 )
(
(CxW1 )
∗)2+2(CxW1 )∗(3α3(CxW1 )(Cx|W |22 )+α2(Cx|W |2W3 )) . . .
+α2
(
2(CxW1 )(Cx
|W |2W
3 )
∗+(Cx|W |
2
2 )
2
))
−2BKcτRe(ν3)
(
α1(Cx
|W |2
2 ) . . .
+2α2(CxW1 )(Cx
W
1 )
∗
)
−2x|W |22 Re(ν3),
(F.4b)
F |W |
2W 2
4 ≡ BKc
(
2(CxW1 )
∗
(
(CxW1 )
(
2α4(CxW1 )
2+3α3(CxW
2
2 )
)
+α2(CxW
3
3 )
)
. . .
+(CxW1 )(3α3(Cx
W
1 )(Cx
|W |2
2 )+2α2(Cx
|W |2W
3 ))+2α2(Cx
W 2
2 )(Cx
|W |2
2 )
)
e−2iω
cτ . . .
−2BKcν3τ
(
α1(CxW
2
2 )+α2(Cx
W
1 )
2
)
e−2iω
cτ −2ν3xW 22 ,
(F.4c)
F |W |
2
4 ≡ B
(
2α2
(
(CxW1 )
∗(Kc(CxW3 )+δ2(Cx
W
1 ))+K
c(CxW1 )(Cx
W
3 )
∗) . . .
+α1δ2(Cx
|W |2
2 )
)
−2BKcτRe(λ3)
(
α1(Cx
|W |2
2 )+2α2(Cx
W
1 )(Cx
W
1 )
∗
)
. . .
−2x|W |22 Re(λ3),
(F.4d)
FW
2
4 ≡ B
(
α2(CxW1 )(2K(Cx
W
3 )+δ2(Cx
W
1 ))+α1δ2(Cx
W 2
2 )
)
e−2iω
cτ . . .
−2BKcλ3τ
(
α1(CxW
2
2 )+α2(Cx
W
1 )
2
)
e−2iω
cτ −2λ3xW 22 .
(F.4e)
F.3 O(ε5) 193
F.3 O(ε5)
The list of forcing terms is:
−∂x3
∂ t2
+ τB
[
−α1δ2C ∂x1∂ t2 (t− τ)−3α3K
c(Cx1(t− τ))2C ∂x1∂ t2 (t− τ) . . .
−2α2KcCx2(t− τ)C ∂x1∂ t2 (t− τ)−2α2K
cCx1(t− τ)C ∂x2∂ t2 (t− τ) . . .
−α1KcC ∂x3∂ t2 (t− τ)+
1
2
τα1KcC
∂ 2x1
∂ t22
(t− τ)
]
. . .
+2α2KcB(Cx1(t− τ))(Cx4(t− τ))+2α2KcB(Cx2(t− τ))(Cx3(t− τ)) . . .
+α1δ2B(Cx3(t− τ))+3α3KcB(Cx1(t− τ))2(Cx3(t− τ)) . . .
+3α3KcB(Cx1(t− τ))(Cx2(t− τ))2+2α2δ2B(Cx1(t− τ))(Cx2(t− τ)) . . .
+4α4KcB(Cx1(t− τ))3(Cx2(t− τ))+α3δ2B(Cx1(t− τ))3+α5KcB(Cx1(t− τ))5.
(F.5)
For the purpose of this study, we are interested only in the explicit expressions of the
resonant forcing terms in equation (5.40), which read:
FW5 ≡ α1δ2B(CxW3 )e−iω
cτ −Bα1λ3τ
(
(δ2(CxW1 )+K
c(CxW3 ) . . .
−1
2
Kcλ3τ(CxW1 )
)
e−iω
cτ −λ3xW3 ,
(F.6a)
F |W |
2W
5 ≡ B
(
3α3δ2(CxW1 )|CxW1 |2+2(CxW1 )∗(3α3Kc(CxW1 )(CxW3 ) . . .
+α2Kc(CxW
2
4 )+α2δ2(Cx
W 2
2 ))+K
c(CxW3 )
∗
(
3α3(CxW1 )
2+2α2(CxW
2
2 )
)
. . .
+ 2α2(Kc(CxW1 )(Cx
|W |2
4 )+K
c(Cx|W |
2
2 )(Cx
W
3 )+δ2(Cx
W
1 )(Cx
|W |2
2 )) . . .
+α1δ2(Cx
|W |2W
3 )
)
e−iω
cτ −Bτ
(
δ2(CxW1 )α1ν3+K
c((Cx|W |
2W
3 )α1λ3 . . .
+2(CxW1 )(Cx
|W |2
2 )α2λ3+α1ν3((Cx
W
3 )− (CxW1 )λ3τ))+Kc(2(Cx|W |
2W
3 )α1 . . .
+4(CxW1 )(Cx
|W |2
2 α2− (CxW1 )α1ν3τ)Re(λ3)3Kc(CxW1 )α3|CxW1 |2(λ3+2Re(λ3)) . . .
+2Kc(CxW
2
2 )α2(Cx
W
1 )
∗(λ3+2Re(λ3))
)
e−iω
cτ . . .
−2x|W |2W3 Re(λ3)−ν3xW3 −λ3x|W |
2W
3 ,
(F.6b)
194 Forcing terms
F |W |
4W
5 ≡ BKc
(
12α4(CxW1 )(Cx
|W |2
2 )|CxW1 |2+2(CxW
2
2 )
∗ (2α4C2(CxW1 )3 . . .
+3α3(CxW1 )(Cx
W 2
2 )+α2(Cx
W 3
3 )
)
+(Cx|W |
2W
3 )
∗
(
3α3(CxW1 )
+2α2(CxW
2
2 )
)
. . .
+2(CxW1 )
∗(3α3((CxW1 )(Cx
|W |2W
3 )+(Cx
W 2
2 )(Cx
|W |2
2 ))+α2(Cx
|W |2W 2
4 )) . . .
+3
(
(CxW1 )
∗)2 (4α4(CxW1 )(CxW 22 )+α3(CxW 33 )))+BKc(3α3(CxW1 )(Cx|W |22 )2 . . .
+2α2(CxW1 )(Cx
|W |4
4 )+2α2(Cx
|W |2
2 )(Cx
|W |2W
3 )
)
e−iω
cτ
−1
2
BKcτ(ν3+2Re(ν3))
(
2α1(Cx
|W |2W
3 )+4α2(Cx
W
1 )(Cx
|W |2
2 ) . . .
−α1ν3τ(CxW1 )+4α2(CxW
2
2 )(C(x
W
1 )
∗)+6α3(CxW1 )(Cx
W
1 )(C(x
W
1 )
∗)
)
e−iω
cτ
−2x|W |2W3 Re(ν3)−ν3x|W |
2W
3 .
(F.6c)
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