6th Symposium on Launcher Technologies, November 8th to11th, 2005, Munich

 

Resonance Frequencies and Damping in Combustion Chambers with Quarter Wave Cavities

 

 

Zoltán FARAGÓ

Michael OSCHWALD

Institute of Space Propulsion, DLR Lampoldshausen, D 74239 Hardthausen, GERMANY

 

 

 

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 line-width 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 Institute of Space Propulsion, in DLR Lampoldshausen acoustical experiments are carried out using common research chambers for both, hot fire [1] and cold flow tests. The experiments presented in this work do not involve combustion or injection; rather they use a model combustion chamber filled with air at room temperature to allow easier and more fundamental characterization of the acoustic processes at work,

 

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 αn,m as the eigenvalues of the Bessel function, m-1 and n being the mode numbers of radial and tangential oscillation. The radius of the chamber is r. The axial-transverse-combination-frequency 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(I1/I2)                                                                                                                                (5)

 

with I1 and I2 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

 

αnm

 

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

 

 

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