[asme asme turbo expo 2010: power for land, sea, and air - glasgow, uk (june 14–18, 2010)] volume...

Proceedings of ASME Turbo Expo 2010: Power for Land, Sea and AirGT2010

June 14–18, 2010, Glasgow, UK

GT2010-23577

THERMOACOUSTIC INSTABILITIES IN AN ANNULAR RIJKE TUBE

Jonas P. Moeck∗

Markus Paul

Christian Oliver PaschereitInstitut fur Stromungsmechanik und Technische Akustik

Technische Universitat Berlin

10623 Berlin, Germany

Email: [email protected]

ABSTRACT

Thermoacoustic instabilities are a major concern in the de-

sign of gas turbine combustors. Most modern combustion cham-

bers have an annular shape with multiple circumferentially ar-

ranged burners and, accordingly, suffer often from azimuthal in-

stability modes. However, due to the complexity of a full annu-

lar system with a large number of burners, most experimental

and numerical studies focus on single burner systems with es-

sentially purely longitudinal acoustics. In the present work, we

therefore introduce a thermoacoustic surrogate system – an an-

nular Rijke tube – which, albeit its simplicity, possesses the ba-

sic mechanisms to feature unstable azimuthal modes. As in a

conventional Rijke tube, the sources of mean and unsteady heat

release in our set-up are electrically driven heating grids. Differ-

ent azimuthal instability modes are observed in the experiment,

and the effect of two types of circumferential variations of the

power input is investigated. A full suppression of the unstable

modes is achieved by the application of an elementary feedback

controller. The experimental investigations are accompanied by

corresponding calculations with a low-order system model. The-

oretical and experimental results are found to agree well.

[Keywords: combustion instability, thermoacoustic,

azimuthal modes, annular Rijke tube]

∗Address all correspondence to this author.

NOMENCLATURE

A cross-sectional area

a modal coefficient

c speed of sound

ck, cτ low-pass parameters

D duct prolongation matrix

F heat release transfer function

f , g down-/upstream traveling wave amplitude

hm,c|s circumferential basis function

I identity matrix

Im imaginary part

i√−1

K feedback gain matrix

k wave number, eigenfunction index

m azimuthal mode order

N number of heat release elements

P electric power

p acoustic pressure

q heat release rate

Re real part

r duct radius

T transfer matrix

T temperature

u axial particle velocity

X characteristic impedance ratio matrix

x coordinate vector

.

...

1 Copyright c© 2010 by ASME

Proceedings of ASME Turbo Expo 2010: Power for Land, Sea and Air GT2010

June 14-18, 2010, Glasgow, UK

GT2010-23577

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greek and calligraphic letters

α acoustic mode index

δi j Kronecker delta

φ angular coordinate

γ ratio of specific heats

ω angular frequency

ψ acoustic eigenfunction

ρ fluid density

σ asymmetry parameter

ξ characteristic impedance ratio

ζ damping coefficient

A admittance matrix

F scaled heat release transfer function matrix

R reflection coefficient matrix

S system matrix

Z impedance matrix

..

sub- and superscripts

(·)d downstream of heat source

(·)u upstream of heat source

(·)n in tube n¯(·) mean quantityˆ(·) Fourier transform, amplitude

INTRODUCTION

For reasons of compactness and of having a uniform tem-

perature distribution at the turbine inlet, modern gas turbine com-

bustion chambers mostly have an annular shape with multiple cir-

cumferentially arranged burners. This geometrical set-up hosts

azimuthal acoustic modes, which may become unstable under

certain conditions. If the heat release provided by the flame

responds dynamically to acoustic perturbations, energy can be

added to the acoustic field, and the pressure fluctuations grow

until limited by nonlinear effects [1]. Although investigations on

annular thermoacoustic systems are significantly more complex

than in the purely longitudinal single burner case, a number of

experimental, numerical, and modeling studies focusing on az-

imuthal modes has been performed over the last decade. We give

a brief overview in the following.

Seume et al. [2] applied a feedback control scheme using

modulated pilot fuel to suppress unstable azimuthal modes in a

full scale annular combustion chamber with 24 burners. They

observed 2nd and 4th order circumferential modes. The un-

stable acoustic modes in this type of combustor have been fur-

ther analyzed by Krebs et al., Berenbrink and Hoffmann, and

Kruger et al. [3–6] based on test rig measurements, finite ele-

ment computations, and low-order Galerkin and network type

models. From the experimental data [4], first and second order

azimuthal modes, dominantly standing as well as rotating, were

identified from the magnitude and phase patterns of the acous-

tic pressure along the circumference of the annular combustion

chamber. Introducing a circumferential variation by means of

cylindrical and asymmetric burner outlets applied to part of the

burners was found to have a stabilizing influence [5, 6].

Evesque et al. [7, 8] developed a low order network model

for annular combustion chambers allowing for asymmetries in-

troduced by the burner transfer matrices. They considered an

annular model combustion chamber geometry with no sources

of unsteady heat release and compared eigenvalue computations

with a finite element solver; good agreement was found. A simi-

lar approach was used by Kopitz et al. [9] to model longitudinal

and circumferential instabilities that were observed in an annular

model combustion chamber.

The Rolls-Royce/Cambridge network code was used in a

number of modeling and control studies considering annular

combustor configurations [10–13]. Stow and Dowling [13] im-

plemented the effect of Helmholtz resonators attached to the cir-

cumference of the combustion chamber and studied the associ-

ated modal coupling. Optimal resonator placement for maximum

damping of unstable azimuthal modes was deduced. In [12], the

same authors extended the approach to account for nonlinearities

in the flame model. By using describing function techniques, the

limit cycle amplitude of thermoacoustic oscillations in an annu-

lar configuration could be calculated. Stow and Dowling [11]

extended the approach to time domain simulations based on the

impulse response of heat release perturbations to those in ap-

proach flow velocity and a nonlinear flame model. The final limit

cycle solution of unstable azimuthal modes were always of rotat-

ing type in accordance with the theoretical considerations in [14]

(see below). Morgans and Stow [10] used the same network code

to develop model-based control strategies for thermoacoustic in-

stabilities in an annular model combustor. In the simulations,

they were able to suppress decoupled azimuthal modes in a rota-

tionally symmetric set-up but also coupled modes resulting from

non-identical burners.

Based on a time domain network model, Schuermans et

al. [14] investigated thermoacoustic instabilities in a rotationally

symmetric annular combustion chamber. They observed that,

while growing, the unstable mode was of standing type (resulting

from axisymmetric white noise excitation). However, the final

limit cycle solution was always of dominantly rotating type with

no preference of either spinning direction. It was argued that

the saturation nonlinearity in the flame destabilizes the standing

wave mode at finite amplitude and therefore promotes the travel-

ing wave solution. For a simpler model system consisting of only

two first order azimuthal wave components, this mechanism was

investigated in more detail. Depending on the initial conditions,

the growing wave could be arbitrarily made up of azimuthally

traveling or standing components, but the final periodic solution

was again always of rotating type. It was shown that there ex-

ist two types of equilibrium solutions at finite amplitude in wave

amplitude phase-space, one with equal clockwise and counter-

clockwise traveling wave components and the other purely spin-



ning. The former, corresponding to a standing wave, was associ-

ated with a saddle point and therefore unstable. The two purely

rotating solutions were found to be stable fixed points.

Staffelbach et al. [15] performed large eddy simulations of

a full annular helicopter combustion chamber. They observed a

dominantly rotating first order azimuthal wave at 740 Hz with a

clockwise/counterclockwise amplitude ratio of 3. The dominant

effect of the acoustic waves on the heat release rate was found

to result from a modulation of the axial mass flow through the

burners. The same configuration was investigated by Sensiau et

al. [16]. Finite element computations based on the Helmholtz

equation with feedback from the heat release (see also [17]) and

non-trivial boundary conditions were used with mean fields and

flame response data obtained from corresponding LES solutions.

Studying thermoacoustic instabilities in annular combustion

chambers is a complex task (from an experimental as well as

from a computational point of view). Therefore, in contrast to

thermoacoustic instabilities in purely longitudinal geometries,

less information regarding azimuthally unstable modes is avail-

able. In this work, we present a mock-up system which is of sig-

nificantly lower complexity than an annular combustion chamber

but has all essential ingredients so that thermoacoustic instabili-

ties associated with circumferential acoustic modes can be stud-

ied.

The generic system set-up we consider is as follows. N

straight tubes (the “burners”) are connected to an annular duct

(the “combustion chamber”) with their downstream end. In each

tube, there is an element that provides a mean heat release and

responds dynamically to fluctuations in axial velocity. A princi-

pal difference to an annular combustion chamber in a gas turbine

is that, in the set-up considered here, the sources of heat release

do not respond to azimuthal velocity fluctuations in the down-

stream annulus but only to an axial perturbation in the tubes. We

assume the latter effect to be generally dominant, as was found

in Ref. [15].

EXPERIMENTAL SET-UP

The annular Rijke tube has 12 tubes connected to an annular

duct at their downstream end. Figure 1 shows a schematic of the

set-up with the basic geometrical dimensions; a photograph of

the experimental arrangement is presented in Fig. 2. The tubes

have an inner diameter of 60 mm with an associated cut-on fre-

quency for the first azimuthal mode beyond 3 kHz at room tem-

perature. The annular duct is 400 mm in length and has a mean

diameter of 720 mm with a “hub–tip ratio” of 0.8. All parts are

made of aluminum with a wall thickness of 10 mm.

As in a conventional Rijke tube (see, e.g., Refs. [18] or [19]),

the sources of mean and unsteady heat release are electrically

driven heating grids. Obviously this type of heat source is much

simpler than a premixed flame, however, both have a qualita-

tively similar frequency response to axial flow perturbations –

800O640O

60O

720O

265

400

Figure 1. SCHEMATIC OF THE ANNULAR RIJKE TUBE WITH THE

BASIC DIMENSIONS

low-pass characteristics with an associated phase-lag. The axial

extent of the grids is 15 mm, thus clearly complying with the as-

sumption of a compact heat source. The heating grids are pow-

ered by three independent DC sources, each with a maximum

output of 1.5 kW. In the nominal set-up, 4 heating grids are con-

nected in parallel. It was taken special care that all heating grids

are identical so that the nominal system would be as symmetric

as possible. The relative difference in the cold electrical resis-

tance of the 12 heating grids was less than 0.5%. Unless men-

tioned otherwise, all heating grids are supplied with equal elec-

trical power. A photograph of one of the heating grids is shown

in Fig. 3.

No external source is used to drive a mean flow. As in the

original Rijke tube, the mean flow is solely convection induced.

This restricts the parameter space (power input and mean flow

velocity cannot be varied independently) but allows for essen-

tially noise-free measurements. Moreover, the acoustic bound-

ary conditions, which have a significant effect on stability and

microphonesheating grids

speaker

Figure 2. PHOTOGRAPH OF THE ANNULAR RIJKE TUBE



Figure 3. PHOTOGRAPH OF A HEATING GRID (TOP VIEW)

oscillation amplitudes [20], are well defined.

Microphones and speakers can be mounted to the annular

Rijke tube via 48 ports in total, 24 each at the tubes and at the

annular duct. In this study, only 12 microphones, mounted in the

tubes, upstream of the heating grids, were used. For the appli-

cation of feedback control, three speakers were attached to the

tubes downstream of the heat sources (see Fig. 2).

LOW-ORDER SYSTEM MODEL

In this section, an analytical system model for the annular

Rijke tube is presented. The model is based on a frequency do-

main representation of the acoustic field coupled to the heat re-

lease transfer functions of the unsteady heat sources. Although

the heat sources considered in this work are electrically driven

heating grids, the principal methodology can be applied to annu-

lar combustors as well.

The basic strategy in setting up the system model will be

to represent the acoustic response up- and downstream of the

elements of heat release separately and then combine the two

by making use of the appropriate coupling relations across the

heat sources. Figure 4 displays the notation that will be used

for the acoustic pressures and velocities immediately up- and

downstream of the heat sources. pu/d.n and uu/d.n denote acoustic

pressure and axial particle velocity up-/downstream of the heat

source in duct n. The ducts are counted in anti-clockwise di-

rection when looking downstream. The acoustic pressure is as-

sumed to be scaled by the (local) characteristic impedance ρc

(the product of mean density and speed of sound). We assume

that the axial extent of the elements of heat release as well as the

diameters of the ducts are much shorter than the relevant acoustic

wavelengths. In this case, we can consider the zone of heat re-

lease as compact, and the acoustic field in the ducts can be treated

as one-dimensional.

The model is completely linear. Accordingly, we only intend

to identify unstable modes and do not attempt to quantify the

limit cycling amplitude. For the latter, detailed knowledge on the

nonlinear response of the heat sources would be required [12, 21]

which was, however, not available.

xxx x(p, u)u.2

(p, u)d.2

(p, u)u.1

(p, u)d.1

(p, u)u.N

(p, u)u.N

flow

direction

Figure 4. NOTATION FOR ACOUSTIC PRESSURES AND VELOCI-

TIES UP- AND DOWNSTREAM OF THE HEAT SOURCE

Up- and downstream acoustic response

To model the acoustics up- and downstream of the heat

sources, we first consider each part separately. Since we as-

sume plane wave propagation in the tubes, the whole action of

the acoustic field downstream of all heat sources can be repre-

sented as a generalized impedance in the sense that it maps the

velocity fluctuations downstream of the heat sources to the pres-

sures, viz.

pd = Z ud, (1)

where Z is an impedance matrix such that element (i, j) is the

pressure response downstream of the heat source in duct i to a

unit excitation in acoustic velocity in duct j while all other acous-

tic velocities are set to zero. Hence, the elements of Z are given

by Zi j = pd.i/ud. j, with ud.k = 0 for all k , j.

Although the acoustic fields in the tubes upstream of the heat

sources are not directly connected, we construct the model to al-

low for such a coupling, too (in the presence of a plenum, for in-

stance). Analogous to the downstream part, we define an admit-

tance matrix A, which relates the acoustic velocities upstream of

the heat sources to the pressures by

uu =A pu. (2)

Now, as mentioned above, in the present case considered, the

upstream ends of the tubes are not connected to each other so

that A is a diagonal matrix. (This holds true only approximately,

see below.) Moreover, since the tubes are all of the same length,

the diagonal entries are all identical and simply correspond to

the admittance of a duct with an unflanged open end. Hence, we

have

Ai j = −1 − i ZLS tan kLu

ZLS − i tan kLu

δi j, (3)

where k = ω/c is the wave number (ω and c denoting angular

frequency and speed of sound, respectively), δi j represents the

Kronecker delta, and ZLS = (kr)2/4+ i 0.6133 kr is the long-wave



Levine–Schwinger open end impedance [22]; r denotes the tube

radius and Lu the length upstream of the heat source. In principal,

Lu could be different for each tube, which corresponds to the case

where the heating grids are at different axial locations. Then A

would still be diagonal but not isotropic. Modeling the upstream

admittance matrix by Eq. (3) is only an approximation for two

reasons. First, the upstream ends of the ducts are not ideally un-

flanged due to a supporting ring (see Fig. 2). Second, since there

is some sound radiation out of the tube inlets, A is not strictly

diagonal. Both effects can, however, be assumed negligible in

the case of small tube Helmholtz numbers.

The acoustic response matrix of an annular duct.

The impedance matrix of the downstream part, Z, can be ob-

tained by making use of an eigenfunction expansion of the

Greens function of the Helmholtz equation. Since the eigenfunc-

tions for an annular duct are known analytically, this method is

flexible and fast. The approach is described in detail in Refs. [23]

and [24]; we only briefly reproduce the essentials below.

In terms of the eigenfunctions of the Helmholtz equation,

the pressure response to a velocity excitation on a surface S at

the boundary can be written as

p(x) = −ik∑

α

ψα(x)

k2 − iζα − k2α

∫

S

ψ∗α(x0)uν(x0)dA0, (4)

where α = {l,m, n} is a multiindex ordered such that kα ≤ kα+1, l,

m, and n are longitudinal, azimuthal, and radial mode indices, re-

spectively, and ψα is the eigenfunction associated with the eigen-

value kα. ζα denotes a modal damping coefficient. With a con-

stant forcing amplitude at the inlet section on an area correspond-

ing to one of the tubes, the impedance matrix can be computed

from (4) if the series is truncated at sufficiently large values of α.

Since the annular Rijke tube has a hub–tip ratio close to one

(as is the case for most gas turbine combustion chambers), we do

not include any radial modes when computing (4). We simply

use the mean radius for the eigenvalues, which is a good approx-

imation for hub–tip ratios close to one [25].

Prolongation of the annular duct response along

an array of straight tubes. The acoustic response matrix

Z needs to be known at the reference locations immediately

downstream of the heat source. The eigenfunction expansion

presented above, however, only allows to calculate Z directly

at the annular duct. A transformation of the reference locations

along an array of straight tubes (which do not necessarily need to

have identical lengths), such as in Fig. 4, can be achieved in the

following way.

Consider the impedance matrix of an annular duct (no tubes

attached). We denote the up- and downstream traveling plane

wave components at coupling area i as gi and fi. Then the ele-

ments Ri j of the reflection coefficient matrix R can be defined

as Ri j = gi/ f j, with fk = 0 if k , j. Since, for plane waves,

we have pi = fi + gi and ui = fi − gi, the impedance matrix re-

lates incident and reflected plane wave components according to

f + g = Z( f − g). Hence, the reflection coefficient matrix can be

computed from the impedance matrix via

R = (Z + I)−1(Z − I), (5)

where I is the N×N identity matrix. This is a generalization of

the case of a plane wave in a single tube.

Denoting the length of the tubes as ∆xi, the up- and down-

stream traveling waves at the heat source locations, gq.i and fq.i,

say, can be related to those at the inlet of the annular duct as

gq.i = e−ik∆xi gi and fq.i = eik∆xi fi. Accordingly, we have for

the reflection coefficient matrix with reference planes at the heat

sources

Rq.i j = e−ik(∆xi+∆x j) Ri j. (6)

Finally, the impedance matrix at the desired reference location,

Zq, can be calculated using the equivalent of (5)

Zq = (I −Rq)−1(I +Rq). (7)

With (6) and (7), Eq. (5) can be reformulated to directly relate

the impedance matrix at the heat sources to that at the inlet of the

annular duct, viz.

Zq =[

(Z + I)D−1 − (Z − I)D]−1 [

(Z + I)D−1 + (Z − I)D]

.

(8)

In (8), the matrix D has elements Di j = e−ik∆xi δi j.

Heat release dynamics

In each tube, the relations between the up- and downstream

acoustic variables are determined by the dynamic response of

the heat source (in that duct). In a general linear framework, this

relation can be expressed by a 2×2 transfer matrix in frequency

domain

[

pd.n

ud.n

]

= Tn

[

pu.n

uu.n

]

. (9)

For the type of compact heat source we consider here, and for

a vanishing mean flow Mach number, it is reasonable to assume



that the pressure loss is negligible. Furthermore, the heat source

induces a jump in acoustic velocity via [26]

A(ud.n − uu.n) =γ − 1

γP0

qn, (10)

where qn and γ denote the heat release rate and the ratio of spe-

cific heats, respectively, A is the cross-sectional area, and P0 rep-

resents the mean pressure.

We introduce the heat release transfer function as the nor-

malized response of heat release fluctuations to normalized per-

turbations in the upstream velocity

Fn =qn

uu.n

uu.n

qn

. (11)

Then the transfer matrix of the heat source in duct n can be writ-

ten as

Tn =

[

ξn 0

0 1 +(

Td.n

Tu.n− 1)

Fn

]

, (12)

where ξn =√

Td.n/Tu.n is the ratio of characteristic impedances

up- and downstream of heating element n. As proposed by

Lighthill [27], we model the heat release transfer function as a

first order low-pass

Fn =ck.n

1 + iωcτ.n, (13)

with gain and phase parameters ck.n and cτ.n. Note that we ex-

plicitly allow for different temperature jumps and heat release

transfer functions in each tube. For later use, we define diag-

onal matrices X and F with elements Xi j = ξiδi j and Fi j =

(Td.i/Tu.i − 1)Fiδi j.

Submodel coupling and dispersion relationTo couple the acoustic models of the up- and downstream

parts, we use the dynamic heat release characteristics given by

Eq. (12). Hence, we have pd = X pu, i.e., the pressure is contin-

uous across the heat source, and the jump in acoustic velocity is

given by

ud =(

I +F)

uu. (14)

Combining Eqs. (1), (2), (14), and the pressure continuity con-

dition, and dropping the subscript u of the vector of pressures

upstream of the heat source, we obtain

X p = Z(

I +F)

A p. (15)

From Eq. (15), it follows that for non-trivial solutions to exist,

the dispersion relation

det S = 0 (16)

must be satisfied, where the system matrix S is given by

S = Z(

I +F)

A − X. (17)

Solutions ωk of the dispersion relation are the system eigenfre-

quencies, Re(ωk) representing the (angular) oscillation frequency

and −Im(ωk) the growth rate. The purely acoustic eigenfrequen-

cies can be obtained for the case with no dynamic heat release as

the solutions of |ZA − X| = 0.

It is interesting to note here that Eq. (16) with S given in

(17) is again a generalization of the simple 1D case of a sin-

gle Rijke tube. In this case, the impedance and admittance are

scalar functions which relate acoustic velocity to pressure and

vice versa. The dispersion relation for this configuration can be

written as Z(1+F )A−ξ = 0 [24], where A and Z are admittance

and impedance immediately up- and downstream of the source of

heat release.

If an eigenvalue ωk of (16) has been obtained, the corre-

sponding pressure pattern in the N tubes, at the heat source po-

sitions, can be computed as the null space of the system matrix

evaluated at ωk, hence

pk = ker S(ωk). (18)

Note here that for a distinct eigenvalue, we have dim ker S(ωk) =

1, whereas for a degenerate one, such as in the case of a fully

symmetric configuration, where two mode shapes share one char-

acteristic value, the eigenspace is two-dimensional.

RESULTS AND DISCUSSION

Identifying the modal structure of the pressure fieldTo assess the azimuthal structure of the experimental and the

model data, we determine the coefficients am,c|s of the m-th order

cosine and sine modes by means of a discrete projection of the N

pressures on an angular Fourier basis hm,c|s(φ), viz.

am,c|s =

N∑

n=1

pnhm,c|s(φn), (19)

where the φn correspond to the circumferential locations of the

tube center axes. In the case of even N, we use

hm,c = bm,c cos(mφ), m = 0 . . .N/2, (20a)

hm,s = bm,s sin(mφ), m = 1 . . . (N/2 − 1), (20b)



0 1 1 2 2 3 3 4 4 5 56 0 1 1 2 2 3 3 4 4 01 1 2 2 3 3 5 5 4 4 6200

300

400

500

600

700

800

900

1000

azimuthal mode order m

freq

uen

cyin

Hz

low-order model

FEM computation

Figure 5. ACOUSTIC EIGENFREQUENCIES AND CORRESPONDING

AZIMUTHAL MODE ORDER m; MODEL AND FEM RESULTS ARE

COMPARED.

were the coefficients bm,c and bm,s are chosen such that the basis

has unit norm in the discretized N-space. Note here that, with this

normalization, the amplitudes of the modal coefficients are larger

than the actual pressure amplitudes by about a factor of√

N. The

azimuthal mode order of the eigenfunctions computed from the

model (Eq. (18)) can be determined in the same way. Using pk in

(19) yields the Fourier transformed modal coefficients am,c|s (the

modal amplitudes) of the k-th eigenfunction.

Eigenfrequencies without dynamic heat releaseThe eigenfrequencies of the annular Rijke tube, computed

with the model presented in the previous section, for a temper-

ature jump of 50 K across the heat source, are shown in Fig. 5.

No transfer function for the dynamic heat release was included

here. To assess the accuracy of the low-order model, the results

obtained with a commercial finite element code are also plot-

ted. The latter computes the eigenfrequencies and corresponding

eigenfunctions from the Helmholtz equation

∇·(

c2∇p)

+ ω2 p = 0. (21)

In the finite element model, we used pressure node boundary

conditions at the inlets of the tubes and at the downstream end

of the annular duct. To account for the end correction of the

open tubes, we extended the part upstream of the heat source ac-

cordingly. Radiation from the open end as well as from the outlet

of the annular duct was not included.

Since the boundary conditions at the up- and downstream

end of our configuration are open, all modes have at least a half-

wave structure in axial direction. Moreover, only the axisymmet-

ric mode and those with azimuthal orders being an integer mul-

tiple of 6 are not degenerate, since we have a 12-fold rotational

Figure 6. PRESSURE PATTERNS CORRESPONDING TO THE

ACOUSTIC MODES OF THE ANNULAR RIJKE TUBE AT EIGENFRE-

QUENCIES 541, 554, AND 557 Hz (FROM LEFT TO RIGHT)

symmetry (with additional mirror symmetries). The fact that cer-

tain azimuthal modes are distinct is a direct result of that symme-

try structure [28, 29]. By deliberately introducing asymmetries,

we can also split modes which are degenerate in the nominal,

symmetric case. This is demonstrated in a later section.

We also note that the modal density between 500 and 600 Hz

is particularly high (Fig. 5). Azimuthal modes of order 4, 5, and

6 in this frequency range were found to be unstable in the exper-

iment (see below). The pressure patterns in the annular duct cor-

responding to these modes are shown in Fig. 6 (computed with

the Helmholtz solver). It should be noted, however, that the an-

gular location of the pressure nodes and antinodes is arbitrary for

azimuthal orders 4 and 5, since these modes are degenerate.

Linear stability

The linear stability characteristics for the experimental sys-

tem were computed with the low-order model including the heat

release transfer function (13). The complex eigenvalues, deter-

mined from Eq. (16), are shown in Fig. 7. Negative imaginary

parts correspond to instability. All eigenvalues with frequencies

larger than 600 Hz were strongly damped and are thus not shown.

The dominant feature is a group of eigenvalues around 550 Hz

with zero or small negative imaginary part. These eigenvalues

correspond to azimuthal mode orders m = 4, 5, and 6 (with in-

creasing frequency) and can be expected to be essential in the

experiment.

Instabilities in the experiment

Consistent with the linear stability analysis based on the

low-order model, azimuthal modes of orders 5 and 6 were ob-

served in the experiment for a certain range of input powers.

The modal amplitudes were computed from the 12 pressure sig-

nals using Eqs. (19) and (20). Figure 8 shows the average

amplitude of the modal coefficients for a variation in the in-

put power. For azimuthal orders 1–5, the effective amplitude

(a2m,c+a2

m,s)1/2 is plotted. All heating grids were driven with iden-

tical power. No instability is observed for input powers smaller

than 1500 W. With increasing heating power, the first 6th order

azimuthal mode becomes unstable. The amplitude grows with



200 300 400 500 600−0.5

0

0.5

1

1.5

2

2.5

3

3.5

Re (ωk)/(2π) in Hz

Im(ω

k)/

(2π

)in

s−1

0

1

2

3

4

5 6

0 1

2

Figure 7. EIGENVALUES OF THE ANNULAR RIJKE TUBE COM-

PUTED WITH THE LOW-ORDER MODEL. NUMBERS INDICATE AZI-

MUTHAL MODE ORDER.

an increase in the electrical power up to about 1700 W. However,

the first 5th order azimuthal mode also starts to grow and dom-

inates for higher power values. At an input power of 2000 W,

the m = 6 mode is almost completely suppressed. For larger

powers, however, this mode remains significant, although the 5th

order mode is clearly dominant. The strong suppression of the

m = 6 mode around 2000 W was a reproducible feature. Also

note that the power dependence of the amplitude of the 5th order

azimuthal mode clearly corresponds to a supercritical bifurca-

tion. The modes with azimuthal order lower than 5 only have a

small contribution to the measured pressure signals. When the

heating grid power is switched on, the m = 6 mode is actually

growing much faster and is limited and then reduced in ampli-

tude through the 5th order azimuthal mode (see also Fig. 17 in

the section on feedback control).

For exemplary purposes, the amplitude spectrum of one of

1000 1500 2000 2500 3000 3500 4000 45000

250

500

750

1000

1250

1500

1750

2000

total power in W

mo

dal

amp

litu

de

inP

a

m = 0 . . .4

m = 5

m = 6

Figure 8. MEASURED MODAL AMPLITUDES FOR VARYING TOTAL

POWER INPUT

0 250 500 750 1000 1250 1500 1750 200040

60

80

100

120

140

frequency in Hz

spec

tral

amp

litu

de

ind

B

Figure 9. PRESSURE AMPLITUDE SPECTRUM FOR A TOTAL INPUT

POWER OF 3200 W

the pressure signals is shown in Fig. 9. The data corresponds to a

total input power of 3200 W. A strong peak, extending more than

four orders of magnitude above the background noise, is visible

around 560 Hz. At this scale, the frequencies corresponding to

the 5th and 6th and the stable 4th order azimuthal modes cannot

be distinguished. A few harmonics are also present.

A more detailed view of the dominant frequency content of

the pressure oscillations is given in Fig. 10. The data corre-

sponds to total input powers of 1700 and 2100 W (cf. Fig. 8).

A peak with small magnitude can be observed between 520 and

530 Hz. This corresponds to the m = 4 mode that was found to

be marginally stable in the low-order model. Two stronger fre-

quency components are found around 540 and 550 Hz. The lo-

cation of the peaks slightly increases with input power due to the

larger speed of sound downstream of the heat source. The spac-

ing of the two dominant peaks is only 3–4 Hz, which also com-

pares well with the model results. For a power input of 1700 W,

the peak with the larger frequency is about an order of magnitude

stronger. On comparison with Fig. 8, we find this spectral com-

ponent to be associated with the m = 6 mode. At 2100 W, the 5th

order azimuthal mode clearly dominates.

The simultaneous oscillation of two instability modes is not

often observed in thermoacoustic instability investigations, the

common explanation being that the oscillation of the “more un-

stable” mode suppresses the growth of the weaker one. In our

case, the presence of two unstable modes with significant ampli-

tudes clearly is related to the close proximity of their associated

oscillation frequencies (see also Fig. 10). A closer inspection of

the temporal evolution of the modal amplitudes showed, how-

ever, that the weaker m = 6 mode did not have a constant am-

plitude. It was growing and decaying, seemingly randomly, on

a much larger time scale than the actual oscillation. A growth

in amplitude was always associated with a decay of the m = 5

mode. Thus, there was a continuous competition between the



500 520 540 560 580 60040

60

80

100

120

140

frequency in Hz

spec

tral

amp

litu

de

ind

B

P = 1700 W

P = 2100 W

Figure 10. PRESSURE AMPLITUDE SPECTRA FOR INPUT POWERS

OF 1700 AND 2100 W WITH THE m = 6 AND m = 5 MODE BEING

DOMINANT, RESPECTIVELY

two modes rather than two coexisting oscillations with constant

amplitudes.

The azimuthal instability modes observed in annular com-

bustion chambers can be either standing or rotating in angular

direction or be a mixture of both [4, 14, 15]. In the present case,

this is more difficult to asses due to the two coexisting oscillation

modes. However, a careful investigation of the time traces of the

12 pressure sensors showed that the m = 6 mode was always of

standing type. This was expected, since this mode is not degen-

erate and, therefore, rotating wave solutions do not exist in the

linear regime. In Ref. [14], it was shown that the saturation in

the heat release response at high amplitudes favors rotating wave

solutions. We conjecture, however, that this does not apply to

azimuthal modes which are not degenerate.

The 5th order azimuthal mode is degenerate so that spin-

ning and standing oscillation patterns are possible. No informa-

tion can be gained from the stability analysis, since all linear

combinations of the two basis functions spanning the degener-

ate eigenspace are equivalent. The postprocessing of the exper-

imental data showed that the m = 5 mode was not of definite

standing or rotating type. In terms of the cos(mφ), sin(mφ) basis

(Eq. (20)), azimuthally standing components are associated with

in-phase modal coefficients am,c and am,s. Conversely, a purely

rotating solution has modal coefficients which are in quadrature

but have identical magnitudes [8]. Thus, the nature of the 5th or-

der azimuthal mode can best be visualized by means of a phase

plane with the coefficients of the cosine and sine modes (am,c and

am,s) as coordinates. Then a circumferentially standing wave is

represented by a straight line whose slope determines the angular

location of the pressure nodes, and a purely spinning wave traces

out a circle.

Figure 11 shows four exemplary results, which correspond

to the same operating conditions (total power of 4500 W) but dif-

−2

−1

0

1

2

a5,s

(t)

ink

Pa

(a) (b)

−2 −1 0 1 2

−2

−1

0

1

2

a5,c(t) in kPa

a5,s

(t)

ink

Pa

(c)

−2 −1 0 1 2

a5,c(t) in kPa

(d)

Figure 11. PHASE PLANE TRAJECTORIES IN a5,c–a5,s SPACE. COL-

ORS FROM BLUE TO RED VIA WHITE INDICATE INCREASING TIME.

THE LENGTH OF EACH TRAJECTORY CORRESPONDS TO A TIME

INTERVAL OF 49 MILLISECONDS, AND THE DATA IS SAMPLED AT

213 HZ. TOTAL ELECTRIC POWER 4500 W.

ferent periods of time. Apparently, the 5th order azimuthal mode

is neither distinctly spinning nor standing. In fact, it does not

exhibit any stationary characteristic at all. Figure 11 (b), for in-

stance, clearly corresponds to a circumferentially standing wave,

whereas Fig. 11 (c) represents an essentially rotating mode. Tran-

sitions between different oscillation patterns can be observed in

frames (a) and (d). Qualitatively similar characteristics were also

present at other input powers, for which the m = 5 mode was un-

stable.

The effect of an asymmetric power distribution on ther-moacoustic stability

An azimuthal variation of burner and flame properties in

an annular combustion chamber, either through geometrical

changes [5, 6] or burner group staging [30], can have a stabi-

lizing influence on the pressure oscillations. We mimic this ef-

fect in the annular Rijke tube through an azimuthal variation of

the power supplied to each heating grid. Changing the geome-

try of a burner or the equivalence ratio with which it is supplied

typically changes the flame response. An analogous effect is ob-

tained when varying the power of a heating grid. In general, such

a variation has an impact on the gain as well as on the phase of

the heat release transfer function. For the heating grid, a larger

power leads to an increase in the downstream temperature and in



540 550 560 570 580 590 600−1

0

1

2

3

C12 C4 C3


Im(ω

k)/

(2π

)in

s−1

Figure 12. VISUALIZATION OF THE NOMINAL POWER DISTRI-

BUTION AND THE TWO STAGING PATTERNS WITH ASSOCIATED

EIGENVALUES; DARKER CIRCLES INDICATE HIGHER ELECTRIC

POWER. THE EIGENVALUES OF THE C4 AND THE C3 CASE COR-

RESPOND TO σ = 1 AND σ = 1.5, RESPECTIVELY.

the mean velocity (because it is convection induced). Since the

heat release response of the grid is essentially a function of the

Strouhal number, the effect of a change in the supplied power can

be estimated based on measured temperatures and mean veloci-

ties for different heating powers.

An azimuthal variation of the heat release transfer functions

does not necessarily have a positive effect on thermoacoustic sta-

bility. Nominally degenerate azimuthal modes can be split if the

system’s symmetry group is reduced in a particular way through

the introduced circumferential variation [28]. To illustrate this

effect, we investigate two patterns of azimuthal “staging”: one

has a 4-fold rotational symmetry (denoted by C4) with reduced

power at every third heating grid, and the other one has a 3-fold

rotational symmetry (denoted by C3) with groups of two heating

grids alternating in lower and higher electric power. The stag-

ing patterns are visualized in Fig. 12. In all results shown in this

section, the total electric power is held constant at 2900 W. To

quantify the strength of the circumferential modulation, we de-

fine the asymmetry parameter σ as

σ =∆P

Pmean

, (22)

where ∆P is the difference in electric power of the two types of

heating grids in one staging pattern, and Pmean is the mean power

(which is held constant).

We discuss the model results for the two staging patterns

first. The system eigenvalues for the reduced symmetry groups

C4 and C3 and the corresponding staging patterns are shown in

Fig. 12. For comparison, the eigenvalues for the original fully

symmetric case (C12) are also plotted. Only the frequency range

between 540 and 600 Hz is shown in Fig. 12 for clearness. Con-

sider the effect of the C4 staging pattern first. Here, σ = 1 has

been used for the computations. The m = 4 mode, degenerate

in the baseline case with a frequency of 546 Hz and marginally

stable, splits into two distinct modes as a result of the reduction

in symmetry. As it turns out, this is an essentially destabilizing

effect, because one of the split modes has a larger imaginary part,

whereas the other moves down into the unstable half plane. The

modes of azimuthal order 5 and 6, dominant in the fully sym-

metric case, are slightly stabilized. The mode with a frequency

of 593 Hz in the baseline case also splits, because it is of 2nd

azimuthal order. However, both m = 2 split modes remain sta-

ble. The removal of the degeneracy of the 4th and 2nd order

azimuthal modes is consistent with the splitting rule stated by

Perrin [28]. Thus, based on the linear analysis, we summarize

the effect of the C4 staging pattern as a slight stabilization of the

nominally unstable m = 5 and m = 6 modes at the cost of a

significant destabilization of one of the m = 4 split modes.

The results of the C3 staging pattern are univocal; all modes

are either stabilized or affected only insignificantly (also those

not shown in Fig. 12). However, the damping of the 5th or-

der azimuthal mode is only slightly increased so that it remains

marginally stable. It is important to note here that no destabi-

lizing splitting of the 6th order azimuthal mode occurs, because

this mode is already distinct in the baseline case due to the 12-

fold rotational symmetry.

To assess the effect of the reduced circumferential symme-

try in the experiment, measurements were performed with a con-

stant total power of 2900 W while varying the distribution to the

12 heating grids according to the C3 and C4 staging patterns.

The measured modal amplitudes as a function of the asymme-

try parameter σ are shown in Figs. 13 and 15. At σ = 0, only

the 5th and 6th order azimuthal modes are oscillating, with the

former dominating. This corresponds to the case already con-

sidered before. Increasing the asymmetry according to the C4

pattern (Fig. 13) leads to a continuous decrease in the amplitudes

of the 5th and 6th order azimuthal modes. At σ = 0.75, both of

these modes are stabilized. However, at the same time, the m = 4

mode is destabilized and grows in amplitude with a further in-

crease of σ. These results correspond well with the eigenvalue

analysis for σ = 1 (Fig. 12).

Since, for the C4 staging pattern, the m = 4 modes are not

degenerate any more, they have a distinct pressure pattern along

the circumference. Hence, we can compare the modeled circum-

ferential pressure distribution (Eq. 18) corresponding to the un-

stable m = 4 mode with the measured one. This comparison is

presented in Fig. 14. For completeness, the pressure pattern of

the stable split mode is also shown. The measured and modeled

pressure patterns both correspond to σ = 1. The experimen-



0 0.25 0.5 0.75 1 1.25 1.50

250

500

750

1000

1250

1500

1750

2000

asymmetry σ

mo

dal

amp

litu

de

inP

a

m = 0 . . .3

m = 4

m = 5

m = 6

Figure 13. MEASURED MODAL AMPLITUDES AS A FUNCTION OF

THE ASYMMETRY PARAMETER σ; TOTAL POWER INPUT 2900 W.

tal observations confirm the model results. Clearly, the selected

pressure pattern identified in the measured data corresponds to

the computed unstable mode. The reason for this pattern be-

ing more unstable is that the pressure nodes are arranged such

that they coincide with the locations of the heating grids with

reduced power. The measured pressure values represent an arbi-

trary snapshot. Note, however, that the amplitude of the modeled

pressure distribution is undetermined, because it is an eigenfunc-

tion. In Fig. 14, the amplitude of the computed pressure pattern

has been scaled such that it matches the experimentally deter-

mined magnitudes.

An increase in σ according to the C3 pattern (Fig. 15) results

in a decrease in the amplitude of the 5th order azimuthal mode.

In contrast, the m = 6 mode is not damped by the asymmetry. In

fact, it increases in amplitude, quite significantly aroundσ ≈ 0.6,

0 2 4 6 8 10 12−500

−250

0

250

500

tube number n

pin

Pa

pexp Re ( pmod,u) Re ( pmod,s)

Figure 14. MODELED AND MEASURED PRESSURE DISTRIBU-

TIONS IN THE TUBES CORRESPONDING TO THE 4TH ORDER AZI-

MUTHAL MODE FOR σ = 1. HEATING GRIDS IN TUBES 2, 5, 8, AND

11 HAVE A REDUCED POWER INPUT. THE STABLE (pmod,u) AND THE

UNSTABLE (pmod,s) m = 4 SPLIT MODES ARE SHOWN.

0 0.25 0.5 0.75 1 1.25 1.50

250

500

750

1000

1250

1500

1750

2000

asymmetry σ

mo

dal

amp

litu

de

inP

a

m = 0 . . .4

m = 5

m = 6

Figure 15. MEASURED MODAL AMPLITUDES AS A FUNCTION OF

THE ASYMMETRY PARAMETER σ; TOTAL POWER INPUT 2900 W.

and settles on a constant level for σ = 0.7–1. Increasing the

asymmetry parameter further, both modes are eventually stabi-

lized at σ = 1.3. The fact that both modes become stable at the

same value of the asymmetry parameter indicates significant in-

teraction even at low amplitudes. The eigenvalue analysis for this

staging pattern with σ = 1.5 (Fig. 12) showed the m = 6 mode to

be stabilized and the m = 5 mode to be marginally stable. This

does not fully correspond to the experimental data but points in

the same direction. (See the discussion on model accuracy at the

end of the results section.)

Feedback control

In this section, we show that, in addition to studying self-

excited instabilities, the annular Rijke tube can also be used to

test active control schemes. Devising an elaborate model-based

control algorithm, such as, e.g., in Refs. [10, 23], is beyond the

scope of the present paper. Instead we present the effect of sim-

ple proportional control on the unstable modes. We used three

speakers, mounted to the tubes at the pressure ports immediately

downstream of the heat sources (see Figs. 1 and 2) at equidistant

azimuthal angles.

To account for the effect of control in the model, we add a

linear feedback term in the jump conditions (14), viz.

ud =(

I +F)

uu +K pu, (23)

where K is a constant N×N gain matrix that basically maps the

input to the output locations. Equation (23) models the loud-

speaker action as an additional volume source at the heating grid

location. This does not represent the experimental feedback con-

figuration exactly, since the actuators are located 65 millimeters

downstream of the heat source. Also, the feedback signal is not



the pressure directly at the heat source but 45 millimeters up-

stream. On the other hand, the pressure field does not change

significantly over these short distances, and, in addition to that,

we are interested in the resonance frequencies only, at which the

complete pressure field in one tube is essentially in phase. There-

fore, we can consider (23) as a reasonably accurate representa-

tion of the feedback configuration in the experiment.

We will consider two types of proportional feedback: (i) a

single sensor signal is fed back to all three actuators and (ii) the

actuators feed back the pressure signal from their respective tube.

For case (i), the gain matrix takes the form

K(i) = [c 0], (24)

where cT = [κ 0 0 0 κ 0 0 0 κ 0 0 0] and κ is a negative constant.

In the experiment, κ has to be tuned to compensate for input and

output amplifiers and the speaker transfer function. For multiple

inputs (case (ii)), the gain matrix K(ii) is diagonal with

K(ii)nn =

κ, n ∈ {1, 5, 9},0, else.

(25)

The effect of these types of control on linear stability can

be assessed by computing the system eigenvalues based on the

modified jump conditions (23). In this case, the system matrix

accounting for the effect of feedback control takes the form

S = Z(

I +F)

A − X +ZK(i,ii). (26)

The influence of control on the stability of the dominant

modes is shown in Fig. 16. The results from type (i) and type

(ii) control are compared to the uncontrolled case; a feedback

gain κ = −0.03 has been used for the computations. In case of

type (i) control, when only a single pressure signal is fed back to

all three actuators, only the 6th order azimuthal mode at 564 Hz

is stabilized. (The damping rate of the stable m = 0 mode at

572 Hz is also increased.) In contrast, type (ii) control stabilizes

all modes; the damping rates of the m = 6 and m = 0 modes co-

incide with those for type (i) control. This is because for axisym-

metric and 6th order azimuthal modes, it makes no difference if

only one signal is fed back; the pressure in all of the actuator

tubes is identical for these modes. However, since the m = 5

mode also shows strong oscillations in the uncontrolled case, we

expect type (ii) control to be superior in suppressing the instabil-

ity.

Results from the experimental application of the two types

of proportional control are presented in Fig. 17. The amplitudes

of the 5th and the 6th order azimuthal modes are plotted versus

time. Here, the temporal evolution of the modal amplitudes was

520 540 560 580 600−1

0

1

2

3


Im(ω

k)/

(2π

)in

s−1

no control

type (i) control

type (ii) control

Figure 16. THE EFFECT OF TWO TYPES OF CONTROL ON THE

SYSTEM EIGENVALUES

computed as the magnitudes of the filtered analytic signals of the

modal coefficients am. A few seconds after the power is turned

on, the amplitudes of the two unstable modes start to grow ex-

ponentially, the m = 6 mode increasing distinctly faster. As

soon as the m = 5 mode has grown to a significant amplitude,

the 6th order azimuthal mode actually decreases, and both settle

on approximately constant levels. The strong overshoot of a6 is

obviously a result of the nonlinear interaction of the two unsta-

ble modes, probably associated with the close proximity of their

oscillation frequencies. A detailed investigation of this nonlin-

ear interaction is, however, beyond the scope of the present pa-

per. Type (i) control is activated from t = 20 s to approximately

t = 30 s. Only a small effect can be observed. There is no signif-

icant reduction of the oscillations, but the amplitude a6 seems to

settle on a somewhat smaller level and shows less variations. The

linear analysis based on the model predicted a stabilization for

the 6th order azimuthal mode using this type of control (Fig. 16).

On the other hand, since the m = 5 mode is still oscillating with

0 5 10 15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

time in s

mo

dal

amp

litu

de

inP

a

m = 5 m = 6

poweron

type (i)control on

type (i)control off

type (ii)control on

Figure 17. TEMPORAL EVOLUTION OF MEASURED MODAL AM-

PLITUDES WITH THE EFFECT OF FEEDBACK CONTROL; TOTAL

POWER INPUT 3200 W.



finite amplitude, the linear analysis may not be applicable. When

activating type (ii) control (based on three pressure sensors) at

approximately t = 40 s, the oscillation diminishes immediately.

This is fully consistent with the results from the linear analysis

(Fig. 16) where all modes in the unstable frequency range were

predicted to be stabilized by the control action.

The amplitude spectra of one of the measured pressure sig-

nals (Fig. 18) shows that type (ii) control completely stabilizes

the system. In the uncontrolled case, the spectral peak corre-

sponding to the instability has a magnitude of almost 140 dB.

Applying type (ii) control, this component is reduced to below

60 dB, resulting in a peak reduction of four orders of magnitude.

The pressure signal used for the spectra in Fig. 18 does not cor-

respond to one of the tubes with actuator.

0 250 500 750 1000 1250 1500 1750 200040

60

80

100

120

140

frequency in Hz

spec

tral

amp

litu

de

ind

B control off

type (ii) control

Figure 18. PRESSURE AMPLITUDE SPECTRA FOR THE BASELINE

AND THE CONTROLLED CASE. TOTAL POWER INPUT 3200 W.

Discussion on model accuracy

The low-order model used in the present work provided

good results that were in accordance with the experimental data.

Unstable modes could be identified, and the effects of asymme-

tries in the circumferential heat release distribution and propor-

tional feedback were fairly well predicted. On the other hand,

we do not claim that the model can be considered quantitatively

correct in every aspect. An exact prediction of the growth rates,

e.g., and, along the same lines, an accurate determination of sta-

bility boundaries, is likely beyond its capabilities. Such quan-

titative predictions are inherently difficult, even with a model

built exclusively from experimental data [31]. One potentially

important aspect was not included in the low-order model. We

assumed a constant temperature downstream of the heat source

for simplicity. Pointwise temperature measurements in the ex-

perimental set-up showed that this was not true. Noticeable vari-

ations in axial as well as in angular direction could be found.

A non-uniform temperature distribution may well have an effect

on thermoacoustic mode stability [32, 33]. Assessing the effect

of an inhomogeneous temperature field in the annular duct on

the basis of the eigenfunction expansion approach used in the

present work is not straightforward, but two potential remedies

have been identified and are subject of ongoing work.

SUMMARY AND OUTLOOK

We presented an experimental thermoacoustic surrogate sys-

tem that facilitates an in-depth study of instabilities associated

with circumferential acoustic modes. For a range of input pow-

ers, we observed thermoacoustic oscillations corresponding to

the 5th and 6th azimuthal mode order. A low-order model repre-

senting the experimental arrangement was set up and gave results

fully consistent with the measured data. The effect of a symme-

try reduction through circumferential modulations of the heating

power was investigated. With increasing asymmetry, the 5th and

6th order azimuthal modes were stabilized, but, for one of the

staging patterns, a mode of 4th circumferential order became un-

stable at the same time. Also in this case, the model was capable

of predicting the effect of the introduced asymmetry.

The control approach presented in this work was, though

effective, rather elementary. A more advanced control scheme,

based on a time-domain model and utilizing a modal decom-

position of all 12 sensor signals in combination with feedback

through 6 speakers, has been successfully applied and is subject

of a forthcoming publication. Extension of the model and an

investigation of different damper configurations is the focus of

ongoing and future work.

ACKNOWLEDGMENT

Financial support from the German Research Foundation

through the Collaborative Research Center 557 is gratefully ac-

knowledged.

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