Acoustic power dissipation on radiation through duct terminations: Experiments

Journal of Sound and Vibration (1983) 91(4), 479-502

ACOUSTIC POWER DISSIPATION ON RADIATION THROUGH DUCT TERMINATIONS: EXPERIMENTS? M. SALIKUDDIN Lockheed-Georgia

AND

K. K. AHUJA

Company, Marietta, Georgia 30063,

U.S.A.

(Received 30 September 1982, and in revised form 17 February 1983)

This paper describes the acoustic transmission characteristics of ducts, nozzles, orifices, and perforated plates, studied under an experimental program using an acoustic impulse technique. In this technique high intensity pulses, generated by discharging a capacitor across a spark gap, were used as the sound source. The test conditions include heated and unheated flows, with and without simulated flight. Results for a straight round duct, three convergent nozzles, a suppressor nozzle, 12 orifice plates, and 10 perforated plates are presented. A low frequency acoustic power loss phenomenon was observed for all configurations at all test conditions including the no flow condition. It was suspected that the power loss phenomenon at the no flow condition could be due to the conversion of acoustic energy into vertical energy due to non-linear propagation of high intensity pulses. However, a small amount of low frequency power loss was noticed even when tests were repeated with low intensity sound. Detailed flow visualization results were also obtained to complement the acoustic results.

1. INTRODUCTION This paper presents a number of selected results on acoustic transmission characteristics of various duct terminations obtained from studies carried out at Lockheed-Georgia Company between 1977 and 1982 which have so far been described mostly in various

reports and presented at conferences [l-6].. These studies were carried out by using an acoustic impulse technique developed by Salikuddin et al. [7], in which a spark discharge generating high intensity pulses was used as the sound source. A rather unusual and novel result of these studies was that significant transmission loss in acoustic power loss was noticed at low frequencies for all the duct terminations at all flow conditions, and most surprisingly, at the no flow condition also. This is the main theme of the present paper. The low frequency range in the present discussion extends up to about 1 kHz (the corresponding non-dimensional frequency, based on the supply duct radius, is about 1). 1.1. CONTENT OF THIS PAPER 1. Acoustic transmission losses for various duct terminations, as derived experimentally by using a spark discharge generating high intensity pulses as the sound source are presented. The effect of (i) size and geometry (shape) of the terminations, (ii) mean flow, (iii) flight velocity, and (iv) temperature, are demonstrated. Emphasis is given to the power dissipation results for no flow conditions. t This work was co-sponsored by Lockheed’s IRAD program and by NASA-Lewis under Contract NAS320797. A portion of this paper appeared in AIAA Paper 81-1978, presented at the 7th Aeroacoustic Conference, 198 1. 479 0022--460X/83/240479+24

$03.00/O

@ 1983 Academic Press Inc. (London) Limited

480

hl

\+.LIKtiDDIN

AND

K.

h

.\Hl:JA

2. Acoustic dissipation results, derived for low intensity sine waves, by using the impedance tube technique, are also presented to study the extent of the non-linearities associated with the intensity of sound source. 3. To understand the mechanism of low frequency power loss in the absence of mean flow, typical flow visualization results, obtained by using an impulsive sound source, are also presented. Since a number of results presented in this paper are new, a summary of the related studies carried out by other researchers is given. This is followed by a short discussion on the contributions of this work to the state of the art. 1.2. SUMMARY OF PREVIOUS WORK The acoustic energy absorption which is observed to occur when an acoustic wave is transmitted out through an air jet issuing from a pipe has been studied by several workers. Bechert’s experimental and theoretical results [S] and Howe’s theoretical analyses [9, lo], for example, offer persuasive evidence that the absorption mechanism is the conversion of acoustic energy into vertical energy at a sharp edge, and the subsequent dissipation of this into heat, without further substantial interaction with the acoustic field. In addition, in the presence of mean flow, it has been established, both theoretically [8] and experimentally [ll], that the nozzle exit area determines the radiation properties. Inspired by the results reported in references [3-61, Cummings and Eversman [12] conducted a theoretical study, the outcome of which indicated that high amplitude effects were responsible for the energy dissipation. Cummings [13] also reported the results of a further study where good agreement was noted between measured power loss, and predictions made on the basis of a non-linear resistance term arising from vortex shedding effects. Evidence of non-linear effects other than acoustic power dissipation have also been reported earlier by, for example, Sivian [14], who showed that the acoustic resistance of an orifice in a plate depends on the strength of the incident sound wave. A similar non-linear behavior of orifice resistance was observed by Bolt et al. [15]. Ingard [16] studied the amplitude dependence of the non-linearity and the dependence on the orifice geometry, and found a non-linear dependence of the resistance on the velocity amplitude in the orifice at higher sound pressure levels. Flow visualization tests to study the implication of internal noise on jets have also been carried out in the past [17, 181. Heavens [17] studied the response of a subsonic round jet to internal excitation by an acoustic wave using flow visualization and noise measurements. The jet was subjected to pulse excitation and also to harmonic excitation. In both cases, large-scale vortex ring structures were observed in the jet mixing layer. With pulse excitation, interactions between the vortex rings were observed. Heavens did not study any configuration in the absence of flow. However, such flow visualization studies in the absence of mean flow were carried out by Whiffen and Ahuja [18] using harmonic excitation signals. They found a well defined vortex generated at the nozzle exit by the harmonic sound generated upstream of the nozzle exit, even in the absence of mean flow. No such flow visualization results are available in the open literature for an upstream impulsive sound source operated in the absence of mean flow. 1.3. THE PRESENT WORK Even though the related previous work cited above [B-18] deals with the acoustic power dissipation phenomena and non-linear impedance characteristics of orifices, yet the present paper contains a considerable amount of new information, especially for no fEow conditions. Even for with flow conditions this paper contains results showing the

POWER

LOSSES

IN RADIATION

OUT

OF

DUCTS

481

effect of mean flow Mach number, flight velocity, flow temperature, etc., on the acoustic power dissipation phenomena which are not available elsewhere. The low frequency acoustic transmission loss phenomena in the absence of mean flow, first pointed out by the authors and their colleagues [l-6] is described in greater detail in what follows. In this context, the effect of termination exit area and termination geometry (shape) on acoustic transmission loss, in the absence of mean flow, is systematically discussed. In addition, the mechanism of acoustic transmission loss is also described, as acertained by using flow visualization techniques. Most of the results on high intensity sound propagation available elsewhere have been derived for discrete frequency sine waves. In this paper, on the other hand, the effects of high intensity pulses are considered. The experimental procedure, measurement technique, and the data analysis scheme are described in the next section. In section 3, the experimentally evaluated results are presented and discussed. Finally, the important observations are summarized in section 4. 2. EXPERIMENTAL

SET-UP

AND DATA

ANALYSIS

2.1. THE DUCT TERMINATIONS A photographic view of all the duct terminations used in the present study is shown in Figure 1. The salient details of these terminations are as follows. 1) The nozzle configurations include a straight 10 cm diameter duct, three conical nozzles with exit diameters of 6.2, 5.0 and 2.5 cm, and a multilobe-multitube suppressor nozzle (conveniently called a daisy lobe nozzle) with an equivalent exit diameter of 6.2 cm.?

Figure 1. Photographic view showing the straight duct, the conical nozzles, the suppressor nozzle, the orifice plates, and the perforated plates used as duct terminations.

(2) The straight duct and the two conical nozzles with diameters of 5 cm and 2.5 cm were tested with and without flow at ambient temperature. (3) The daisy lobe nozzle and the 6.2 cm diameter conical nozzle were tested both for heated and unheated flows, with and without simulated flight. (4) The 12 orifice plates and 10 perforated plates, with open area to duct cross sectional area ratios AR, ranging from 0 to 1, were tested only for theno-flow condition. (5) The impedance tube tests were carried out only for the straight circular duct in the no-flow condition, t These are the nominal values and are used in the text. The corresponding 5.08 and 2.54 cm.

exact values are 10.16, 6.198,

482

M. SALIKUDDIN

AND

K. k

At3Lf.A

2.2. EXPERIMENTAL C‘ONFICiURATION The experimental configuration as shown in Figure 2 consisted of a spark noise source located at the centerline of a 10 cm diameter pipe about 6 m upstream of the duct exit termination, which is located inside an anechoic room. A pressure transducer was mounted through the wall of the pipe, 76.2 cm upstream of the termination, to measure the in-duct signals. Provision was made to mount various nozzle, orifice plate, and perforated plate configurations. Far field signals were measured on a plan arc of 2.44 m radius with 0.635 cm diameter Briiel and Kjaer (B&K) microphones, extending from 0” to 120” with respect to the jet axis. Most of these microphones were placed at 10” intervals and a few, in the rear arc, were placed at 5” intervals. IO cm diameter Absorbent llnlng

Orlflce plate

I

I,

Jet 0x1s

In-duct

/’

e

‘\ Polar radius R, = 2.44 m

Figure 2. In-duct and far field measurement configuration.

2.3. TEST PROCEDURE The basic test procedure consisted of (1) discharging the capacitor across the spark gap located inside the duct at the source section, and (2) measuring the resulting incident and reflected pressure pulses by the in-duct transducer, and the transmitted pulse by the far field microphones. An example of typical time histories of the incident, the reflected, and the transmitted pulses for a 10 cm diameter straight duct termination is shown in Figure 3. The transmitted pulse shown in this figure was measured at a polar angle of 60”. The in-duct and all the far field pulses were recorded simultaneously on a 28-track tape recorder. Thus, subsequent analysis of each pulse was achieved conveniently on a dual channel, digital FFT signal analyzer with transient capture facilities. The spectral content of the incident, the reflected and the transmitted pulses were obtained from the Fourier transforms of the pulses produced by the FFT signal analyzer. Examples of such spectra obtained from the pulses shown in Figure 3 are plotted in Figure 4. The complex transfer function operation is also performed by the FFT analyzer. This operation between reflected and incident pulses is identically equal to the complex spectral power reflection coefficient of the termination. Similarly, the transfer function between incident in-duct and far field acoustic signals gives the transmission coefficient

POWER

LOSSES

IN RADIATION

OUT

483

OF DUCTS

Time hs)

Figure 3. Time histories of (a) the incident pulse, (b) the reflected pulse, and (c) the transmitted measured at 0 = 60” for a 10 cm diameter duct termination in the absence of mean flow.

401 0

Ii I

2

Non-dlmenslonal

3 frequency,

4

pulse

I 5

k,Q,

Figure 4. The spectral distributions of the pulses shown in Figure 3. -, pulse; - - -, transmitted pulse.

Incident

pulse; - - -, reflected

of the duct-nozzle system. The variation of far field sound pressure levels as a function of polar angle in the radiated field defines the directivity as a function of frequency. This pressure directivity is used to compute radiated power in the far field by integration over a spherical surface centered on the exit plane. 2.4. TEST SET-UP AND EXPERIMENTAL PROCEDURE FOR FLOW VISUALIZATION TESTS The flow visualization tests were carried out by using a smoke wire technique. A steel wire of 0.5 mm diameter was placed diametrically at the duct/nozzle exit plane (see Figure 5). A thin coating of oil was put over this wire from one end to the other. The wire was electrically heated just before the escape of the acoustic pulse from the duct. Therefore, at the instant when the pulse was transmitted out, the wire would generate smoke outside the termination due to the heating of the oil covering the wire. By using a synchronized light source for a short duration and a camera with the shutter open, a photograph of the smoke outside the termination, oriented according to the acoustic field, was taken. This photograph would indicate if a vortex was formed due to the transmission of the acoustic pulse.

484

IV. SAl.IKCJDDIN

AND

ti

h.

AHVJA

tilgh voltage powersuppI)

SWI closed Open -

SW2 closed -

Chargrq clrcuft co High voltage power supply Isolated: smoke wre clrcult complete Camera opens Spark discharge takes place Tape recorder

Figure

5. Schematic

showing

the spark discharge,

smoke generation,

and camera

operation

circuits.

The existing free-jet facility used for the acoustic tests could not be used for flow visualization tests in the absence of mean flow. Since the flow duct in this facility was horizontal, the smoke generated in the absence of mean flow tended to rise upwards due to its lower density compared with the ambient air. Therefore, another set-up was built especially for the tests at zero flow. In this case the flow duct was kept vertical. The source was located about 2 m upstream of the termination. The opposite end of the duct was terminated in a form pad located at about 1 m from the source. The circuit diagram for the operation of spark discharge, smoke generation, and camera operation is shown in Figure 5. When Switch 1 (SWl) was closed, the charging circuit for the capacitor across the spark gap was complete. Once the charging was done, Switch 1 (SWl) was opened, which isolated the high voltage power supply from the circuit, and completed the smoke generation circuit. When Switch 2 (SW2) was closed, the camera was opened and the spark discharge took place. The pulse generated by the spark discharge was recorded by an in-duct transducer, located about one meter upstream of the termination. This signal was used to trigger the stroboscopic light with appropriate time delay. The transient pulse, measured by the pressure transducer inside the duct was recorded on an analog tape recorder. The Fourier transform of the incident pulse in the 10 cm duct cross section was used to compute the pulse intensity which was used as a parameter in all the how visualization tests. 2.5. POWER CALCULATIONS To derive the power absorption results, the incident reflected and far-field acoustic power spectra are needed. The incident power WI and renected power WR are derived from the sound pressure level spectra as measured by the in-duct transducer by using the following expressions (under the assumption of plane wave propagation): Wr = (&I&G&I

+&A*,

WZZ= (&/PL&)P?z(~

-MD)*.

(172)

In these equations p0, CD,AD and MD are the mean flow density, speed of sound, cross sectional area of the duct, and the Mach number in the duct, respectively, and pr and

POWER

LOSSES

IN RADIATION

OUT

485

OF DUCTS

pR are the incident and reflected pressure magnitudes, respectively. The transmitted power W,, as expressed below, is the difference between the incident and the reflected

power:

WT= (ADIP~D)[eI2(1+Mg)*--P~(l -%)*I.

(3)

The radiated sound power in the far field W,, as expressed below, is calculated by the summation of the products of sound intensities measured by individual microphones and the corresponding elemental areas dSi. The sound intensities are derived from the far field sound pressure levels. Thus, WF = 1 DoC0

i

i=l

(pc)iASi,

(4) ‘(&+1-&*)/2,

ASi =

4rR2, sin (Si) sin (d6,/2), 27Ri[l_COS

0 < Bi
(AOi/2)],

I’

A6i =x

e2-81, ,8”

-6-1,

l
1.

In these equations PO and co are the mean density and speed of sound in the anechoic chamber, respectively, R, is the radius of the polar arc, Bi is the polar angle for the ith microphone measured from the jet axis, ABi is the elemental angle for the ith microphone, and it is the total number of far field microphones. The power loss, or power absorption, is the difference between the far field power WF, and the transmitted power WT..In a decibel scale the power absorption (LT is defined as 4T

=

10

lo&o

(wF/wT),

dB.

(5)

3. EXPERIMENTAL RESULTS Presentation of the experimental results showing the low frequency power absorption is the basic purpose of this paper. However, for the sake of completeness, the far field sound pressure level spectra and the far field power spectra for a few representative test configurations are also presented here. The important observations, as derived from the experimental results, are described. Some physical interpretations are also incorporated to explain the results. In presenting the results, various parameters like wave number k, nozzle exit diameter D, jet Mach number MI, free jet Mach number MT, exit area ratio (ratio of exit open area to duct cross sectional area) AR, and temperature T, are used to identify their effects. 3.1.

GENERAL

RESULTS

3.1.1. Far field sound pressure levels Effect of nozzle exit area on far field sound pressure levels. Typical far field sound pressure level spectra, normalized with respect to the incident sound pressure levels, in the absence of flow, for a 10 cm diameter straight duct and two conical nozzles with exit diameters of 5 cm and 2.5 cm at various polar angles are presented in Figure 6(a). The supply duct radius, RD, is used in these plots as the non-dimensionalizing parameter for the frequency. The normalized sound pressure level spectra shown in Figure 6(a) are replotted in Figure 6(b) with the nozzle exit radius RE, instead of the duct radius Ro, used as the non-dimensionalizing parameter for the frequency. This exercise is carried out to determine if the nozzle exit radius RE is a suitable scaling parameter for the frequency scale. In fact, as seen in Figure 6(b), the parameter RE scales down the

486

M.

Non-dlmenslonol

SALIKUDDIN

frequency, kRD

AND

K.

h.

AHUJA

Non-dlmens~onal

frequency, kRE

Figure 6. Effect of nozzle exit area on normalized far field sound pressure level spectra at various polar angles 8, at the no flow condition. -0, 10 cm diameter straight duct; - - -0, 5 cm diameter conical nozzle; --A, 2.5 cm diameter conical nozzle. (a) kRD normalized frequency; (b) kRE normalized frequency.

frequency parameter for the nozzles, and therefore, the normalized sound pressure level spectra for different nozzles collapse well. Similar behavior was also observed at various flow Mach numbers by Salikuddin and Plumblee [6]. Eflect of flow on fur field sound pressure levels. The far field sound pressure level spectra normalized with respect to the incident sound pressure levels, at various Mach numbers, for the 2.5 cm diameter nozzle are plotted in Figure 7. As seen in Figure 7, the normalized far field sound pressure level increases with increasing Mach number at polar angles of 30” and 60”. However, the trend is reversed for the forward arc as seen at polar angles of 90” and 120”. Similar behavior was also observed with the results for the 10 cm diameter duct and the 5 cm diameter conical nozzle [6]. In view of the results presented here and those observed in other Lockheed studies [l-6], the important observations can be summarized as follows: (1) the exit area of the nozzles appears to determine the shape and levels of the far field sound pressure level spectra (see Figure 6(a)); (2) the sound pressure level spectra for different nozzles collapse well when the nozzle exit radius RE is used as the non-dimensionalizing parameter for frequency (see Figure 6(b); (3) the sound pressure levels increase with increasing Mach number in the rear arc; however, the trend is reversed in the forward arc (see Figure 7). 3.1.2. Far field power The details of the effect of nozzle geometry and the effect of flow, on the nozzle transmission behavior, can be demonstrated by examining the far field sound pressure level spectra at all the polar angles. However, since the purpose of this paper is not to dwell upon these details, a better way to indicate this effect, on an overall basis, is to express the far field sound pressure levels in the form of far field acoustic power WF. In this study, since the tests were conducted with a spark discharge sound source, the

POWER

LOSSES

IN RADIATION

Non-dmenslonal

OUT

frequency,

OF DUCTS

487

kRE

Figure 7. Effect of flow on normalized far field sound pressure level spectra for a 2.5 cm diameter conical nozzle at various polar angles. -, M, = 0.0; - -, M, = 0.18; - - -, M, = 0.36; - - -, M, = 0.9.

incident power could have been different from one test to another. Therefore, to compare the far field power spectra between several test conditions, far field powers WF were normalized with respect to the corresponding incident power WI. In a decibel scale, this far field power transfer function (&) is expressed as follows: *r = 10 log,, (W,/W,).

(6)

Eflect of nozzle exit area on fur field power. Typical far field power spectra, normalized with respect to incident power, for the 10 cm diameter straight duct and the two conical nozzles, with exit diameters of 5 cm and 2.5 cm, at various flow Mach numbers, are shown in Figure 8(a). As seen in this figure, the normalized far field power is higher, at each frequency, for the duct as compared with the nozzles, at MJ = 0 and kfJ = O-36, M, being the jet flow Mach number. The normalized far field power for the 5 cm diameter nozzle is higher compared with that for the 2.5 cm diameter nozzle at all flow conditions including the M. = 0.9 case. Clearly, the radiated power at each frequency is directly dependent on the exit area of the duct/nozzle. The normalized far field power spectra shown in Figure 8(a) are replotted in Figure 8(b) with the nozzle exit radius RE, instead of the duct radius RD, used as the non-dimensionalizing parameter for the frequency. It can be seen in Figure 8(b) that except for very low frequencies the normalized far field power spectra for different nozzles collapse reasonably well at all Mach numbers.

488

M. SALIKUDDIN

2

AND

h

h

.AHClJA

I

4

Non-dimensional

frequency,

kRo

Figure 8. Effect of nozzle exit area on far field power normalized with respect to incident power at (i) A4, = 0, (ii) A4, = 0.36 and (iii) A4, = 0.9, M, being the jet Mach number. -0, 10 cm diameter straight duct; - - -0,5 cm diameter conical nozzle; --A, 2.5 cm diameter conical nozzle. (a) kRD non-dimensional frequency; (b) kRE non-dimensional frequency.

Eflect offrow on fur field power. The far field power spectra normalized with respect to incident power, at various Mach numbers, for the 10 cm diameter duct and for the 2.5 cm diameter nozzle are shown in Figure 9. As seen in Figure 9(a), the effect of flow on the far field power for the duct seems to be less dominant. However, further examination of the 2.5 cm nozzle data indicates that, in general, the normalized far field power increases with Mach number (see Figure 9(b)). It should be noted that due to the inability of the 10 cm diameter straight duct to supply a jet at IV, = 0.9 at the time these experiments were performed, these data are not shown in Figure 9 and are also absent in a few subsequent figures. The important observations are as follows: (1) the exit area of the nozzles appears to determine the levels of far field acoustic power; at low frequencies, the far field acoustic power is always less than the incident power (see Figure 8(a)); (2) the far field power spectra for different nozzles collapse well when plotted against kRE instead of kRD, RD and RE being the duct and the nozzle exit radii, respectively (see Figure 8(b)); (3) an increase in far field acoustic power with increasing Mach number is observed (see Figure 9). 3.2. POWER ABSORPTION RESULTS The power absorption results are described under two categories, (1) high acoustic intensity results, and (2) low acoustic intensity results. All of the results presented above, however, were derived from the high acoustic intensity tests.

POWER

LOSSES

IN RADIATION

K

0

I

2

Non-dtmenslonol

OUT

3

frequency,

OF

4

DUCTS

489

5 kRD

Figure 9. Effect of flow on far field power spectra normalized with respect to incident power for (a) a 10 cm diameter straight duct and (b) a 2.5 cm diameter conical nozzle. -, M, = 0; --, M, = 0.18; - - -, M, = 0.36, - - -, M, = 0.9.

3.2.1. Power absorption for high acoustic intensities In this section the acoustic power absorption results derived by using high intensity pulses (about 150-155 dB) for a straight duct, conical nozzles, suppressor nozzle, orifice plates, and perforated plates are presented. The optical results for these terminations for no-flow conditions are also presented along with the corresponding power absorption results. Emphasis is given to the low frequency power absorption, especially, at no-flow conditions. The results for a 10 cm diameter straight duct, two conical nozzles with exit diameters of 5 cm and 2.5 cm, orifice plates and perforated plates are presented as functions of a non-dimensional narrow band frequency, with the duct radius RD used as the non-dimensionalizing parameter. However, the results for the suppressor nozzle and the equivalent conical nozzle with 6.2 cm exit diameter are mostly presented as functions of l/3 octave band mid-frequencies. Tests with heated flow and with simulated flight were conducted only for the suppressor nozzle and the 6.2 cm diameter conical nozzle. (a) Effect of termination open area on power absorption Figure 10 shows the power absorption spectra for a 10 cm diameter duct and for three conical nozzles with exit diameters of 6.2, 5 and 2.5 cm. The maximum amount of low frequency power loss is observed for the 2.5 cm diameter nozzle, which has the smallest exit area. With increasing exit area, the low frequency power loss decreases. With increasing frequency the amount of power absorption reduces for each nozzle and asymptotically a power balance is reached (before the first radial cut-on frequency, kRD = 3.83).

The flow visualization results for different nozzle terminations are shown in Figure 11. These photographs were taken 20 ms after the pulse propagated out of the termination exit. The intensity of the incident pulse for all the terminations in the measurement location was about 147 dB. It can be clearly observed that the presence of an acoustic pulse is the cause of the formation of the vortex ring. Since the incident pulse intensities

490

M.

SALIKlJDDIN

AND

ti

h

/

I

I

I

2

3

Non-dlmensoxl

Figure 10. Effect of nozzle exi: area on acoustic duct; .O, 6.2 cm diameter conical nozzle; --A--,

frequency,

AtIU.IA

1 4

2

kf?,

power absorption without flow. -0, 10 cm diameter straight 5 cm diameter conical nozzle; - + (2.5 cm conical nozzle.

Figure 11. Effect of nozzle exit area on the vortex formation at the nozzle exits without any flow at incident pulse intensity= 147dB, photographed 20ms after the pulse propagated out of the exit plane; (a) 1Ocm diameter straight duct, (b) 6.2 cm diameter conical nozzle, (c) 2.5 cm diameter conical nozzle.

in the upstream 10 cm diameter duct section for all the terminations were the same, one might expect that the intensities of the vortices formed at each termination would be identical. However, this is not the case. With decreasing open area, one can see a more dominant vortex; also, the vortex moves away from the termination exit plane as the open exit area decreases. The reason for this is as follows. For the straight duct, the incident pulse would propagate as it was, whereas for the conical nozzle the intensity of the propagating incident pulse would increase due to the reduction in cross sectional area, and would be sufficiently different at the exit compared to that for the straight duct termination. The formation of the vortex is basically controlled by this pulse at its final stage. Therefore, the vortex for the nozzle with the smallest exit open area would be the strongest. Therefore, upon comparing the power absorption results of Figure 8(b) with these optical results, it can be concluded that the amount of power absorption increases with the increasing vortex strength.

POWER

LOSSES

IN RADIATION

OUT

491

OF DUCTS

Therefore, a higher intensity pulse generates a stronger vortex ring and the propagation speed of the ring is relatively higher due to the higher particle velocity of the pulse; and the amount of acoustic power absorption increases with increasing particle velocity. The effect of orifice plate (Figure 12(a) and perforated plate (Figure 12(b)) openings on the power absorption is found to be quite similar to that observed for the nozzles.

i’ -401 0

I I

I 2 Non-dlmensionol

I 4

I 3 frequency,

(b) 5

kRo

Figure 12. Effect of (a) orifice and (b) perforated plate opening on acoustic power absorption (a) -0, AR = 1; .*.O, AR = 0.5; --&&=0.2;-++,AR=0~1111;-~~,AR=0*0352;--~, AR = 0.0156; (b) -_o, AR = 1.0; --a, AR = 0.5065, . . .A, AR = 0.237; - .., AR = 0.0716; -

at

no

flow

condition. 0.0529;

-

. . a, AR =

. A, AR = 0.0313.

At low frequencies, the amount of power absorption increases monotonically with decreasing area of the orifice plate opening or the equivalent opening of the perforated plate. These results are described in detail in references [4] and [6]. This is also demonstrated in Figure 13, where the formation of vortices is shown for various orifice plates with different open areas. The intensity of the incident pulse was 143 dB in this case, and all the pictures were taken 20 ms after the pulse propagated out of the termination exit. The propagation speeds of the vortices for the orifice plates with AR = O-0625 and 0.0156 are so high that they travelled beyond the image area of the photograph. The optical results for the no flow condition indicate multiple vortex formation. The weaker vortices following the initial vortex ring, in this case, were due to the reflections of the main pulse from the termination at the upstream end of the duct.

492

M. 54LIKUDDlN

AND

K

h

AHUJA

Figuw 1.3. Effect of orifice plate t,pcnmg on the vortex lormatnm
any flow; plane: (a)

It can be concluded that the magnitude of the low frequency power absorption depends upon the size of the exit open area. However, unlike the far field sound pressure level spectra or normalized far field power spectra shown in Figures 6(b) and 8(b), the power absorption spectra for different size terminations will not collapse if plotted against non-dimensional frequency with equivalent exit radius (RE) as the non-dimensionalizing parameter. This is evident from Figures 10 and 12(a) and (b), since the spectra in each figure run parallel to one another in the lower frequency range, instead of converging to a single point at zero frequency. (b) Effect of termination geometry (shape) on power absorption To establish if the duct termination shape was an important factor in determining the amount of power loss, results for different termination shapes, but with constant exit open areas, are compared in Figures 14(a) and (b). The power absorption spectra for a perforated plate, an orifice plate, a conical nozzle, and the suppressor nozzle, each with the same exit area ratio, AR = 0.37 (equivalent exit diameter of 6.2 cm), are plotted in Figure 14(a). The low frequency power absorption seems to be the same for all the terminations for a constant area ratio. The power absorption results were somewhat different when the area ratio AR was reduced considerably from 0.37 to 0.0625. These results for the latter value of AR (equivalent exit diameter = 2.5 cm) are compared in Figure 14(b) for a perforated plate, an orifice plate, and a conical nozzle. It can be seen in this case that, in the very low frequency range, the perforated plate shows the maximum power loss and the conical nozzle shows the least amount of power loss. The flow visualization results for different termination shapes but with fixed exit open area (i.e., AR = 0*37), corresponding to the power absorption results shown in Figure 14(a), are presented in Figure 15. These photographs were taken for a 147 dB incident pulse 20 ms after it propagated out of the exit. Even though the amount of low frequency power loss seems to be the same for each termination with equal exit open area, the vortex structures are quite different for each termination. While doughnut shaped vortices were formed for the conical nozzle and for the orifice plate, the vortex shapes for the suppressor nozzle and for the perforated plate were quite complex. Even the doughnut shaped vortices for the conical nozzle and the orifice plate are not the same in shape and size. The diameter of the vortex ring generated by the conical nozzle is larger compared to that of the orifice plate. However, the propagation speed

POWER LOSSES IN RADIATION

I

2

Non-dlmenslonol

493

OUT OF DUCTS

4

3

frequency,

5

kRD

Figure 14. Effect of termination geometry on acoustic power absorption with a fixed area ratio at no flow condition. -Cl, Perforated plate; . . . 0, orifice plate; --A, Conical nozzle; -+, daisy lobe nozzle. (a) AR = 0.37 (equivalent exit diameter 6.2 cm); (b) A, =0.0625 (equivalent exit diameter 2.5 cm).

Figure 15. Effect of termination geometry on the vortex formation at the termination exits, without any flow, with a fixed area ratio, A, = 0.38 (equivalent exit diameter of 6.2 cm); incident pulse intensity = 147 dB; photographed 20 ms after the pulse propagated out of the exit plane. (a) Conical nozzle; (b) suppressor nozzle; (c) orifice plate; (d) perforated plate.

494

M

SALIKUDDIN

AND

K. k

AHIJJA

seems to be lower for the conical nozzle vortex compared with that of the orifice plate. It appears, therefore, that the shape and the structure of the vortex formed at a termination exit is controlled by the geometry of the termination. The overall conclusion that can be drawn from these results is that except for very small porosity (compared with the total duct cross section) the low frequency power absorption primarily depends upon the magnitude of the exit open area and not as much on the shape of the termination. When the equivalent exit area compared to the total duct cross section is small, the shape of the duct termination 2:ppears to influence extremely low frequency results. The shape and the structure of the vortex generated by a termination is controlled by its geometry. However, the shape and the structure of the vortex do not appear to influence the amount of acoustic power loss significantly. (c) Power absorption for a discrete frequency sine wave To further confirm frequency sine wave is is presented in Figure sine wave propagation

the power loss phenomena which also occurs when a discrete propagated out of a termination exit, a typical schlieren photograph 16. This photograph shows the vortex fortiation due to a discrete through a nozzle exit in the absence of flow.

Figure 16. Photographically averaged schlieren picture of vortex formation sound [ 121, excitation frequency = 300 Hz, excitation level = 135 dB.

at the nozzle exit due to upstream

Figure 16 is reproduced from reference [ 181 and was obtained by using an improved laser schlieren system incorporating the technique of photographic averaging. A brief description of this technique and typical results are given here. Most shadowgraphs and schlieren photographs, particularly those for axisymmetric flows, display a certain degree of confused detail resulting from small scale turbulence in the jet, and from thermal convection in the ambient air. A method of removing these sometimes unwanted details is the application of a photographic averaging technique and is an effective method for revealing large scale structure. The method consists of repeated triggering of the light source and superposition of all the schlieren images on a single photographic film. By this means, the images of a periodic structure associated with the trigger signal are reinforced and those from the random turbulence tend to cancel.

POWER

LOSSES

IN RADIATION

OUT

495

OF DUCTS

This set-up was used by Whiffen and Ahuja [18] to obtain schlieren pictures of the vertical activity at the exit of a nozzle for the no flow condition. Under normal conditions, the no flow jet visualization by use of the schlieren system can be achieved by seeding the flow with helium. Since injecting helium would have essentially provided a low velocity jet, and would have defeated the purpose of providing a true no flow condition, a heated Nichrome wire was placed along a diameter of the nozzle exit. The changes in density along this diametrical plane thus made possible the visualization of the vortex formation due to the high level sound at the nozzle exit. The photograph shown in Figure 16 was obtained by exposing the film to the flow field 30 times and using the photographic averaging technique. The source of sound was an electroacoustic driver generating sound of approximately 135 dB at a frequency of 300 Hz. Clearly, a well defined vortex is generated at the nozzle exit even in the absence of mean flow by sound generated upstream of the nozzle exit. This obviously is related to (and confirms) earlier discussion of conversion of acoustical energy to vertical energy and the associated power loss. (d) Eflect of flow on power absorption Three sets of plots, each at various flow Mach numbers, indicating the power absorption for a 10 cm diameter duct and for two conical nozzles with diameters of 5 cm and 2.5 cm, are shown in Figure 17. A substantial increase in power absorption is observed due to the initiation of flow for the 10 cm diameter duct in the frequency range of kRD = 0.3

IO

I

I

I

I

(0)

I

I

I

I

I I

I 2

I 3

I 4

(b)

E 2

II

-20 -

_; 0

I I

I 2

I 4

1 3

0

I

5

2

Non-dlmenslonol

Figure 17. Effect of flow on power absorption conical nozzie and (c) a 2.5 cm diameter conical ---, MJ =0.9.

0

3

frequency,

4

5

5

kR,

for (a) a 10 cm diameter straight duct, (b) a 5 cm diameter nozzle. -, A4, = 0; --, IU, = 0.18; -. -, M, = 0.36;

496

Il.

5ALIKLJDDIN

h.

ANI)

h

\tiC’lA

to kK,, = 1.5 (see Figure 17(a)). For the two nozzles. the power absorption seems to increase initially with increasing Mach number and then starts decreasing with further increase in Mach number (see Figures 17(b) and (C)I. Similar trends are observed for nozzles with complicated geometry. Figure 18 illustrates one such example of power absorption for a 12 lobe, 24 tube suppressor nozzle at various flow Mach numbers. More results for the suppressor nozzle are available in references [3] and [5].

-30 0.2

0-e

3.15

l/3

Figure 18. Effect of flow on power

absorption

octave

frequency

for a suppressor

12.5

50

(ktiz)

nozzle. 0, hf, = 0; 9, hf, = 0.4; a, kf, = 0.8.

3.2.2. Discussion of the high intensity power absorption results To explain the low frequency power loss phenomena, analytical studies were also carried out at Lockheed-Georgia [12], which have been summarized in a companion paper by Cummings and Eversman [19]. It appears from this analytical study that the acoustic pressure difference Ap, between two sections along the direction of the one dimensional sound propagation, which consists of a mean flow term and a non-linear term, can be expressed as follows [12]: (7) mean flow term------J

L---non-linear

term

In this expression I?? is the mean flow velocity and u is the particle velocity associated with the intensity of the propagating sound. In the present discussion Ap is the pressure difference between the section before the beginning of the termination contraction and the section at the vena contracta in the jet. The power absorption function W,l Wr, as derived by Cummings and Eversman [12] is indirectly related to the pressure difference Ap expressed in equation (7). The non-linear term U* that appears in equation (7) can be neglected for low intensity sound. It cannot, however, be neglected for high intensity pulses, as used in the present study. The implications of equation (7) in relation to the present results are further discussed in what follows. (a) Power absorption in the absence of mean flow The power absorption for the no flow condition should not occur at all according to the linear theory if no surface losses are accounted for. Since in the present study high intensity pulses were used linear theory was strictly not applicable. Therefore, it was postulated that the acoustic energy, in the absence of mean flow, could be dissipated by

POWER

LOSSES

IN RADIATION

OUT

OF DUCTS

497

the same mechanism as occurring in the presence of flow if the acoustic velocity were high enough to cause the conversion of acoustic energy into vertical energy. Thus one can attempt to explain the results on the basis of the non-linear behavior of high intensity sound, and in particular its ability to generate vorticity. For high intensity sound propagation, the pressure difference dp, expressed in equation (7) would reduce to ApoC2.

(8)

One can now discuss the possibility that the non-linear term u2, in equation (8), contributes to the power absorption demonstrated in Figures 10-15. The non-linear behavior responsible for power absorption can be physically interpreted in the following manner. The high intensity sound generates a high enough particle velocity u, comparable to a very low velocity nozzle flow [12]. Since in the presence of a subsonic nozzle flow the emission of low frequency sound from a jet pipe involves a transfer of energy from the acoustic wave to essentially incompressible vortex waves on the jet [8,9], a similar phenomenon may thus take place when high intensity sound propagates out of the jet pipe, even in the absence of mean flow, and produce a net attenuation in the transmitted sound [9, lo]. The acoustic particle velocity for most of the data presented here was of the order of 3 m/s. For such high particle velocities, conversion of the acoustical energy into vertical energy was therefore quite likely. Evidence in support of the proposition that this phenomenon was probably responsible for the low frequency power absorption at the no flow condition can be found in the results showing that the low frequency power absorption increased with decreasing exit open area. This was associated with increasing particle velocity with decreasing exit open area. The intensity of a given pulse increases on propagating from a larger cross sectional area into a smaller area at the exit. Since in the present study the incident pulse amplitude in the upstream supply-duct section was kept constant, the smaller duct terminations inevitably had higher particle velocities at their exits. Thus the configuration associated with higher particle velocity should display larger power absorption. This indeed was the observation made in Figures 10-15. (b) Power absorption in the presence of mean flow The low frequency power absorption for high intensity pulse propagation in the presence of flow, as shown in Figures 17 and 18, can also be explained on the basis of equation (7). Mathematically, the power loss in this situation is contributed both by the non-linear term u2, and the mean flow term 2& as shown in equation (7). However, with increasing mean flow u, the non-linear term u2 becomes insignificant compared to the mean flow term 21!?u, in equation (7). Therefore, in the presence of flow, the low frequency power absorption as shown in Figures 17 and 18 should be mostly due to the mean flow effect, the situation being comparable with that where low intensity sound is used with the same amount of mean flow. There have been reports by other researchers indicating that the non-linear effects are negligible in the presence of mean flow. For example, references [20] and [21] show that the measurements at higher mean flow velocities lead to a substantially linear dependence of orifice resistance on the velocity amplitude. The above effects in the presence of mean flow are further demonstrated in Figure 19 where the power loss results for a 5 cm diameter nozzle, from the present study, at A& = O-7 are compared with the similar results obtained by Bechert [22] for a 4 cm diameter nozzle but for low intensity sound. In this figure, for proper comparison, both the results are expressed with the corresponding nozzle exit radius (RE) used as the non-dimensional parameter instead of the duct radius (RD). Bechert’s results show slightly

498

M. SALIKUDDIN

‘IL

5

-20

v,,,_

0

0.5

AND

I.0

k.

k

AHlJJA

__~_I

l-5

2.0

2.5

Non-dlmenslonal frequency, kff, Figure 19. Comparison of power absorption results between the present study (for a 5 cm diameter nozzle) and Bechert’s measurement (for a 4 cm diameter nozzle) at jet Mach number IV, = 0.7. -, Present study (high intensity results); 0, Bechert’s results [15].

higher power loss beyond kR E = 1 compared to the results from the present study, because of the smaller nozzle dimension (4 cm diameter) used by him compared to that for the present study (5 cm diameter). (c) Eff*ct of temperature on power absorption Figure 20 illustrates the effect of temperature on power absorption for a 6.2 cm diameter conical nozzle at M, = 0.8. The power absorption increases with increasing temperature at low frequencies- up to about 400 Hz. At higher frequencies, the power

Figure 20. Effect of temperature (T) on power absorption for a 6.2 cm diameter conical nozzle at jet Mach number M, = 0.8. 0, Ambient temperature; A, T = 600 K.

absorption seems to decrease on heating the jet. This behavior is also evident from the analytical expressions developed by Bechert [8], Cummings and Eversman [12, 13, 191 and others. For example, the simple expression for power absorption developed by Bechert (see equation (15) of reference [B]) is W,lw~

= ((1 +‘$~:,)/~&W?E/~D)~,

(9)

where w is the radian frequency. For a fixed M D, if the mean flow temperature is increased, then ED will increase. This will reduce the amount of acoustic power absorption, WF/ W,, in the above expression, as indeed observed in the present results above 400 Hz.

POWER

LOSSES

IN RADIATION

OUT

499

OF DUCTS

Below 400 Hz the increase in acoustic power absorption due to heating is not explained by equation (9) and therefore needs further study. Detailed results for the heated jet tests are available in a Lockheed report [3] and in an AIAA paper [5]. (d) Effect of simulatedflight on power absorption The anechoic free jet facility at Lockheed-Georgia Company was used for conducting flight simulation tests for the suppressor nozzle and the 6.2 cm diameter conical nozzle. Both nozzles had the same exit area. The facility is capable of providing continuous free jet flow of velocities up to 95 m/s. For the present study, simulated flight tests were conducted at three free jet Mach numbers, namely, 0.08,0*16 and 0.24 with and without the main jet flow. A detailed description of the flight simulation tests and results are presented in references [3] and [5]. However, a typical result showing the effect of flight on the power absorption for the 6.2 cm diameter conical nozzle is shown in Figure 21. It appears from this figure that the low frequency power absorption is present even in the flight situation, and it, in fact, increases.

0.8

3.15

l/3 octave Figure 21. Effect of simulated flight on power absorption 0, MT = 0; A, MT =0*24; (MT = free jet Mach number.)

12.5

frequency

50

(kHz)

for a 6.2 cm diameter

conical

nozzle without

flow.

3.2.3. Power absorption for low acoustic intensities Substantial low frequency power loss has been shown to occur when high intensity sound propagates out through a duct termination, in the absence of flow. This phenomenon can also be explained on the basis of non-linear behavior of the high intensity sound as indicated in equation (8). For low intensity sound, when the non-linear term u* in equation (8) is ignored, the linear theory thus developed predicts acoustic power balance. To determine if an acoustic power balance does occur, in reality, when relatively low intensity sound propagates out through a duct termination, a series of tests were conducted with sine waves of about 115 dB SPL by using the impedance tube technique. A small amount of power absorption was still found to exist. These results for a 10 cm diameter duct are plotted in Figure 22 along with the high intensity pulse data. Significant improvement is noticed in the power conservation results. However, up to 10 dB power absorption at low frequencies still exists in these relatively low intensity test results. The power conservation results for relatively low intensity sound and the no flow condition as reported by other researchers [8, 22, 231 also show a small amount of low frequency power absorption. A small amount of low frequency power loss at the no flow condition is observed in Bechert’s result [8, 221, which he attributes to the “possible

500

.X4. \Al.IKUDDIN

AND

Non-dmensionol

K.

h

AHCIJA

frequency,

kR,

Figure 22. Low frequency power absorption for a straight duct with no flow. 0, Impedance ments with low intensity sound (present study); -, impulse experiment with high intensity study); - - -, Chung and Blaser’s results [23].

tube measurepulse (present

experimental error”. The power conservation spectrum presented by Chung and Blaser [23], for a duct without any flow, indeed shows a low frequency power loss. Chung and Blaser’s results are compared with impedance tube data, obtained in the present study, in Figure 22. Excellent agreement is found between these two sets of results. Therefore, it seems that the low frequency acoustic power loss will disappear only at intensities appreciably lower than 115 dB SPL.

4. CONCLUSIONS

The most important conclusion of this study is that by direct observations it has been shown that the conversion of sound energy into vertical energy, even in the absence of mean flow, can account for the low frequency power loss. The other important conclusions are outlined below. 4.1.

POWER

ABSORPTION

FOR

HIGH

ACOUSTIC

INTENSITIES

4.1.1. Effect of termination geometry on power absorption 1. Power absorption at low frequencies is observed for the duct, nozzles, orifice plates, and perforated plates in no flow conditions. 2. The amount of power absorption appears to be controlled primarily by the open area at the exit. However, the magnitude of the power loss is not directly proportional to the exit area. 3. The strength and the propagation speed of the vortices formed at the termination exit are controlled by the termination open area, and the shape and the structure of these vortices are controlled by the termination geometry. 4.1.2. Effect of flow on power absorption 1. Power conservation results with flow are similar to those for the no flow case. With flow the power absorption, for the duct and conical nozzles, initially increases and then decreases with increasing Mach number. 2. Power absorption results for a suppressor nozzle are similar to those for a conical nozzle. 3. The high intensity sound “non-linear effect” on power absorption is insignificant in the presence of flow.

POWER

LOSSES IN RADIATIONOUT OF DUCTS

501

4.1.3. Effect of temperature on power absorption Power absorption also occurs for heated jets. Compared with the results for unheated jets, however, more power loss at very low frequencies and less power loss at higher frequencies (above 400 Hz) are observed with heating. Further study is essential to explain the increase in power loss with heating at very low frequencies. 4.1.4. Effect of simulated flight on power absorption Low frequency power absorption is noticed for flight conditions also, and, in fact, it increases with flight. 4.2. POWER ABSORPTION FOR LOW ACOUSTIC INTENSITIES Even for relatively low intensity sound a small amount of low frequency absorption exists.

power

ACKNOWLEDGMENTS The authors are particularly thankful to Mr W. H. Brown for his help in obtaining the flow visualization results. Many useful discussions with Drs H. K. Tanna, H. E. Plumblee, Jr, A. Cummings and W. Eversman are gratefully acknowledged.

REFERENCES 1. P. D. DEAN, M. SALIKUDDIN,K. K. AHUJA, H. E. PLUMBLEE, JR and P. MUNGUR 1979 NASA CR-159698 Volume 1. Studies of the acoustic transmission characteristics of coaxial nozzles with inverted velocity profiles. 2. M. SALIKUDDINand H. E. PLUMBLEE, JR 1979 Lockheed-Georgia Company Engineering Report LG79EROlSl. Effect of blow-in doors and centerbody plugs on the acoustic radiation characteristics of modal superconic inlets-an experimental study. 3. K. K. AHUJA, M. SALIKUDDIN, R. H. BURRIN and H. E. PLUMBLEE, JR 1980 NASA CR- 165 144. A study of the acoustic transmission characteristics of suppressor nozzles. 4. M. SALIKUDDIN and H. E. PLUMBLEE, JR 1980 American Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-0991. Low frequency sound absorption of orifice plates, perforated plates, and nozzles. 5. K. K. AHUJA, M. SALIKUDDIN and H. E. PLUMBLEE, JR 1980 American Institute of Aeronautics and Astronautics 6th Aeroacoustics Conference Paper No. 80-1027. Characteristics of internal and jet noise radiation from a multi-lobe, multi-tube suppressor nozzle tested statically and under flightconditions. 6. M. SALIKUDDINand H. E. PLUMBLEE, JR 1980 Lockheed-Georgia Company Engineering ReportLG80ERO204. Internal noise radiation characteristics of baffles, nozzles, orifice plates and perforated plates used as duct terminations. and K. K. AHUJA 1980 Journal of 7. M. SALIKUDDIN,~. D. DEAN,H. E. PLUMBLEE,JR Sound and Vibration 70, 487-501. An impulse test technique with application to acoustic measurements. 8. D. W. BECHERT 1980 Journalof Sound and Vibration 70,389-405. Sound absorption caused by vorticity shedding demonstrated with a jet flow. 9. M. S. HOWE 1980 Journal of Sound and Vibration 70, 407-411. The dissipation of sound at an edge. 10. M. S. HOWE 1979 Journal of Fluid Mechanics 91, 209-229. Attenuation of sound in a low Mach number nozzle flow. 11. C. IMELMANN 1978 Diplomarbeit, Technische Universitiit Berlin (West). Einflujl der Stromung auf den Schalldurchgang durch Rohrmundungen [Influence of flow on the sound transmission through orifices]. 12. A. CUMMINGS and W. EVERSMAN 1980 Lockheed-Georgia Engineering Report LG80ER0164. An investigation of acoustic energy loss in radiation from ducts to the far field at low frequencies, low Mach numbers, and high sound pressure levels.

502

M. SALIKUDDIN

AND

ii.

k

AHlJJA

13. A. CUMMINGS 1981 American Society of Mechanical Engineers Paper Xl-WA/NCA- 10. High-amplitude acoustic power losses in perforated materials. 14. L. J. SIVIAN 1935 Journalof the Acoustical Society of America 7,94-101. Acoustic impedance of small orifices. 15. R. H. BOLT, S. LABATE and U. INGARD 1949 Journal of the Acoustical Society of America 21, 94-97. The acoustic reactance of small circular orifices. 16. U. INGARD 1953 Journal of the Acoustical Sociefv of America 25, 1037-1061. On the theory and design of acoustic resonators. 17. S. N. HEAVENS 1980 Journal ofF/uidMechanics 100,185-192. Visualization of the acoustic excitation of a subsonic jet. 18. M. C. WHIFFEN and K. K. AHUJA 1983 Journal of Sound and Vibration 86, 99-105. An improved schlieren system and some new results on acoustically excited jets. 19. A. CUMMINGS and W. EVERSMAN 1983 Journal of Sound and Vibration 91, 503-518. High amplitude acoustic transmission through duct terminations: theory. (A portion of this paper was presented at the 7th Aeroacoustics Conference, 1981: Paper No. 81-1979 “Acoustic power dissipation on radiation through duct terminations-theory”.) 20. D. A. BIES and 0. B. WILSON JR 1956 Journal of the Acoustical Society of America 29, 711-714. Acoustic impedance of a Helmholtz resonator at very high amplitude. 21. F. BARTH&L_ 1958 Frequenz 12, l-11. Untersuchungen iiber nichtlineare HelmholtzResonatoren (Investigations on non-linear Helmholtz resonators). 22. D. BECHE’RT, U. MICHEL and E. PFIZENMAIER 1978 American Institute of Aeronautics and Astronautics 4th Aeroacoustics Conference Paper No. 77- 1278. Experiments on the transmission of sound through jets. 23. J. Y. CHUNG and D. A. BLASER 1980 Journal of the Acoustical Society of America 60, 1570-1577. Transfer function method of measuring acoustic intensity in a duct system with flow.