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STABILITY LIMITS AND NONLINEAR CHARACTERIST
ICS OF A SELFEXCITED COMBUSTION INSTABILITY
Roel A. J. Müller, Jakob Hermann
IfTA Ingenieurbüro für Thermoakustik GmbH, 82194 Gröbenzell, Germany, www.ifta.com
Wolfgang Polifke
Lehrstuhl für Thermodynamik, Technische Universität München, 85747 Garching, Germany
Selfexcited combustion instabilities of a bluffbodystabilized premixed ﬂame are investigated.
The burner is situated in a duct of rectangular crosssection with closed/open acoustic boundary
conditions, representing a quarterwave acoustic resonator. Fuel gas (methane) is injected
through small holes a short distance upstream of the ﬂame, such that it burns in partially
premixed mode. Operating points (i.e. thermal power and equivalence ratio) where instability
occurs are identiﬁed. Variation of amplitude and frequency while operating conditions are
changed are discussed for the dominant mode of oscillation, as well as its second and third
harmonic. For certain operating points, the harmonics of the ﬁrst mode seem to couple with
higher independent modes.
Pressure oscillations during limit cycle operation for one constant operating point are measured
along the combustor length. The shape and amplitude of several independent thermoacoustic
modes, as well as the most signiﬁcant higher harmonics, are reconstructed, identiﬁed and
interpreted in relation to the combustor geometry. The phase shift between pressure variation at
the ﬂame and heat release (Rayleigh index) is discussed for the independent modes, as well as
the higher harmonics.
1. Introduction
The EUfunded Marie Curie research project LIMOUSINE (
Lim
it cycles of thermoac
ous
tic
oscillations in gas turb
ine
combustors) aims to strengthen the fundamental scientiﬁc work in the ﬁeld
of thermoacoustic instabilities in combustion systems. A bluffbodystabilised, partially premixed test
combustor was built to provide experimental data for validation of numerical and analytical results.
This paper presents the combustion instabilities as they occur in this test combustor.
Operating points (thermal power
P
th
and equivalence ratio
Φ
) where instability occurs are
identiﬁed and discussed. During limit cycle operation, the pressure amplitude of the dominant
oscillation, as well as its higher harmonics, are recorded. For a reference operation point (
P
th
= 40kW
and
Φ = 0.7
), the pressure signal was measured along the length of the combustor. Pressure proﬁles
were reconstructed from these measurements. The following mode shapes are identiﬁed: the ﬁrst
thermoacoustic mode, which is the dominant oscillation (
I,1
), its second and third harmonic (
I,2
and
I,3
)
and the next two independent modes (
II,1
and
III,1
). These mode shapes are discussed, as well as their
phase shift compared to the heat release represented by the OH* chemiluminescence signal from the
ﬂame.
ICSV19, Vilnius, Lithuania, July 8–12, 2012 1
19
th
International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
2. Setup
2.1 Combustor
Fig. 1 gives an overview of the LIMOUSINE combustor as installed at IfTA GmbH. The burner
is situated in a duct of rectangular crosssection with closed/open acoustic boundary conditions,
approximating a quarterwave acoustic resonator. The rectangular cross section with relatively large
aspect ratio leads to an approximately 2D ﬂow in the x,y plane.
50
40
25
x = 0
272
227
+780
50
100
200
50
100
200
250
750
133
B
C
150
Flow direction
300
350
700
400
450
500
550
600
650
(a) Cross section of the combus
tor: inner dimensions (in mm)
on the left, locations of holes for
sensor access on the right
Detail C
8
2
4×11×
1
3
Detail B
1
14
2
× 31×
7
19
Flame
holder
Combustion
chamber
Plenum
3
x
y z
(b) Details of ﬂame holder (above) and air inlet
(below)
(c) The combustor as in
stalled in the laboratory at
IfTA
Figure 1. Overview of the combustor with coordinate axes. The origin of x is the top of the ﬂame holder. The
inner width of the duct in z direction is 150mm.
Air enters the bottom of the combustor at the front and at the back through choked oriﬁces, which
guarantees an acoustically hard boundary condition, after which perforated tubes distribute the air
along the span (
z
direction) of the combustor. The prismatic ﬂame holder, triangular in cross section,
rests at about one quarter of the height of the combustor. The fuel gas (methane) is also passed through
oriﬁce plates and injected through 62 holes along the ﬂame holder. A partially premixed ﬂame forms,
stabilised at the top of the ﬂame holder, where quartz glass windows on three sides provide optical
access to the ﬂame. The window in the front wall is replaced by a steel plate to allow mounting of
acoustic sensors here as well.
There are holes along the length of the front combustor wall for sensor access, which are usually
closed. The holes are placed at
50mm
intervals where space allows. In the vicinity of the ﬂame holder
the holes deviate from this pattern for constructional reasons. A Photo Multiplier Tube (PMT) views
the ﬂame along z direction from the back.
2
19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
2.2 Measurement equipment
2.2.1 Pressure sensors
Two sensors, PCB Model 106B50, have been used during this measurement. Throughout the
experiments, watercooled sensor adaptors were used to protect the sensors against the high combustor
wall temperature. Before the measurement on the combustor, the difference in frequency response
between the sensors was determined in phase and amplitude. For
f < 500 Hz
the sensor signals differ
less than one per cent. The data presented for the mode shapes are compensated for this difference in
sensor response.
2.2.2 Photo Multiplier Tube (PMT)
A PMT measured the overall OH* chemiluminescence of the ﬂame from the back of the
combustor. The phase shift between this signal and the local pressure is used as a qualitative measure
of the modal Rayleigh index
Ri( f ) = R
ˆp( f ) ˆq ( f )
∗
. If
Ri( f ) > 0
, the ﬂame acts as an acoustic
source at frequency
f
. The heat release rate in the frequency domain
ˆq( f )
is represented by the PMT
signal. ˆq( f )
∗
denotes its complex conjugate.
1,2
2.2.3 Frequency domain and correlation analysis
IfTA’s Argus Oscillation Monitoring and Diagnostics System (OMDS) recorded the various
signals with a sampling frequency of
5120Hz
. The signals of the sensors are cut in segments of
0.2s
with an overlap of
50%
. After application of a Blackman window, the signal is transformed into
the frequency domain using a Fast Fourier Transform (FFT). Here, the sign of the phase is deﬁned
such that a positive phase shift corresponds to a shift ahead in time. Auto and cross power spectral
density are computed and averaged over 200 of these cycles according to Welch’s method.
3
One such
measurement takes
20s
, and delivers a spectrum for
f ∈ [0, 2000]Hz
. Based on the Auto Spectral
Density (
ASD ∈ R
) and Cross Spectral Density (
CSD ∈ C
) of two arbitrary signals
X
and
Y
, Argus
OMDS delivers amongst others:
• An overprediction of the transfer function
4
from X to Y : TF
+
X,Y
=
ASD
Y
CSD
∗
X,Y
• An underprediction of the transfer function from X to Y : TF
−
X,Y
=
CSD
X,Y
ASD
X
• The coherence between both signals: Coh
X,Y
=

CSD
X,Y

2
ASD
X
ASD
Y
=
TF
−
X,Y
TF
+
X,Y
•
Amplitude and frequency of the dominant oscillation within set frequency bands, both of which
are compensated to overcome the usual frequency resolution restriction (“picket fence effect”)
of the FFT.
3. Stability depending on operating conditions
3.1 Method
For these measurements, a sensor was located at
x = −100 mm
. Measurement at this position
does not require a cooling adapter, which makes comparative measurements on other burners easier.
For three (ﬁxed) fuel ﬂow rates, the equivalence ratio
Φ
was varied
(0.5 < Φ < 1)
by increasing and
subsequently decreasing the air ﬂow rate. The combustor had been running before the start of the
experiment, and the operating conditions were changed slowly (around
2min
between
Φ = 1
and
Φ = 0.5) to reduce the effects of transient cooling and heating of the structure.
3
19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
3.2 Results
Fig. 2 gives an overview of the amplitude and frequency of the fundamental pressure oscillation in
the combustor for three discrete values of the thermal power
P
th
, while the equivalence ratio decreases
and subsequently increases. The edge colour of the markers indicate the thermal power, the marker
orientation and ﬁlling differentiate between increasing or decreasing equivalence ratio. Generally, for
a given equivalence ratio, higher power will lead to a higher amplitude and higher frequency. There is
a difference in amplitude and frequency between the oscillation during decreasing and increasing
Φ
.
Possible causes are path dependence (hysteresis) of the ﬂame stability state, but also transient heating
of the combustor.
5,6
The heat capacity of the combustor causes the temperature to react slowly to the
operating conditions, which Subsection 4 will discuss as well.
Φ [−]

ˆp
I,1

[Hz]
I: incr. Φ, C: decr. Φ
20, 30, 40 kW
Φ [−]
f
I,1
[Hz]
I: incr. Φ, C: decr. Φ
20, 30, 40 kW
Figure 2. Amplitude

ˆp
I,1

(left) and frequency f
I,1
(right) of the dominant mode for various thermal powers,
plotted against equivalence ratio. Sensor location: x = −100 mm.
The combustion is stable at lower equivalence ratios. For all but the lowest thermal powers,
the transition from stable to unstable combustion, or vice versa, takes place very abruptly
(≈ 0.1 s)
between
Φ = 0.5
and
0.6
. As a result, the left side of Fig. 2 shows just a few points between 1000
and
2000Pa
. Between
Φ = 0.6
and 0.7 the amplitude remains approximately constant. For higher
equivalence ratios there is again a trend to higher amplitudes. In the case of
P
th
= 40kW
more than
5000Pa
are reached. For
P
th
= 20 kW
, the transition between stable and unstable combustion or vice
versa, is much smoother. There is no trend to higher amplitudes near Φ = 1.
The dominant frequency
f
I,1
depends on thermal power, where higher power leads to higher
frequency due to a higher temperature of combustion products. The dependence on
Φ
is not monotonic,
with a maximum around
Φ = 0.8
. For lower values of
Φ
, the excess air reduces the adiabatic ﬂame
temperature, leading to a lower speed of sound. For
Φ > 0.8
slow mixing could cause a lower
temperature in the ﬂame region.
3.2.1 Increasing versus decreasing equivalence ratio Φ
Fig. 3 shows two intensity plots, showing the inﬂuence of the equivalence ratio
Φ
on the spectrum
of the pressure signal at
x = −100 mm
for
P
th
= 40kW
. The transition between stable oscillation and
limit cycle is seen as a sharp line near the bottom of both plots, at
Φ ≈ 0.56
and
Φ ≈ 0.60
respectively.
Again, path dependence (hysteresis) of the ﬂame stability state as well as transient heating of the
combustor could cause this difference. Both effects were observed on the LIMOUSINE combustor.
Besides the
Φ
dependent curves, there are two straight vertical lines in Fig. 3, around
570Hz
and
680Hz
respectively. Their apparent temperatureindependence suggests these modes are structural, or
originate in cold sections of the setup. Subsection 4 corresponds to a mode at
672Hz
whose pressure
proﬁle describes half a wave in the (cold) plenum, which likely causes this peak. Altunlu et al.
7
mention a structural mode at 673Hz as well, but none near 570Hz.
4
19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
f [Hz]
← Φ [−]

ˆp

[Pa]
f [Hz]
Φ [−] →

ˆp

[Pa]
Figure 3. Spectral intensity for changing equivalence ratio, at P
th
= 40kW. Decreasing Φ on the left,
increasing Φ on the right. Sensor location: x = −100 mm.
3.2.2 Relative amplitudes of the higher harmonics
Besides the fundamental frequency, the higher harmonics of the dominant oscillation mode
are clearly visible as an equidistant array of peaks. Looking at the raw time signal of the pressure
ﬂuctuation, these higher harmonics are (for most operating points) phaselocked to the fundamental
oscillation, leading to a relatively constant wave form over time.
The amplitudes of the ﬁrst two harmonics, relative to the fundamental amplitude,

ˆp
I,2


ˆp
I,1

and

ˆp
I,3


ˆp
I,1

respectively, are shown in Fig. 4. As these plots show, the ﬁrst harmonic is rather
small in amplitude, while the second harmonic can reach signiﬁcant amplitudes. It tends to be stronger
at higher equivalence ratios, but depending on thermal power, the exact location of the peak varies
without obvious trend.
Φ [−]

ˆp
I,2


ˆp
I,1

I: incr. Φ, C: decr. Φ
20, 30, 40 kW
Φ [−]

ˆp
I,3


ˆp
I,1

I: incr. Φ, C: decr. Φ
20, 30, 40 kW
Figure 4. Relative amplitude of the second (I,2 left) and third (I,3 right) harmonic of the ﬁrst mode for various
powers plotted against equivalence ratio. Sensor location: x = −100mm.
Plotting the relative amplitudes of the harmonics against their respective frequencies on the
other hand, does show a clear trend. Irrespective of thermal power, the higher harmonics are stronger
when their frequency is either close to
570Hz
or
680Hz
. These are the same frequencies as noted in
Subsection 3.2.1 and do not depend on the operating point.
Since proximity to these two frequencies has such a strong inﬂuence on the amplitudes of the
higher harmonics, it is hard to identify other inﬂuences, such as those of thermal power or equivalence
ratio.
4. Pressure proﬁles
4.1 Method
The measurements described in this section were conducted at the operating point of
P
th
= 40kW
and
Φ = 0.74
. Before the measurements started, the combustor had been running at constant operating
conditions for
10min
to reduce the effect of transient heating. One pressure sensor was positioned
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19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
f
I,2
[Hz]

ˆp
I,2


ˆp
I,1

I: incr. Φ, C: decr. Φ
20, 30, 40 kW
f
I,3
[Hz]

ˆp
I,3


ˆp
I,1

[−]
I: incr. Φ, C: decr. Φ
20, 30, 40 kW
Figure 5. Relative amplitude of the second (I,2 left) and third (I,3 right) harmonic of the ﬁrst mode for various
powers plotted against frequency. Sensor location: x = −100 mm.
at the most upstream hole (
x
ref
= −200 mm
) while the other one traversed along the length of the
combustor (going downstream along the combustion chamber ﬁrst, followed by the plenum). To
construct the pressure proﬁles in the LIMOUSINE combustor, the correlation quantities as described in
Subsection 2.2.3 were calculated in real time by the Argus OMDS, following the method of Hermann
8
.
4.2 Results
Fig. 6 gives an overview of the pressure ﬂuctuation against position and frequency
p( f ,x)
. The
plotted intensity shows the square root of the ASD of the pressure signal per sensor access hole, which
is the
L
2
norm (averaged over the analysis segments) of the absolute value of the Fourier transformed
pressure signal. The signal is clipped between
1
and
100Pa
to adequately show the weaker resonances.
f [Hz]
x [mm]
p
ASD
p
≈

ˆp

[Pa]
I,1 II,1 I,2 III,1 I,3 I,4 I,5 I,6
I,7
Figure 6.
p
ASD
p
( f ,x), which is the L
2
norm of the Fourier transformed pressure signal, per frequency bin,
averaged over 200 cycles.
4
The plot gives an overview of the amplitude of the pressure ﬂuctuation as a function
of frequency and position at the operating conditions P
th
= 40kW and Φ = 0.74.
The plot shows two phenomena clearly. Firstly, vertical bright lines show the resonant frequencies.
Drift in temperature during the experiments deforms these lines, especially at higher frequencies.
Secondly, there is a pressure node at the outlet, and at integer half wavelengths upstream of the inlet.
These are seen as hyperbolic blue bands. There should be a pressure antinode at the inlet, though
there are not enough measurement points to show this clearly. At a quarter wave length (plus an integer
of half wave lengths) downstream of the inlet there are pressure nodes as well.
Fig. 7 shows
p
ASD
p
( f )
measured at the reference sensor at
x = −200mm
, both at start and end
of the experiment. The dominant peaks are harmonics of the ﬁrst mode. Contrary to the behaviour at
x = −100 mm
, discussed in Section 3, at the current position the amplitude of the second harmonic
ˆp
I,2
is higher than the third,
ˆp
I,3
. The frequency of the second mode of oscillation
II,1
is hard to pinpoint,
since it is so close to
f
I,1
. The third mode of oscillation
III,1
is seen as a separate peak between the ﬁrst
and second harmonic (I,2 and I,3) of the dominant mode.
6
19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
4.2.1 Pressure proﬁles
f [Hz]

ˆp

[Hz]
x [mm]

ˆp
I,1

[Pa]arg ˆp
I,1
[−]
Figure 7. Left: Pressure spectrum
p
ASD( f ) at the ﬁrst and last analysis run of the experiment (around 10
respectively 30 minutes after ignition of the combustor). Peaks corresponding to the mode shapes plotted in the
following ﬁgures are indicated. Sensor location:
x = −200mm
. Right: Pressure proﬁle of the dominant mode of
oscillation, I,1 of the LIMOUSINE combustor at P
th
= 40kW and Φ = 0.74.
The pressure proﬁles are presented in the form of amplitude and phase plots, based on the
correlation data as discussed in Subsection 2.2.3. The amplitude at
x
ref
= −200 mm
is based on
the average between the peaks of the two spectra in Fig. 7. At the other positions the amplitude is
determined via the transfer functions calculated by the Argus OMDS. Red dots indicate the estimated
transfer function
TF
≈
=
√
TF
−
TF
+
, while the range between
TF
−
and
TF
+
is shaded. The phase
plots show the pressure signal as red dots, relative to OH* chemiluminescence indicated by a plus.
The
1
/4
wave pressure proﬁle of the dominant mode of oscillation is shown on the right in Fig. 7.
The discontinuity in cross section at the ﬂame holder and the acoustic effect of the ﬂame cause a
cusp in the proﬁle. The OH* signal has a small phase shift compared to pressure, indicating the
thermoacoustic interaction of the ﬂame with the dominant mode generates ﬂuctuation energy, in
agreement with the Rayleigh index (
Ri
I,1
> 0
). Fig. 8 shows the proﬁles associated with the second and
third harmonic of the dominant mode. Both proﬁles show standing waves obeying the zero pressure
ﬂuctuation boundary condition around
x = 800 mm
, i.e. near 0.3 times the hydraulic diameter behind
the outlet.
9
The proﬁle shape in the upstream section and the behaviour at the ﬂame holder, are harder
to interpret. The phase between OH* signal and pressure is more than
π
/2
, which indicates that these
harmonics, although forced by higherorder, harmonic components of the nonlinear ﬂame response,
dissipate ﬂuctuation energy due to their negative modal Rayleigh indices Ri
I,2
< 0 and Ri
I,3
< 0.
x [mm]

ˆp
I,2

[Pa]arg ˆp
I,2
[−]
x [mm]

ˆp
I,3

[Pa]arg ˆp
I,3
[−]
Figure 8.
Pressure proﬁle of the second and third harmonic of the ﬁrst mode (
I,2
on the left and
I,3
on the right)
of the LIMOUSINE combustor at P
th
= 40kW and Φ = 0.74.
The geometry of the combustor suggests the existence of
3
/4
wave,
5
/4
wave and higher modes
as well. A proﬁle reminiscent of
3
/4
wave is found around
f
II,1
≈ 380Hz
. At
x = −200 mm
its peak
is hidden by spectral leakage of the dominant oscillation, but at locations further downstream it is
seen more clearly. This mode is shown on the left in Fig. 9. For this mode, there is a pressure node
7
19
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International Congress on Sound and Vibration, Vilnius, Lithuania, July 8–12, 2012
close to the location of the ﬂame, which might explain why this mode has such a low amplitude. Also,
the pressure node at the ﬂame holder means a velocity antinode. The energy of this mode is likely
dissipated efﬁciently by aerodynamic resistance of the bluff body ﬂame holder.
x [mm]

ˆp
II,1

[Pa]
arg ˆp
II,1
[−]
x [mm]

ˆp
III,1

[Pa]arg ˆp
III,1
[−]
Figure 9. Pressure proﬁles of the second and third fundamental mode. II,1:
3
/4 wave on the left and III,1:
5
/4
wave on the right. P
th
= 40kW and Φ = 0.74.
The
5
/4
wave mode
f
III,1
, discussed by Tufano et al.
10
, has its peak around
670Hz
. The proﬁle,
on the right in Fig. 9, shows a pressure antinode, and therefore a velocity node at the ﬂame holder.
Following the reasoning before, the relatively high amplitude of this mode seems reasonable. For II,1
and especially
III,3
, the phase shift between the OH* and pressure signals is small again, indicating a
positive Rayleigh index for these frequencies; Ri
II,1
> 0 and Ri
III,1
> 0.
5. Conclusion
Selfexcited combustion instabilities of the LIMOUSINE combustor were investigated. The
pressure proﬁles associated with the dominant oscillation, its second and third harmonic, and of
two higher independent modes are measured, identiﬁed and discussed. The relative strength of the
individual modes can be explained by the geometry of the combustor. The higher harmonics of the
ﬁrst mode are far stronger than those of the following modes. The strength of the third harmonic of the
ﬁrst mode is assumed to be inﬂuenced by coupling with the
5
/4 wave mode.
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