PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Thermoacoustic instabilities: physical mechanisms andmathematical modelling
Maria HECKL
School of Chemical and Physical Sciences, Keele University, U.K.
m.a.heckl@keele.ac.uk

ABSTRACT
If a flame is put into an acoustic resonator, an escalating interaction between the flame's heat release andthe acoustic field can occur, giving rise to intense pressure oscillations. This phenomenon is termed"thermoacoustic instability". It occurs in combustion systems that have a continuously burning flame, suchas gas turbines, jet aeroengines, boiler and heating systems, furnaces and rockets. Thermoacoustic
instabilities are a serious problem because they can lead to excessive structural vibrations, fatigue, and evencatastrophic hardware damage. The escalating flame-sound interaction occurs in tandem with otherphysical mechanisms, leading to a complex web of interactions, most of which are nonlinear. They include
flame-vortex interactions, flame response to fluctuations in fuel concentration, entropy waves, flamestructure interactions, and others.
The development of low-pollution combustion systems is very important for our environment.
However, such combustion systems are particularly prone to thermoacoustic instabilities. Progress with
developing combustion systems that are immune to these is hampered by insufficient physical insight.
Efforts to gain further insight are going on by researchers worldwide using experimental, analytical andnumerical tools. This talk aims to give an overview of the key physical mechanisms involved inthermoacoustic instabilities and will present mathematical modelling approaches inspired by physicalinsight.
Keywords: flame models, tailored Green's function, low-pollution combustion systems

1. INTRODUCTION
The most basic thermoacoustic system is a Rijke tube [1,2]. It is an open-ended tube, held vertically,
with a horizontal wire gauze in the lower half of the tube. If the gauze is sufficiently hot, the tube emits aloud sound, typically at a frequency that corresponds to the fundamental mode of the tube. This sound isgenerated by thermoacoustic feedback, which is the interaction between the following two processes:(1) The heat release rate of the hot gauze fluctuates, leading to fluctuations in density and temperature of
the air surrounding it; as a consequence, a sound wave is generated.
(2) The sound wave impinges on the hot gauze, leading to oscillations in the air flow and hence in the heattransfer rate; as a consequence, the heat release rate from the hot gauze fluctuates.
A feedback loop between these two processes is established because any sound waves generated by the
hot gauze are reflected at the tube ends back into the tube and towards the flame. This is shown
schematically in Figure 1.

sound waves

Aach en ICA 1 .cd rheat release rateoscillations

Figure 1 – Feedback between hot gauze and sound field9

acoustic pressure [Nm-2]

An example of an escalating thermoacoustic feedback, i.e. a thermoacoustic instability, can be seen in
Figure 2, which shows the time history of the pressure in a Rijke tube [3, p. 49]. At first, the pressure
amplitude increases exponentially (as indicated by the dashed line), then it grows less rapidly, andeventually the amplitude reaches a constant value.

time [s]Figure 2 – Time history of the acoustic pressure during a thermoacoustic instability in a Rijke tube
This thermoacoustic feedback occurs not only in the Rijke tube, but more importantly in any acoustic
resonator, which houses a flame that responds to fluctuations in the surrounding flow field. This includes
all combustion systems that have a continuously burning flame, such as gas turbine engines, jetaeroengines, boiler and heating systems, furnaces and rockets. Here, the basic instability mechanism is alsoa thermoacoustic feedback loop. This, however, does not occur in isolation, but is coupled with otherphysical mechanisms, leading to a complex web of interaction, which tend to be nonlinear.
Combustion is responsible for a number of pollutants of the environment, in particular NOx and CO2. Inorder to minimise NOx production, green combustion technologies use lean premixed flames and burn at
low temperatures. CO2 production can be minimised by adding hydrogen to the fuel and/or using biofuels.
Unfortunately, combustion systems based on these approaches are particularly prone to thermoacoustic
instabilities. These can occur suddenly and spontaneously, and when they occur, the resulting oscillations
may be so violent that they cause structural damage; for example, Goy et al [4, chapter 4] report thedestruction of a combustion liner and a burner assembly.
Progress with developing combustion systems that are immune to thermoacoustic instabilities ishampered by insufficient physical insight. Efforts to gain new insight are made by researchers worldwide,using experimental, analytical and numerical approaches. One can get an overview in the review articles ofCandel [5], Lieuwen [6] and Huang & Yang [7], in the books by Lieuwen & Yang [4] and Poinsot &Veynante [8], in the text by Culick [9], and in the recent review paper by Heckl [10].
The aim of this paper is to consider a generic combustion system, to describe the most important
physical processes that occur in it, and to explain how these can be modelled mathematically. The paper is
structured as follows. Section 2 will give an overview of the key physical mechanisms involved inthermoacoustic instabilities. Section 3 will introduce a new method to model linear and nonlinear flames
mathematically. The advantages of using the tailored Green's function for thermoacoustic modelling
purposes will be demonstrated in Section 4. Section 5 will present a case study, where the concepts from
Sections 3 and 4 are applied. Conclusions will be given in Section 6.

10

2. PHYSICAL MECHANISMS
We consider a generic combustion system, as shown in Figure 3(a). It comprises a mixing tube, acombustion chamber and an exhaust nozzle. Air and fuel enter the upstream end of the mixing tube throughseparate inlets. They mix as they move downstream, and form a more or less homogeneous mixture thatenters the combustion chamber as a jet. The shear layer surrounding the jet is unstable; this leads to theformation of vortices, which are shed periodically, as illustrated in Figure 3(b). The flame, which isanchored near the upstream end of the combustion chamber, is perturbed by the vortices; its surface areafluctuates, and as a consequence, the rate of heat release fluctuates; this is shown in Figure 3(c). As
explained in Section 1, a heat source with a fluctuating rate of heat release acts as a sound source. Hencethe flame emits acoustic waves, which travel away from either side of the flame, see Figure 3(d). The
acoustic waves are reflected at the resonator boundaries and form a standing wave with a frequency that isclose to one of the resonance frequencies of the combustion system.
The acoustic wave induces several secondary feedback loops:(1) One of these involves mixture inhomogeneities, which are generated because the oscillating pressure in
the mixing tube modulates the air flow and/or fuel flow. This leads to perturbations in the equivalence
ratio I . These are swept by the mean flow towards the flame, where they induce further fluctuations in
the heat release rate. This establishes a feedback loop, where the I fluctuations travel with the bulk
flow – at low Mach number – from the air/fuel inlet to the flame, and the sound waves travel in the
opposite direction with the speed of sound, from the flame to the air/fuel inlet; Figure 4(a) illustrates
this.
(2) Another secondary feedback loop involves modulations of the flow of the premix, which are generated
by the oscillating pressure at the combustion chamber inlet. The modulated flow perturbs the heatrelease rate of the flame, which in turn creates a new sound wave. In this feedback loop, the flow
modulations travel with a speed similar to that of the mean flow from the combustion chamber inlet tothe flame, and the sound waves travel with the speed of sound in the opposite direction from the flame
to the combustion chamber inlet, see Figure 4(b).
(3) A further secondary feedback loop involves vortices, which are generated upstream of the flame,
typically at a sharp edge that forms part of the flame holder. The velocity fluctuations imposed by thesound wave at the sharp edge synchronises the vortex shedding. This establishes a feedback loop, where
vortices travel the length of the flame and on their way disturb the surface area and heat release rate of
the flame. The flame responds by generating a sound wave, which reinforces the synchronised vortex
shedding. The vortices travel with a speed similar to the mean flow in the combustion chamber from thebase to the tip of the flame, and the sound waves travel with the speed of sound in the opposite
direction. This is shown schematically in Figure 4(c).
Each of these three feedback loops are characterised by a frequency that is determined by the time ittakes to complete one cycle of the loop. Given that the mean flow travels much more slowly than the soundwave, the time to complete one cycle is dominated by the speed of the mean flow. The periodic oscillations
driven by these secondary feedback loops are termed "intrinsic modes"; they have received much attention
recently [11, 12, 13]. Their frequencies are independent of the resonator frequencies. However, if thefrequency of an intrinsic mode happens to be close to one of the resonator frequencies, coupling between
an acoustic and intrinsic mode ensues, and they may adopt a common frequency. This phenomenon iscalled "lock-on" [14, 15].
Although the flame is the primary sound source in the combustion chamber, it may not be the only
sound source. The following process can activate a secondary sound source: Apart from generating soundwaves, an unsteady flame also generates hot-spots that are swept downstream towards the exhaust. These
travelling hot-spots are called "entropy waves" [16, 17]; their propagation speed is similar to that of themean flow. As long as the hot-spots travel with a constant speed, they produce no sound. However, if they
are accelerated, which happens when they pass through a constriction (e.g. the exhaust nozzle or throughthe gap between turbine blades) they turn into a sound source. A new feedback loop ensues: a hot-spot
travels from the flame to the exhaust where it gets accelerated; the sound waves generated by this process
travel from the nozzle back to the flame, where they disturb the heat release rate and induce yet more hotspots. Figure 4(d) visualises this feedback loop.

11

fuel
air

combustion chamber

exhaustAache n IC A 3.cdr

mixing tube

mean
flow
Figure 3(a) – Mixing of fuel and air to form a premix travelling with constant speed (the mean flow speed).

fuel
air

combustion chamber

exhaustmixing tube
Aache n ICA fi gure 3 b.cdr

jetvortices
Figure 3(b) – Vortex shedding near the upstream end of the combustion chamber

fuel
air

combustion chamber
oscillating flame
surface area

Aa chen ICA fi gure 3 c.cd r

mixing tube

exhaustvortices

Figure 3(c) – Fluctuations of the flame surface area induced by vortices

fuel
air

combustion chamber

oscillating flame
surface area

Aa chen ICA fi gure 3 d.cdr

mixing tube

exhaustFigure 3(d) – Generation of acoustic waves travelling towards the boundaries of the combustion system

12

fuel
air

exhaustcombustion chamber

Aache n IC A figu re 4a.cdr_mixing tube
uc
mixtureinhomogeneity

Figure 4(a) – Feedback loop formed by right-travelling mixture inhomogeneities (speed similar to u ) andleft-travelling sound waves (speed c )

fuel
air

exhaustcombustion chamber
mixing tube

Aache n IC A figu re 4b.cdr_ucmodulated
flow

Figure 4(b) – Feedback loop formed by right-travelling flow modulations (speed similar to u ) and lefttravelling sound waves (speed c )fuelexhaustcombustion chamber

air
_
u

Aache n IC A figu re 4c.cdr

mixing tubecvortexFigure 4(c) – Feedback loop formed by right-travelling vortices (speed similar to u ) and left-travellingsound waves (speed c )

fuel
air

exhaustcombustion chamber_uhot-spotc

Aa chen ICA fi gure 4 d.cdr

mixing tube

Figure 4(d) – Feedback loop formed by right-travelling hot-spots (speed similar to u ) and left-travellingsound waves (speed c )13All the feedback mechanisms illustrated in Figure 4 combine to form a web of multiple interactions.
They may be supplemented by further interactions: a swirler (typically near the downstream end of themixing tube) would generate a flow field with different velocity components; a row of heat exchanger tubes
would act as a heat sink and also redistribute the acoustic field; flexible combustion chamber walls would
be able to vibrate and so absorb acoustic energy from the sound field.

3. MATHEMATICAL MODELLING OF FLAMESFor low-order modelling of a combustion system, the flame is considered as an input-output system anddescribed in the frequency ( Z ) domain. The input is a velocity fluctuation, uˆ (Z) , and the output is afluctuation in the heat release rate, Qˆ (Z) (the hat ^ denotes complex amplitudes). The flame is described by
the flame transfer function (FTF), denoted here by T (Z) and defined by
T (Z)Qˆ (Z) / Q,uˆ(Z) / u(1)

where Q is the mean heat release rate and u is the speed of the mean flow. The FTF can be measured by
the setup shown in Figure 5.

Figure 5 – Setup to measure the FTF
The mean flow u gives rise to the mean heat release rate Q . The oscillating velocity comes from anexternal source (e.g. a loudspeaker); it excites the flame with velocity uˆ (Z) and induces fluctuations Qˆ (Z)in the heat release rate. Qˆ (Z) is measured indirectly from the light emitted by a particular radical produced
during the chemical reaction. The frequency Z is varied during the measurement in the required range.
Figure 6 shows a typical result for the gain, | T (Z) | , and phase, T (Z) , of a premixed flame.

Aachen ICA figure 6.cdr

T (Z )|T (Z)|
1Z

Z

Figure 6 – Typical spectrum for the gain and phase of a FTF.
If the inverse Fourier transform is applied to Eq. (1), the following result is obtained,
Q '(t )Qf

³ h(W)

fu '(t  W)
dW ,
u

(2)

where h(W) is the inverse Fourier transform of T (Z) . The dashes indicate time-domain quantities: Q ' isthe fluctuating part of the heat release rate in the time domain, and u ' is the fluctuating part of the velocity.

14

The function h(W) has a straightforward physical meaning: it is the impulse response of the flame, as canbe seen easily by puttingu '(t ) u G(t )(3)into Eq. (2). The FTF and the impulse response are mathematically equivalent, and therefore they containexactly the same physical information: the FTF is the representation in the frequency domain, and theimpulse response is the representation in the time domain. A typical result for h(W) is shown in Figure 7.

h(WWAachen ICA figure 7.cdr

0Figure 7 – Typical time history of a flame's impulse response
This time history has a succession of maxima and minima, which fade for large times. This is in linewith the physical observations described in section 2: there are several feedback loops involving transport
processes, and each of these transport processes introduces its own time delay. The maxima and minima in
Figure 7 are not infinitesimally narrow, like the delta function in Eq. (3), but are surrounded by a
distribution. The finite width of these distributions is due to various factors, such as dispersion during thetransport processes (section 7 in [10]). A detailed analysis of the dispersion in one such transport process(swirl waves) has been made by Albayrak et al [18].
For mathematical modelling purposes, is useful to have an analytical representation of the FTF and
impulse response of a given flame. A method has been developed by Gopinathan and Heckl [19], which
exploits the fact that the individual maxima and minima in a flame's impulse response have the shape of abell and so can be fitted with Gauss curves. Gopinathan and Heckl use a superposition of Gauss curves,h (W)

n1
e
2SV1



( WW1 )2
2V12



n2
e
2SV2



( WW2 )2
2V22



n3
e
2SV3



( WW3 )2
2V32

!

(4)

to approximate a given impulse response obtained from FTF data, which may come from numericalsimulations or experimental measurement. The following quantities are treated as fitting parameters anddetermined numerically from a nonlinear least-square fitting routine:n j - measure for the height of the Gauss curve; this can be positive (maximum) or negative (minimum)
W j - position of the centre of the Gauss curve along the time axis, i.e. prominent time-lag
V j - width of the Gauss curve

The integer j 1, 2, 3, ! denotes the jth Gauss curve in the sum in Eq. (4).
The frequency-domain equivalent of Eq. (4) is the Fourier transform, which can be evaluatedanalytically. The result isT (Z)

n1e



Z2V32
Z2V12
Z2V22


2 eiZW1  n e
2 eiZW2  n e
2 eiZW3  ! .
2
3

(5)

The approach described here is a physically motivated mathematical tool to represent a set of FTF data
points by analytical expressions for the FTF and its corresponding impulse response. These expressions
involve a small number of fitting parameters (typically 6 or 9), which are the same in the frequency domain
and in the time domain.

15

The concept of the FTF has been extended into the nonlinear regime by Noiray et al [20]. They
measured the FTF with input signals uˆ(Z) of several velocity amplitudes, denoted here by a , and so
obtained an amplitude-dependent FTF, called "flame describing function" (FDF),
T (Z, a)Qˆ (Z, a ) / Q.
uˆ(Z, a) / u(6)

Just like its linear equivalent, the FDF can be represented analytically by a superposition of Gauss
curves as described above. The fitting parameters n j , W j , V j ( j 1, 2,3,! ) then become amplitudedependent. An example is shown in Figures 8, 9 and 10 below. These figures are for a methane-hydrogen
flame (80% CH4, 20% H2) with an equivalence ratio of 0.95. Figure 8 shows the gain and phase of the
FDF; the markers indicate the original data, and the solid curves give the corresponding analyticalrepresentations, using three Gauss curves. The following five amplitudes are shown:H a / u 0.02, 0.04, 0.06, 0.08, 0.1 . Figure 9 shows the corresponding inverse Fourier transforms; thedashed curves are for the inverse Fourier transforms of the original data, and the solid curves denote thecorresponding analytical representations. The curves are extremely similar and hard to distinguish from one
another. Figure 10 illustrates the amplitude-dependence of the fitting parameters (the amplitude range
depicted here includes higher values that are not shown in Figures 8 and 9).

Figure 8 – Gain and phase of the FDF of a CH4 - H2 flame. Markers: original data from numericalsimulation. Solid curves: analytical representations (from [19]).

16

Figure 9 – Inverse Fourier transform of the FDF in Figure 8. Dashed curves: inverse Fourier transform of
the original FDF data. Solid curves: analytical representations (from [19]).

Figure 10 – Amplitude-dependence of the fitting parameters for the CH4 - H2 flame in Figures 8 and 9(from [19]).

17

4. THERMOACOUSTIC MODELLING WITH THE TAILORED GREEN'S FUNCTION
The Green's function is an impulse response. In an acoustic context, it is the response to a point sourceGGat position x * , emitting an impulsive sound signal at time t * , measured by an observer at position x attime t . The tailored Green's function is the response in a configuration, where the source and observer are
G G
both inside an acoustic resonator. We denote the Green's function by G ( x , x*, t  t*) . Its governing
equation is the nonhomogeneous wave equation,
1 w 2G
2

2

c wt

G G
G( x  x*)G(t  t*) ,  2G(7)

where c is the speed of sound. It also has to satisfy the boundary conditions of the acoustic resonator, as
well as the causality condition
G G(8)G ( x , x*, t  t*) 0 for t  t * .
The general form of the tailored Green's function is a superposition of resonator modes; for a 1-D
geometry, this can be written as
G ( x, x*, t  t*)f

H(t  t*) ¦ Gn ( x, x*) eiZn (t  t*) ,

(9)

n 1

where n is the mode number, H is the Heaviside function (this guarantees that the causality condition (8) issatisfied), Zn is the frequency (which may be complex with a negative imaginary part if the mode isdamped) of mode n , and Gn ( x, x*) is its amplitude. Zn and Gn ( x, x*) can be calculated analytically for
configurations, that can be divided into uniform 1-D sections; in principle, they can also be measured.
The tailored Green's function is a useful mathematical tool for the solution of partial differential
equations (PDE) that have a forcing term. One such PDE is the acoustic analogy equation [21],1 w 2M2c wt

2

 2M U( J  1) wq '
,
wt
c2

(10)

written here for the velocity potential M . The left hand side is the well-known part of the standard wave
equation, while the right hand side represents the forcing term. This contains the quantity q ' , which is thefluctuating part of the local heat release rate (heat release rate per unit mass); the quantities U (meandensity) and J (specific heat ratio) are constants.
Equation (10) expresses the fact (mentioned in the second paragraph of the introduction) that a heatsource with a fluctuating rate of heat release is a source of sound. By combining the two PDEs (7) and (10),and assuming that the flame is compact, i.e. its heat release rate can be described by
G
G G
(11)
q '( x , t ) q(t )G( x  xq ) ,an integral equation can be derived for the acoustic field. This derivation involves several mathematicalsteps, which can be found in [22]. The resulting equation for the 1-D case isu 'q (t )



J 1
c

2

t

³

t* 0

wG
q(t*) dt * ,wx x xq(12)

x ' xqwhere u 'q (t ) denotes the acoustic velocity at the flame position xq . This equation can be interpreted in astraightforward manner. To this end, we write it in terms of the acoustic pressure p 'q at the flame position(omitting the constant term with U and J ),p 'q (t )t³ q(t*) G (t  t*) dt * .

t* 0

18

(13)

G (t  t*) is the Green's function, i.e. the response at time t to a unit impulse emitted at time t * . Hencethe response to an impulse emitted at t1 with peak value q1 is q1 G (t  t1 ) . This is illustrated in the top two
sub-figures of Figure 11. The response to an impulse emitted at t2 with peak value q2 is q2 G (t  t2 ) , andso on. The time history of the source term q(t*) can be approximated by a series of impulses in quick
succession, with varying peak values q1 , q2 , ! , as shown in the bottom left sub-figure of Figure 11. The
responses to these individual impulses superimpose to form the combined response
q1 G (t  t1 )  q2 G (t  t2 )  ! , as shown in the bottom right sub-figure. This sum is simply the discretizedversion of the integral in Eq. (13).

qq1G(t-t1)

q1
t1
t1

t

t

impulseimpulse response
q1G(t-t1)+q2G(t-t2)+ ...

t1 t2
t1 t2

t

t

Aachen ICA figure 11.cdr

q
q2
q1

sum of impulse responses

series of impulses

Figure 11 – Interpretation of the integral equation (13)In order to be able to solve Eq. (12), it is necessary to know how the heat release rate depends on the
acoustic field. A suitable flame model provides this information. We assume that it is given by a functionalin the generic formq(t )F (uq (t )) .

(14)

In the simplest case, (14) would be a time-lag law with coupling coefficient n and time-lag W ,q(t )n uq (t  W) .

(15)

In the general case, the functional F can be determined from the FTF or FDF of a specific flame.
A governing integral equation for the velocity uq (t ) is obtained by inserting (14) into the integrand of
Eq. (12),u 'q (t )



J 1
c

2

t

³

t* 0

wG
F (uq (t*)) dt * .
wx x xq(16)

x ' xqThis equation can be solved numerically with a straightforward iteration, stepping forward in time; details
can be found in Bigongiari and Heckl [23].

195. CASE STUDY: MODE COUPLING IN A HALF-WAVE RESONATOR WITH ANONLINEAR FLAME
The configuration in this case study is an open-ended tube with a compact swirl flame at x
Figure 12). There is a jump at x

xq (seexq in mean temperature, from T1 to T2 , and a corresponding jump in themean density from U1 to U2 as well as in the speed of sound from c1 to c2 . The flame is a compact swirl
flame, burning methane in the lean combustion regime ( I 0.77 ). This configuration was studied in [24,25], and the highlights are reported here.

x = xq

x=0

open
end

flame __
T2, U2, c2_ _T1, U1, c1

Aache n IC A figu re 12.cdr

open
end

x=L

Figure 12 – Schematic illustration of the modelled configuration
The FDF was determined from CFD simulations [24]. An analytical approximation was found by themethod described in Section 3, involving a superposition of two Gauss curves in the time domain,fª fºK « n1 ³ uq (t  W) D1 (W) dW  n2 ³ uq (t  W) D2 (W) dW » ,«¬ W 0
»¼
W 0

q(t )

(17)

Q
(this is a conversion factor between the local and global heat release rate; S is the crossSu2 U2sectional are of the tube), and the distributions

with K

D1 (W)

2
V1 2S

( WW1 )2e

2V12, D2 (W)2
V 2 2S( WW2 )2

e

2V22

.

(18)

The fitting parameters n1 , W1 , V1 and n2 , W2 , V2 were determined as described in Section 3. All of themturned out to decrease with amplitude. Their amplitude dependence was therefore approximated by thefollowing linear functions (best fit),n1

7.4  14.94

a
,
u

n2

4.7  14.94

a
u

(19)

W1

4.13  6.56

a[ms] , W2
u6.31  1.89

a
[ms]
u

(20)

V1 1.94  3.16

a[ms] , V2
u1.21  1.54

a
[ms]
u

(21)

The tailored Green's function was written as the sum of the first two resonator modes,
G ( x, x*, t  t*)H(t  t*) ªG1 ( x, x*) eiZ1 (t t *)  G2 ( x, x*) eiZ2 (t t *) º .
¬
¼

(22)

The frequencies Z1 , Z2 were calculated from an eigenvalue approach; they depend on the parameters
c1 , c2 , xq and L . The Green's function amplitudes, G1 , G2 , depend on the same parameters. Details can befound in [25].
By solving the integral equation (16) numerically with the time stepping method outlined in Section 4,
the time history uq (t ) was obtained over a long time interval, t > 0, 10s @. This is shown in Figure 13; therelevant parameters are given in the figure caption.

20

uq
u

time [s]Figure 13 – Time history of uq (t ) in an open-ended tube with length L 1.2m and flame at xq

0.20 m ;there is a cold region upstream (300 K for 0 d x  xq ) and a hot region downstream (1700 K for
xq  x d L ). The initial condition is uq (t0) / u0.01 ; the Green's function includes the first two modes;

the factor K was K

1.41 u 106 W s kg -1

The amplitude increases exponentially at first, and then grows less rapidly; beyond about t 2s , a limit
cycle is established with final amplitude a / u 0.1 . In order to investigate whether or how the modal
amplitudes change during the evolution, the frequency spectrum was calculated for an early time window,[0, 1s] (see Figure 14(a)), and for a late time window, [9s, 10s] (see Figure 14(b)). These spectra reveal thatmode 1 is dominant during the growth stage of the evolution (small amplitudes), and that mode 2 takes overat the limit cycle stage (large amplitudes).

(a)

(b)

Figure 14 – Fourier transform of the time history shown in Figure 13.
(a) Time-window [0, 1s]. (b) Time-window [9s,10s].

21

6. CONCLUSIONSThis presentation has given an overview of the key physical mechanisms involved in thermoacoustic
instabilities. At the core is a thermoacoustic feedback involving a flame in an acoustic resonator. In tandemwith this primary feedback loop, there are secondary feedback loops; these involve low-Mach number
transport processes perturbing the flame, which in turn emits an acoustic wave.
The two core elements of a low-order mathematical approach are a model for the flame and a model forthe acoustic resonator. A method to model a flame analytically was introduced, based on experimental or
numerical data for its flame describing function. This method is inspired by the physical fact that different
transport processes are associated with different time-lags. Unsurprisingly, given this physics-inspired
mathematical approach, the accuracy of the analytical flame models is very good and requires only a small
number of fitting parameters.
Our model for the acoustic resonator is based on the tailored Green's function of that resonator. This isan uncommon approach; network models ending up in eigenvalue calculations are usually the method of
choice for an analytical approach. However, the Green's function approach is more versatile. It can easily
be combined with any flame model and predictions can be made, no matter whether the flame model islinear or nonlinear. The tailored Green's function is made up the actual resonator modes, so it is a natural
tool for studying the interaction between two or more thermoacoustic modes. The most challenging part ofthis approach is the calculation of the tailored Green's function. However, once this has been determined
for a particular configuration, a variety of scenarios can be simulated in a straightforward manner, andparameter studies can be made with a minimum of numerical effort. It is a very useful tool for designing
new combustion systems that are immune to thermoacoustic instabilities.

REFERENCES[1] Rijke, P.L. (1859) Notice of a new method of causing a vibration of the air in a tube open at both ends.
London Edinburgh and Dublin Philosophical Magazine and Journal of Science 17, 419-422.
[2] Raun, R.L., Beckstead, M.W., Finlinson, J.C. & Brooks, K.P. (1993) A review of Rijke tubes, Rijke
burners and related devices. Progress in Energy and Combustion Science 19, 313-364.
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