6th Symposium on Launcher Technologies,
November 8th to11th, 2005,
Resonance Frequencies and
Damping in Combustion Chambers with Quarter Wave Cavities
Zoltán FARAGÓ
Michael OSCHWALD
Abstract – In the present study, the eigenmodes of a cylindrical
chamber without and with coupling to an absorber cavity are taken under examination.
The spectrum of eigenmodes is determined and the damping of modes is
characterized by the linewidth of the resonances. It is found that usually
damping for a given acoustical eigenmode is connected to an increase of the
intensity of another one. For the acoustically coupled system of a cylindrical
resonator and an absorber cavity it is shown that the eigenfrequencies and
other properties of the acoustical eigenmodes deviate from those of the
uncoupled cylindrical resonator. The damping is investigated as a function of
resonator length and the optimal length for efficient damping is discussed. The
damping can be increased by the application of capillary volume at the rear end
of the absorber.
1 – Theoretical
background
In rocket engines,
undesirable oscillation of combustion is usually caused by tangential modes. At
the
In the present
study, the abbreviation for the axial, radial and tangential modes is L, R and
T, respectively. The number before the abbreviation enables the mode identification.
Exemplarily, 1L means the axial or length basic tone, 2L the first axial harmonic,
3L the second axial harmonic, and so on.
The oscillation
frequency for the length mode of a half wave tube (a tube with two open or two
closed ends) can be predicted as
(l = 1, 2, 3 …) (1)
with f (Hz) as the frequency, c (m/s) as the speed of the sound, L (m) as the tube length and l as the mode number. The frequency of
the resonance in axial direction in a quarter wave tube (a tube with one open
and one closed end) is
(l
= 1, 2, 3 …). (2)
The frequency of
transverse modes in cylindrical chamber can be calculated as
(m
= 1, 2, 3… and n = 0, 1, 2…) (3)
with α_{n,m} as the eigenvalues of the Bessel
function, m1 and n being the mode numbers of radial and tangential
oscillation. The radius of the chamber is
r. The axialtransversecombinationfrequency is
(4)
with l, m, n as the order of the L, R and T
modes.
The spectral energy
density or intensity of the different modes can be predicted from the
definition of the unit decibel as
dB = 10·log(I_{1}/I_{2}) (5)
with I_{1} and I_{2} as the spectral energy density of the oscillation of
the eigenmodes 1 and 2.
2 – The
experimental procedure
The experimental
rig is exhibited in figure 1. The dimensions of the cylindrical chambers and
resonators are shown in table 1.
FIGURE 1: Sketch of the experimental device
Chamber No. 
Chamber 
Cavity 

Diameter D 
Length L 
Number 
Diameter d 
Length l 

1 
200 
42 
1 
12.3 and 10 
0 to 180 
2 
220 
44 
42 
9 
0 to 90 
TABLE 1: Dimensions of the experimental device
While exciting the
cylindrical chamber, the microphone voltage signal shows the pressure
oscillation. The fast Fourier transform (FFT) analysis of the microphone signal
exhibits the frequency distribution. The chamber is excited by a loudspeaker.
For excitation a Visaton K 50 WP 50 Ohm full range speaker was used with a
frequency response of 180 – 17000 Hz and a response frequency of 300 Hz. The
sound signal is measured by a Microtech Gefell measuring microphone MV 302. For
acoustical analysis the common software HobbyBox® 5.1 has been used. The single
sinusoidal signal was generated by the function generator Yokogawa FG 220, the
MLS signal, a kind of repeatable white noise sequence, by HobbyBox® itself.
The identification
of the modes is carried out by comparison of the frequency distribution of the
measured signal to the predicted mode frequencies. Table 2 shows the mode
identification for the experiments with chamber No. 2 for c = 345 m/s.
No. 
n 
m 
α_{nm} 
Mode 
Calculated frequency (Hz) 
Measured frequency
(Hz) 
Relative energy density Imax = 100% 
Intensity distribution Σ =100 % 
1 
1 
1 
1.8410 
1T 
919 
930 
100 
41.6 
2 
2 
1 
3.0541 
2T 
1525 
1530 
50 
20.8 
3 
0 
2 
3.8318 
1R 
1913 
1910 
20 
8.3 
4 
3 
1 
4.2013 
3T 
2097 
2100 
40 
16.6 
5 
4 
1 
5.3175 
4T 
2654 
2660 
16 
6.7 
6 
1 
2 
5.3320 
1R1T 
2661 
2670 
4.5 
1.7 
7 
5 
1 
6.4160 
5T 
3203 
3210 
5 
2.1 
8 
2 
2 
6.7085 
1R2T 
3349 
3350 
2.5 
0.9 
9 
0 
3 
7.0155 
2R 
3502 
3500 
0.25 
<0.1 
10 
6 
1 
7.5018 
6T 
3745 
3740 
0.3 
0.1 
11 
 
 
 
1L 
3920 
3910 
0.1 
<0.1 
 
3 
2 
8.0146 
1R3T 
4001 
not found 
 
 
12 
 
 
 
1L1T 
4026 
4010 
1.6 
0.4 
13 
 
 
 
1L2T 
4206 
4200 
1.3 
0.25 
 
1 
3 
8.5363 
2R1T 
4261 
very weak 
 
 
 
7 
1 
8.5781 
7T 
4282 
not found 
 
 
 
 
 
 
1L1R 
4362 
very weak 
 
 
14 
 
 
 
1L3T 
4446 
4430 
1.3 
0.25 
15 
4 
2 
9.2825 
1R4T 
4634 
4620 
<0.1 
<0.1 
16 


 
1L4T 
4734 
4720 
<0.1 
<0.1 
17 
8 
1 
9.6475 
8T 
4816 
4820 
<0.1 
<0.1 
TABLE 2: Acoustical modes presented in figure 3; speed
of sound c = 345m/s
The measuring
procedure contains two steps: 1) The chamber is acoustically excited by a white
noise. This step permits to determine at least the first 20 acoustical modes.
However, the signal quality of this method is weak and the result is useable
for frequency determination only. 2) The acoustical mode in question is excited
a second time by its eigenfrequency. This step results in a high quality signal
which enables the determination of all acoustical quantities. The duration of
the single sinus signal was 50 ms, the output level was 2V. Figure 2
demonstrates the raw signal of the first tangential mode excited by its eigenfrequency.
Origin of the FFT analysis is the sudden interruption of the excitation signal.
FIGURE 2: Signal of excitation and decay of the 1T
mode
Chamber No. 2; cylindrical chamber without cavity
3 – Hierarchy
of acoustical modes in a cylindrical chamber
Acoustical modes
can be excited by different acoustical signals, among others by white noise or
by a single sinusoidal signal. The sinus signal may have the eigenfrequency of
the mode to be excited, but it can also be a signal having a different frequency.
According to linear acoustics there is no energy exchange between different
acoustical modes. Our results show, however, that linear acoustic theory cannot
explain all features we have found in our experiments. We observe energy
transfer between modes, a phenomenon, which is beyond linear theory. The goal
of the following experiments is the determination of the eigenfrequencies of
the combustor coupled with quarter wave tubes and to investigate the mode conversion
process.
The experiment
presented in figure 3 shows the FFTresult of the pressure oscillation in the
frequency range of 800 to 5000 Hz using white noise excitation. In this
frequency range different basic tints, as the first tangential, the first
radial and the first axial modes, can be seen including their overtones. Table 2
contains the first 21 tones being presented in figure 3. In table 2 are,
however, only 17 tones numbered, since four overtones indicate only very weak
signal. The 1T mode (mode No. 1 in table 2) has the highest intensity among all
acoustical modes, followed by the second (No. 2) and the third tangential ones
(No. 4) which show an intensity of about 50 and 40 % of the 1T mode,
respectively. The first three tangential modes are followed by the 1R (No. 3)
and 4T (No. 5) modes having a relative intensity of 20 and 16 % compared to the
1T mode, respectively. The sixth place in the ranking possesses the 5T mode
(No. 7) with a relative amplitude of about
5 % followed by
the 1R1T (No. 6) and 1R2T (No. 8) combination modes with a relative intensity
of about 4 and 2.5 %, respectively. The
energy density of the 1L1T (No. 12), 1L2T (No. 13) and 1L3T (No. 14) modes is
just above the 1% level of the 1T mode. The term “pedestal intensity” is explained
in figure 3. The Full Width of Half Maximum (FWHM, see figure 4) of a mode is
overestimated for pedestal intensities below
10 dB and strongly
overestimated for values below 5 dB.
FIGURE 3: Frequency distribution at white noise
excitation
Figures 2 – 9
present experiments with chamber No. 2 without cavity. The mode identification
numbers on these figures refer to table 2.
Figures 4 to 6
show the frequency distribution for excitation of different eigenmodes by their
eigenfrequencies. As can be seen, the relative energy density of the excited
modes decreases with increasing excitation frequency. Figure 4 presents the frequency
distribution of the 1T mode excited by its eigenfrequency. As can be observed,
the intensity of the 1T mode is over 35 dB higher than that of the following 2T
mode. This means that the intensity of the 2T mode (No. 2) is less than 0.03 %
of the 1T mode’s intensity. Figure 5 presents the frequency distribution for
the 5T mode excitation (No. 7). The experiment in figure 6 shows the
frequency distribution of the 8T mode excitation (No. 17). The difference
intensity, as shown in figure 5, enables to predict the relative energy density
of a mode using equation (5). The energy density then is the basis for the calculation
of the intensity distribution as presented in table 2.
FIGURE 4: Frequency distribution at excitation with 1T
eigenfrequency (No. 1, 930 Hz)
FIGURE 5: Frequency distribution at excitation with 5T
eigenfrequency (No. 7, 3210 Hz)
FIGURE 6: Frequency distribution at excitation with 8T
eigenfrequency (No. 17, 4820 Hz)
Looking to figure
6 it can be seen that the excitation of the 8T mode (No. 17) by its
eigenfrequency excites all lower eigenmodes. The excitation of the lower order
modes in figure 6 is even of higher quality than the white noise excitation
presented in figure 3. Thus, the energy of the 8T eigenmode converts into all
other lower order acoustical eigenmodes.
FIGURE 7: Microphone voltage of the acoustical modes
excited by their eigenfrequencies
Figure 7
demonstrates that the microphone voltage decreases with increasing excitation
frequency. This experimental finding can be explained by the fact that the high
frequency excitation energizes many modes rather than excitation at lower
frequencies. Since the oscillation of the multiplicity of the excited modes is
not synchronized, the pressure peak is lower than that of an excitation by
lower frequencies.
FIGURE 8: Intensity of the acoustical modes excited by
their eigenfrequencies
Figure 8 presents
the decrease of the intensity of the excited modes with increasing excitation
frequency, while 100 % is the sum of all intensities. This experimental result
is an evidence of the energy conversion from higher order into lower order
modes.
FIGURE 9: Intensity of the 1T, 2T, 1R and 3T modes
while exciting other modes
Figure 9 exposes
very strong evidence for the mode to mode conversion and for the hierarchy of
acoustical modes. The intensity of the 1T, 2T, 1R and 3T modes is plotted
VERSUS the excitation frequency. The plotted modes are not excited by their
eigenfrequencies but by the frequency of other modes. It can be observed that
the intensity of the nonexcited modes increases with increasing excitation
frequency: Thus, the incoming energy into the nonexcited eigenmodes increases
with increasing excitation frequency.
Furthermore, we obtain the hierarchy of the acoustical modes for a
cylinder without quarter wave cavities: The highest intensity has the first
tangential mode, followed by the second and the third tangential modes. Thus,
higher order modes are emitting energy to lower order ones. The higher the
eigenfrequency of the emitting mode, the higher will be the amount of the
emitted energy. The lower the eigenfrequency of the receiving mode, the higher
will be the received energy amount.
4 –
Hierarchy of acoustical modes in a cylinder coupled with a quarter wave cavity
In the experiment
presented in figure 10, the chamber was excited by the eigenfrequency of the
coupled system. The coupled frequency changes stepwise with the increasing
resonator length as can be observed in figure 10. First step: For low resonator length the frequency decreases very
slowly with increasing resonator length: The system frequency seems to cling to
the calculated cylinder eigenfrequency. Second
step: When the resonator length gets close to a value at which the
eigenfrequency of the coupled system converges to the calculated λ/4requency,
the system eigenfrequency seems to cling to the calculated frequency of the λ/4tube.
Third step: The coupled system
eigenfrequency begins to converge to the value of a lower cylindrical mode. This
phenomenon was first described by Searby et al. [2].
If the resonator
length is leading to a λ/2tubefrequency which equals a cylinder
frequency of the chamber, thus, if equations (1) and (3) are leading to the
same value, the acoustical properties of the given eigenmode of the coupled
cylinderresonatorsystem are equal to the proper eigenmode properties of the
cylinder without resonator. In figure 10, the dotted line crosses the measured
frequencies of the coupled system. Exemplary, f_{2T, L=0} = f_{1R, L=105}
means that all acoustical properties of the 1R mode at L = 105 mm are identical
to those of the 2T mode at L = 0.
If the λ/4tubefrequenc
of the resonator is equal to the coupled frequency of the chamberresonatorsystem,
the experiment shows very low amplitude and high damping. In figure 10, the
dash line crosses the measured frequencies. The damping, in these cases, is affected
by the mode to mode conversion.
FIGURE 10 (upper plot): Coupled frequency of the
chamberresonatorsystem
FIGURE 11 (lower plot): Acoustical properties of the coupled system, f_{excitation} = f_{2T}
Chamber No. 1, trumpet shaped resonator, resonator
diameter D = 12.3 mm
At 72 mm resonator
length, the following events can be observed from figures 10 and 11: The 2T
mode resonance frequency of the coupled system equals the λ/4frequency of
the resonator (figure 10). The FWHM of the 2T mode has a maximum (solid
triangle, fig. 11). The pedestal intensity (see figure 3) becomes a minimum
(empty triangle, figure 11). Caused by the very low pedestal intensity, the measured
FWHMwidth of the 2T mode might be overestimated. In the resonator length range
of roughly 60 to 80 mm a halfmaximumwidth of about 50 Hz seems to be more
realistic. The intensity of the 1T frequency has a maximum at +20 dB (empty
circle, figure 11): This means that the spectral energy density of the nonexcited
1Tσ mode is about 100 times higher than that of the excited 2T. The intensity
of the 1R frequency has a maximum at +13 dB (empty square, figure 11): Thus, the
intensity of the nonexcited 1R mode is about 20 times higher than that of the excited
2T.
FIGURE 12:
Mode conversion map; chamber No. 1, trumpet shaped resonator, D = 12.3
mm
Figure 12 presents
the areas at which the coupled resonator prevents pressure oscillation for the
1T and
2T
modes via mode conversion.
The arrows show the direction of energy movement. The thickness of the arrows
indicates the energy flux.
5 –
Increasing the acoustical damping of a chamber coupled with a quarter wave
cavity
Several
experiments have been carried out to understand the influence of the cavity
shape on the acoustical damping. The goal is to find geometry constellations
with high acoustical damping to prevent high frequency combustion oscillation. Figure
13 shows results of measured FWHM distribution for the first and second tangential
mode of a sharp edged and a trumpet shaped resonator as shown in figure 1.
FIGURE 13:
Measured FWHM values for the 1T (left) and 2T (right) modes; Chamber No.
1; Trumpet shaped (empty symbols) and sharp edged resonators (solid symbols), D
= 12.3 mm
Surprisingly, the
experiments with trumpetshaped quarter wave tube are leading to higher
acoustical damping in comparison to sharp edged ones. This experimental finding
is in contrast to the expectation that velocity fluctuation at the inlet of the
resonator is a key process for acoustical damping.
FIGURE 14: Increase of the 3dBwidth VERSUS gap size
at the closed end of the
resonator for nondimensional resonator length of L / R = 0.85,
f_{excitation} = f_{1T}
Chamber No. 1, Sharp edged resonator, D = 10 mm.
A tunable
resonator has always an infinitesimal gap between the tube and the closing
piston (see figure 14). However, even a very small gap at the closed end of a
quarter wave resonator produces a strong increase of the acoustical damping, as
can be seen in figure 14. At ordinate = abscissa = 0 the FWHM of the chamber
without resonator equals 18 Hz. For the resonator length of L = 85 mm and the
relative gap size of 0.2 %, the FWHM amounts to 21.5 Hz, thus the increase
equals 21.5 – 18 = 3.5 Hz. While increasing the relative gap size, the increase
of the FWHM grows steadily. A relative gap size of about 1.5 % leads to a doubling
of the FWHMincrease compared to a hermetically closed end of the resonator. A
relative gap size of about 4 % leads to an increase of the damping gain by a
factor of 10. Possible implementations of this effect to oppress combustion
oscillation of rocket engines are described in a patent application [3]
6 – Optimizing the resonator length for one
coupled quarter wave cavity
The common way to
determine the optimal resonator length is the evaluation of the socalled
transferfunction. An important experimental finding in this study is the
observation that for the resonator length leading to the highest obliteration
of a given cylindrical eigenmode, the eigenmodes of the coupled system closest
to the unwanted cylindrical resonance show a strong symmetry according
amplitude, damping and intensity. Figure 15 shows examples for some
particularities of the acoustical properties when the resonator length is
optimized to suppress the first tangential eigenmode of the combustion chamber.
The optimized length of L = 85 mm is
marked by a dashline. In figure 10, the same length is marked, too. In figure
10, this is the length when the hyperbola of the lambdaquartertube frequency
crosses the 1T cylindrical frequency of 1003 Hz. At this frequency, the right
hand sides of the equations (2) and (3) are equal, thus equation (6) is true.
_{} (6)
For l = m = n = 1 and r = 0.1m, thus
for chamber No. 1, equation (6) is true at L = 85.3mm. For about the same
resonator length, namely at L = 84
mm, the FWHM for the excitation of both the 1T and the 2T eigenmodes is the
same (FWHM_{1T} =FWHM_{ 2T} = 36 Hz). And for about the same
length, at L = 85 mm, the microphone
voltage of the excitation of above eigenmodes is the same, too (U_{1T}
=U_{ 2T}). Further, at this resonator length, the pedestal intensity of
both signals is the same, too.
When the hyperbola of the lambdahalftube frequency crosses the 1T
cylindrical frequency in figures 10 and 12, the right hand sides of the
equations (1) and (3) equal, thus equation (7) is true.
_{} (7)
For chamber No. 1 this is the case at the length of about 170 mm. At
this resonator length, the 1T eigenfrequency
is roughly f_{1T} ≈ 500 Hz, thus 2·f_{1T} ≈ f_{2T}.
For about this length, namely in the length range of 160 < L < 170 mm,
the 1TFWHMwidth for sharpedged resonator has a minimum. For trumpet shaped
resonator, however, the 3dBwidth increases steadily with the resonator length
in the same length range, as can be observed in figure 13, left. The
disagreement between the FWHM development for trumpet shaped and sharp edged
resonators cannot be explained and needs further examination, in order to
provide resonator design with optimized damping capacity.
FIGURE 15: FWHM
width, microphone voltage and pedestal intensity at optimized resonator length
to suppress the 1T mode oscillation; Chamber No. 1, Sharp edged resonator, D =
12.3 mm.
Equation (6) permits a
quick determination of the optimized resonator length for one quarter wave
cavity coupled to a cylindrical resonator. Since the length of the quarter wave
tube us not identical to the so called effective tube length, a fine tuning of
the resonator is required to achieve best damping. The equality of the acoustical
properties of the 1T and 2T modes at the optimized length, as shown in figure
15, leads to symmetry of the frequency distribution of above modes, too. Taking
this symmetry has enabled the development of a quick and very effective measuring
procedure leading to a patent application [4].
7 – Conclusion and outlook
In order to highlight the acoustical behavior of the more important
tangential modes of combustion instabilities, the presented experiments were carried
out on cylindrical chambers with low axial length. For studying nonlinearity of
acoustical excitation, cylindrical chambers without resonator and coupled with
one resonator were tested. The experiments certify that for cylindrical
chambers without resonator, energy from eigenmodes with high eigenfrequencies
emerge into modes in the lower frequency range. For coupled cylinderresonator
systems, the energy of excitation emerges from eigenfrequencies close to
satisfy equation (6) into eigenfrequencies close to satisfy equation (7). The
experiments exposed symmetries according the acoustical properties of the 1T and 2T modes for λ/4tubes having
optimized resonator length. The detection of the symmetry led to a procedure for
optimizing the resonator length described in a patent application [4]. The
investigation of the effect of an infinitesimal gap at the closed end of a
resonator led to another patent application for the increase of acoustical
damping of combustion chambers [3].
FIGURE 16: Influence of a cavity ring on the
eigenfrequency; Chamber No. 2,
42 Resonators; Left: experiment; Right: calculation
using FlexPDE^{®}
Presently, the
focus of investigation regards the effect of a cavity ring on the acoustical
properties. The experiments are accompanied by calculations using FlexPDE^{®} common software, a finite element
method for the solution of partial differential equations. A comparison of
calculation and experiment for a cavity ring of a steam generator is presented in
figure 16. The steam generator to be examined has a geometry configuration
close to rocket engines. For experiment and calculation plotted in figure 16,
the axial length is 44 mm. The ongoing examination includes, however, the axial
length of the combustion chamber with a cavity ring containing 42 quarter wave
resonators. The goal of the investigation of the cavity ring as damping
equipment is to proof if and how much the optimized shape and length of one
quarter wave tube coupled to a cylindrical resonator are identical to those of
a cavity ring.
8 –
References
[1] B.
Knapp, M. Oschwald, S. Anders: Untersuchung der tangentialen Moden von
hochfrequenten Verbrennungsinstabilitäten in Raketenbrennkammern; DGLR –
Jahrestagung 2005, Paper No. 189, September 2005
[2] G. Searby and F. Cheuret: Laboratory scale
investigations of acoustic instability in a cylindrical combustion chamber, Proceedings
of 7th FrenchGerman Colloquium on Research in Liquid Rocket Propulsion, 1718
September 2002, Orléans
[3] Z. Faragó: Resonator
device to adjust acoustical properties of a combustion chamber; Patent application,
German Patent
Office, German Aerospace Establishment (applicant), Application No. 10 2005 050
029.3, October 2005
[4] Z. Faragó: Procedure to
adjust acoustical properties of a combustion chamber; Patent application,
German Patent
Office, German Aerospace Establishment (applicant), Application No. 10 2005 035
085.2, July 2005