EXCERPT FROM:
Damping characteristics of
combustion chambers coupled with
acoustic elements
Zoltán Faragó
Sebastian Markgraf
DLR – LA – HF – RP – 020
DLR
Deutsches Zentrum für Luft- und Raumfahrt
Sebastian Markgraf
A. Abstract
This work deals with the influence of damping elements on the acoustically excited pressure oscillations in a laboratory scale combustion chamber and a steam generator coupled with up to 42 resonator tubes. The main task is to point out, how damping in a combustion chamber works and how it can be influenced. The effects of different chamber setups are pointed out and several excitation methods are investigated.
The text focuses on effects that cannot be explained by the theory of linear acoustics. The phenomenon of mode-to-mode conversion, i.e. the energy transfer between modes is described for combustion chambers coupled with acoustic elements and the existence of a hierarchy of acoustical modes in cylindrical chambers is shown. The hierarchy describes the interaction of modes of lower and higher frequency in reference to the damping.
B. Acknowledgment
First and foremost I have to thank my supervisor Mr. Zoltán Faragó for his patience and willingness to answer even the most trivial questions. His colorful explanations were always of great help to understand difficult tasks. Furthermore I have to appreciate his helpfulness not only during problems at work but also during leisure time.
Also I have to thank the team of the propulsion institute for their courteousness and the theoretical background they offered to me.
Table of contents
A. Abstract 3
B. Acknowledgment 3
C. Table of contents 4
D. Overview over used abbreviations and symbols 5
E. Overview over used software and equipment 6
1. Introduction 7
1.1. Pressure distribution in a cylindrical resonator 8
1.2. Quarter / half wave tube and Helmholtz resonator 10
1.3. Eigenmodes and eigenfrequencies 11
1.4. Damping and Lorentzian line profiles 13
1.5. Non-linear acoustics and influence of the resonator length 14
on the eigenfrequencies of the steam generator
1.6. Optimal resonator length for damping certain modes in 19
cylindrical chambers coupled with resonators
2. Experimental procedure 20
2.1. Test bench and used equipment 20
2.2. Proceeding and signal types 24
2.3. Measured variables 26
2.4. Calibration of the amplifiers 28
2.5. Optimizing the signal quality 29
2.5.1. Influence of the angle of the loudspeaker 30
2.5.2. Distance between loudspeaker and nozzle 31
2.5.3. Diameter of the loudspeaker 32
2.5.4. Summary of chapter 2.5. 34
3. Results and discussion 35
3.1. Hierarchy of modes 35
3.2. Pressure distribution along the circumference of the cavity ring 45
3.3. Mode-to-mode conversion due to extension of resonator length 51
3.3.1. Results concerning the cavity ring 51
3.3.2. Results concerning steam generator coupled with 59
cavity ring
3.4. Optimization of the resonator length for chambers with 72
different numbers of resonators
3.4.1. Concerning 1T mode 73
3.4.2. Concerning 2Tmode 79
4. Conclusion 84
4. Bibliographical reference
The French German coop research and technology program “High frequency combustion instabilities” engages in the investigation of the oscillations in combustion chambers caused by acoustic excitations. The evolved pressure fluctuations can lead to problems like increased noise and unsteady combustion behavior, but can also cause structural damages through excessive oscillations or even the destruction of the engines. Other effects are variations in thrust vector, oscillatory propellant flow rates and high heat transfer rates. Participants in the program are also CNES, CNRS, EADS and SNECMA.
For the experiments several common research chambers (CRC) are used whereas two of them are located in Lampoldshausen. One of them is examined by Bernhard Knapp [1] under real combustion conditions. The other one is used by several editors under the supervision of Zoltán Faragó [2]…[6] for tests at room temperature and without injection and spraying.
First measurements were done by Eunan J. McEniry [3]. The quintessence of these tests was the good comparability of different excitation methods. It was shown that the acoustical excitation by the use of a loudspeaker and therefore the placement of another damping element doesn’t affect the acoustical properties compared to the mechanical hammer excitation.
Alexander N. Uryu’s [4] works showed the coherency between the measured full width at half maximum of acoustic modes and the pedestal intensity of the peaks and pointed out the systematic errors caused by the gauging of the 3 dB width of weak modes.
The report of Thibaut A. Barbotin [5] is focused on the feature of mode-to-mode conversion in a CRC coupled with one resonator and the influence of resonator shape and length on damping and energy content of the acoustic field.
Finally Guillaume Dellea’s [6] task was to broaden the field of application from a one-dimensional CRC to a steam generator (SG) where the influence of the length was not negligible and the extension of the number of resonators up to 42.
After the introduction to the basic theory of acoustics, the measurement principles and the design of the test bench, this study deals with the influence of various numbers of damping elements on the pressure oscillations in a steam generator (SG). During the work different methods for exciting the acoustical modes are examined. The main task is to point out how damping in the chamber works and how it can be biased. Former measurements are approved and non-linear effects like mode-to-mode conversion are investigated. The number of resonators attached to the chamber is changed from zero to 42 and their length is altered from zero to 105 mm. The length and shape of the chamber itself is varied from 44 mm and cylindrical shape to 684 mm with exit nozzle.
Figure 1.5.2: Frequency and FWHM of acoustical eigenmodes in system of CRC
coupled with resonator cavity, CRC radius 100 mm [5]
circle = 1T, triangle = 2T, quad = 1R, diamond = 3T (empty = frequency, solid = FWHM)
 |
(14)
 |
(15)
Crossing for resonance frequency between experiment and λ/2-hypebolas appears if equation (14) is fulfilled. Anti-crossing between experiment and λ/4-hypebolas appears if equation (15) is fulfilled. In this the λ/4-hypebolas (solid hyperbolas in figure 1.5.3) do cross the cylindrical transverse modes.
The “avoided crossing” or “anti-crossing” in figure 1.5.2 can be explained in the following way: If equation (15) is true, both requirements should be fulfilled: 1) The quarter wave tube should show a velocity profile as presented in figure 1.2.1, and 2) the pressure distribution in the cylindrical chamber should be like in figure 1.1.1. These pressure and velocity distributions, however, suspend each other.
A crossing point between a radial mode and the λ/4-osciallation of the coupled resonator cannot be realized because the λ/4-oscillataion requires a pressure node at the resonator inlet, and, at the same time and at the same location, the radial cylindrical mode requires a pressure anti-node at the radial position r/R = 1 in the cylindrical chamber.
Similarly, a crossing point between a tangential mode and the λ/4-oscillation of the coupled resonator cannot be realized either because the λ/4-oscillation requires a radial velocity fluctuation at the orifice of the resonator, but the tangential eigenmode requires an azimuthal oscillation at the same location.
In the anti-crossing regions in figure 1.5.2 we can find two eigenmodes close to each other. The one of them has a slightly higher and the other one a slightly lower frequency than that of the belonging parental chamber modes. Denoting the avoided crossing regions according the denotation of the parental modes, the two eigenmodes are called + (plus) and – (minus) mode. An example can be found in figure 1.5.2 around the resonator length region of ≈0.8 < L/R < ≈0.9 in the frequency range of ≈900 < f < ≈1100 Hz for the 1T mode. The lower mode (red circles) is named 1T- mode and the upper one (violet triangle) 1T+ mode. Both show similarities to the 1T mode, however, the 1T- has a pressure node inside the resonator and the 1T+ a pressure node in the chamber in front of the resonator.
Figure 1.1.1: Pressure distribution in the chamber
for the acoustical modes 1T, 2T (top), 1R and 3T (bottom) [8]
Figure 1.2.1: Velocity distribution in lambda-quarter and lambda-half tubes
Figure 1.5.3: Fundamental chamber frequencies and cavity eigenfrequencies
 |
Resonance frequency for λ/2-hypebolas
 |
Resonance frequency for λ/4 hyperbolas
 |
Resonance frequency for cylindrical chamber modes
For αn,m values see table 3.1.1
 |
Figure 1.5.6: Mode to mode conversion in a cavity ring coupled with one resonator [5]
The amplitude of the pressure oscillation in the
chamber is low and the damping of the eigenmode is high when the frequency of
the coupled system is close to one of the resonator eigenfrequencies
and far from the cylindrical chamber eigenfrequencies
. In this case energy of the oscillation with the frequency of 
transforms to
oscillations of transverse cylindrical modes with the frequency of 
. Thus, the
oscillation frequency of the coupled system, 
, is suppressed but the chamber is
not protected from pressure oscillation with the frequency of as can be seen
in figure 1.5.6.
For the 2T parental mode this is the case at L/R ≈ 0.71. At this resonator length the measured system frequency crosses the λ/4 hyperbola in figure 1.5.2.
Figure 1.6.1.: Optimal length for damping 1T mode, CRC radius 100 mm [2]
The pressure oscillation is effective suppressed in the chamber for the transverse mode of order m and n when, for a given constellation of l, m and n, equation (15) is satisfied. In this case the chamber is protected against the pressure oscillation of the acoustical eigenmode satisfying equation (15), but the chamber is not necessarily protected against oscillation of other eigenfrequencies as can be seen in figure 4.1 and table 4.1 [12]. Equation (15) defines the “anti-crossing region” in figure 1.5.2, and the belonging resonator length is called the “optimized resonator length” for protection against the eigenfrequency covered up by equation (15).
Figure 1.6.1
presents the acoustical properties of the modes 1T- and 1T+ for l=m=n=1 in equation (15). The anti-crossing for the frequencies is connected
with a crossing of all acoustical properties of the concerning modes, thus
equation (15) describes an “exceptional point”. The special symmetry at this
condition leads to a symmetrical frequency distribution as can be seen in
figures 3.3.3, 3.4.2, 3.4.3 and 3.4.7. The exceptional point enables to adapt a
procedure to adjust acoustical
properties of a combustion chamber as described in [10].
Figure 2.1.1: Geometry of steam generator, and equivalent cylindrical geometry assuming the steam generator being a lambda-quarter tube
The equivalent length to determine the length modes is higher than the geometrical length. It is not the same for the different length modes (see table 3.3.3)!
Figure 2.1.2: SG, test bench Figure 2.1.3: SG, cavity ring
Figure 2.1.4: Geometry of cavity ring
Figure 2.1.5: Scheme of configuration 1-2 and 1-7
Position 1-2: both ends closed
Position 1-7: one end closed, the other end open
|
|
geometric |
equivalent |
||
|
Length lg |
Radius Rg |
Length le |
Radius Re |
|
|
position 1-2 |
44 mm |
110 mm |
44 mm |
110 mm |
|
position 1-7 |
684 mm |
110 mm |
704,2 mm |
108,5 mm |
Table 2.1.1: Geometric and equivalent dimensions of the steam generator and cavity ring
 |
Figure 2.1.7: Scheme of signal flow
|
|
Chapter |
Examined task |
Description |
|
1 |
1.5. |
Influence of resonator length on acoustical properties |
position 1-2 (figure 2.1.5), one resonator 0 < L/R < 1.8 Summarization of previous experiments [2 - 5] |
|
2 |
3.1. |
Hierarchy of modes |
position 1-2 (figure 2.1.5), resonator L/R = 0 |
|
3 |
3.2. |
Pressure distribution along circumference of cavity ring |
position 1-7(figure 2.1.5), 42 resonators, resonator length adjusted for optimal damping of 1T mode (L/R = 0.85), rotation of microphone and speaker in steps of 3,75° |
|
4 |
3.3.1. |
Mode-to-mode conversion due to extension of resonator length |
position 1-2, 42 resonators, resonator length 0 < L/R < 1, microphone and speaker on metal sheet on top |
|
5 |
3.3.2. |
position 1-7, 42 resonators, resonator length 0 < L/R < 1, microphone and speaker on metal sheet on top |
|
|
6 |
3.4.1. |
Optimization of resonator length for chambers with different numbers of resonators |
1T mode examined; position 1-2 and 1-7; one and 42 resonators at length for optimal damping of 1T mode (0.81 < L/R < 0.87 compared to L/R = 0) |
|
7 |
3.4.2. |
2T mode examined; position 1-2 and 1-7; one and 42 resonators at length for optimal damping of 2T mode (L/R = 0.49 compared to L/R = 0) |
Table 3.1: Overview of enforced test series
800 – 5000 Hz
Figure 3.1.1: Pulse response and frequency distribution after MLS excitation, position 1-2,
mode identification: table 3.1.1
|
No. |
n |
m |
αnm |
Mode |
Calculated frequency (Hz) |
Measured frequency (Hz) |
Relative energy density Imax = 100% |
Spectral energy density 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 |