Thermo-Acoustic System Modelling and Stability Analysis

CHAPTER 1

THERMO-ACOUSTIC SYSTEM MODELLINGAND STABILITY ANALYSIS –CONVENTIONAL APPROACHES

1.1 INTRODUCTION

In science and technology, the term instability characterizes situations where small(”input”) perturbations of initial or boundary conditions result in a violent re-sponse, such that system (”output”) variables change drastically or grow to verylarge amplitudes. After a transition period, a new system state is established, whichis usually quite di!erent from the initial state. Indeed, a complete breakdown ofthe initial state is often observed. Many instabilities involve large-amplitude spatio-temporal oscillations of the system variables; in this case a so-called limit cycle maybe established as a final state.

A wide variety of instabilities are known in mechanics, hydrodynamics, plasmaphysics, geophysics, control theory, aeronautics, or even economics. Well-knownexamples are acoustic feedback in public address systems (electro-mechanical feed-back), the collapse of the Tacoma Narrows bridge (aeroelastic flutter), the oscilla-tions of the London Millennium bridge (an instability involving ”positive biofeed-back”), the formation of large scale vortices in shear layers (Kelvin-Helmholtz in-stability; Karman vortex street), blade flutter or surge and stall in the compressorof gas turbines (aeroelastic flutter or pressure rise / mass flow instability), etc.

Thermo-acoustic instabilities arise primarily from an interaction of acoustic wavesand unsteady heat release. In general thermo-acoustic instabilities are undesirable,because they can produce excessive noise, limit operational flexibility and can evenresult in structural damage. This is particularly true for combustion instabilities,

Nonnormality in Fluid Systems. By XXXISBN A-BBB-CCCCC-D c!2009 John Wiley & Sons, Inc.

1

Invited lecture presented at workshop ''Advanced Instability Methods'' IIT Madras, Chennai, India, January 2009

WOLFGANG POLIFKE

2 THERMO-ACOUSTIC SYSTEM MODELLING AND STABILITY ANALYSIS –CONVENTIONAL APPROACHES

which are a cause for concern in applications as diverse as small household burners,process heaters, gas turbines, or rocket engines.

Unlike other flow instabilities, thermo-acoustic instabilities are not a local phe-nomenon in the sense that stability properties are not determined solely by thedynamics of the heat source and the flow field in its immediate surrounding. In-stead, acoustic waves travel back and forth across the extent of the system, andas a consequence acoustic boundary conditions far away from the heat source – atthe top of the chimney, say – may strongly influence stability properties. In (pre-mixed) flames, convective transport of fuel inhomogeneities from the fuel injectorto the flame, or of entropy inhomogeneities (”hot spots”) from the flame to theexhaust, are other non-local phenomena which influence system stability. All thesee!ects are linked together in feedback loops, as illustrated in Fig. 1.1. for a premixcombustor. It should be obvious that the complete and accurate description of athermo-acoustic instability can be a daunting task, since the system under consid-eration may involve a variety of fluid-dynamic and physico-chemical phenomena,covering a wide range of space and time scales.

Whether the interaction of the fluctuations of heat release rate, pressure, velocity,fuel concentration, etc., actually gives rise to an instability depends in an essentialmanner on the relative phases, and in particular on the phase between fluctuationsof heat release and pressure at the heat source. This has been known since Rayleigh(1878); the stability criterion named after him is discussed in the next section, wherethe basic ”physics” of thermo-acoustic instabilities is outlined. It is also shownwhy the Rayleigh criterion represents a necessary, but not su"cient criterion forinstability.

Heat Release Fluctuations

Turbulence Flame

Combustion Chamber

Acoustic Waves

Air / Fuel Supply

Equivalence Ratio Fluctuations

Coherent Structures

Burner Pressure Loss

Volume Flow Velocity

Pressure Entropy waves

Figure 1.1. Interactions between flow, acoustics and heat release in a combustor as anexample of a thermo-acoustic system which may exhibit instability.

In Section 3, an overview of various methods for stability analysis is given. Sinceresearchers in thermo-acoustics often come from di!erent backgrounds, it shouldnot come as a surprise that a wide variety of methods have been proposed. Anintroduction to possible alternatives for modelling ”the system” comes next, rangingfrom use of full-scale computational fluid dynamics (CFD) to simple, low-ordernetwork models as they are popular in duct acoustics. These models are somewhatlimited in geometrical flexibility and may seem simplistic – nevertheless they often

THE PHYSICS OF THERMO-ACOUSTIC INSTABILITIES 3

produce non-trivial results and are most helpful to develop insight into instabilitymechanisms.

More details on network models are given in Section 4. It is shown how a math-ematical description of network model elements may be derived from fundamentalprinciples for simple geometries. A few example computations obtained with sim-ple network models are presented and discussed. In the last section CFD-basedapproaches are discussed in more detail.

In these notes, it has not been attempted to give reference to, let alone discussin adequate detail all publications which have contributed to the development ofthe field. Suggestions for further reading are given in the short review sections atthe end of some chapters – but the selection is admittedly not free from personalbias and features the work of the combustion dynamics groups at ABB CorporateResearch and TU Munchen more prominently than would be adequate for a properreview of the subject.

1.2 THE PHYSICS OF THERMO-ACOUSTIC INSTABILITIES

Consider flow of a gas past a heat source, e.g. a premix or di!usion flame, or asurface with convective heat transfer to the gas (the so-called ”stack” in a thermo-acoustic machine, or the wire in a Rijke tube, see below.). Mass conservation insteady state requires that

(!u)c = (!u)h. (1.1)

With the density !h on the downstream side (index h for ”hot”) being lower thanthe density of the upstream side (index c for ”cold”), the velocity u, i.e. the volumeflux, must increase across the heat source. Now, if the rate of heat released by thesource fluctuates, the volume ”produced” will also fluctuate, thereby generatingsound1 – just like a loudspeaker box with its oscillating membrane.

The heat release rate in a flame, say, may be perturbed by turbulent fluctuationsof the velocity field upstream of the flame front. This gives rise to combustion noise,e.g. a camping burner or a blow torch which ”hisses” or ”roars”. Combustion noiseoften exhibits a broad band frequency distribution, which derives from the sizedistribution of the turbulent eddies perturbing the flame. Combustion noise mayalso be generated by fairly large scale, vortical coherent structures, originating fromhydrodynamic instability of the base flow (e.g. a shear layer or swirling flow). Inany case, if one speaks of combustion noise, it is usually implied that there is nosignificant feedback from the sound emitted back to the flow fluctuations whichperturbed the heat release in the first place.

However, if the heat source is enclosed in a chamber that acts as an acousticresonator, sound will in general be reflected back to the flame such that a feedbackloop is established (see again Fig. 1.1.). If the phase between the sound field estab-lished in the chamber and the fluctuations of heat release is just right, a self-excitedfeedback instability may occur. Then small (infinitesimal) perturbations are am-plified ever more, until eventually some kind of saturation mechanism kicks in anda limit cycle is established. For saturated thermo-acoustic combustion instabilities,limit cycle velocity fluctuations often exceed the mean flow velocities, amplitudes of

1Note that sound production by heat release with its monopole character is a very strong soundsource and – if present – usually dominates other source processes.


!

"

#$

%$

&$

#'

&'%'

(

(

(

! )*+

,

,

#$%$

&$

-

-

-

#' %'

&'

Figure 1.2. Thermodynamic interpretation of Rayleigh’s stability criterion. Left: p-v diagram of isentropic compression / expansion ( —— ), with heat addition in-phasewith pressure fluctuations (1-2’-3’-4’) and out-of phase (1-2”-3”-4”). The dashed gray linesrepresent lines of constant entropy. Right-top: Fluctuations of pressure ( —— ) and velocity( - - - ) in a standing wave, Right-middle/bottom: in-phase / out-of-phase fluctuations ofheat release rate.

pressure fluctuations can reach more than 120 dB in atmospheric flames, and sev-eral MPa in rocket engines. Damage to the combustion equipment can then resultvery quickly due to excessive mechanical or heat loads. If such thermo-acousticinstability occurs, the frequency spectrum of the resulting pressure and velocityoscillations typically exhibit one or several distinct peaks, with frequencies often(but not necessarily) close to the acoustic eigenfrequencies of the enclosure (or thecomplete combustion system) without unsteady heat release.

Rayleigh (1878) has proposed a criterion which tells us when the phases betweenthe sound field and the heat release fluctuations are ”just right”: For instability tooccur, heat must be released at the moment of greatest compression. A more generalformulation of the Rayleigh criterion states that a positive correlation betweenthe fluctuations of heat release and pressure, respectively, is required for thermo-acoustic instability to be possible:

!p! Q! dt > 0. (1.2)

Here the integration runs over one period of the oscillation, p! denotes fluctuationsof pressure at the heat source, Q! the fluctuations of the heat transfer (or release)rate. There is also a straightforward generalization of this ”Rayleigh integral” forspatially distributed heat release rate, which includes a spatial integration

". . . dV

over the volume where heat is released.The Rayleigh criterion (1.2) may be made plausible by analogy with a thermo-

dynamic cycle. Consider a small volume of gas, which is compressed and expandedby a standing acoustic wave. Sound waves are isentropic, so in a p-v diagram thevolume moves back and forth on an isentrope (see the line in the diagram on theleft side of Fig. 1.2.). What happens if heat is added and extracted periodically tothe gas? Well, addition of heat will result in a comparative increase of the specificvolume v of the gas, and if the heat addition is in-phase with pressure fluctuations,the state of the gas volume moves clockwise around a thermodynamic cycle (curve1-2’-3’-4’ in the diagram). So this is a ”thermo-acoustic heat engine”, which feedsmechanical energy into the sound wave – and a self-excited instability may occur,

THE PHYSICS OF THERMO-ACOUSTIC INSTABILITIES 5

as Rayleigh suggested. If fluctuations of heat addition are not perfectly in phasewith pressure fluctuations, then the area of the cycle 1-2’-3’-4’ will be smaller, andthe e"ciency of the engine will decrease. In the extreme case where fluctuations ofheat release rate Q! are opposite in phase with p!, the system moves counterclock-wise through the cycle 1-2”-3”-4” – and mechanical energy is extracted from theacoustic wave.

The mechanical work performed by the thermodynamic cycle resulting fromacoustic perturbations with heat release illustrated in Fig. 1.2. may be formulatedas follows:

!p! dv! = ! v

"p

!p! dp! +

!p! dv!(Q) = 0 +

!p!

dv!(Q)

dtdt "

!p! Q! dt. (1.3)

Here the changes v! in specific volume have been split into an isentropic part, forwhich v! = !vp!/"p (where " is the ratio of specific heats), and a part v!(Q) whichis due to heat addition (or removal). The rate of change of this term with time isproportional to fluctuations of the rate of heat addition. One may conclude thatthe work done by the ”thermo-acoustic engine” is positive (energy is added to theacoustic field), if the integral of p! Q! over one period of oscillation is positive – justas Rayleigh has argued.

Rayleigh’s criterion is ”necessary, but not su"cient” for instability to occur2.This means that a system is guaranteed to be thermo-acoustically stable, if thestability is not fulfilled – but if the criterion holds, it is not guaranteed that aninstability will indeed develop. The reason for this limitation is the following: ifthe criterion is fulfilled, Rayleigh’s thermo-acoustic engine feeds mechanical energyinto the acoustic field. However, oscillation amplitudes will only grow, if losses ofacoustic energy, which may occur elsewhere in the system, do not exceed the rateof energy generated by the fluctuating heat source. More complete stability criteriahave been developed, they are discussed in Section 1.3.

Let’s conclude this section with an important observation: Rayleigh’s criterion isstated in terms of fluctuations of heat release rate and pressure at the heat source.Nevertheless, acoustic boundary conditions far away from the heat source can andin general will determine stability properties in a decisive manner. Why is that?To answer this question, one must consider that convective heat transfer rates aswell as the heat release rate in a premix flame usually respond much strongerto changes in velocity u! than in pressure p!. The phase between fluctuations ofpressure p! and u! at the flame is determined by the acoustics of the system, e.g. theacoustic impedance at the boundaries, and the wave pattern – perhaps with partialreflections and transmissions – inside the enclosure. As a very simple example,consider the fundamental #/2-mode in a straight duct of length L with two openends and a compact heat source at position x = L/4 as shown in Fig. 1.3.. Such asystem has been known to exhibit thermo-acoustic instability for a long time andis known as a”Rijke tube” Rayleigh (1878); McManus et al. (1993); Heckl (1988).Acoustic fluctuations of velocity in the left half of the tube are opposite to those inthe right half of the tube (velocity u! is ahead of pressure p! by $/2 in the left halfand lags behind by !$/2 in the right half). If we assume that a heat source placedin the tube responds to changes in velocity, then clearly the sign of the Rayleigh

2Other stability criteria found in the literature, which are based in a similar manner only on phaserelationships, su!er also from this limitation.


!"

# $% # $%

& ' ( )

Figure 1.3. Rijke tube of length L with two ”open end” boundary conditions (inlet ”i”,exit ”x”) and a compact heat source Q at xc = L/4 with xh ! xc " L. Distribution offluctuating velocity and pressure / density at one particular instant during the oscillationare indicated by arrows and shading, respectively.

TECHNISCHE UNIVERSITÄT MÜNCHEN

Growth rate of instabilities

Q = 385 ± 7 W; vmean = 0.0218 ± 0.0002 m/s ( )1ft

gr+

Figure 1.4. Time trace of pressure in an electrically heated Rijke tube. After a limitcycle is established, a plug is pulled from the tube, such that the instability collapses.

Integral#

p!Q!dt will depend on the position of the heat source: a heat source,which gives rise to self-excited instability when placed in the left half of the tube,will not go unstable when placed in the right half, and vice versa. This is explainedin more detail in the following.

It follows that it is always the combination of heat release dynamics and systemacoustics which controls stability. System acoustics determines largely both theimpedance at the heat source (the phase between p! and u!) as well as the total lossesof acoustic energy (dissipation inside the system, also radiation to the environmentat the boundaries). For this reason stability analysis without a model for theacoustics of the enclosure – a system model – is in general not possible for thermo-acoustic systems.

STABILITY ANALYSIS 7

1.3 STABILITY ANALYSIS

A system (or a system state) may be called stable against a perturbation, if sometime after the perturbation is imposed, the initial system state is re-established.It follows quite naturally from this definition that in order to investigate the sta-bility properties of a system, one must determine the response of the system toperturbations.

Often it is fairly easy to compute the response to perturbations with infinites-imally small amplitudes. In this case the governing equations may be simplifiedby neglecting the so-called non-linear terms, i.e. terms which are higher order inperturbation amplitudes. A system is accordingly called linearly stable, if it returnsto its initial state after a slight perturbation. Note that it is not specified whatkind of perturbation is imposed, except that the amplitude be small in some sense.

In linearly unstable systems, a dominant or most unstable mode – i.e. a distinctpattern of vibration with a particular frequency % – develops in the very early stagesof the instability. This most unstable mode has – as the term implies – a growth ratethat is larger than that of any other unstable mode. After a while, the most unstablemode dominates the behaviour of the system such that the whole system oscillatesat the frequency % of the mode. In this stage, the frequency spectrum typicallyshows a single peak at the frequency %. The oscillation amplitudes keep growingexponentially, until saturation of amplitude sets in and eventually a limit cycle,i.e. oscillation with large, but finite amplitudes, is established. Significant growthof amplitudes is associated with linear instability, correspondingly the departurefrom exponential growth is due to non-linear terms, i.e. those terms of higher orderin oscillation amplitudes, which were neglected in the linearized analysis. Duringthe non-linear phase of the evolution of the instability, the frequency spectrumtypically shows more than one peak – often a dominant peak at the frequency %and lower-amplitude, but still rather distinct peaks at integer multiples n % of thefundamental frequency (harmonics).

This scenario of linear instability # growth of dominant mode # non-linearsaturation# limit cycle is a well established paradigm in instability theory (Drazin2002; Keller 1995; Dowling 1995). Experimental results for a Rijke tube obtainedby Lumens (2006) are shown in Fig. 1.4.. Nevertheless, there is evidence that thisscenario does not always apply. For example, some linearly stable thermo-acousticsystems operate stably for a long time, but exhibit strong instability after su!ering astrong external perturbation. This kind of non-linear instability behaviour is calledtriggering. It has been observed in thermo-acoustic systems ranging from Rijketubes to rocket engines (Culick 1989, 1994; Flandro et al. 2007). More recently, therelevance of non-normal e!ects for thermo-acoustic instabilities has been broughtto attention by Balasubramanian and Sujith (2007a, b).

Furthermore, many combustion systems, while not exhibiting a strong thermo-acoustic instability, show significant fluctuation levels (of pressure, velocity, heatrelease rate) over a band of frequencies (often in the vicinities of combustor eigen-frequencies). Such a state should perhaps not be described as the limit cycle of aninstability, but rather as resonant amplification of combustion noise by the com-bustion chamber.


1.3.1 Unsteady simulation

Probably the most straightforward method to assess the stability of a thermo-acoustic system makes use of unsteady simulation in the presence of small (initial)perturbations: If the perturbations grow in amplitude with time, the system isunderstood to be unstable, and stable otherwise. Provided that non-linear e!ectsare taken into account properly by the simulation model – which is in principlepossible for many models in use – a limit cycle is established once the simulationis continued for a su"ciently long time,

As an example of this strategy, Fig. 1.5. shows the temporal development of thevelocity just upstream of the heat source in a generalized Rijke tube, i.e. a resonatorwith closed-open boundary and heat source with time lag placed in the center. Thecomputational model used to generate the data shown in Fig. 1.5. was based onsimplified 1-D conservation equations for mass, energy and momentum in laminarcompressible flow. ”Lumped parameter” source terms for energy and momentumwere used to model the heat source, see Polifke et al. (2001b). Obviously, thesimulation exhibits instability: the oscillation amplitudes grow until a limit cyclewith negative velocities, i.e. reverse flow past the heat source during part of theoscillation cycle, is established3.

The use of unsteady simulation for stability analysis is conceptually a straight-forward approach, and has been used by a number of authors (Janus and Richards1996; Dowling 1997; Veynante and Poinsot 1997; Peracchio and Proscia 1998; Po-lifke et al. 2001b; Pankiewitz and Sattelmayer 2003b; Sung et al. 2000). This doesnot imply, however, that this approach is easy to implement or generally superiorto alternative strategies. The following drawbacks may be identified:

• Unsteady simulation is a ”brute force” approach which can be computation-ally very expensive. This is in particular the case if a computational fluiddynamics simulation model for (turbulent, reacting), incompressible flow isused, i.e. the unsteady Reynolds-averaged Navier-Stokes (URANS) or LargeEddy Simulation (LES).

• In general, only the most unstable mode is identified. Stable modes, or modesthat are unstable, but have smaller growth rates than the most unstable mode,cannot be observed. In this sense, the stability assessment based on unsteadysimulation is not comprehensive.

• In some instances, it is not easy to discern a spurious numerical instabilityfrom a genuine physical instability. A related problem is that it can be di"-cult to numerically generate the initial state, from which the instabilities aresupposed to develop.

• Results can depend on initial conditions, i.e. details of the initial perturbation(typically a random perturbation of very small amplitude) can influence whichmode develops. For example Evesque et al. (2003), a Finite-Element basedmodel of linear acoustics, showed that for non-plane acoustic waves in anannular combustor, the initial conditions can determine whether a standing or

3It is straightforward to interpret this result: if hot gases are swept back to the heat source,the temperature di!erence and therefore also the heat transfer between the hot wire and the gasdecreases, thereby reducing with increasing fluctuation amplitude the gain of the transfer functionand the Rayleigh integral until the instability saturates.


30

20

10

0

-10

vc [

m/s

]

0 .70 .60 .50 .40 .30 .20 .10 .0

t [s]

Figure 1.5. Temporal development of velocity vc upstream of the heat source in ageneralized Rijke tube. Time lag ! of the heat source was adjusted to fulfill the Rayleighcriterion.

a rotating mode develops. Further analysis showed that rotating modes withclockwise and counterclockwise sense of rotation are degenerate eigenmodesfor this problem. This explains the sensitivity against initial conditions inthis particular case.

• In general, the acoustic impedance at the boundary of the computationaldomain is a complex-valued function Z(%) which depends on frequency %.The implementation of appropriate acoustic boundary conditions in the timedomain is a non-trivial problem. See Schuermans et al. (2005); Huber et al.(2008) for recent advances on this problem in the context of CFD-based mod-els of thermo-acoustics in the presence of mean flow.

1.3.2 Determination of eigenmodes and eigenfrequencies

The stability of a system can be determined by identifying the eigenmodes and inparticular the eigenfrequencies of the system. This classical method, known also asdynamic stability analysis, is very popular and can be applied to a wide variety ofproblems.

What is an eigenmode and an eigenfrequency? The German prefix Eigen canbe translated as own, peculiar to, characteristic or individual. An eigenfrequencyis then a frequency that is easily excited in a system. Once excited, the system,left to itself, will continue to oscillate for some time at that frequency. The corre-sponding pattern of vibration, e.g. the spatial distribution of nodes and anti-nodes,is identified as the shape of the eigenmode. Typically, a dynamic system has morethan one – even infinitely many – eigenmodes and corresponding eigenfrequencies.

Trying for a bit more mathematical rigor, the eigenvectors &xm of an operator Aare those vectors which are left unchanged under the operator up to factors 'm,the eigenvalues of the operator:

A &xm = 'm &xm. (1.4)

This should be familiar from linear algebra. The notions of ”eigenvector” (or ”eigen-function” or ”eigenmode”) and ”operator” may be interpreted in a rather generalsense. For example, consider acoustic oscillations of pressure p! in a duct of length


L with two open end boundary conditions p!(0) = p1(L) = 0, obeying the 1DHelmholtz-equation,

1c2

(2p!

(t2! (2p!

(x2= 0. (1.5)

The eigenmodes are the well-known standing waves

p!(x, t) " exp(i%mt) sin(kmx).

The wave numbers km, which are restrained to the values km = n$/L in orderto fulfill the boundary conditions, determine the shape of each eigenmode. TheHelmholtz-equation then is re-written as

(2p!

(t2= !%2

m p!, (1.6)

for the eigenfrequencies %m = kmc. When formulated in this way, the analogy with(1.4) should be obvious: The eigenmodes correspond to particular oscillations ofthe gas column in the duct, such that the second derivative of pressure with respectto time is simply the shape of the pressure distribution scaled by a factor. Thisfactor, the eigenvalue, turns out to be !%2

m, where %m is the angular frequency ofthe oscillation. The fact that the time derivative operator is Hermitian4 leads toseveral important properties, such as that the standing wave patterns are orthogonalfunctions.

For ideal systems without dissipative e!ects, an eigenmode oscillates withoutdecay, corresponding to a purely real eigenfrequency % $ R. If damping (due toviscous losses or radiation losses at the system boundary) is non-negligible, oscil-lation amplitudes decrease in time. If amplitude levels are small, non-linear termsmay be neglected, then an exponential decrease of amplitudes is typical. Assumingharmonic time dependence " exp(i%t) an eigenfrequency %m with positive imagi-nary part %(%m) > 0 describes this situation5. Unstable modes in systems whichcan exhibit self-excited instability – this is of course the case most interesting inthe present context – are characterized by eigenfrequencies with negative imaginarypart %(%m) < 0.

A growth rate or cycle increment of an eigenmode m can be derived from itseigenfrequency:

# & exp$!2$

%(%m)'(%m)

%! 1. (1.7)

The growth rate indicates by which amplitude ratio a mode increases or decreasesper cycle. For example, a growth rate of # = 0.2 indicates an increment of theamplitude of 20% per cycle.

To illustrate this method of stability analysis, it is shown in the next subsec-tion how the eigenfrequencies of a Rijke tube can be determined from a low-orderthermo-acoustic model.4In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one thatis its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix isone which is equal to its own conjugate transpose. By the finite-dimensional spectral theoremsuch operators have an orthonormal basis in which the operator can be represented as a diagonalmatrix with entries in the real numbers (from Wikipedia).

More details on ”Sturm-Liouville theory” (the theory of linear di!erential equations, di!erentialoperators and their eigenvalues / eigenfunctions) are found in many mathematics text books.5Note that often the opposite convention " exp(#i!t) for the time dependence is chosen. In thiscase, a negative imaginary part indicates a decaying eigenmode.


1.3.2.1 Eigenfrequencies of a Rijke tube The Rijke tube is a simple, yet interestingexample of a system that can exhibit self-excited thermo-acoustic instability. It hasbeen discussed previously by many authors for its ”pedagogical” value (McManuset al. 1993; Dowling 1995; Polifke et al. 2001b; Polifke 2004). Here it is shown howthe eigenfrequencies and thereby the stability characteristics of a Rijke tube canbe determined from the characteristic equation (also dispersion relation), which inturn is derived from an approximate low-order model of the Rijke tube.

We consider a system as shown in Fig. 1.3., i.e. a heat source placed in a duct.It is assumed in the following that there is an acoustically closed end (u! = 0) atthe left boundary. This little trick simplifies the analysis a bit. Also, the position ofthe heat source is not at position xc = L/4, as it is for a ”real” Rijke tube. Instead,the length of the duct to the left of the heat source is denoted as lc, while to theright we have lh = L! lc.

It is assumed that the reader is familiar with basic acoustics, so instead of starting”from scratch”, we just introduce some notation and go into medias res quickly:Riemann Invariants f and g (also called p+ and p" by some authors) are relatedto the primitive acoustic variables as follows,

f =12

&p!

!c+ u!

', g =

12

&p!

!c! u!

'. (1.8)

One may think of these ”Riemann invariants” (also called ”characteristic waveamplitudes”) simply as acoustic waves propagating in the down- and upstreamdirection, respectively6.

Now consider the length of duct to the left of the heat source, i.e. the ”cold”part of the Rijke tube. A downstream traveling wave f undergoes a change in phaseexp{i%lc/cc} as it travels with the speed of sound c across the distance lc = xc!xi

from the inlet to the cold side of the heat source. Similar for the wave g travelingin the upstream direction. Along the length of the duct, there is no interactionbetween the waves f and g. From these coupling relations the transfer matrix ofthe duct is readily obtained:

&fc

gc

'=

&e"ikclc 0

0 eikclc

' &fi

gi

'. (1.9)

with a wave number kc & %/cc for plane waves without mean flow and dissipativee!ects. For the ”hot” duct to the right of the heat source, the transfer matrix is ofthe same form, but of course c =

("RT and l may be di!erent, so the coe"cients

of the transfer matrix may be di!erent, too.For the heat source (which is a wire mesh or ”gauze” in a Rijke tube), one

observes that the pressure drop across the heat source is negligible for su"cientlysmall mean-flow Mach number, p!c = p!h, while for the velocity we assume that

u!h(t) = u!c(t) + nu!c(t! )). (1.10)

This is the famous n ! ) model for a heat source with time lag. This simplemodel has played an important role in the development of the theory of combustion

6A note on nomenclature: we make in these notes no explicit distinction between acoustic variablesin the time domain and in the frequency domain. Most of the time we are in frequency space,and the f ’s and g’s are to be understood as complex-valued Fourier transforms of time series.


instabilities in rocket engines, is used frequently as a pedagogical example (as inthese notes), and is even relevant as a building block for models of ”real-world”turbulent premix flames. One can show (see e.g. Polifke (2004); Poinsot andVeynante (2005)) that the second term on the r.h.s. appears if the heat release ofa flame (or some other source of heat) responds to a change in velocity u with acertain time lag ) . The important thing about the time-lag is that it may allow somephase-alignment between fluctuations of pressure and heat release, respectively,even with standing acoustic waves (where the phase between p! and u! is ±$/2).So, for given impedance Z = p!/u! at the heat source, the time lag ) in combinationwith the frequency % controls the sign of the Rayleigh integral and therefore systemstability. For the interaction index n, one finds under certain assumptions – whichhold true for the gauze of a Rijke tube, but not necessarily for a premix flame –that n = Th/Tc ! 1, i.e. n is determined by the increase in mean temperature Tacross the heat source.

In terms of the Riemann invariants, the coupling relations across the heat sourceare

!hch

!ccc(fh + gh) = (fc + gc), (1.11)

fh ! gh =(1 + n e"i!"

)(fc ! gc). (1.12)

At the left boundary u! = 0 and therefore fi!gi = 0, while at the right boundary,p! = 0 and correspondingly fx + gx = 0. This completes the construction of a low-order model for the Rijke tube: there are eight equations (2 boundary conditions,2)2 equations from the ducts on both sides of the heat source, and 2 equations fromthe heat source) for the eight unknowns fi, gi, . . . gx. In matrix & vector notation,this reads *

+Matrix

ofCoe"cients

,

-

*

.+fi...

gx

,

/- =

*

.+0...0

,

/- . (1.13)

As indicated, the system is homogeneous (the right hand side is a vector with all0’s), so a non-trivial solution exists only if the determinant of the system matrix Svanishes, i.e. if

Det(S) = 0. (1.14)

How can this condition be satisfied? Inspection of the coupling relations (1.9)- (1.12) shows, that the coe"cients of the system matrix depend on geometricaland physical parameters (length l, speed of sound c, density ! in the ducts plusthe interaction index n and the time lag )), which are fixed for a given system.However, the frequency %, which appears in several of the coe"cients, is not fixeda priori, so we may hope to find one or indeed several eigenfrequencies %m of thesystem such that Det(S)|!=!m

= 0. In this sense Eq. (1.14) is the equivalent tothe characteristic equation Det(A! 'I) = 0 in linear algebra.

For the Rijke tube with * & !hch/!ccc = cc/ch,

Det(S) = 4{cos(kclc) cos(khlh)! * sin(kclc) sin(khlh)(1 + n e"i!"

)}, (1.15)

see McManus et al. (1993) and Appendix 1.7 for the derivation.The characteristic equation Det(S) = 0 is a transcendental equation, i.e. it

cannot be solved explicitly for eigenfunctions %m. In general one has to resort


1 2

m=0

m=1

m=2

m=3

3 !0

Figure 1.6. Stability map of the first four modes in a ”Rijke” tube with one closed endand weak coupling n # 0. Gray regions indicate stability, i.e. positive imaginary part ofthe eigenfrequency "m.

to numerical root finding to determine the eigenfrequencies (see the Appendix fora simple implementation in Matlab). An approximate analytical solution for thespecial case lc = lh, !h = !c and cc = ch = c has been derived by McManus et al.(1993). Introducing non-dimensional variables with L and L/c as characteristiclenght- and time-scales, respectively, the characteristic equation (1.15) reduces forthis configuration to

cos(%)! n e"i!" sin2(%/2) = 0. (1.16)

For n = 0, i.e. in the absence of thermo-acoustic coupling between velocity andheat release at the gauze, the solutions to this equation are the familiar quarter-waveduct eigenmodes with wave lengths # = L/4, 3L/4, . . . and frequencies

%(0)m =

$

2,3$

2,5$

2, . . . = (2m + 1)

$

2; m = 0, 1, 2, . . . . (1.17)

These eigenfrequencies are real-valued, so there is no amplification or damping ofthe eigenmodes.

For non-zero, but small interaction index n * 1, the eigenfrequencies can bedetermined approximately as %m + %(0)

m + %!m, where the deviations %!m from theeigenfrequencies %(0)

m are assumed to be small, %!m * %(0)m . Then, to first order in

%!m,

cos(%(0)m + %!m) + (!1)m%!m,

sin2(%(0)m + %!m) + 1

2+

(!1)m

2%!m.

Inserting this in Eq. (1.16) and retaining only terms that are 1st order in n or %!mone obtains

%!m + (!1)m+1 n

2e"i!m" or (1.18)

%m + %(0)m + (!1)m+1 n

2e"i!(0)

m " , (1.19)

where %(0)m = (2m + 1)$/2, see Eq. (1.17). Obviously, coupling between acoustics

and heat release (non-zero interaction index n) results in complex-valued eigenfre-quencies, i.e. and eigenmodes that asymptotically grow or decay in time. Adoptingthe convention " exp (i%t) for the time dependence, an eigenmode is unstable if


0.5 1 1.5 2 2.5 3!

1.45

1.5

1.55

1.6

1.65

1.7

1.75

Re!w"

0.5 1 1.5 2 2.5 3!

-0.15

-0.1

-0.05

0.05

0.1

0.15

0.2

Re!w"

Figure 1.7. Comparison of eigenfrequencies determined with numerical (——) andapproximate analytical solutions (dashed lines) of the dispersion relation (1.16) withinteraction index n = 0.3 for the fundamental mode m = 0. With increasing n, m thedi!erences between numerical and analytical solutions will increase (not shown).

the imaginary part of its eigenfrequency is negative,

%(%m) + (!1)m n

2sin(%m)) < 0, (1.20)

because in this case the amplitude grows with time as exp (!%(%m)). The stabilitylimits for the time lag ) resulting from this criterion for eigenfrequencies accordingto Eq (1.19) are shown in Fig. 1.6. for modes 0 to 3. For almost any value ofthe time lag ) , one or several of the first four eigenmodes is unstable. In a morerealistic model, viscous losses and especially radiation of acoustic energy to theenvironment at the open end should be taken into account. Both e!ects tend toincrease stability, and they both tend to increase with frequency, so that the widthof the stable regions in Fig. 1.6. would increase, and especially so for the highermodes (Indeed, in a real Rijke tube, one usually observes only the fundamentalmode m = 0). Note that the Rayleigh criterion, which cannot take into accountthe beneficial e!ect of losses on system stability, would then be overly pessimistic.

The real part of %!m shows that the frequency of an unstable mode will in generaldi!er slightly from the eigenfrequency of a corresponding stable mode (with n = 0):

'(%!m) + (!1)m+1 nc

2Lcos %m), (1.21)

This is due to the phase change introduced by the time-lagged response of the heatsource.

It is not di"cult to solve the dispersion relation numerically for larger values ofn (and indeed for arbitrary values of the parameters l, c, n, ) as well as arbitraryreflection coe"cients at the duct ends). A simple Mathematica script is given inthe Appendix, results are shown in Fig. 1.7..

1.3.2.2 Further remarks on dynamic stability analysis

• What has been done here ”by hand” for the simplistic model of a Rijke tubecan be formalized and cast into a numerical algorithm – a network model.With this approach, the thermo-acoustic system is modelled as a collectionof network elements. Each network element is represented mathematicallyby its transfer matrix (or scattering matrix ), derived from coupling relationsakin to Eq. (1.9). With given network structure and a library of network


!"#$%# &'()% *+),"-.+#/0#12"33'4

&"%'12"33'4

Figure 1.8. Network model of a combustion system.

elements, the system of equations (1.13) (or the system matrix) representingthe complete network may then be generated automatically by the computerwithout any ”paper and pencil” work. This can be done for networks ofarbitrary topology, representing combustion systems with fuel and air supplyducts, cooling channels, etc. as suggested in Fig. 1.8.. Of course, the searchfor eigenfrequencies is then also performed numerically.

• The computation and inspection of eigenfrequencies as presented above for theRijke tube is compatible with Rayleigh’s criterion: If one determined from anetwork model the relative phases between fluctuations of pressure p! and heatrelease Q! at the heat source and evaluated the Rayleigh integral (1.2), onewould obtain results in complete agreement with Eq. (1.20). However, this isonly so because we neglected losses and assumed ideal boundary conditionswith acoustic reflection coe"cients |r| = 1. Because loss mechanisms caneasily be taken into account with a low-order model, the dynamic stabilitycriterion is more powerful than Rayleigh’s criterion.

• The dynamic stability criterion is in principle applicable to any homogeneouslinear system. However, for complicated systems, the roots of the charac-teristic equation (1.14), i.e. the eigenfrequencies %m, can in general not bedetermined analytically, because the system of equations (1.13) is simply toolarge for paper-and-pencil work. Instead, iterative numerical algorithms forroot-finding are required. The iterative search for the roots gives rise to anumber of problems: In order to assure that a system is stable with respectto self-excited thermo-acoustic oscillations, it is necessary (and su"cient) toverify that all eigenfrequencies %m are located in the upper half of the com-plex plane. However, numerical root-finding always starts from an initialguess and moves iteratively by ”trial and error” towards a root. The rootsmay be located anywhere in the complex plane; the basins of attraction ofthe various roots may be quite di!erent in size, and some roots may indeedbe very hard to find. So one can never be sure that one has really found alleigenfrequencies – but one unstable root that goes unnoticed may render anotherwise stable system unstable!

• Sometimes the coe"cients of the system matrix are known only for real-valued frequencies (e.g. if transfer matrices are determined by experiment orfrom computational fluid dynamics). In this situation, one cannot solve forcomplex-valued roots of the characteristic equation, because the determinantof the system matrix away from the real axis is not known.

In the next section, an alternative approach making use of Nyquist diagrams asthey are familiar from control theory is presented, which does not su!er from someof the shortcomings mentioned. In particular, iterative searches for eigenfrequenciesin the complex plane are not required with that approach.


G(s)

1

x1

x0

Figure 1.9. Open loop system with unit negative feedback.

1.3.3 The Nyquist Criterion

The Nyquist stability criterion is a well-known tool in control theory. It allows totest for stability of a closed-loop system by inspection of the Nyquist plot of theopen-loop transfer function (OLTF) (Jacobs 1993). In this section, a brief review ofthe method is given, then the application to thermo-acoustic systems is discussed.

1.3.3.1 Nyquist plots in control theory Consider a system as shown in Fig. 1.9.with open loop transfer function G(s) and unit negative feedback H(s) = 1, suchthat

x1 = G(s) (x0 ! x1), or x1 =G(s)

G(s) + 1x0. (1.22)

The characteristic equation, from which stability can be deduced, is then

G(s) + 1 = 0. (1.23)

The system is stable, if all the roots sn of this equation are located in the lefthalf of the complex plane. In that case, all perturbations of the system will decayexponentially " est with time.

To locate all the roots can be tedious, therefore alternative methods to assess thestability have been developed. For example, in control theory the transfer functionis usually a fraction of polynomials,

G(s) =PN (s)PD(s)

=ansn + . . . + a0

bmsm + . . . + b0. (1.24)

The Routh-Hurwitz stability criterion exploits this fact and allows to determinesystem stability from the polynomial coe"cients (Jacobs 1993).

Alternatively, one can deduce stability from the Nyquist plot, i.e. a polar plotof the imaginary axis7 mapped through the open loop transfer function. The asso-ciated Nyquist stability criterion is based on Cauchy’s argument principle, whichstates the following: Consider an analytical function f(z) with a number Z of zerosf(z) = 0 and a number P of poles f(z) # , within a simple closed contour C inthe complex plane. Then the winding number of the image curve of the contour Cmapped f(z) around the origin 0 + i0 is equal to Z ! P .

7Strictly speaking, the Nyquist plot is the image of the Nyquist contour, which is a closed contourencompassing the right half of the complex plane. Starting from #i$ one moves along theimaginary towards +i$ and then in a clockwise circular arc with radius r %$ back to the point#i$.


For the characteristic equation (1.23) and the Nyquist plot, Cauchy’s argumentprinciple implies that the number Z of zeros of the OLTF G(s) in the right halfof the complex plane is related to the number N of positive encirclements (i.e. inclockwise direction) of the ”critical point” (!1 + 0i) and the number of poles P ofG(s) as follows:

N = Z ! P. (1.25)

For a stable system, no roots of the characteristic equation (1.23) should be on theright side of the s-plane, i.e. Z = 0. Nyquist’s criterion follows with Eq. (1.25):for stability, the number N of anticlockwise encirclements about the critical point(!1 + 0i) must be equal to P , the number of open loop poles in the right half plane.

1.3.3.2 Application of the Nyquist criterion to thermo-acoustic systems It is by nomeans obvious how the Nyquist criterion can be employed to analyze the stabilityof thermo-acoustic systems. A network-model-based approach for generation of aNyquist plot has been proposed by Polifke et al. (Polifke et al. 1997; Sattelmayerand Polifke 2003a): To establish an analogy to control systems and to define theequivalent to the OLTF, the network model must be ”cut open”. This can be donefor networks of arbitrary topology by introducing a ”diagnostic dummy” elementin the network.

As indicated in Fig. 1.10., two of the four variables of this network elementare simply ”shortened out”, e.g. gd = gu, while the second pair of variables isnot connected at all. Instead, a fixed value is assigned to one of those ports, e.g.fd = 1. Note that by inserting the diagnostic dummy, the homogeneous systemof equations (1.13) changes into an inhomogeneous system S! &x ! = b with r.h.s.b = (0, . . . , 1, . . . , 0). Now one can define the OLTF of the thermo-acoustic systemwith diagnostic dummy as

G(%) = ! fu

fd

. (1.26)

The minus sign is introduced to maintain close analogy with negative feedbackcontrol systems.

In general, i.e. for arbitrary frequency %, the solution &x !(%) of the inhomoge-neous system with diagnostic dummy will not be equal to any of the eigenmodes ofthe original system Eq. (1.13). In this case the values of the unconnected variablesacross the diagnostic dummy will not be equal, i.e. fd -= fu. However, for everyeigenfrequency %n, the solutions &x !(%n) of the system with diagnostic dummy willbe identical to the corresponding eigenvector &xn of the original system (up to anarbitrary scaling factor), and the acoustic variables will match across the ”cut”,fd = fu. It follows that Eq. (1.26) defines a mapping, which maps every eigen-mode of the homogeneous system Eq. (1.13) to the critical point (-1 + 0i), i.e.G(%n) = !1 for every eigenfrequency %n. This important property of the OLTFwill be exploited in the following.

Although the equivalent to the open loop transfer function is now defined, theclassical Nyquist criterion, as it was formulated in the previous section, is not di-rectly applicable to a thermo-acoustic system: As already mentioned, low-ordermodels for thermo-acoustic stability analysis are commonly formulated with har-monic time dependence " ei!t, and the imaginary part of the angular frequency %determines stability. For a stable system, no eigenfrequencies %n must be located in


fdˆ

gdˆguˆ

fu 1

Figure 1.10. Diagnostic dummy

the lower half of the complex plane. This implies for the Nyquist plot, that the realaxis of the %-plane is mapped by the open loop transfer function to the G(%)-plane.

Once these modifications are taken into account, one could in principle proceedwith application of the criterion in complete analogy to control theory. However,transfer functions in thermo-acoustics are in general not polynomials in % or frac-tions thereof, but involve harmonic or exponential functions8. The identification ofpoles is in this case no easier than the determination of eigenfrequencies by iterativenumerical solution of the characteristic equation (1.23). Moreover, the coe"cientsof acoustic transfer matrices – the building blocks of network models – are notalways given in analytical form. In this case, the OLTF cannot be evaluated forcomplex-valued frequencies with imaginary part %(%) -= 0. These di"culties arealso discussed by Polifke et al. (1997); Sattelmayer and Polifke (2003a).

Therefore, a modified rule for the interpretation of Nyquist plots has been putforward, which is more suitable for application to thermo-acoustic systems (Polifkeet al. 1997; Sattelmayer and Polifke 2003a). The proposed rule is based on aproperty of analytical functions: An analytic function is conformal, i.e. it preserveslocal angles or ”handedness”, at any point where it has a nonzero derivative (Janich1983). Consider now the open loop transfer function G(%) as a conformal mappingfrom the %-plane onto the G-plane, see Fig. 1.11.. The real axis %(%) = 0 in the%-plane (left graph) and its image in the G-plane (right graph) are indicated by thethick dashed line with arrow head. According to Eq. (1.26), the eigenfrequencies(roots of the characteristic equation (1.23)) %n are mapped to the critical point(-1 + 0i). Because the conformal mapping % # G(%) preserves handedness, thecritical point will lie to the left (right) of the image curve of the real axis if thecorresponding root %n lies in the upper (lower) half of the complex %-plane. Thesituation shown in the Fig. 1.11. corresponds to an unstable mode.

These deliberations suggest the following modified Nyquist criterion: Considerthe image curve of the positive half of the real axis % = 0 # , under the OLTFmapping in the G(%)-plane, as shown in Fig. 1.11.. As one moves along the imagecurve in the direction of increasing frequency %, an eigenmode with eigenfrequency%n is encountered each time the image curve passes the critical point (!1 + 0i) (InFig. 1.11., only one sweep past an eigenfrequency is shown). If the critical pointlies to the right of the image curve, the eigenmode is unstable, because then itsfrequency %n is located below the real axis in the %-plane. On the other hand, ifthe critical point is located to the left, the eigenmode is stable. If the image curvepasses through the critical point, the mode is neutrally stable.

In comparison to the classical Nyquist stability criterion, it is perhaps easierto foster an intuitive understanding of the modified Nyquist rule. However, it isadmitted that no strict mathematical proof for the modified criterion is known. It

8Incidentally, for this reason application of the Routh-Hurwitz criterion to thermo-acoustic sys-tems is not possible.


!n

Figure 1.11. Conformal mapping " # G(") of the real axis under the OLTF for anunstable eigenfrequency.

has been validated successfully for a number of cases, where eigenfrequencies andgrowth rates can be determined by solution of the characteristic equation (Polifkeet al. 1997; Sattelmayer and Polifke 2003a). Nevertheless, one must concede thaterroneous predictions may be obtained if the derivative of the OLTF vanishes forsome real-valued frequency % $ R (the mapping is then not conformal). Further-more, if an eigenfrequency has a large imaginary part such that the OLTF curvepasses the critical point at a large distance, the proposed criterion may fail becauseconformality is a local property, i.e. it holds only in a finite-size neighborhood ofthe point considered. Fortunately, very large growth rates with %(%) * 0 are notobserved in realistic network models, while very large damping rates with %(%). 0correspond to strongly damped modes, which are of no concern for the overall sta-bility of a combustion system.

1.3.3.3 Identification of eigenfrequencies and growth rates from a Nyquist plot Con-formality, i.e. the local preservation of angles under a mapping f : z # f(z), impliesthat an orthogonal grid of lines with constant real or imaginary part, respectively,is mapped to an orthogonal grid of lines in the image plane (with the exceptionof points where the derivative of f is zero). In other words, under a conformalmapping, the neighborhood of any point is rotated and stretched or shrunk, asillustrated in Fig. 1.11. (Janich 1983).

This interpretation of conformality implies that it is possible to estimate thefrequency of an eigenmode as well as its rate of growth or decay from the imagecurve of the OLTF:

1. the real part of the eigenfrequency is approximately equal to the frequency %for which the distance between the image curve and the critical point attainsa local minimum.

2. the imaginary part %(%n) is approximately equal to the minimum distancefrom the critical point to the OLTF image curve divided by the scaling factorof the mapping.

The scaling factor can be roughly determined by evaluating how an interval ('(%n)!$%;'(%n)+$%), is mapped to a segment G('(%n)!$%)# G('(%n)+$%) of theOLTF image curve. The scaling factor is then estimated as the arc length dividedby 2$%.


A more precise way of deducing the growth rate from the image curve has beenproposed by Sattelmayer and Polifke, based on the identity theorem of functionalanalysis (Sattelmayer and Polifke 2003b). The theorem assures that a polynomial fitfor the OLTF, which approximates the transfer function G(%) with good accuracyfor a range of purely real frequencies %1 / % / %2, with %,%1, %2 $ R will locallyapproximate the OLTF also for complex-valued frequencies % $ C.

This consideration suggests to determine the growth rate of an eigenmode asfollows: generate a polynomial fit PG(%) = gm%m + . . .+g0 which approximates theOLTF curve close to the critical point, i.e. in the vicinity of an eigenmode %n, seeFig. 1.11.. Then determine with a numerical root finding algorithm the frequency%# for which the approximating polynomial PG(%#) = !1. If the eigenmode %n isnot too far away from the real axis, then %n + %#.

It is remarkable that in this way complex-valued eigenfrequencies of an acousticalsystem can be determined from the OLTF image curve, which is computed forpurely real frequencies % $ R. This is very convenient when analytical expressionsfor transfer matrix coe"cients are not known, which is usually the case for transfermatrices or response functions determined from experiment.

In concluding this section we remark that with the proposed rule for interpre-tation of Nyquist plots, it is obvious that the frequency at which the OLTF curvecrosses the real axis should not be identified with an eigenfrequency. Indeed, it hasbeen shown by example that this popular, but incorrect ”heuristic” version of theNyquist criterion does lead to erroneous predictions (Lundberg 2002; Sattelmayerand Polifke 2003a).

1.3.4 Acoustic energy budget

The physical interpretation of the Rayleigh criterion implies that correlated fluctu-ations of heat release and pressure generate acoustic energy, such that self-excitedthermo-acoustic instabilities may develop. On the other hand, loss of acousticenergy – through viscous dissipation, through transfer of acoustic energy to hydro-dynamic or evanescent modes, or through radiation losses at the system boundary– tends to stabilize a thermo-acoustic system.

Stability considerations based on a balance between generation and loss of acous-tic energy may appear to be an almost self-evident generalization of Rayleigh’scriterion9. However, quantitative methods for stability analysis based on this ideahave been proposed only recently (Ibrahim et al. 2006, 2007; Nicoud and Poinsot2005; Giauque et al. 2006, 2007).

Ibrahim et al. (2006, 2007) base their analysis of thermo-acoustic instability in ageneric premix combustor on an expression for the instantaneous density of acousticenergy,

e =!

2

0&p!

!c

'2

+ u!iu!i

1. (1.27)

The first term is the potential acoustic energy, the second term (with summationover repeated indices implied) the kinetic acoustic energy. Morfey (1971) suggeststhat in the presence of mean flow, there should be additional terms of the form

9Recall that the Rayleigh criterion is ”necessary, but not su!cient for instability to occur”,because loss mechanisms are not taken into account.


p!uiu!i/c2 – a kind of ”convective” acoustic energy density – but these terms havenot been considered by Ibrahim et al.

The conservation of acoustic energy is then formulated as a transport equationfor the energy density e,

(e

(t+

(

(xi(p!u!i) +

(

(xi(eui) = %. (1.28)

The second and third term on the l.h.s. represent the flux of acoustic energy, withconvection by the mean flow &u retained. The source term % comprises thermo-acoustic and viscous contributions,

% =" ! 1!c2

p!q! ! (u!i(xj

) !ij , (1.29)

where the former is obviously related to the Rayleigh integral.Introducing an average over the oscillation period T ,

0. . .1 =1T

2 T

0. . . dt,

integrating over the domain considered and employing the divergence theorem, theenergy balance can be written as follows

(

(t

2

V0e1dV = !

2

A0p!u!i1dAi !

2

A0eui1dAi +

2

V0%1dV. (1.30)

From the balance equation (1.30), an amplification coe"cient + is then defined,

+ &!

"A0p

!u!i1dAi !"

A0eui1dAi +"

V 0%1dV.

2"

V 0e1dV. (1.31)

The first term in the numerator represents boundary work, the second term rep-resents advection of acoustic energy across the boundary by the mean flow, thethird term is the net production (the thermo-acoustic source vs. the viscous sink).Some of these contributions to the overall amplification increase the acoustic en-ergy and are denoted +amp, while the magnitude of contributions that represent aloss of energy are denoted +damp, such that for the overall amplification coe"cient+ =

3+amp !

3+damp. Separation of destabilizing and damping e!ects in this

way allows to introduce a stabilizing factor

S(%) =3

+amp3+damp

. (1.32)

Ibrahim et al. (2006, 2007) then argue that if an acoustic eigenfrequency %m of thecombustor with S(%m) > 1 exists, that eigenmode will be unstable.

The appealing feature of this method is that it should allow the combustordesigner to investigate quickly and inexpensively a wide variety of potential designconfigurations and operating conditions. Validation studies carried out by Ibrahimet al. (2006, 2007) met moderate success. It was possible to rationalize experimentalresults to some extent. However, it was also found that results are very sensitiveagainst the model for the heat release employed.


Ibrahim et al. (2006, 2007) used in their validation studies rather simplistic as-sumptions concerning the acoustic wave field in the combustor, i.e. the acousticimpedance at the location of heat release. Furthermore, some of the models pro-posed for the dynamics of the heat release in a premix flame are quite unreasonablein our opinion (although it is of course interesting to study the sensitivity of pre-dictions against model details). In any case, acoustic impedance at the heat sourceand the dynamics of the heat source are, as explained above, decisive for thermo-acoustic stability. More dependable predictions with the energy-budget approachproposed by Ibrahim et al. require more accurate models for the dynamics of theheat source (i.e. its transfer function) as well as the combustor acoustics.

A conceptually related approach has been proposed by Nicoud and Poinsot(2005); Giauque et al. (2006, 2007). It is suggested to derive the acoustic energybudget from simulation data generated with large eddy simulation (LES). Indeed,with computational fluid dynamics a complete closure of the energy budget shouldbe possible, and all terms appearing in the budget may be analyzed in great de-tail. However, Nicoud and Poinsot (2005) realized that this hitherto not availablelevel of detail leads to unexpected di"culties: it is by no means obvious what theappropriate definition of acoustic energy or fluctuation energy should be. Giauqueet al. (2006) have identified ”additional and significant energy density, flux, andsource terms, thereby generalizing the recent results of Nicoud and Poinsot (2005)to include non-zero mean flow quantities, large amplitude disturbances, and varyingspecific heats. The associated stability criterion is therefore significantly di!erentfrom the Rayleigh criterion in several ways. The closure of the exact equation isperformed on an oscillating 2-D laminar flame. Results show that in this case thegeneral equation can be substantially simplified by considering only entropy, heatrelease, and heat flux terms. The first one behaves as source term whereas thelatter two dissipate the disturbance energy. Moreover, terms associated with thenon-zero baseline flow are found to be important for the global closure of the bal-ance even though the mean Mach number is small.” (Quotation from the abstractof the paper).

The most recent results (Giauque et al. 2007) confirm the importance of theentropy fluctuations and the non-linear terms. However, it is also pointed out thatto account for these terms is analytically tedious and numerically challenging. Itremains to be seen whether stability analysis for systems of practical interest canprofit from these developments.

1.3.5 Suggestions for further reading

Stability criteria which are like the Rayleigh criterion based on relative phasesbetween fluctuations of velocity, pressure, fuel concentration, heat release rate, en-tropy, etc., have been developed and used by Richards and Janus (1998); Lawn(2000); Polifke et al. (2001a). Such criteria are certainly helpful for qualitativeanalysis of relevant feedback mechanisms, but one must be aware of their limita-tions! As discussed above, phase-based criteria can be overly pessimistic regardingthe stability of a combustion system, because they cannot include the stabilizing ef-fects of loss of acoustic energy. Furthermore, Polifke et al. (2001a) have shown thatit is not always possible to determine the relative phases involved with su"cientaccuracy without a full network model of the combustion system.

SYSTEM MODELLING 23

In Fig. 1.6. a simple stability map is shown, i.e. the stability limits vs. the valueof the time lag ) . Similar maps can be constructed in many ways to illustrate howthe stability properties of a system depend on one or more system parameters. Ofcourse, one can do more than merely indicate parameter regions of stable/unstablebehavior. For example, one may plot frequency of limit cycle oscillation, instabilitygrowth rates, ”mode switching”, etc. as a function of time lags, spread of time lags,combustor residence times, or Helmholtz resonator location, to name but a fewpossible arrangements (Richards and Janus 1998; Lieuwen and Zinn 1998; Polifkeet al. 2001a; Flohr et al. 2001; Stow and Dowling 2001, 2003).

A comparison of the dynamic and ”diagrammatic” approaches for system sta-bility analysis has been carried out by Sattelmayer and Polifke (2003a). It is foundthat the approach proposed by Polifke et al. (1997), which has been presentedabove, produces the same results as the search for eigenmodes and eigenfrequen-cies, while the earlier rules for interpretation of the Bode diagram (Deuker 1995;Kruger et al. 2000) are often in error.

1.4 SYSTEM MODELLING

Miscellaneous methods of stability analysis for thermo-acoustic systems have beendiscussed in the previous section. It should have become apparent that stabilityanalysis in general requires an underlying system model, which represents the rel-evant thermo-fluid- dynamic and acoustic phenomena. A wide variety of systemmodels have been proposed for thermo-acoustic stability analysis, ranging fromsimple estimates for phase lags based on convective time scales to CFD simulation– notably Large Eddy Simulation (LES). In this section, it is attempted to give anoverview of the most important types of system models, see also Fig. 1.12..

Some system models – notably CFD-based approaches – include sub-models forall relevant processes as they may occur in (turbulent, reacting) compressible flowin complicated geometries. However, due to the high numerical e!ort associatedwith this ”brute force” approach, a strategy of divide and conquer is often pre-ferred, where the propagation (and dissipation, and reflection) of acoustic waves ismodelled explicitly, while information on the dynamics of the heat source is sup-plied as input to the model. This input data is obtained from experiment or CFDmodels, and supplied to the system model in terms of a flame transfer function, ora burner matrix, say.

1.4.1 Finite-Volume / Finite-Element models

1.4.1.1 Computational fluid dynamics Modelling of unsteady (turbulent, reacting)compressible flow is a powerful tool, particularly attractive for thermo-acoustic in-stabilities because advanced CFD tools can capture in principle all relevant pro-cesses. An introduction to unsteady flow modelling is given in companion lecturesof this lecture series by Nicoud (2007), the book by Poinsot and Veynante (2005)is recommended for further reading.

An obvious drawback of this approach is the high computational cost, which iscertainly one of the reasons why it has been used only rarely until now (Hantschkand Vortmeyer 1999; Murota and Ohtsuka 1999; Wall 2005; Schmitt et al. 2007).


Time Domain Frequency Domain

CFDnonlinear PDEs -

Navier-Stokes

Computational Acousticslinearized PDEs -

often extended Helmholtz

Galerkin MethodsODE

Network Modelsalgebraic equations

Nonlinear Linearized Equations

Fin

ite

Vo

lum

e,

Fin

ite

Ele

me

nt

Mo

de

-Ba

se

d

Figure 1.12. Overview of models for thermo-acoustic systems.

If CFD is used for thermo-acoustic stability analysis, a significant technical dif-ficulty is the implementation of boundary conditions which enforce not only therequired mean and turbulent flow conditions, but also the appropriate acousticboundary conditions (impedance or reflection coe"cient). See Selle et al. (2004);Polifke et al. (2006); Schuermans et al. (2005); Huber et al. (2008) for recent pub-lications on this problem.

A third shortcoming of the CFD approach is a point that is not always appre-ciated by practitioners of that art: interpretation of CFD results in the thermo-acoustic context is usually not straightforward! Insight into a thermo-acoustic in-stability mechanism is often gained only after sophisticated post-processing of CFDdata, usually supplemented with acoustic analysis. See the publication of Martinet al. (2006) for an example along these lines.

Note that ”brute force” simulation of transient system behaviour in the presenceof small perturbations is not the only possible way of applying CFD to the studyof thermo-acoustic instabilities. CFD can also be used as an element in a divideand conquer strategy. In such a framework, flow simulation is often used to deter-mine the flame transfer function or a transfer matrix (Sklyarov and Furletov 1975;Deuker 1995; Kruger et al. 1998; Bohn et al. 1998; Polifke et al. 2001b; Polifke andGentemann 2004; Gentemann et al. 2004; Armitage et al. 2004; Gentemann andPolifke 2007; Zhu et al. 2005; Giauque et al. 2008; Huber and Polifke 2008). It isalso possible to use flow simulation to compute the OLTF of a (sub-)system, whichis then used to generate a Nyquist plot (see section 1.3.3). Explicit determinationof a flame transfer function or burner transfer matrix is not required, which canbe an advantage for some system configurations (Kopitz and Polifke 2005, 2008;Neunert et al. 2007).

SYSTEM MODELLING 25

1.4.1.2 Computational acoustics In this section, system models for thermo-acousticanalysis that make use of a finite-element- or finite-volume-formulation of linearizedperturbation equations are discussed.

From the equations for conservation of mass and momentum in flow of an idealgas, the linearized Euler equations for small fluctuations of velocity &u!, pressure p!,and density !! can be derived

(!!

(t+ ui

(!!

(xi+ u!i

(!

(xi+ !

(u!i(xi

+ !!(ui

(xi= 0, (1.33)

(u!i(t

+ uj(u!i(xj

+ u!j(ui

(xj+

1!

(p!

(xi! !!

!2

(p

(xi= 0, (1.34)

(p!

(t+ ui

(p!

(xi+ u!i

(p

(xi+ "p

(u!i(xi

+ "p!(ui

(xi= (" ! 1) q!. (1.35)

Here q denotes a volumetric rate of heat release, " is the ratio of specific heats.The variables without apostrophe ! represent a mean flow state, and may dependon position &x, but not on time.

One can show – see e.g. Chu and Kovasznay (1958); Pierce (1981) – that everyperturbation of the mean flow state described by this system of equations can beinterpreted as a superposition of three di!erent modes: an acoustic mode, a vorticitymode, and an entropy mode. If the mean flow is homogeneous and heat sourcesare absent, then the three modes propagate independently from each other. In thatcase, the acoustic mode represents propagation of pressure waves (with the speedof sound), the vorticity mode represents production and transport of fluctuationsof vorticity, the entropy modes describes production and transport of ”hot spots”(regions of temperature and density di!erent from the mean value). The latter twomodes are transported convectively.

If the mean fields are not spatially homogeneous, coupling between the threemodes is to be expected. This can give rise, e.g., to low-frequency oscillations incombustors, where so-called entropy waves are an important element of the thermo-acoustic feedback loop (Keller et al. 1985; Polifke et al. 2001a; Eckstein 2004).

The linearized Euler equations Eqs. (1.33) – (1.35) represent a very generaldescription of the generation and propagation or transport of small disturbances incompressible flow. This set of coupled, partial di!erential equations is – althoughlinearized – notoriously di"cult to handle numerically. Therefore, the linearizedEuler equations are not often used for computational acoustics in the form shownabove .

Ewert and Schroder (2003) have shown how a set of acoustic perturbations equa-tions (APEs) can be derived from from Eqs. (1.33) – (1.35), where after suitablevariable decomposition, each equation represents one of the perturbation modes.This formulation is a good basis to introduce simplifications and isolate the phe-nomena of interest in a specific problem setup.

In FE/FV-based system models for thermo-acoustic analysis, even simpler for-mulations are typically employed. Often the vorticity mode is neglected completely,while entropy waves are considered only in a rudimentary manner (if at all). Forexample, Pankiewitz and Sattelmayer (2003a, b) have developed a finite-element-based model of acoustic wave propagation in 3D geometries in the presence of heatrelease with non-constant speed of sound and density. The model is based on a


0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40.2

0.3

0.4

0.5

0.6

0.7

0.8

!

f

(1,0,0) (1,1,0) (1,0,0)

Figure 5. FREQUENCIES AND CORRESPONDING MODE TYPES

FOR DIFFERENT DELAY TIMES. (!,") LOW ORDER MODEL, (•,!)TIME DOMAIN SIMULATION. (!,•) UNSTABLE MODE, (!,") STABLEMODE.

domain simulation, from the time evolution of the acoustic quan-

tities the frequency and the cycle increment (and thus the growth

rate) can be determined.

For a comparison of the two techniques we carried out cal-

culations for a model combustor with a geometry similar to fig. 1

with twelve burners. We employed a simple heat release model

adopted from Dowling [8]. At one burner j this model reads

1

"0dq"jdt

+ q"j = #u"B, j(t# !) , j = 1, . . . ,12 , (7)

where # = ¯q/uB is the ratio of the mean heat release rate to themean burner exit velocity and "0 is the cut-off frequency of the

low-pass filter. For the calculations with the low order model

eq. (7) is transfered to the frequency domain.

Figure 4 shows the frequencies (given in a non-dimensional

form) as a function of the delay time ! for a given value of " 0.

Additionally the type of the (unstable and least stable, resp.)

mode is given. Either the pure axial mode (1,0,0) or the first

combined axial-circumferential mode (1,1,0) (see fig. 4) is ob-

served in distinct regions for the value of !.Both the low-order model (model A) and the time domain

simulation predict the same type of mode for the same values

of !. The corresponding frequencies are similar in both models,though in the time domain simulation the values are somewhat

lower. However, in the low-order model the calculated level of

instability is generally higher. Therefore some modes which are

stable in the time domain simulation, are predicted to be unstable

in the low-order model.

The differences in frequency and stability can be attributed

to the sudden change of temperature from the plenum/burner–

section to the combustion chamber. This discontinuity at the mo-

ment is treated differently in the two approaches. So already

the purely acoustical eigenfrequencies (not shown here) differ,

which affects the stability analysis. We are confident that a more

elaborate treatment of this problem will remove the discrepan-

cies and lead to a not only qualitatively but quantitatively good

agreement.

CONCLUSIONS

We have presented two novel techniques for the thermoa-

coustic analysis of annular combustors. The low-order model is

especially attractive for its speed of execution. The time-domain

simulation has the advantage that it can readily be applied to ar-

bitrary geometries and does not rely on any assumptions about

the propagation and coupling of the acoustic modes. First results

indicate that both approaches are suitable for a stability analysis

and have the potential to be developed to design tools.

REFERENCES

[1] S. Hubbard and A. P. Dowling. Acoustic Instabilities in Pre-

mix Burners. AIAA Paper AIAA-98-2272, 1998.

[2] T. Sattelmayer. Influence of the Combustor Aerodynamics

on Combustion Instabilities from Equivalence Ratio Fluctu-

ations. ASME Paper 2000-GT-0082, 2000.

[3] F. E. C. Culick. Some Recent Results for Nonlinear Acous-

tics in Combustion Chambers. AIAA Journal, 32(1), pp.

146–169, 1994.

[4] T. Murota and M. Ohtsuka. Large-Eddy Simulation of Com-

bustion Oxcillation in the Premixed Comubustor. ASME Pa-

per 99-GT-274, 1999.

[5] U. Kruger, J. Huren, S. Hoffmann, W. Krebs, P. Flohr, and

D. Bohn. Prediction and Measurement of Thermoacous-

tic Improvements in Gas Turbines With Annular Combus-

tion Systems. Journal of Engineering for Gas Turbines and

Power, 123, pp. 557–566, 2001.

[6] S. R. Stow and A. P. Dowling. Thermoacoustic Oscilla-

tions in an Annular Combustor. ASME Paper 2001-GT-0037,

2001.

[7] W. Krebs, G. Walz, and S. Hoffmann. Thermoacoustic Anal-

ysis of Annular Combustor. AIAA Paper AIAA 99-1971,

1999.

[8] A. P. Dowling. Nonlinear self-excited oscillations of a

ducted flame. Journal of Fluid Mechanics, 346, pp. 271–

290, 1997.

4

Figure 1.13. Eigenfrequency of dominant modes in an annular combustor computedwith an FE (triangles) and a low-order model (circles), respectively. Filled symbols indicatedunstable modes. As the frequency increases, the non-plane (1,1,0) mode becomes dominant.

generalized wave equation for pressure fluctuations

1c2

D2p!

Dt2! !

(

(xi

&1!

(p!

(xi

'=

" ! 1c2

Dq!

Dt. (1.36)

In low-Mach-number flow, the substantial derivative operator D . . . /Dt may bereplaced by the standard partial derivative w.r.t. time ( . . . /(t. Then it is apparentthat Eq. (1.36) is a generalized Helmholtz equation, with spatial dependence ofspeed of sound c and density ! taken into account, and a source term due tofluctuations of the heat release rate q!.

For the heat source, models based on time lags between the acoustic velocityu!b (computed from the gradient of pressure p!) at a reference position and themomentary heat release rate in a certain region of the combustion chamber areemployed. In the simplest case, a formulation equivalent to the famous n!) modelis used

q!(&x, t)q(&x)

= nu!b(t! )(&x))

ub. (1.37)

More advanced formulations are discussed by Pankiewitz (2004). The interactionindex n – which is a measure of the strength of the thermo-acoustic coupling – andthe distribution of time lags )(&x) cannot by computed by the FE-model, but mustbe determined by other means (experiment, CFD, . . . ) In this sense, the approachof Pankiewitz and Sattelmayer (2003a, b) is one example of the ”divide and conquerstrategy” mentioned above.

With a time-domain, finite-element based solver for Eq. (1.36), the investigationof system stability is now in principle a straightforward task, if one follows thestrategy outlined in Section 1.3.1: first a reference solution for Eq. (1.36) withoutthe source term is computed, then the response to small initial perturbations issimulated. Results obtained for the eigenfrequency of the dominant mode in an an-nular combustor are shown in Fig. 1.13.. This figure is taken from Pankiewitz et al.(2001). It is seen that with varying time lag ) of the heat release model, di!erentplane (mode (1,0,0)) and non-plane (mode (1,1,0)) eigenmodes are dominant. Asexpected, the stability of the modes also depends on the time lag.

SYSTEM MODELLING 27

the swirler. Since circumferential modes of the combustor are not purely azimuthal and comprise not only

a longitudinal component in the combustor but also in the plenum. Thus, through this component these

modes are influenced by the introduction of the swirler as it can be seen on Fig. 2.6 and Fig. 2.7. For this

harmonics, the role of the swirler can be sgnificant particulary when acoustic flame is taken into account: in

the configuration without swirler, the peak of pressure is located in the flame zone so the Rayleigh criterion

[11] is potentially high whereas in the case with swirler, the amplitude of the mode is far less important in

this zone.

Z

X

Y

Figure 2.6: 1st circumferential mode of the combustor in the configuration without swirler, f 836hz, acoustic

pressure modulus.

Z

X

Y

Figure 2.7: 1st circumferential mode of the combustor in the configuration with swirler, f 753hz, acoustic

pressure modulus.

14

Figure 1.14. Mixed axial/azimuthal acoustic mode in an annular combustor. Nounsteady heat release (Courtesy of L. Benoit).

Note that local nonlinearities in the response of the heat source to large ampli-tude oscillations can be implemented in the formulation of Pankiewitz et al.. Ifdissipative e!ects are also included – e.g. by imposing non-ideal acoustic bound-ary conditions – a limit cycle can then be observed as an asymptotic state of thesimulation. Of course, the formulation of physically reasonable, let alone accuratemodels for non-linear e!ects and loss mechanism remains a formidable challenge.

Pankiewitz (2004) has also implemented a frequency-domain solver for Eq. (1.36)in the presence of a harmonic forcing term. This allows to compute the response ofa thermo-acoustic resonator to an external excitation. From pressure distributionsobtained in this way, the transfer matrix of geometrically non-trivial elements canbe computed.

A frequency-domain formulation that determines eigenfrequencies %m to assessthe stability – see Section 1.3.2 – of a thermo-acoustic system has been proposed byBenoit et al. (Benoit 2005; Benoit and Nicoud 2005; Martin et al. 2004; Selle et al.2006). A wave equation for small pressure perturbations (Poinsot and Veynante2005) is considered,

(2p!

(t2! (

(xi

&c2 (p!

(xi

'= (" ! 1)

(q!

(t, (1.38)

which is – for small Mach numbers and homogeneous mean pressure p – equivalentto Eq. (1.36). In the simplest case, the flame is modelled as a purely acousticelement. Neglecting the e!ects of local turbulence, chemistry or heat loss, the heatrelease rate responds to fluctuations of velocity at a reference position, see again Eq.(1.37). For the boundary conditions, the acoustic impedance must be expressed inthe form

1Z

=+1

%+ +2 + +3 % with % $ C. (1.39)

Using a classical Galerkin finite element method to discretize Eq. (1.38) on afinite element mesh with m nodes, one obtains an algebraic system of equations

[A][P ] + %[B][P ] + %2[P ] = [D(%)][P ]. (1.40)


Here [P ] is the vector of unknowns p!, the matrix [A] represents the spatial-derivative operator on the r.h.s. of Eq. (1.38), [B] represents the boundary terms.If the source term due to unsteady heat release is neglected, [D] = 0, a suitablevariable transformation yields a problem of dimension 2m that is linear in the fre-quency %. This problem can be solved by a direct method, e.g. QR-based, or anArnoldi method. The latter is preferable, because only the first few modes areusually of interest.

If the source term on the r.h.s. of Eq. (1.38) is included, the resulting problemcannot be solved by standard methods of linear algebra. Benoit and co-workerspropose two methods to find eigenfrequencies

1. an expansion for ”weak thermo-acoustic interaction” with an expansion pa-rameter

, & 1V

2

Vn(&x) dV.

Eigenmodes (%m, p!m) are sought as first-order expansions around the modes(%(0)

m , p!(0)m ) without heat release fluctuations,

%m = %(0)m + ,%(1)

m + O(,2), (1.41)p!m = p!(0)m + ,p!(1)m + O(,2). (1.42)

After some algebra, an explicit expression for ,%(1) in terms of the pressurefield p!(0)m (&x), the heat release fluctuations q!(&x) and the boundary condi-tions is obtained. The imaginary part of %(1) then indicates whether thethermo-acoustic coupling destabilizes the system, see Benoit (2005); Benoitand Nicoud (2005) for details.

A validation of this approach against a resonator with local, time-delayed heatsource – again a generalized Rijke tube – gives excellent agreement for veryweak thermo-acoustic coupling (interaction index n = 0.01 and expansionparameter , = 0.003). However, with coupling strength representative ofcombustion, i.e. interaction index n = 5 and expansion parameter , = 1.6significant deviations in the computed growth rates are observed.

2. The non-linear eigenvalue problem Eq. (1.40) can be solved iteratively for asequence of eigenfrequencies %(k)

m , k = 1, 2, . . . from the quadratic problem4[A]! [D(%(k"1))]

5[P ] + %(k)[B][P ] + (%(k))2[P ] = 0. (1.43)

Martin et al. (2004) report that usually less than 5 iterations are enough toachieve convergence (starting from the homogeneous problem [D] = 0 for thefirst step of the iteration. Using this system model in combination with LESresults, Martin et al. (2004) discuss acoustic energy and modes in a turbulentswirled combustor.

The computational acoustics models based on linearized PDEs for perturbationsin compressible flow with heat release presented in this section have not yet beenvalidated in a comprehensive, quantitative manner against experimental data. Asignificant di"culty is the implementation of realistic models for the fluctuatingheat release q!(&x).

SYSTEM MODELLING 29

1.4.2 Galerkin method

In the remainder of this section, a system modelling approach that relies on a modalrepresentation of the acoustic field is introduced: the Galerkin method. Anothernoteworthy approach that makes use of a modal expansion is the state-space ap-proach developed by Schuermans and co-workers (Schuermans et al. 2002, 2003;Schuermans 2003). Although that approach appears to be quite flexible, powerfuland e"cient, it is not discussed further in these introductory notes.

The Galerkin method is a classical method, that can be interpreted as a par-ticular variant of a weighted residual method (Fletcher 1991). The application ofGalerkin methods is by no means restricted to thermo-acoustic problems – on thecontrary the method is applicable to a wide variety of di!erential equations. Thepresentation here follows the discussion of Culick (1989) and is restricted to a one-dimensional situation for ease of presentation. Note, however, that the Galerkinapproach may very well be applied to geometries of applied interest, e.g. annulargas turbine combustion chambers (Krebs et al. 1999). Other recent applications arethose of Deo and Culick (2005); Tyagi et al. (2007); Huang and Baumann (2007).

Consider wave propagation in the absence of a fluctuating heat source, repre-sented by the Helmholtz equation (like Eq. (1.36), but with zero right-hand side).In the one-dimensional case with 0 < x < L and with constant speed of sound,eigenmodes -m(x) of the pressure perturbations satisfy the equation

(2-m

(x2+ k2

m-m = 0, (1.44)

with wave number km = %m/c = m$/L for a chamber open at both ends, p! = 0 (achamber closed at both ends, u! = 0, has the same set of possible wave numbers).The solutions to this homogeneous problem are the well-known orthogonal modes,i.e.

-m(x) = sin(kmx), (1.45)

where orthogonality implies2 L

0-m-n dx = E2

m .mn. (1.46)

The modes are orthogonal, because – simply speaking – the Laplace operator in theHelmholtz-equation is Hermitian. More mathematical details on ”Sturm-Liouvilletheory” (the theory of linear di!erential equations, di!erential operators and theireigenvalues / eigenfunctions) are found in many text books.

The essential feature of the Galerkin method is that the acoustic field in thepresence of the heat source is expressed as superposition of these ”homogeneouseigenmodes” -m,

p!(x, t) =6

m

/m(t)-m(x), (1.47)

u!(x, t) =6

m

/m(t)"k2

m

d

dx-m(x). (1.48)

Substituting this expansion into the governing equations (1.36), multiplying with abasis function -n, integrating over the domain

" L0 , and exploiting the orthogonality


property (1.46), a system of ordinary di!erential equations for the amplitudes /m(t)is obtained:

d2/m

dt2+ %2

m/m = Fm, (1.49)

with a forcing term

Fm =" ! 1E2

m

2 L

0q!(x) -m(x) dx. (1.50)

Note that there may be an additional term that stems from the boundary con-ditions, see (Culick 1989). Also, non-linear terms, which may arise from localnon-linearities in the response of the heat source to large amplitude flow perturba-tions, or from non-linear acoustics, can be integrated in the formulation. In thatcase nonlinear terms, which may take the form

3l aml/m/l or similar, appear in

Eq. (1.49).From the Galerkin expansion, a system of ordinary di!erential equations for

the amplitudes /m(t) is obtained. Naturally then, stability analysis proceeds by”unsteady simulation” as outlined in Section 1.3.1. Note that the shape and thefrequency of the dominant mode, which develops in the course of the simulation,need not be equal (not even similar) to one of the eigenmodes -m of the homoge-neous problem. In general, an eigenmode of the inhomogeneous problem (with heatsource) is a collective phenomenon, to which many normal modes -m contribute.Especially in the case of nonlinear interactions, very complex behaviour with trans-fer of energy between normal modes during the time evolution of the ODE (1.49)is to be expected.

When complex geometries of technical interest – an annular gas turbine combus-tor – are to be considered, the normal modes -m can no longer be determined an-alytically. Instead, they are determined by computational acoustics, usually basedon an FE formulation (and of course without the heat source term). The projectionof the partial di!erential equation to generate the ODEs for the amplitudes /(t),which requires integration over the computational domain with the acoustic vari-ables and the -m’s as integrands, is then also performed numerically (Krebs et al.1999). One must make certain that from such a procedure indeed a normal set ofbase functions -m is produced. This property is often taken for granted – but asBenoit (2005) has shown, non-trivial boundary conditions are enough to lead to anon-normal set of eigenmodes for the homogeneous problem!

1.5 NETWORK MODELS OF ACOUSTIC SYSTEMS

In section 1.3.2.1, a low order model of the Rijke tube has been constructed. Thismodel represents the Rijke tube as an assembly of three elements – the upstreamduct, the gauze, the downstream duct – plus the boundary conditions. Assuminglinear acoustics and harmonic time dependence " exp{i%t}, a mathematical modelof the relevant thermo-acoustic processes is obtained in terms of a linear system ofequations, see Eq. (1.13). The unknowns are the (Fourier-transformed) acousticvariables10 of pressure p! and velocity u! at the junctions – the ports – of theelements.10Instead of the primitive acoustic variables p! and u!, the so-called Riemann invariants f , g areoften used, but this is a matter of convention and convenience.

NETWORK MODELS 31

Many thermo-acoustic systems can be represented in an approximate manner assuch a network of acoustic elements, with each element corresponding to a particularcomponent of the system, e.g. a duct, a nozzle or orifice, a burner, a flame, somekind of acoustic termination, etc., see Fig. 1.8.. The coupling relations for theunknowns across an element are combined into the transfer or scattering matrix ofthe element. The transfer matrix coe"cients of all network elements are combinedinto the system matrix S of the network. What has been done ”by hand” for theRijke tube, can also be done ”by software”, which is particularly useful if an networkcomprises many elements. The numerical tools that build up a representation ofthe system as an assembly of interconnected elements, construct the correspondingsystem matrix from this network, and then solve the algebraic problem, are oftencalled ”network models” or ”low-order network tools” in the acoustics community.

In the EU Research and Training Network AETHER, TU Munchen is responsiblefor ”System Modelling and Stability Analysis”. The network tool ”taX” will bemade available to network members. A structured training event is planned inMarch 2008, where AETHER researchers will have the opportunity to learn how touse the software. In the following, previous work on network models is reviewed andsuggestions for further reading are given. Then, a few simple examples of networkelements and results obtained with network models are discussed.

1.5.1 Review of previous work

An abundant literature exists on low-order models of acoustic wave propagation inducts, good introductions are found in Munjal (1986) and Poinsot and Veynante(2005) (the latter with consideration also of combustion instabilities).

The use of low order models for thermo-acoustic stability analysis has been pio-neered by Merk (1956). The low-order method has been applied to study combus-tion instabilities in afterburners (Bloxsidge et al. 1988), Rijke tubes (Heckl 1988)and lean premixed stationary gas turbines (Keller 1995). Validations of 1D low-order models against experimental data from a single burner test rig have beenpublished by Bloxsidge et al. (1988) and Schuermans et al. (2000).

For the particular case of annular gas turbine combustors, the low-order methodhas been extended to quasi 2D or 3D geometries. Under the guidance of J. J.Keller, Curlier (1996) developed a numerical tool to study the propagation of acous-tic waves in annular geometries in the presence of a mean flow. The underlyingapproach to represent the acoustic wave field goes back to Tyler and Sofrin (1962).Kruger et al. (1999) used a network of four-pole acoustic elements to investigate thestability limits of azimuthal modes in a heavy-duty annular gas turbine combus-tor. Their particular approach, however, seems to be limited inasmuch as it is notcapable of handling mixed modes – see Fig. 1.16. below – in a reasonable manner.

The interaction of entropy waves with acoustics in an annular combustor witha choked exit has been investigated by Polifke et al. (1999), details of the methodand additional results have been published by Polifke et al. (2001a); Polifke (2004).An extension of the method, capable of describing the e!ect of burner-to-burnervariability (”broken symmetry”) and mode coupling has been proposed and vali-dated by Evesque and Polifke (2002). This is important, because Berenbrink andHo!mann (2000) showed experimentally that judiciously introduced di!erences be-tween burners could be exploited as a means of passive control. The di!erence


xu xd

Au

Ad

l

Figure 1.15. Acoustically compact contraction.

and significance of spinning vs. standing modes in annular combustors has beendiscussed by Evesque et al. (2003).

Bohn and Deuker (1993) were perhaps the first to propose the implementationof low-order methods as a network tool, where a library of acoustic elements is gen-erated, each of them represented by its respective transfer matrix, and combinedin a flexible and to some extent automated manner to represent the dynamic prop-erties of complicated systems. Since then, network tools have been implementedby the research groups at RWTH Aachen for Siemens Power Generation (Krugeret al. 1999), at ABB Corporate Research in Dattwil (Polifke et al. 1997, 2001a)and more recently also at the University of Cambridge (Stow and Dowling 2001).Stow and Dowling (2001, 2003) also implemented a low-order model for annularcombustors. Particular attention was paid to the boundary conditions appropriatefor a gas turbine combustor, otherwise the formulation is in essence identical tothe method proposed by Keller and co-workers above Keller (1995); Curlier (1996);Polifke et al. (2001a).

A new, qualitatively di!erent and seemingly quite powerful variant of the networkapproach has been proposed recently by Schuermans et al. (2003); Schuermans(2003), which makes use of a state space approach.

1.5.2 Compact element with inertia and loss

Consider the propagation of acoustic perturbations through a contraction as shownin Fig. 1.15. with length l = (xd ! xu) such that kl * 1 (acoustically compact).A transfer matrix for such an element may be derived in an approximate mannerfrom momentum and mass conservation.

Conservation of momentum is exploited to derive a coupling relation betweenacoustic perturbations p!, u! on both sides of the contraction by integrating theunsteady Bernoulli equation (Hirschberg 2001):

(

(xi

&(0

(t+

u2

2+

"

" ! 1p

!

'= 0. (1.51)

along a streamline of the mean flow from xu to xd. First consider the term withthe velocity potential 0:

(

(t

2 xd

xu

(0

(xidxi =

(

(t

2 xd

xu

uidxi +(

(tuu

2 xd

xu

Au

A(x)dx. (1.52)

The approximation u(x) + ucA(x) is valid because it was assumed that the el-ement is acoustically compact – then the flow through the element is e!ectivelyincompressible.

NETWORK MODELS 33

Introducing an extended length

lext &2 xd

xu

Au

A(x)dx, (1.53)

and assuming harmonic time-dependence, one obtains:

(

(t

2 xd

xu

(0

(xidxi = i % lext u!u. (1.54)

The remaining two terms in (1.51) are simply evaluated at xu and xd. Linearizationand dividing by c yields

i k lext u!u +7M u! +

p!

! c

8d

u

= O(kl)2, (1.55)

with [v]du & vd ! vu for any variable v. This result is exact to first order in theHelmholtz number k l, because the time derivative in (1.52) introduces a factor k l,so that any deviations of u(x) from ucA(x) appear as second order terms in (1.55).Neglecting terms of first order in Mach number, this result may be simplified

p!

! c

9999d

=p!

! c

9999u

! i k lext u!u + O(kl)2 + O(M). (1.56)

Note that lext may be significantly larger than the geometrical length l of theelement if the area A(x) < Au, Ad between the up- and downstream terminations.

Mass conservation between upstream (”u”) and downstream (”d”) demands that

d

dt

2! dV +

2!ui dAi = 0. (1.57)

For the geometry considered (see Fig. 1.15.) and assuming quasi-one-dimensionalflow, this can be simplified as follows

d

dt

2 xd

xu

!(x)A(x) dx + [!uA]du = 0. (1.58)

For the density, !!(x) = !!u+O(kl) and with the assumed harmonic time dependence

i %!!u

2 xd

xu

A(x) dx + [(!!u + !u!) A]du = O(kl)2. (1.59)

Diving by the mean density ! and defining a reduced length

lred &2 xd

xu

A(x)Ad

dx (1.60)

one obtains:

i k lred Adp!

! c

9999u

+7&

u! + Mp!

! c

'A

8d

u

+ 0. (1.61)

Neglecting terms of first order in M , this simplifies further to

u!dAd = u!uAu ! i k lred Adp!

! c

9999u

+ O(kl)2 + O(M). (1.62)


Finally, the transfer matrix for a compact acoustic element with loss is approxi-mated as follows:

*

+p!

!cu!

,

-

d

=&

1 !i k le! ! 1M!i k lred +

' *

+p!

!cu!

,

-

u

(1.63)

This is a combination of (1.56) and (1.62), with an area ratio + & Au/Ad and apressure loss term 1M . The loss term 1M has been introduced to account in anapproximate manner for loss of acoustic energy due to, e.g., flow coupling. Withthis term included, a change in volume flow rate u!A through the element resultsin a change in pressure drop p!u ! p!d across the element.

• The loss coe!cient 1 for the acoustic pressure that appears in Eq. (1.63)is in general not equal to the hydrodynamic loss coe"cient 1d for the totalhead. It is fortunately not di"cult to establish a relation between the twoloss coe"cients, which may be expected to hold with reasonable accuracy a)in the quasi-stationary limit and b) if the relevant loss mechanisms flow andacoustics are the same.

Consider Bernoulli’s equation from position ”u” upstream to position ”d”downstream with a loss term:

pd +!

2u2

d = pu +!

2u2

u !!

21du

2d. (1.64)

By convention, the loss is expressed in terms of the downstream velocity ud.Continuity implies that ud = + uu with + = Au/Ad, and the downstreampressure pd can be expressed in terms of upstream pressure pu and velocityuu as follows:

pd = pu +!

2u2

u(1! +2(1 + 1d)). (1.65)

Linearization and division by !c yields

p!d!c

=p!u!c! [+2 (1 + 1d)! 1]Mu u!d. (1.66)

The term in angular[. . .] brackets appears as the acoustic loss coe"cient 1 inEq. (1.63).

For a sudden change in flow cross-sectional area, conservation of momentumimplies for the loss coe"cient 1d = (1! 1/+)2 (neglecting the vena contractae!ect for contractions). Then 1 = 2+(+! 1), which is small and negative for0 < + < 1 and increases rapidly for area ratios + > 1 (contractions).

• The e"ective length introduced in (1.63) combines the extended length – seethe definition (1.53) – with end corrections lec, which may appear at sharpcorners,

le! & lext + lec,u + lec,d (1.67)

For an element with significant ”intermediate contraction”, A(x) < Au, Ad

over some length, the following inequalities should hold:

lred < l < lext < le!.

NETWORK MODELS 35

(a) Axial mode (1,0) (b) Azimuthal mode (0,1) (c) Mixed mode (1,1)

Figure 1.16. Plane and higher order eigenmodes in a thin annular duct.

Physically, the e!ective length accounts for the inertia e!ects, i.e. a change inpressure di!erence p!u ! p!d will accelerate the fluid between the two referencepositions ”u” and ”d”, resulting in a gradual change of velocity u! throughthe element, which leads to a phase di!erence between p!u ! p!d and u!.

An expression similar to (1.63) has been introduced by Paschereit and Polifke(1998) to approximate the transfer matrix of a premix burner (without flame). Itturns out that this form is indeed quite general and applicable to all sorts of compactelements, e.g. sudden changes in cross sectional area, an orifice, etc. (Gentemannet al. 2003). Computational fluid dynamics may be used to determine the valuesof the coe"cients appearing in (1.63), see e.g. Flohr et al. (2003)

1.5.3 Ducts

In this section, we present transfer matrices for a few non-trivial duct elements.

1.5.3.1 Thin annular duct The equation (1.9) given above for wave propagationin a duct with wave number k = %/c is valid only for plane waves without meanflow. In general, non-plane waves (”higher-order modes”) may propagate in a ductat su"ciently high frequency (above the cut-o! frequency). Of particular interestfor gas turbine combustion are thin annular ducts, with the height of the annulussignificantly smaller than the radius R or the length L, such that the amplitude ofacoustic waves depends on the axial coordinate x and the circumferential position2, but not on the radius r,

p! " exp (i%t! ikxx! ik$R2) , (1.68)

and similarly for velocity u! or the Riemann invariants (but see the discussion ofstanding vs. rotating modes by Evesque et al. (2003)). Di!erent mode shapes insuch a thin annulus are shown in Fig. 1.16.. For the azimuthal or perpendicularcomponent of the wave vector, k$ & ±m/R with mode index m, because the wavefield must be periodic in the azimuthal direction. If one plugs the expression (1.68)into the 2-D form of the convective wave equation,

((

(t+ &u ·2)2 p! ! c222 p! = 0, (1.69)


one obtains the following condition for the axial component of the wave vector

kx± =%/c

1!M2

*

+!M ±

:

1!&

k$%/c

'2

(1!M2)

,

- . (1.70)

Here M & U/c is the mean flow Mach number for uniform mean flow in the axialdirection with velocity U . The transfer matrix for non-plane wave propagation ina thin annular duct with mean flow is similar in form to the previous result (1.9):

&fd

gd

'=

&e"ikx+l 0

0 e"ikx!l

' &fu

gu

'. (1.71)

Note, however, that the relation between Riemann invariants and the primitiveacoustic variables is a bit more complicated,

p!

!c= f + g, u! = 3+f + 3"g. (1.72)

with a coe"cient3± &

kx±

k !Mkx±

. (1.73)

For zero Mach number and plane waves m = M = 0, this reduces to the previousresults (1.9) with kx± = ±%/c.

With this formalism, it is possible to describe e"ciently the acoustic wave fieldin a modern gas turbine with an annular combustor from the compressor exit tothe turbine inlet, say.

1.5.3.2 Ducts with varying cross-sectional area A duct with varying cross sectionalarea – also an annular duct with curved walls, like a gas turbine combustion chamber– can be e!ectively modelled by ”stair-stepping” the contour of the duct, i.e. asan alternating succession of short straight duct elements and small discontinuouschanges in cross sectional area. The transfer matrices of these ”building blocks”have been introduced above (a change in cross-sectional area can be described as acompact element as in 1.5.2), the transfer matrix of the complete duct is obtainedas the matrix-product of the individual transfer matrices. Note that in general foreach sub-element, the cross-sectional area A and the radius R may be di!erent,therefore the wave vector (kx, k$), the Mach number and the area ratio will not beconstant. Experience shows that geometries of applied interest can be representedwith su"cient accuracy with about one dozen of sub-elements.

1.5.3.3 Joints and forks In networks with non-trivial topology, c.f. Fig. 1.8.,elements are needed to describe a situation where two flow paths are joined intoone, or one flow path splits into two. In principle, this requires a 3 ) 3 transfermatrix to interconnect the six Riemann invariants at the three ports of such anelement. In the simplest case, one may assume equal pressures, i.e. p!i = p!j = p!kand conservation of mass, i.e. u!iAi + u!jAj + u!kAk = 0 to construct the transfermatrix. Of course, if end corrections, loss coe"cients, etc. must be considered, thecoupling relations are more complicated, and one has to take into account meanflow direction – but there is no essential di"culty in constructing transfer matricesfor such elements.

NETWORK MODELS 37

1.5.4 Transfer matrix of a compact flame

In this section a closure formulation for the transfer matrix of a compact flame (moregenerally speaking: a compact source of heat) is presented. ”Compact” meansthat the heat release is concentrated in a region much smaller than acoustic wavelengths. In this approximation, a premix flame front is considered as a discontinuityof negligible thickness, which adds a certain amount of heat q per unit mass (unitsJ/kg = m2/s2) to the flow of an ideal gas.

For steady flow of gas, Chu (1953) has shown that conservation of energy, massand momentum across a thin source of heat leads to the following Rankine-Hugoniotrelations:

c2h = c2

c + (" ! 1) q + O(M2), (1.74)

uh = uc + (" ! 1) Mcq

cc+ O(M2), (1.75)

*

&p

!c

'

h

=&

p

!c

'

c

! (" ! 1) Mcq

cc+ O(M2), (1.76)

with * & !hch/!ccc. Here the index ”c” stands for the cold (upstream) side of thediscontinuity, ”h” stands for the hot (downstream) side A detailed derivation ofthese relations (using the present notation) is given in (Polifke et al. 2001a). Auseful result follows from Eq. (1.74):

Th

Tc! 1 = (" ! 1)

q

c2c

. (1.77)

The linearization of the Rankine-Hugoniot relations (1.74) - (1.76) in the presenceof (small) acoustic fluctuations shows how fluctuations of pressure and velocity up-and downstream of the flame are related to fluctuations of the heat release rate:

*

&p!

!c

'

h

=&

p!

!c

'

c

!&

Th

Tc! 1

'ucMc

0u!cuc

+Q!

Q

1, (1.78)

u!h = u!c +&

Th

Tc! 1

'uc

0Q!

Q! p!c

pc

1. (1.79)

where Q & !uq is defined as rate of heat addition per unit area, and terms of orderM2 or higher are neglected. Again, a detailed derivation of these results is foundin (Polifke et al. 2001a).

Eqns. (1.78) and (1.79) do not yet allow to write down the transfer matrix of theflame, which would relate acoustic perturbations on both sides of the flame sheet toeach other, because it is not yet clear how acoustic waves influence the fluctuationsQ! of the heat release rate – we have a closure problem! In general, the heat releaserate Q! may respond to perturbations of pressure or velocity at the flame front, orat the burner mouth, or at the location of fuel injection, as discussed above. Aquantitative model for the flame response,

Q! = Q! (u!h, u!c, u!i, . . . , p

!i, . . . ,2

!;L, M, 2, SL, . . .) , (1.80)

is obviously needed to achieve closure, i.e. formulate a consistent network model.As indicated, the heat release rate will in general not only depend on velocity


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Figure 1.17. Frequencies (left) and growth factors (right) of first two eigenmodes in aduct with area change vs. position of the area change. Loss coe"cient # = 0, boundaryreflection coe"cients r = !1 at both ends.

fluctuations u!c, u!h just up- and downstream of the flame, but may also involve

perturbations of velocity, or pressure, or equivalence ratio 2 elsewhere in the system.As indicated the flame response also depends on length scales (of the burner / theflame), Mach numbers, equivalence ratios, flame speeds, etc.

In the simplest case, the flame reponse may be described in terms of a transferfunction that depends on the upstream (”cold”) velocity u!c,

F (%) & Q!/Q

u!c/uc. (1.81)

Then the corresponding transfer matrix for a compact flame follows from the lin-earized Rankine-Hugoniot relations Eqns. (1.78) and (1.79);

T =

*

+#ccc

#hch!

4ThTc! 1

5Mh (1 + F (%))

!"4

ThTc! 1

5Mc 1 +

4ThTc! 1

5F (%)

,

- (1.82)

1.5.5 Example network calculations

To conclude, we present here a result of a very simple network, which neverthelessproduces non-trivial results. The configuration considered consists of the followingelements:

open end – duct – area change – duct – open end

For the ducts, a transfer matrix as given in Eq. (1.9) or Eq. (1.71) is used. A Machnumber M = 0.1 is assumed, mean flow e!ects on wave propagation are taken intoaccount by setting the wave numbers kx± = 3%/c(1 ± M) (this follows from Eq.(1.70) for plane waves with k$ = 0.) ). The area change is an expansion witharea ratio + = Au/Ad = 1/3, it is described as a compact element according toEq. (1.63). For simplicity the e!ective lengths are set to zero for the exampleconsidered.

Fig. 1.17. shows frequencies and growth factors (computed as exp(!i%(%)) ofthe first two eigenmodes with a loss coe"cient 1 = 0. It should come as a surprisethat growth factors larger than unity are predicted. Clearly, there is no ”thermo-acoustic engine” in this configuration – so what is the source of acoustic energythat is responsible for the predicted growth of amplitudes?

ACKNOWLEDGMENTS 39

The answer is that an overly simplistic model for the boundary conditions wasused: If p! = 0 is assumed at the ”open end” boundaries, the amplitudes of thereflected characteristic wave amplitude g at the downstream end, say, is equal tothe amplitude of the outgoing wave amplitude f . However, this boundary condi-tion does not conserve acoustic energy, because the convective transport of energy,represented by the factors (1 ± M) in the definition of the wave number, is notconsidered. So, a boundary condition with g = f actually loses acoustic energy.In the present configuration with an expansion placed between two open ends, theconvective loss of acoustic energy at the downstream end (where the Mach numberis smaller) is smaller than the inflow of acoustic energy at the upstream end (wherethe Mach number is larger). So in sum energy is added to the system.

A boundary condition that conserves acoustic energy can be derived from Bernoulli’sequation,

p +12!u2 = p%. (1.83)

Here the l.h.s. stands for the conditions at the open end, the r.h.s. for thesurrounding. Assuming that there are no fluctuations of the surrounding pressurep%, linearization of Eq. (1.83) yields to first order in Mach number

p!

!c+ Mu! = 0 or (1.84)

f(1 + M) + g(1!M) = 0. (1.85)

If this boundary condition is used, growth factors equal to unity (constant oscilla-tion amplitude) are computed if the loss coe"cient 1 is also set to zero (not shown).With a non-zero loss coe"cient, the growth factorss are always less than unity, i.e.the eigenmodes are damped, see Fig. 1.18.. The results indicate furthermore thatthe losses vanish whenever the area change, which is the only element where lossesmay occur in the present configuration, is located at a velocity anti-node withu! = 0. This is close to the center of the tube at x/L + 0.5 for the fundamentalmode, and at x/L + 0.25 and x/L + 0.75 for the second mode.

It is also interesting to discuss the e!ect that the position of the area changehas on the value of the frequency of the eigenmodes, see the right graph in Figs.1.17. and 1.18.. The frequencies in these plots are normalized with the frequencyf0 of the fundamental mode in a duct of length L with constant cross-sectionalarea and two open ends. For the fundamental mode, the area change reduces theeigenfrequency of the ducts with area change for x/L < 0.5, while it increases theeigenfrequency for x/L > 0.5. This can be explained by considering – ignoring forthe sake of the argument the loss term – how the acoustic variables u!, p! on bothsides are related to each other, and what this implies for the characteristic waveamplitudes f, g. Details are left to the reader – it is useful to think of the variablesu!, p!, f, g as pointers in the complex plane!

1.6 ACKNOWLEDGMENTS

Thanks to doctoral students Jan Kopitz, Robert Leandro, and Andreas Huber.Discussions with Raman Sujith on non-normal modes in thermo-acoustics are ap-preciated. My apologies to Bruno Schuermans for not discussing his state spaceapproach – time and space did not permit this.


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Figure 1.18. Frequencies (left) and growth factors (right) of first two eigenmodes in aduct with area change vs. position of the area change. Loss coe"cient # = 4, boundaryreflection coe"cient r = !(1±M)/(1$M).

1.7 APPENDIX: CHARACTERISTIC EQUATION OF THE RIJKE TUBE

The following lines of Mathematica code determine the determinant of the ”RijkeTube” network model discussed in this lecture:

ClearAll[A,nEt,deta,"@"];A = {{-1,1,0,0,0,0,0,0}, (* left b.c. *){0,0,1,1,-xi,-xi,0,0},(* pressure coupling across heat source *){0,0,(1+nEt),-(1+nEt),-1,1,0,0}, (* velocity coupling across heat source *){0,0,0,0,0,0,-1,-1}, (* right b.c. *){e1m,0,-1,0,0,0,0,0},(* next four lines: ducts *){0,e1p,0,-1,0,0,0,0},{0,0,0,0,e2m,0,-1,0},{0,0,0,0,0,e2p,0,-1}} ;MatrixForm[A] (* this should output the 8x8 system matrix *)

deta = Det[A]//Simplify (* compute the matrix, simplify *)

(* insert phase propagators for ducts:*)e1p = Exp[I k1]; e1m = Exp[-I k1 ];e2p = Exp[I k2]; e2m = Exp[-I k2 ];deta = ComplexExpand[deta]//Simplify (* voila *)

These commands should produce first a nice 8 ) 8 system matrix and then thefollowing output statement:

Out[...]= e1m (-e2p (-1 + xi + nEt xi) + e2m (1 + xi + nEt xi)) + ...+ e1p (- e2m (-1 + xi + nEt xi) + e2p (1 + xi + nEt xi))

Out[...]= 4 Cos[k1] Cos[k2] - 4 (1+nEt) xi Sin[k1] Sin[k2]

The latter is obviously the determinant of the system matrix (up to an irrelevantfactor of 4).

1.8 FINDING ROOTS

The following bit of Mathematica code searches for the eigenfrequency of the fun-damental mode m = 0 of the Rijke tube. First values are assigned to lengths andphysical parameters (in arbitray units), then the coe"cients k1, k2, nEt so far leftundefined in the variable deta are assigned.

lc = 0.5; lh = 0.5;cc = 1.; ch = 1.0;xi = 1./ch; n = (ch/cc)^2-1;n=0.1;tau = 1.5 Pi; (* most unstable for ground mode m = 0*)

k1 = Pi/2. omega lc / cc;k2 = Pi/2. omega lh / ch;nEt := n Exp [ - I omega tau ];

FindRoot[deta==0.,{omega,1}]

Out[...[] = {omega -> 0.998921 - 0.0278638 i}

Bibliography

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