Non-linear effects in finite amplitude wave propagation through ducts and nozzles

Journal of Sound and Vibration (1986) 106(1), 71-106

N O N - H N E A R EFFECTS IN FINITE AMPLITUDE WAVE PROPAGATION T H R O U G H DUCTS A N D NOZZLES'}" M. SALIKUDDIN AND W. H. BROWN

Lockheed-Georgia Company, Marietta, Georgia 30063, U.S.A. (Received 9 November 1984, and in revised form 20 April 1985) In this paper an extensive study of non-linear effects in finite amplitude wave propagation through ducts and nozzles is summarized. Some results from earlier studies are included to illustrate the non-linear effects on the transmission characteristics of duct and nozzle terminations. Investigations, both experimental and analytical, were carried out to determine the magnitudes of the effects for high intensity pulse propagation. The results derived from these investigations are presented in this paper. They in.clude the effect of the sound intensity on the acoustic characteristics of duct and nozzle terminations, the extent of the non-linearities in the propagation of high intensity impulsive sound inside the duct and out into free field, the acoustic energy dissipation mechanism at a termination as shown by flow visualizations, and quantitative evaluations by experimental and analytical means of the influence of the intensity of a sound pulse on the dissipation of its acoustic power. 1. INTRODUCTION The non-linearity o f finite-amplitude wave propagation is well established. While linearization of the governing equations often leads to mathematically more simple expressions, with solutions showing surprisingly good agreement with experimental results, it is recognized that the validity of the solutions is limited. The non-linear effects of interest in what follows here are those relevant to the sound transmission characteristics of various duct terminations for high intensity impulsive sound waves. During the past several years, research on the transmission of sound through various duct terminations has been carried out at Lockheed-Georgia C o m p a n y [1-6], with use of an acoustic impulse technique developed by Salikuddin et al. [7]. In this technique, high intensity pulses generated by spark discharge were used as the sound source. Some evidences of the presence of non-.linearity in impulse sound propagation have been observed with this technique in the tests described in references [1-6]. The propagation speed was observed to be faster than the speed of sound for compression waves and slower for expansion waves. The reflection coefficient of an unflanged duct is observed to be lower than the analytical result derived from linear theory. Moreover, low frequency power loss was observed in the absence o f m e a n flow in duct terminations and orifice plates. More indications of non-linear behavior of finite amplitude wave propagation have been described by other researchers [8-13]. In reference [8]. Watanabe and Urabe have shown that the propagation speed of a finite amplitude wave is different from the speed of sound. A brief review o f finite amplitude wave propagation has been given by BjclrnO [14]. He has described the histoical development of finite amplitude wave propagation t This work was co-sponsored by Lockheed's IRAD program and by NASA-Lewis under Contract NAS320797. 71 0022-460X/86/070071 +36 $03.00/0 9 1986 Academic Press Inc. (London) Limited

72

M. S A L I K U D D I N

A N D W. H. B R O W N

in fluids, and the theoretical basis for the propagation of plane, cylindrical or spherical finite amplitude waves through lossless, thermoviscous or relaxing fluids, as well as the characteristic features o f the distortion occurring in finite amplitude waves in various propagation regimes. In this p a p e r selected results from the studies described in references [1-6] are presented to illustrate the main non-linear effects of interest. Then, results are presented of systematic experimental and analytical investigations performed to determine the magnitudes o f the effects of high-intensity pulse propagation through duct and nozzle terminations. Results obtained by using flow visualization means are also presented. The experimental procedures are described in the next section. In section 3, the results indicating non-linear behavior extracted from references [1-6] are described. In section 4, the effects of pulse intensity on the acoustic characteristics of duct and nozzle terminations, as determined experimentally, are presented. Sections 5 and 6 are devoted to non-linearities occurring in the propagation of high intensity pulses inside the duct and in free field, respectively. Section 7 contains flow visualization results obtained in a study of the acoustic energy dissipation mechanism at a duct termination. In addition, quantitative results showing the effect o f the pulse intensity on the acoustic power dissipation, as evaluated both experimentally and by analytical means, are described in this section. Finally, the important observations are summarized in section 8. 2. EXPERIMENTAL SET-UP AND DATA ANALYSIS The experimental set-up and a photographic view of all the duct and nozzle terminations used in the present study are shown in Figure 1. These terminations include a straight 10 cm diameter duct, three conical nozzles with exit diameters of 6.2 cm, 5.0 cm, and 2-5 cm, and a multilobe-multitube suppressor nozzle (called a daisy-lobe nozzle) with an equivalent exit diameter of 6.2 cm. Absorbent lining

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Figure 1. (a) Schematic showing the source section and the in-duct and far field measurement system; (b) photograph of a 10cm diameter duct. a suppressor nozzle (the daisy lobe nozzle) with an equivalent diameter of 6-2 cm, and three conical nozzles with exit diameters of 6.2 cm, 5 cm and 2.5 cm.

NON-LINEAR WAVES IN DUCTS AND NOZZLES

73

2.1. EXPERIMENTAL CONFIGURATION, TEST PROCEDURE, A N D DATA ANALYSIS

The experimental configuration as shown in Figure l(a) consisted of a spark noise source located at the centerline of a 10 cm diameter pipe about 6 meters upstream of the duct termination, which is located inside an anechoic chamber. A pressure transducer was mounted through the wall of the pipe 76.2 cm upstream o f the termination to measure the in-duct signals. Provision was made to mount various nozzle configurations. Far field signals were measured on a polar arc o f 2.44 meters radius with 0.635 cm diameter Briiel and Kjaer microphones placed at 10 degree intervals extending from 0 to 120 degrees with respect to the jet axis. The basic test procedure consisted of discharging the capacitor across the spark gap located inside the duct at the source section and measuring the resulting incident and reflected pressure pulses by the in-duct transducer, and the transmitted pulse by the far field microphones. The in-duct and all the far field pulses were recorded simultaneously on a 28-track tape recorder. The spectral content of the incident, the reflected, and the transmitted pulses were obtained by the Fourier transform of each pulse, by using a dual channel F F T signal analyzer. The complex transfer function operation between reflected and incident pulses gives the complex spectral power reflection coefficient of the termination. To illustrate this process consider a typical in-duct time history of an open ended duct, as shown in Figure 2(a) measured by a transducer located at a distance L, upstream of the termination. Since llol

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Figure 2. Illustration of complex transfer function operation between reflected and incident pulses to derive complex spectral reflection coefficient; (a) in-duct time history; (b) edited incident and reflected pulses after propagation time correction.

the incident and the reflected time histories are measured on one transducer, the analysis procedure requires that both the memory channels of the analyzer be loaded with the same in-duct signal. Channel A must be edited to remove all signals except the incident time history and channel B must be similarly edited to remove all except the reflected time history. The incident and the reflected signals thus stored in channel A and B, respectively, have a relative time separation, r, due to pulse propagation time. The reflected

74

M. S A L I K U D D I N

A N D W. H . B R O W N

pulse must therefore be shifted in the time domain to be brought into alignment with the incident pulse as shown in Figure 2(b). This time shift r is equal to the sum of the propagation time of the incident pulse from the transducer location to the termination and the propagation time of the reflected pulse from the termination back to the transducer location. Without this time shift, the pulse propagation time has the effect of producing a frequency-dependent phase shift that is not related to the termination properties. The complex transfer function operation (B/A) then performed by the FFT analysis is identically equal to the complex spectral reflection coefficient (or) of the termination. The radiation impedance of the termination (Z) is computed by using the complex reflection coefficient at the exit plane. Similarly, the transfer function of a far field pulse with respect to the incident pulse gives the transmission coefficient of the termination. Its variation as a function of polar angle in the radiated field defines the directivity of the termination configuration and is a function of frequency..This pressure directivity can be used to compute radiated far field power (Ws) by integrating it over a spherical surface centered on the exit plane. The spectral coefficients thus evaluated were used to compute the power transfer functions with respect to incident power (PTF~) and transmitted power (PTF,). The PTFt is the same as the power difference between the far field (WI) and the transmitted (IV,) powers (i.e., power imbalance). The detailed procedure and the expressions to evaluate the above parameters have been described in references [15, 16]. The analytical expressions used tcr compute the reflection coefficient (o-) and PTF, were developed by Cummings and Eversman [17, 18]. 2.2. T E S T S E T - U P A N D E X P E R I M E N T A L P R O C E D U R E F O R F L O W V I S U A L I Z A T I O N T E S T S The flow visualization tests were carried out by using a smoke wire technique. A photograph of the complete flow visualization instrumentation and the circuit diagram for the flow visualization operation are shown in Figure 3. A steel wire of 0.5 mm diameter was placed diametrically in the duct/nozzle exit plane. A thin coating of oil was put over this wire from one end to the other by using an oil supply syringe at the top. The wire was electrically heated to generate smoke just before the escape of the acoustic pulse from the duct. By using a synchronized light source of short duration and a camera with the shutter open, a photograph of the smoke pattern outside the termination was taken. Such a photograph would indicate if a vortex were formed during the transmission of the acoustic pulse. The existing free-jet facility was used only for flow visualization tests in the presence of mean flow. Since the flow duct in this facility was horizontal, the smoke generated in the absence of mean flow tended to rise due to buoyancy effects. Therefore, a set-up with a vertical duct was built especially for tests at zero flow. The source was located about 2 meters upstream of the termination, and the opposite end of the duct was terminated into a foam pad at about 1 meter from the source. Two types of impulsive sources, one a spark discharge source and the other an acoustic driver, were used for this study. The impulse technique with the acoustic driver as the impulsive source has been described in references [19, 20]. The circuit diagram for the operation of spark discharge, smoke generation, and camera operation is shown in Figure 3(b). When Switch 1 (SW1) was closed, the charging circuit for the capacitor across the spark gap was completed. Once the charging was done, Switch 1 (SW1) 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

75

NON-LINEAR WAVES IN DUCTS AND NOZZLES

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the termination. This signal was used to trigger the stroboscopic light after an appropriate time delay. The same circuit was employed when an acoustic driver was used to generate the pulse. In this case the high-voltage power supply was not operated and the spark gap in the source section was replaced by an acoustic driver. The driver was fed with a repetitive input signal to generate a periodic chain of pulses. Switch 1 (SW1) was opened, in this case, after the driver was switched on. The remaining operations were identical to those when a spark discharge source was used. The incident part of the transient pulse, measured by the pressure transducer inside the 10 cm duct cross-section was Fourier transformed to compute the pulse intensity for use as a parameter in all the flow visualization tests.

3. R E S U L T S I N D I C A T I N G N O N - L I N E A R B E H A V I O R Most o f the results, obtained with the high intensity impulsive sound source, and previously reported [1-6], are considered to be accurate, and accounted for by linear theory. The accuracy o f some of these results was validated by suitable comparisons.

76

M. SALIKUDDIN AND W. H. BROWN

However, a small amount of data indicated non-linear propagation effects. Some of these results are discussed in this section. 3.1. REFLECTION COEFFICIENTS FOR UNFLANGED DUCT TERMINATION

The experimentally determined reflection coefficient amplitudes derived when using high intensity pulses (about 150 dB), are compared with the Levine and Schwinger solution [21] in Figure 4. The Levine and Schwinger analytical result is based upon an exact

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Figure 4. Reflection coefficient comparison between impulse method wtih high intensity pulse (about 148 dB) and Levine-Schwinger theory for a 10 cm diameter duct termination in the absence of mean flow; , impulse method; - - - - , Levine-Schwinger theory.

mathematical analysis o f the sound field within and outside an unflanged circular pipe for a plane wave mode incident on the pipe termination from within the pipe. The experimental data are contaminated by cut-on of the first radial mode (kRo = 3.83, k and Ro being the wave number and the radius of the duct). However, even below this cut-on frequency the comparison is not good, with the measured value decreasing more rapidly with increasing frequency than that predicted by Levine and Schwinger. This behavior indicates a reduction in the refle~ted wave intensity due to a possible non-linear behavior of the high intensity pulses. 3.2. REFLECTION COEFFICIENTS FOR CONICAL NOZZLE TERMINATIONS The reflection coefficient amplitude spectra for the 10 cm diameter straight duct and two conical nozzles with exit diameters o f 5.0 cm and 2.5 cm were evaluated experimentally by using high intensity pulses (the incident pulse intensity being approximately 150 dB) and the results are plotted in Figure 5. Unlike the straight duct reflection coefficient, at very low frequencies (kRo < 0.5) a decrease in reflection coefficient levels is observed for the conical nozzles, and ~hese levels at zero frequency do not tend to the expected value of 1-0 (i.e., zero dB). Moreover, as the nozzle diameter decreases from 5.0 cm to 2.5 cm, the reflection coefficient level is decreased at very low frequencies. This phenomenon can be explained as follows. For conical nozzles, unlike a straight duct, two opposite types of reflections are observed (see Figure 6). One is due to the nozzle contraction and is in phase with the incident pulse, and the other one is due to the open termination and is out of phase with the incident pulse. Therefore, a decrease in reflection coefficient at low frequencies can be

NON-LINEAR 0

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77

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Figure 5. Effect of nozzle exit area on reflection coefficient for various nozzle terminations attached to a 10 cm diameter duct, in the absence of mean flow; , 10 cm diameter straight duct; - - - , 5 cm diameter conical nozzle; . . . . , 2 . 5 c m diameter conical nozzle.

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viewed as due to the cancellation of a portion of the pulse reflected from the nozzle contraction by the reflection from the open end. Since the in phase reflection due to the nozzle contraction takes place earlier than the out-of-phase reflection from the open end, they should not coincide if their propagation speeds are identical. Therefore, the observed cancellation is possible only if the reflections of opposite phases do overlap due to different propagation speeds, which could occur due to non-linear propagation effects. 3 . 3 . A C O U S T I C POWER LOSS

The acoustic power conservation, in the absence of mean flow for the duct and conical nozzle terminations was studied by plotting the far field power normalized with respect to the transmitted power (PTF,= 10 LOG~o (IV//Wt)) in Figure 7. For the ideal situation, when there is no power loss, the power transfer function (PTF,) would have been zero. However, a low frequency power loss was observed for all the terminations. This acoustic

78

M. SALIKUDDIN AND W. H. BROWN i

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Figure 7. Effect of nozzle exit area on acoustic power imbalance in the absence of mean flow; --[3--, 10 cm diameter duct; - -O- -, 6.2 cm diameter conical nozzle; - - A - - , 5 cm diameter conical nozzle; - - + - - , 2.5 cm diameter conical nozzle. p o w e r loss i n c r e a s e s with d e c r e a s i n g nozzle exit area. T h e l o w f r e q u e n c y p o w e r loss p h e n o m e n o n in the a b s e n c e o f m e a n flow c o u l d be a t t r i b u t a b l e to the a c o u s t i c p o w e r d i s s i p a t i o n d u e to n o n - l i n e a r b e h a v i o r o f the high intensity s o u n d waves used. 3.4. REFLECTION COEFFICIENT CONTRADICTS FAR-FIELD ACOUSTIC POWER FOR DAISY LOBE NOZZLE F i g u r e 8 s h o w s the effect o f j e t M a c h n u m b e r on the relative levels o f the i n c i d e n t a n d the reflected c o m p o n e n t s o f the i n - d u c t signal for the d a i s y l o b e nozzle. C l e a r l y , the reflection f r o m the o p e n e n d decreases with i n c r e a s i n g M a c h n u m b e r . A d r a m a t i c c h a n g e in the time histories is n o t i c e d as the j e t M a c h n u m b e r is i n c r e a s e d b e y o n d 0.4. T h e r e is little sign o f reflection f r o m the o p e n e n d b u t t h a t from the h a r d (or solid) p a r t o f the nozzle t e r m i n a t i o n i n c r e a s e s c o n s i d e r a b l y as t h e j e t M a c h n u m b e r increases. At M~ -- 1-2 the i n c i d e n t a n d the reflected signals a p p e a r to b e o f the s a m e a m p l i t u d e . The i m p l i c a t i o n o f these results is t h a t if there were c o n s i d e r a b l e internal n o i s e g e n e r a t e d u p s t r e a m o f

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Figure 8. In-duct time histories for the daisy lobe nozzle termination at various jet Macb numbers, Mj.

NON-LINEAR

79

WAVES IN DUCTS AND NOZZLES

the suppressor nozzle exit then much of it should be reflected back at higher Mach numbers and very little should be transmitted to the far field. The behavior shown in Figure 8 can be better seen in Figure 9 where the reflection coefficients (troL) and the normalized far field power (PTF~)oLfor the daisy lobe nozzle 0 -===.~-o\

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Figure 9. Effect of jet Mach number, Mj, on (a) reflection coefficient and (b) normalized for field power for the daisy lobe nozzle; --O--, M x =0.0; ---~- --, Mj =0-4; --O--, Mj=0.8; - - - F I - - - , Mj = 1.2.

at four jet Mach numbers (M1 = 0, 0.4, 0.8 a n d l . 2 ) are presented as functions of frequency. At low frequencies as the jet Mach number (and so the duct Mach number) is increased, the reflection coefficients first decrease and then increase) (see Figure 9(a)). At supersonic jet Mach numbers, o'ol. appears to approach 0 dB (i.e., that of a rigid termination). This behavior was also observed in the time history plot in Figure 8. If actually complete reflection were taking place then there should be no transmission of acoustic energy out to the far field. However, it can be seen from the far field power spectra (PTF~)oL that a considerable amount of energy has, in fact, been transmitted out (see Figure 9(b)). Figure 9(b) clearly indicates that the far field acoustic power for the daisy lobe nozzle first decreases and then starts increasing with increasing jet Mach number. This behavior certainly does not correlate with the reflection coefficient data because the trends, versus jet Mach number, of far field acoustic power and of reflection coefficient should oppose rather than parallel each other. The behavior of o'oL itself and the contradiction between the croL and (PTFi)oL behaviors are difficult to explain without considerable theoretical and experimental work, but it seems clear that factors involved include the non-linearity due to the high intensity of the pulses, the complicated termination geometry, and the flow conditions encountered by the incident wave in traveling from upstream to the multi-jet exits. 4. EFFECT OF PULSE INTENSITY ON ACOUSTIC CHARACTERISTICS The unusual pattern observed in the results presented in the last section prompted the investigation of the non-linear effects associated with high intensity pulses. To achieve this, several tests were conducted on various duct and nc,zzle terminations at various pulse intensities.

80

M. SALIKUDDIN AND W. H. BROWN

4.1. DUCT WITH PLUGGED TERMINATIONS T h e hypothesis of non-linear propagation of finite amplitude waves was tested by conducting experiments for plugged terminations. In these experiments, terminations of different materials, hard and soft, were used to plug the open end of the 10 cm diameter duct. 4.1.1. Plugged terminations with hard materials Reflection coefficient spectra for rigid plugs of two different materials were determined for incident pulses o f about 145 dB intensity. Figure 10(a) is a typical time history showing the incident and reflected pulses inside a duct with a hard plugged end. The reflected pulse is in phase with the incident pulse; zc is the propagation time of the pulse from the transducer location to theinner surface of the plug and back to the transducer location.

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Figure 10. Pluggedend reflectionfor high intensity pulses (about 148 dB). (a) A typical time history showing incident and reflected pulses; (b) reflectioncoefficientspectra for different plug materials; , aluminum; - - - , wood.

If the propagation o f the pressure pulse were linear then a positive unity (zero dB) reflection coefficient would be obtained at least up to the first radial cut-on frequency (i.e., kRo = 3-83). The reflection coefficient spectra for the duct with a wooden plug and with an aluminum plug are plotted in Figure 10(b) up to a frequency of 6 kHz which is beyond the first radial cut-on frequency for the 10 cm diameter duct used (Le., about 4 kHz) in this case. The reflection coefficient values in the frequency range are close to zero dB. The reflection coefficients for the aluminum plug are closer to the zero dB level than those for the wooden plug, as would be expected siflce wood is comparatively softer than aluminum. However, the deviations of the reflection coefficients from zero dB values for the aluminum plug are attributable to the non-linear propagation and reflection process of the pulse. 4.1.2. Effect of pulse intensity on acoustic characteristics of absorbing material Absorbing materials, like polyurethane foam, behave highly non-linearly when exposed to high intensity sound field. To establish this, and to determine a maximum pulse intensity at which the acoustic properties of the absorbing material remain linear, several tests were conducted with a polyurethane foam sample. A sample of 2.5 cm thick polyurethane foam, mounted on a steel plate, was fitted as a termination of the 10 cm diameter duct. Several tests were conducted with pulses of varying intensity. The reflection coefficient spectra for the sample as shown in Figure 11 indicates that the reflection coefficient amplitude increases with decreasing pulse intensity. However, once the intensity goes

N O N - L I N E A R WAVES IN DUCTS AND NOZZLES ,

81

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Figure 11. Effect o f pulse intensity on reflection coefficient o f a 2.5 cm thick polyurethane foam, mounted on a steel plate plugged at the 10cm diameter duct termination; - - . , 156dB; ~ - - , 148 dB; u _ 140dB; -----,

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Figure 12. Effect of pulse intensity on the impedahce ratio and the absorption coe'fficient of a 2.5 cm thick polyurethane foam, moUnted on a steel plate plugged at the I0 czh diameter duct termination; - - - , 156 dB; , 1 4 8 dB; - - - - , 140 dB, m . __, 133 riB.

82

M. S A L I K U D D I N

A N D ~,V. H. B R O W N

down to a linear level there is no further change in the reflection coefficient value with further decrease in the pulse intensity. The pulse intensity has very little effect on the phase values. The impedance ratio values and the absorption coefficients as derived by using the reflection coefficient results are plotted in Figure 12. The effect of pulse intensity is quite significant for both the resistance and absorption coefficient values; however, its influence is small on the reactance values.

4.2.

EFFECT OF PULSE INTENSITY

ON

PROPAGATION

SPEED

As described in section 2.1, the sum of the propagation time of the incident pulse from the transducer location to the termination and the propagation time of the reflected pulse from the termination to the transducer location, z, must be exactly known to evaluate the complex reflection coefficient and radiation impedance of the termination. This propagation time, ~', can be'easily evaluated once the propagation speeds of the incident and reflected pulses and the physical distance between the transducer and the termination are known. For low intensity sound waves, which do not undergo any non-linear distortion, the propagation speeds of all spectral components of the incident and reflected waves are equal and are identically equal to the isentropic sonic speed (this is true of course only if viscous and thermal diffusion is negligible). However, for high intensity pulses, the propagation speeds depend on the intensity, and the propagation speed varies along the propagation path due to the change in pulse intensity. This variation is very prominent in variable area ducts. In this section, experimental results illustrating the effect of pulse intensity on propagation speeds for compression waves and expansion or rarefaction waves are presented. In principle, the propagation time between the incident and reflected pulses can be determined by locating the leading edges of the incident and reflected pulses in the time domain. However, in practice, it is difficult to locate the leading edge of the reflected pulse precisely, since it is non-sharp in nature. Moreover, since the time-history appears in digitized form, the leading edges for incident and reflected pulses can be located only at discrete times, so the actual propagation time could be higher or lower by a fraction of the incremental time between two successive digitized points. A small error in the propagation time can introduce a substantial error in the radiation impedance values. As an example to show the effect of propagation time correction on the reflection coefficient, the phase, and the radiation impedance, a test was conducted to evaluate the reflection coefficient and the radiation impedance for a thin walled 10 cm diameter pipe without any flow, in which the in-duct transducer was mounted 48.3 cm upstream of the exit. The data analysis was carried out with use of the digital FFT analyzer (SD360) to digitize the complete in-duct time history into 1024 points. Since the time range for this analysis was set at 25 ms, the incremental time, dr, between successive digitized points is 0.02441 ms. Various propagation time corrections in a step of Ar (one digitized point) were applied to the reflected pulse to generate reflection coefficient phase spectra which are shown in Figure 13(a). It can be seen from Figure 13(a) that an error of 0.02441 ms in propagation time correction causes a considerable amount of error in phase values and that the error in phase is directly proportional to the frequency. It can also be seen that the correct reflection coefficient phase spectrum for the duct termination, shown in Figure 13(a), does not coincide with any of the phase spectra generated by applying propagation time corrections in steps. Instead, the correct phase spectrum lies between two spectra generated by applying two successive propagation time corrections. Figures 13(b) and (c), respectively, indicate the effect of propagation time correction on radiation resistance and reactance values. Therefore, it is very important to determine

NON-LINEAR

,8o 9"-:

WAVES IN DUCTS AND NOZZLES

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'. -,-.- ~'-",~..~....~-~---.~..~1----,,,

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.- .... - - - - ~ - - ~

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e=

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oo

1"0

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0

05

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1-0

.. t

L~

".

.

.

.

1"5

I

2"0

2"5

5"0

Non-dimensionol frequency, kRo

F i g u r e 13. S p e c t r a for a n u n f l a n g e d d u c t o f l 0 c m d i a m e t e r for v a r i o u s p r o p a g a t i o n t i m e c o r r e c t i o n s w h e n the t r a n s d u c e r was 48-3 c m u p s t r e a m o f the exit. (a) Reflection coefficient p h a s e ; (b) r a d i a t i o n r e s i s t a n c e ra!io; (c) r a d i a t i o n r e a c t a n c e ratio; ,2-709 ms; - - -, 2-734 ms; - - - , 2.758 ms; - - - - - - , 2.783 ms; - - - - - , 2.807 ms; ,2.832 ms; .... , 2 - 8 5 6 ms; - --, 2.880 ms; , 2 - 7 7 4 ms ( e x a c t p r o p a g a t i o n time).

the propagation time as precisely as possible, which dictates in turn the accurate determination of propagation speed in the duct-nozzle system. From Figure 13(a), it can be seen that the incremental propagation time corrections applied to the reflected pulse are directly proportional to the changes in phase values at each frequency. Therefore, if the appropriate phase values at a given frequency are known, the corresponding propagation time correction can be accurately calculated. Consider the example of the open end duct for which the reflection coefficient phase spectra, as shown in Figure 14, are evaluated with time corrections of rn and ri2 applied to the

180

"'7,

60 0

I 0.5

I I I Prediction I 1.0 1-5 2.0 Non-dimenslonol frequency, kR o

I 2-5

3.0

F i g u r e 14. Reflection coefficient p h a s e s p e c t r a for a n u n f l a n g e d d u c t o f 1O c m d i a m e t e r to i l l u s t r a t e the c a l c u l a t i o n p r o c e d u r e for p r o p a g a t i o n time.

84

M. SALIKUDD1N A N D W. H. BROWN

reflected pulse with respect to the incident pulse. The appropriate phase spectrum corresponding to an accurate propagation time correction, ~', is also plotted in this figure as a long dashed curve. This phase spectrum in this case, is derived from the end correction, lo for an unflanged straight duct as predicted analytically by Levine and Schwinger [21]. The relationship between the phase ~b and the end correction le is derived by defining the complex reflection coefficient tr in the following forms: o-= - I o 1 e 2ikt"= I~1 e i('-=~'~ --- I~le i~.

(1)

q5 = ~r- 2( kRo)( Id Ro)

(2)

Therefore, 6 = 1r-2kl,, or Let ~b be the appropriate reflection coefficient phase at a frequency fl. Let q'H and (~12 be the phases at the same frequency f~, determined from the experiment when time corrections of Tti and ~'12 are applied, respectively. Since the appropriate phase at f~ is ~b, the required propagation time correction between the incident and reflected pulses which would yield a phase value of ~b can be expressed as (7")s, = ~qt+{(~b - ~b,,)/(612- ~bi,)} ( r , 2 - ~'li).

(3)

Equation (3) can be applied to any situation, whether the appropriate phase spectra lie outside or between the two spectra obtained with time corrections of r~ and ~'~,. Equation (3) can be used to determine the values of ~" at a number of frequencies up to the first circumferential cut-on frequency (i.e., kRo = 1.84), since the incident and the reflected waves remain plane up to this frequency. An average value, ~', can then be determined to minimize any possible experimental error. It should be noted that in this procedure the appropriate phase values derived from the Levine and Schwinger analysis [21] for the open end duct is used only to compute the average propagation time, ~-, covering a frequency range up to kRo = 1.84. The experimental phase spectrum, appropriate for the open end duct, for the complete frequency range, as shown in Figures 13(a) and 14, by a thick solid curve, is calculated from the following expression, in which the average propagation time r is used: (r

= (q',,)s + {(~" - ~,,)/(~',~ - ~,,)}{(q',~)s

- (~',,)s}-

(4)

Here (~b)j- is the appropriate phase value at the frequency f, and ( t ~ i l ) s and ((ht2)s are the phase values at the same frequency associated with the time corrections rH and ~'~2, respectively. The propagation speed was thus calculated by using the proper time correction r, in the following manner, for these experiments. Two different terminations with known reflection coefficient phase spectra were used in the present study to determine the effect of pulse intensity on the propagation speeds of both the compression and the rarefaction waves. Several tests were conducted with a polished steel plug as the termination for the 10 cm duct with different pulse intensities to determine the propagation speed of the compression waves. Since, for the plugged end, the reflected wave is also a compression wave, very similar to the incident wave, the propagation speed must be very nearly the same for both incident and reflected waves. The appropriate phase value for a plugged end has to be zero. Therefore, the required propagation time (~'r for the compression wave, which should be the exact correction time to yield a zero phase, can be expressed from equation (3) in the following form: ("Ds, = ~',, - ~i,{(~'i~-

~,,)/(q',.-

~>,,)).

(5)

The propagation speed, 2,.; for the compression wave was then determined by using the distance, L, between the in-duct transducer to the inner surface of the plug and the

NON-LINEAR WAVES IN DUCTS AND NOZZLES

85

propagation time of the pulse, ~-, from the transducer location to the plug surface and back to the transducer:

6cl 6o = (1/n/b--R-T)( 2 L / Tc).

(6)

Here 60, v, R, and T are the speed of sound, the ratio o.f specific heats, the gas constant, and the absolute static temperature, respectively. The ratio 6d6o was calculated for pulses with different intensities and is plotted in Figure 15 as a function of pulse intensity. The ratio 6c/6o for compression waves increases with increasing pulse intensity. 1"02

I

I

I

I

I

I

I

l 160

I

1-01

C

o

r

n

~

1.00

0"99

g o

o

I:L

~

0-98

0-97

t

150

1 140

l

I I 150 Intensity (dB)

170

Figure 15. Effect of pulse intensity on the propagation speed.

To determine the propagation speed, 6,, of the expansion waves (i.e., the open end reflection) several tests were conducted for a straight open end duct at different pulse intensities. The appropriate reflection coefficient phase values were calculated by using the end correction values for a duct as predicted by Levine and Schwinger [21]. Then the propagation time, r, between the incident and reflected pulses for each of the tests was derived by using equation (3). Since the incident pulse was a compression wave and the reflected wave was an expansion wave, the total propagation time can be expressed as ~'=89

or

r=(L/6c)+CL/6,),

(7)

where z,/2 is the propagation time for the expansion wave from the exit plane to the transducer location. The propagation speeds for expansion waves were then computed from

6,16o = ( llx/-~-RT){ LI[ r - (L/6=)]}.

(8)

The values of 6, used in equation (8) were derived from the plot of 6c/6o vs. intensity given in Figure 15. The values of 6~/6o as a function of the reflected pulse intensity are also plotted in Figure 15 and indicate that 6d6o decreases with increasing pulse intensity. 4.3. REFLECTION COEFFICIENT AND RADIATION IMPEDANCE FOR AN OPEN DUCT TERMINATION 4.3.1. Effect of pulse intensity The reflection coefficient amplitude spectra for a 10 cm diameter straight duct termination were experimentally determined for various pulse intensities and are plotted in Figure 16(a). The relative level of the reflection coefficient amplitudes decreases with increasing

86

M. S A L I K U D D I N A N D W. H. BROWN ,

.

.

.

-,5

7

180

.

1.0

(c)

.

.

'

0"75 o

0.5

rr 0-25

o

E

.

,,

:!

,

' ":"

j

,,

~J

/J"

0 \~..

E

~" .... . ~ _ 0

1 Non-dimensionol

2

5 frequency,

4

kRD

Figure 16. Ettect o f p u l s e intensity on reflection coefficient and radiation impedance ratio f o r a 10 cm diameter , 142dB; - - - , 147dB; ~ - - , 149 d B ; ,155 d B ; - - - - - , 158 dB. duct termination.

pulse intensity for a fixed frequency, up to the first circumferential cut-on frequency (i.e., kRo = 1.84). The reflection coefficient phase spectra for each of the pulse intensities were determined by using the corresponding average propagation speed as discussed in section 4.2. The reflection coefficient amplitude and phase as shown in Figure 16(a) were used to compute the radiation impedance for the duct and are plotted in Figure 16(b). The resistance values tend to increase whereas the reactance values tend to decrease with increasing pulse intensity for a given frequency up to at least kRo = 1.5. 4.3.2. Comparison of results for very low and high intensity sounds The experimentally determined values of the reflection coefficient for a 10 cm diameter duct termination are shown in Figures 17(a) and (b), together with those from the Levine and Schwinger [21] solution. The experimental results were derived by using the impulse technique, for both high and low intensity pulses, and also by using the impedance tube technique for low intensity, discrete frequency, pressure waves. The reflection coefficient amplitude results derived by the impulse technique, for a low intensity pulse (about 100 dB), by the impedance tube method, and by the Levine and Schwinger solution agree very well with each other zip to about kRo = 2.5 which is higher than the first circumferential cut-on frequency, kRu = 1-84. The reflection coefficient amplitude derived by using the impulse technique with a high intensity pulse does not agree with the low intensity results. The different result for the high intensity pulse can be attributed to its non-linear behavior. Figures 17(c) and (d) shows the radiation impedance spectra derived from the reflection coefficient results. Except for the high intensity impulse results, the radiation impedance values derived by the three methods agree well up to about kRo = 2-5.

N O N - L I N E A R WAVES IN DUCTS A N D N O Z Z L E S 0

9

_~ I. " " I

,

,

,

, . Second

.

mode,

mode,I

cikR~, rcumderent =Od, le~4i a,l ~ ~" ~ "~'/I

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First I circumferentlol radial l

"~.~,, ~"~

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1 2 3 Non-dimensional frequency,

/

\/1

~

4

kRo

Figure 17. Reflection coefficient and radiation impedance ratio fora 10 cm diameter duct termination evaluated for high and low intensity sound; Q, Impedance tube result with low intensity sound; , impulse result with high intensity pulse (148 dB); - - - - - , impulse result with low intensity pulse (100 dB); - - - , Levine-Schwinger solution.

m

:_a

0-

,

"~ o: ~" g -lO-5

~ t

0

'

" -"

"

I

i 2 3 Non-dlmensional frequency, kRo

4

Figure 18. Reflection coefficient amplitudes for conical nozzle terminations with (a) 2.5 cm diameter and (b) 5.0 cm diameter, evaluated for high (148 dB) and low (I00 dB) intensity pulses. , High intensity; - - -, low intensity,

88

M. SALIKUDDINAND W. H. BROWN

4.4. REFLECTION COEFFICIENTS OF CONICAL NOZZLES FOR VERY LOW AND HIGH INTENSITY PULSES The low intensity reflection coefficient amplitudes for the two conical nozzles with exit diameters of 5.0 cm and 2.5 cm were determined and are compared with the corresponding reflection coefficient amplitudes for high intensity pulses in Figur~ 18. The reflection coefficient amplitudes determined for low intensity pulses are considerably higher than those for high intensity pulses. In addition, the reflection coefficients tend to unity (zero dB) at zero frequency for low intensity pulses, as compared to the very low values obtained with high intensity pulses. 4.5. EFFECT OF SOUND INTENSITY ON POWER ABSORPTION For ducts, nozzles, and orifices, as discussed in references [1-6], low frequency acoustic transmission loss occurred at all flow conditions including the zero flow condition. Since high intensity pulses were being used, it was suspected that the power imbalance in the absence of flow could be due to non-linear propagation of these pulses in the duct. Therefore, tests were conducted by using the impulse technique for both high and low intensity pulses, and also by using the impedance tube technique for low intensity, discrete frequency, sine waves (about 115 dB). The impedance tube power absorption results and

.

.

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Non-dimensional frequency,

o18

1-o

kRo

Figure 19. Low frequency power absorption for a 10 cm diameter duct termination evaluated for using high and low intensity sound. O, Impedance tube result with low intensity sound; , impulse result with high intensity pulse (148 dB); - - -, impulse result with low intensity pulse (100 dB).

the low intensity pulse results are compared with high intensity pulse data in Figure 19. Significant improvement in the agreement with linear theory predictions can be seen in the power conservation results obtained by the impedance tube technique and impulse technique with low intensity pulses, though a small amount of power loss at low frequencies still exists. Power loss levels at low frequencies of about 2 to 3 dB can be expected since the far field power was calculated for the angular range of 0~to 125~with the downstream jet axis instead of 0~ to 180~ This does not, however, account for the remaining 2 to 3 dB loss at the lowest frequency (kRD=0.15). The results presented in this section confirm the presence of non-linear effects for high intensity sound. The non-linear behavior could be influencing the acoustic characteristics at three stages: during the propagation of incident and reflected waves inside the ductnozzle systems; during the propagation of the transmitted wave in the free field; and at the termination where the incident wave is divided into reflected and transmitted waves. Therefore, some studies were carried out to find out the extent of the non-linearities appearing at each of these three stages of the sound transmission.

N O N - L I N E A R WAVES IN D U C T S A N D N O Z Z L E S

89

5. PROPAGATION OF HIGH INTENSITY PULSES INSIDE A DUCT Several tests were conducted to study the distortion caused to a high intensity pulse propagating inside the duct due to non-linear effects. In these tests the axial position, x, of a pressure probe inside a 10 cm diameter duct, was varied. Pulses with a fixed intensity (about 148 dB) were measured by the in-duct probe at several axial locations. The in-duct time histories at each axial location, as captured by the probe, are shown in Figure 20. When the probe was placed just at the exit of the duct (i.e., x/Ro = 0) the pulse seen is a single sharp pulse which is the combination of the incident and the reflected pulses. As the probe was m o v e d inside the duct, the reflected pulse gradually separated out from the incident pulse. At x/Ro = 6, the reflected pulse was clearly isolated from the incident pulse. Further increase in the distance between the duct exit and the probe increased the separation of incident and reflected pulses as expected. Axial in-duct position,x/Ro, J from the duct exit J

. ~

0.25

t

1-o o

g

a

".3

0

I

T

5

10

15

Time (ms) Figure 20. In-duct time histories for a I0 cm diameter duct termination measured at various in-duct axial locations, x, for high intensity pulses (about 148 dB).

The spectra of the incident and the reflected p u l s e s c a p t u r e d at a number of in-duct axial locations for x/Ro>~6 were derived and are plotted in Figure 21. Insignificant differences are observed between the incident wave spectra at different axial positions (less than 1 dB) up to the first radial cut-on frequency, kRo = 3.83. For the reflected wave, the spectral distributions at different axial positions show somewhat greater differences beyond the first circumferential cut-on frequency, kRo = 1.84. This effect is probably due to the generation o f circumferential modes in the reflected wave. Rather similar results were derived for a 2-5 cm diameter conical nozzle attached to a 10 cm diameter duct (these results are not shown here), but the variation with axial position of the reflected sound pressure spectra was greater than for the straight duct. The probable reason for this is that the intensities o f both the incident and reflected pulses undergo greater changes

90

M. SALIKUDDIN

AND

W. H. BROWN

i

i

i

I

I

I

~ -10

--

7

•-2o !

"~.,

.-3o

-'~ t_.,o 0

~,-".. ,;,

" -

, 1

, 2 FrequenCy (kHz}

3

4

Figure 21. Relative sound pressure levels of incident and reflected pulses measured at various axial positions, , x / R o = 6; - - -, x / R o = 12;

x / R n , inside a 10 cm diameter duct for high intensity pulses (about 148 dB); ~ , x / R o = 15, , x / R o = 21; . . . . , x / R o = 36.8.

in the conical nozzle (du e to the area contraction/expansion) and hence distortion due to non-linear effects can become more prominent. In addition, the higher order mode content of the reflected wave is greater for a conical nozzle.

6. PROPAGATION OF HIGH INTENSITY PULSES IN FREE FIELD Since high intensity pulses were used in the tests, it was suspected that the effect o f non-linear propagation might also be present in the free field. In particular, the far field pulses might not follow the inverse square law usually assumed in far field power calculations. To determine whether or not such an effect existed in the free field sound propagation, tests were conducted in which the free field time histories were measured at various distances from the exit of the straight duct at polar angles of 0 ~ 20 ~ and 60 ~ These time histories as shown in Figure 22 have been adjusted for inverse square law decay by adjusting the amplifier gain setting. It can be observed from this figure that the shapes and amplitudes at a particular polar angle are identical for all the free field pulses measured beyond one duct radius from the exit. This is further demonstrated in Figure 23 where sound pressure level spectra o f the free field pulses are plotted after applying the required inverse square law correction. It can be clearly observed that the sound pressure level spectra at a particular polar angle are all identical, at least up to 4 kHz (i.e., the first cut-on frequency for the 10 cm diameter duct), except for x / R o = 1.0. For x / R o = 1-0, however, even according to linear theory, the sound field outside the duct exit would contain a near field component which does not follow the inverse square law.

NON-LINEAR

91

WAVES IN DUCTS AND NOZZLES

O = 0*

O= 2 0 ~

1'

0Sins Time

O= 60 ~

"I

Figure 22. Far field time histories adjusted for inverse square law for a 10 cm diameter duct termination at three polar angles, 8, measured at various polar distances, X/RD, for high intensity pulses (about 148 dB). 20 .

I

.

i

i

.

.

I

~,,20

i

(o) e--o~ 10

10

50

oi (b) 8 = 20*

20 20

10 (c) 8 = 60* =0

0

/

-10

0

' 2

4 Frequency (kHz)

J 6

8

F i g u r e 23. R e l a t i v e levels o f f a r field s o u n d p r e s s u r e s p e c t r a , a d j u s t e d f o r i n v e r s e s q u a r e law, f o r a 10 c m d i a m e t e r d u c t t e r m i n a t i o n at t h r e e p o l a r angles, 0, m e a s u r e d at v a r i o u s p o l a r d i s t a n c e s , x / R o , for h i g h intensity pulses ( a b o u t 148 dB); , x / R o = 1; - - - -, x / R o = 3; - - - - -, x / R o = 6; - - - - , x / R D = 18; - - - - - , x / R o = 30. ( a ) O = 0~ (b) 0 = 20~ (c) 0 = 60 ~

M. SALIKUDDIN AND W. H. BROWN

92

From these results and considerations, it can be concluded that, even though the in-duct pulses showed small effects of non-linear propagation, the free field pulses were effectively free of non-linearity. A possible reason for this is the low intensity o f the free field pulse compared to that of the in-duct pulse, due to the increase in the area associated with the propagating pressure field outside the duct. In the near field region there could be some small non-linear effects but beyond a diameter from the duct exit any such effects are negligible. 7. DISSIPATION OF ACOUSTIC ENERGY AT THE TERMINATION From the results presented in sections 5 and 6 it can be concluded that the non-linear propagation o f a high intensity sound wave (up to about 150 dB), both in the duct itself and outside the duct, in the free field, has only a small effect on the acoustic characteristics o f various duct-nozzle systems. However, it has been demonstrated in section 4 that the sound intensity has a significant effect on the acoustic transmission characteristics of a duct-nozzle termination. Therefore, the non-linearity significantly affecting the acoustic transmission characteristics of the duct-nozzle systems must be occurring in the region of the duct/nozzle termination itself; i.e., the interface between the duct/nozzle system and the free field. Work on the acoustic power dissipation mechanism in the presence of Intensity (dE])

No pulse

151

156

145

147

Figure 24. Effect of pulse intensity on the vortex formation at the exit plane of (a) 10 cm diameter duct and (b) 2.5 cm diameter conical nozzle terminations photographed 20 ms after the pulse propagated out of the exit plane; M~ = 0-0.

NON-LINEAR WAVES IN DUCTS AND NOZZLES

93

flow [24-26] and the work done by Sivian, Bolt et al. and Ingard [27-29], without flow, strongly suggest that the acoustic power dissipation taking place at the termination is due to the conversion of acoustic energy into vortical energy. In this section a systematic study of the dissipative mechanism by visual means, and including a quantitative estimation of energy loss, is described. 7.1. FLOW VISUALIZATION Tests were conducted with a straight duct of 10 cm diameter, two conical nozzles (one with 6.2 cm exit diameter and the other with 2-5 cm exit diameter), and the daisy lobe nozzle with an equivalent exit diameter of 6.2 cm. These tests were conducted to investigate the effects of pulse intensity, nozzle geometry, and flow velocity on whatever vortex rings might be formed at the termination when the acoustic pulses emerged from it. 7.1.1. Flow visualization results in the absence o f mean flow Flow visualization tests in the absence of mean flow were carried out with the vertical set-up described in section 2. To demonstrate the effect of pulse intensity on the vortex formation, two terminations, a 10 cm diameter straight duct and a 2-5 cm diameter conical nozzle attached to the 10 cm diameter flow duct, were chosen. Acoustic pulses of different intensities were used. Figure 24 shows the effect of pulse intensity on the vortex formations at the exit planes of the straight duct and the conical nozzle. These photographs were taken 20 ms after the pulse propagated out of the termination exit. Results in the absence o f the acoustic pulse are also included for comparison. It can be clearly observed that the presence o f an acoustic pulse is the cause o f the formation of the vortex ring. With increasing intensity, one can see a more prominent vortex; also, the vortex position is IriClden| !~lse

(0) Duct

intensity ( ClEf,

(b) Nozzle

'3' '36

8 143

Q.

E 0

147

J

pulse

I'

,ores

"l I"

,ores

"1

Time Figure 25. In-duct time histories for (a) 10cm diameter duct and (b) 2.5 cm diameter conical nozzle terminations at various incident pulse intensities; M1 = 0.0.

94

M. S A L I K U D D I N

A N D W. H. B R O W N

further away from the termination exit plane as the pulse intensity increases. Therefore it can be concluded that a higher intensity pulse generates a stronger vortex ring and the propagation speed of that ring is relatively higher due to its higher particle velocity. The in-duct pulse time histories (the incident pulse and the corresponding reflected pulse from the termination) corresponding to the various pulse intensities are plotted in Figure 25. The incident pulses at a given intensity for both configurations would be effectively identical. However, the reflected pulses are quite different due to the differences in termination geometry. At this stage one might expect that, since the incident pulses for both the terminations are identical (at a fixed intensity), the corresponding intensity (shape and size) of the vortices formed at the termination would be identical. However, this is not the case. The incident pulses shown in Figure 25 were measured in the 10 cm diameter flow duct for both the terminations. For the straight duct, the incident pulse would be effectively the same at the exit, whereas for the conical nozzle the intensity of the propagating incident pulse would increase due to the reduction in cross-sectional area, and hence would be significantly more intense at the exit. The formation o f the vortex is basically controlled by the incident pulse at this final stage. Therefore, the vortex for the nozzle would be stronger than that for tl~e straight duct. This can be seen in Figure 24 where the vortex ring for the nozzle is stronger and moves faster than the corresponding vortex ring for the duct. Intensity (dBI

No pulse

Figure 26. Effectof nozzle geometry(with same equivalent open area) on the vortex formation at the nozzle exits at various pulse intensities photographed 20 ms after the pulse propagated out of the exit plane; Mj = 0.0.

NON-LINEAR

WAVES IN DUCTS AND NOZZLES

95

A second, weaker, votrtex as seen in the optical results for straight duct and conical nozzle terminations in the absence of flow. This second vortex may have been caused by the reflection of the original pulse from the foam pad at the opposite end of the duct which was about a meter from the spark source. Therefore, the reflected pulse arrived at the exit approximately 6 ms after the original pulse. Because the reflected pulse was much weaker than the original pulse, the resultant second vortex generated by the reflection was weaker, and thereby propagated at a slower speed than the original vortex. Therefore, the relative positions of the vortices are not good measures of their relative timing.

Figure 27. Vortex formation at various planes at the exit of a suppressor nozzle for a pulse intensity of 143 d B , p h o t o g r a p h e d 20 m s after the pulse propagated out of the exit plane; M j = 0-0.

Intensity(dB) I

115

109

lOcm

diameter duct

6-2 cm diameter conical nozzle

Suppressor nozzle with on

equivalent

diameter of 6 . 2 cm

2.5 crn diameter conical noz z le

Figure 28. Acoustic field at the exit of various nozzle terminations, created by low intensity acoustic pulses generated from an acoustic driver; M ] = 0.0.

96

M. SALIKUDDIN AND W. H. BROWN

Figure 26 shows the formation o f vortex rings due to acoustic pulses of various intensity for two nozzles of the same exit open area, one being a conical nozzle and the other being a suppressor nozzle. The effect of intensity is quite similar to that observed in Figure 24. However, the vortex ring for the suppressor nozzle is not similar to that for the conical nozzle, in spite of the fact that both the nozzles have the same exit open area. The vortex structures for the suppressor nozzle are dictated by the detailed nozzle geometry, as can be seen from Figure 27. In this figure the vortex structures p h o t o g r a p h e d at two different planes are compared. One plane contains the axis of the duct-nozzle system and covers two lobes (the smoke wire/passing over the center of two lobes) and the other one is an off-axis plane covering several lobes and two tubes (the smoke wire off-centered and passing over two tubes and portions of the lobes). In this figure the vortex structures are quite different at the two nozzle exit planes. The acoustic fields created by low intensity pulses from an acoustic driver and viewed at the exit planes of various nozzle terminations are shown in Figure 28. The vortices formed in this case are very weak. The strength, however, increases with decreasing area of the nozzle. Figure 29 shows the propagation of the vortex rings for a fixed pulse intensity. In the absence of flow, the propagation speed of the vortex is the same as the particle velocity.

Time(ms] I

' f

(0) Duct

(b} Nozzle

I

Nl

20

37

51

Figure 29. Propagation of vortex rings created by a pulse with an intensity of 143 dB for (a) 10 cm diameter duct and (b) 2.5 cm diameter conical nozzle terminations; M~ = 0.0.

NON-LINEAR WAVES IN DUCTS AND NOZZLES

97

The figure shows that, during a given time interval, the vortex ring shed from the conical nozzle propagated much faster than that shed from the straight duct termination. Figure 30 shows the propagation o f vortex rings for the 6.2 cm diameter conical nozzle c o m p a r e d with that for the suppressor nozzle. Even though the vortex structures are different for the two nozzles, the propagation speed seems to be the same at a given intensity. Therefore, the size of the exit open area seems to be the controlling factor for the vortex intensity and travel speed.

Figure 30. Propagation of vortex rings created by a 143 dB pulse for (a) a conical nozzle and (b) a suppressor nozzle with the same equivalent exit open area.

7.1.2. Flow visualization results with mean flow Optical results with m e a n flow were obtained only for very low Mach numbers because it was difficult to generate uniform smoke at high flow velocities. Tests in the presence of mean flow were conducted with the horizontal flow set-up described in section 2. Figure 31 demonstrates the effect of an acoustic pulse propagating out through a straight duct at various flow Mach numbers. The corresponding optical results in the absence of the pulse are also presented here. The propagation speed for the vortex ring, in this case, is the sum of the particle velocity of the pulse and the mean flow velocity. Therefore, the vortex ring formed in the presence of flow has propagated further downstream than that for the no flow condition. Figure 32 is identical to Figure 31 except that the results in this case are obtained for a 6.2 cm diameter conical nozzle.

98

M. SALIKUDDIN

AND

W. H. BROWN

tD O II

O I1

O ii

O II

2

Figure 31. Excitation of the jet by a 143 dB pulse for a 10 cm diameter duct at various Mach numbers, Mj; photographed 6 ms after the pulse propagated out of the duct exit. (a) Without excitation; (b) with excitation.

The effects o f pulse intensity on the formation of vortex rings in the presence of mean flow for a straight duct and a conical nozzle are shown in Figure 33. As observed previously for the no flow condition, the strength of the vortex ring is greater for a pulse with higher intensity, and the vortex ring propagates faster. However, in the presence of flow, the effect of pulse intensity is not as dominant as it is in the absence of flow. This is because the particle velocity o f the pulse is much smaller than the mean flow velocity. Therefore, it is expected that at higher flow speeds the non-linear effects associated with the intensity of the sound pulse itself would be almost negligible [16-18]. No experimental confirmation of this is described in this section since it was not possible t o generate uniform smoke at higher flow velocities with the present smoke technique, but a quantitative demonstration is given in the next section. Figure 34 demonstrates the effect of nozzle exit area on the formation of vortex rings in the presence of flow. In this figure, at a fixed flow condition (Mach n u m b e r Mj = 0.0317) and with a fixed pulse intensity (i.e., 143 dB) the formation of the vortex ring for the 10 cm diameter straight duct is c o m p a r e d with that for the 6-2 cm diameter conical nozzle. As observed previously for the no flow condition, the strength of the vortex ring for the nozzle seems to be greater than t h a t for the straight duct. However, the effect is hot as

N O N - L I N E A R WAVES IN D U C T S A N D N O Z Z L E S

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Intensity (dB)

(a] I0 cm duct

(b) 6"E cm conical nozzle

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Figure 33. Effect of pulse intensity on the vortex formation at the exit of (a) a 10 cm diameter duct at M~ =0-0317 and (b) a 6-2 cm diameter nozzle at Mj =0-0448; photographed 6 ms after the pulse propagated out of the exit plane.

100

M. S A L I K U D D I N A N D W. H. B R O W N

Figure 34. Propagation o f vortex ring in the jet formed by a 143 dB pulse, photographed '6 ms after it propagated out o f the exit plane of (a) a 10 cm diameter duct and (b) a 6-2 cm conical nozzle; Mj = 0.0317.

important as it was in the no-flow condition. The reason discussed above applies here as well. Fore all the results presented in this section, it can be concluded that the acoustic energy transmitted out o f a termination in the form of a pulse is partly converted into vortical energy. This phenomenon is observed with and without mean flow. In the next section, it is shown quantitatively that a power loss occurs in the transmission o f the pulse from in.duct to far field, and that the amount of power loss (or power imbalance) is greater J'or a pulse with higher intensity. 7 . 2 . Q U A N T I T A T I V E E V A L U A T I O N O F P O W E R LOSS

The optical results presented in section 7.1 show the formation of vortex rings when a finite amplitude pulse (or signal) propagates out through the termination. The intensity of the vortex ring increases with increasing pulse intensity. These observations indicate that some portion of the acoustic energy contained in the pulse is converted into vortical energy when the pulse propagates out to the far field. In other words, the acoustic power reaching the far field has to be smaller than the radiated power estimated on the basis of in-duct measurement which does not account for any kind of power loss at the termination. This behavior was distinctly observed in previous investigations [1-6], and also by others [24-26]. In the present study, a systematic investigation was carried out to evaluate the acoustic power loss quantitatively with varying pulse intensity for various terminations at different flow conditions. The quantitative assessment of the power loss was done both experimentally and theoretically, and the results are compared in this section. The acoustic power loss was quantitatively estimated by using the theoretical model developed by Cummings and Eversman [17, 18]. In this model, a simple radiation condition is imposed at the duct end to calculate the reflections and thus the transmitted power. A Kirchhoff type of model with analogous monopole and dipole representations

NON-LINEAR WAVES IN DUCTS AND NOZZLES

101

o f the duct end is used to calculate the radiated field. Non-linear impedance conditions are incorporated in this model. In the model plane wave m o d e p r o p a g a t i o n is assumed in the duct. The incident pulse intensities derived from the experiments were used in the analytical model to estimate the p o w e r loss, for various terminations with and without mean flow. The controlling parameters used here are the intensity o f the pulse, the ratio o f the termination o p e n area to the duct cross-sectional area, and a discharge coefficient (Co). The discharge coefficient used for all the terminations except for the straight duct is 0.61 as justified by C u m m i n g s and E v e r s m a n [17]. In reference [17] a value o f 0.05 was used as the discharge coefficient for the straight duct in the absence o f m e a n flow, and therefore the same value is used in the present study. However, for the duct with mean flow, the discharge coefficient value is f o u n d to be 0.61 on the basis o f g o o d agreement between experiment and prediction. 7.2.1. Effect of acoustic intensity on power loss (a) In the absence o f m e a n flow. Figure 35 shows the effect o f pulse intensity on the power imbalance spectra for the straight duct and the daisy lobe nozzle at M j = 0.0. The experimental p o w e r loss results are c o m p a r e d with the c o r r e s p o n d i n g predicted values and they agree well. The low frequency p o w e r loss increases with the increasing pulse intensity for both the terminations. (b) In the presence o f mean flow. The effect o f pulse intensity on p o w e r imbalance spectra for a 1 0 c m diameter duct at M j = 0 . 1 and 0.2, as s h o w n in Figure 36, is very -lO

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MI=O.I and (b) Mj =0.2, derived experimentally (lines) and by a prediction scheme (symbols).

102

M. SALIKUDDIN AND W. H. BROWN

small. However, at MI = 0-2 the amount o f power loss seems to be less for the lower intensity pulses. This reduction is not entirely due to the effect of intensity. Most likely, at Mj = 0.2, for the straight duct, the signal (far field) was contaminated by the jet mixing noise, which in turn introduced some error into the results mainly in the low intensity cases where the signal to noise ratio was poor. The predicted power loss values agree well with the experimental results for both Mach numbers. They both show that, in the presence of flow, the pulse intensity (and hence the non-linear behavior associated with it) has a negligible effe.ct on the power loss mechanism. Similar comparisons are made to show the effect of pulse intensity on the power loss for a 6.2 cm conical nozzle at M~ = 0. I and Mj = 0.4 (see Figure 37). The conclusions here I

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are similar to those for the straight duct. That is, in the presence of mean flow, the non-linear effects of pulse intensity are negligible. The predicted results h.ere agree very well with the experimental results for Mj = 0.1. For M~ = 0.4, the prediction shows higher power loss than the experiment. 7.2.2. E f f e c t o f n o z z l e e x i t a r e a on p o w e r loss The power loss spectra evaluated at a fixed pulse intensity o f 147 dB and Mj = 0 . 0 for conical nozzles with 2.5 cm and 6.2 cm diameters are c o m p a r e d with those for a 10 cm diameter straight duct in Figure 38. The smaller nozzle exhibits more power loss than the larger nozzle. The predicted results and the optical results behave in the same manner. -10

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NON-LINEAR

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Similar results evaluated for a 6.2 cm diameter conical nozzle are compared with those for a 10cm diameter straight duct for a fixed pulse intensity o f 143 dB at Mj =0.1 in Figure 39. The power loss spectra for both the terminations are quite alike except at the lower frequencies where the nozzle exhibits more power loss than the straight duct. The predicted results agree well with the measurements. lO

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Figure 39. Comparison of power imbalance spectra between a 6.2 cm diameter conical nozzle and a 10 cm diameter duct with incident pulse intensity of 143 dB evaluated experimentally (lines) and by a prediction scheme (symbols) (a) ~,lj = 0-032 and (b) Mj = 0-1; , II, 6.2 cm nozzle; - - -, O, 10 cm duct.

7.2.3. Effect o f nozzle geometry on power loss The measured and predicted power loss spectra for a conical nozzle are compared with those for a daisy lobe nozzle for two ditterent pulse intensities at Mj = 0.0 in Figure 40. The open exit area for both the nozzles was 30.2 sq cm (i.e., a diameter of 6-2 cm), but the shape of the daisy lobe nozzle was quite different from the conical nozzle. The 10

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104

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7.2.4. Effect o f mean flow on power loss The measured power loss spectra for the 10 cm diameter duct and the 6.2 cm diameter nozzle at various Mach numbers are plotted in Figure 41. The effect of Mach n u m b e r on power loss seems to be very small. For the straight duct, the power loss at lower frequencies increases slightly with Mach number. However, at higher frequencies there is no apparent difference in the power loss levels. This is consistent with the prediction which shows increasing p o w e r loss with Mach number, especially at lower frequencies. -10

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Figure 41. Effect of flow on the power imbalance spectra evaluated experimentally(lines) and by a prediction scheme (symbols) for (a) a 10 cm diameter duct ( I , ----,MI = 0-0; O, - - -, M~= 0-032; ,t, - - -, M~= 0-045; +, , M~ =0-055; @, ~ . - M~ =0.1; x , - - - - - , Mj = 0-2) and (b) a 6.2cm diameter conical nozzle ( I , ~ , M~ =0.0; ~, ---, M~--- 0.032; ~k, - - - , Mj =0.045; +, ~ - - , Mj =0.055; e , - - . - - , Mj =0.2; x, ~ - ~ , M, = 0.4). The effect of Mach number on acoustic power loss is more distinct for the conical nozzle. The low frequency power loss increases with increasing Mach number. Results of the prediction program show the same trend. The agreement between experiment and prediction is very good except at higher Mach numbers where the prediction shows more power loss. 8. CONCLUSIONS The important observations are summarized in this section. All conclusions, of course, apply strictly only to the range o f sound pulse intensities (up to about 160 dB) and duct/nozzle configurations used in these studies. (1) Compression waves (incident pulses) propagate faster than the expansion waves (reflected pulses). (2) Due to the non-linear behavior of high intensity pulses, the reflection coefficient amplitude for an open termination decreases with increasing pulse intensity. (3) Non-linear propagation leads to a slight change in the pulse shape in the direction of propagation within the duct. Therefore, the acoustic properties measured at different axial locations in the duct are slightly different. (4) Far field sound propagation remains effectively linear even if the incident (induct) pulse is slightly non-linear in nature. (5) Low frequency power loss in the absence of mean flow increases with sound intensity due to the non-linear behavior o f sound transmission through the duct/nozzle

NON-LINEAR WAVES IN DUCTS AND NOZZLES

105

system termination (i.e., vortex generation). This p h e n o m e n o n remains, though less significantly, even for lower intensity pressure waves (down to about 100 dB in these studies). 9(6) Acoustic properties of absorbing materials are sensitive to the intensity of the sound field. However, when the intensity drops below a non-linear threshold level the acousiic properties of absorbing materials remain unaffected. (7) The presence of an acoustic pulse (signal) causes the formation of a vortex ring at the termination exit. (8) The strength and the propagation speed o f t h e vortex ring generated at a termination increases with the intensity of the incident pulse. (9) The vortex structure is dependent on the termination geometry. (10) For a fixed pulse intensity, the amount of power absorption (or power loss) appears to be controlled primarily by the open area at the exit and not so much by the shape of the termination. (11) Power absorption at a duct termination increases with increasing mean flow Mach number. (12) The non-linearity associated with the amplitude of a high intensity pulse becomes considerably less important in the presence o f mean flow, insofar as the acoustic transmission charalcteristics o f the duct/nozzle system are concerned. ACKNOWLEDGMENTS The authors are particularly thankful to Dr H. K. Tanna and Dr H. E. Plumblee, Jr. for their deep involvement in this study. The authors also wish to acknowledge the contributions of Dr K. K. Ahuja who helped in planning and conducting some Of the experiments. REFERENCES l. P. D. DEAN, M. SALIKUDDIN,K. K. AHUJA, H. E. PLLIMBLEE,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. SALIKUDDIN and H. E, PLUMBLEE, JR. 1979 Lockheed-Georgia Company Engineering Report LG79EROI51. Effect of blow-in doors and r plugs on the acoustic radiation characteristics of modal supersonic inlets--an experimental study. 3. K. K. AHUJA, M. SALIKUDD1N, R. H. BURRIN and H. E. PLUMBLEE,JR. 1980 NASA CR-165144. 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 6 th Aeroacoustie Conference AIAA Paper 80-0991. Low frequency sound absorption of orifice plates, perforated plates, and nozzles. 5. K. K. AHUJA, M. SALIKUDDIN and H. E. PLUMBLEE,JR !980 American Institute of Aeronautics and Astronautics 6th Aeroacoustic Conference AIAA Paper 80-1027. Characteristics of internal and jet noise radiation from a multi-lobe, multi-tube suppressor nozzle tested statically and under flight conditions. 6. M. SALIKUDDIN and H. E. PLUMBLEE,JR. 1980 Lockheed-Georgia Company Engineering Report LG8OER0204. Internal noise radiation characteristics of baffles, nozzles, orifice plates and perforated plates used as duct terminations. 7. M. SALIKUDDIN, P. D. DEAN, H. E. PLUMBLEE, JR., and K. K. AHUJA 1980 Journal of Sound and Vibration 70, 487-501. An impulse test technique with application to acoustic measurements. 8. Y. WATANABE and Y. UP,ABE 1978 Acoustical Society of America and Acoustical Society of Japan, Joint Meeting, Honolulu, Hawaii. Propagation of high intensity impulsive sound in air generated by the wire explosion. 9. T. NAKAMURA,A. NAKAMURA and R. TAKEUCHI 1977 Acustica 38, 331-333. Simulation for nonlinear propagation of finite amplitude sound wave through a circular pipe.

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10. T. KAMAKURA and K. IKEGAYA 1978 Acoustical Society of America and Acoustical Society of Japan Joint Meeting, Honolulu, Hawaii. Propagation of spherical acoustic noise of finite amplitude. 11. T. NAKAMURA, A. NAKAMURA and R. TAKEUCHI 1978 Acoustical Society of America and Acoustical Society of Japan Joint Meeting, Honolulu, Hawaii. Measurement of nonlinear reflection of N wave at open end of a circular pipe. 12. A. NAKAMURA and R. TAKEUCHI 1972 Acustica 26, 42-50. Reflection and transmission of an acoustic shock wave at a boundary. 13. R. T. SMITH, L. BJIDRN~ and R. W. B. STEPHENS 1978 Acustica 39, 124-129. Weak shock propagation in liquid filled tubes. 14. L. BJIDRNO 1979 A G A R D Report No. 686. Special Course on Acoustic Wave Propagation. Finite amplitude wave propagation. 15. M. SALIKUDDIN and P. MUNGUR 1983 Journal of Sound and Vibration 86, 497-522. Acoustic radiation impedance of duct-nozzle system. 16. M. SALIKUDDIN and K. K. AHUJA 1983 Journal of Sound and Vibration 91,479-502. Acoustic power dissipation on radiation through duct terminations-experiments. (Also presented at 1981 7 th Aeroacoustics Conference AIAA Paper 81-1975.) 17. A. CUMMINGS and W. EVERSMAN 1980 Lockheed-Georgia Engineering Report LG80ERO164. 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. 18. A. CUMMINGS and W. EVERSMAN 1983 Journal of Sound and Vibration 91, 503-518. High amplitude acoustic transmissin through duct terminations: theory. (A portion of this paper was presented at the 7th Aeroacoustics Conference 1981 AIAA Paper 81-1979, as Acoustic power dissipation on radiation through duct terminations-theory.) 19. M. SALIKUDDIN, W. H. BROWN, R. RAMAKRISHNAN and H. K. TANNA 1982 NASA CR-3656. Refinement and application of acoustic impulse technique to study nozzle transmission characteristics. 20. M. SALIKUDDiN, K. K. AHUJA and W. H. BROWN 1984 Journal of Sound and Vibration 94, 33-61. An improved impulse method for studies of acoustic transmission i n flow ducts with use of signal synthesis and averaging of acoustic pulses. (Also 1981 M. NALIKUDD1N,R. RAMAKRISHNAN, K. K. AHUJA and W. H. BROWN AIAA Paper 81-1978, 7th Aeroacoustics Conference. A unique method to study acoustic transmission through ducts using signal synthesis and averaging of acoustic pulses.) 21. H. LEVINE and J. SCHWINGER 1948 Physical Review 73, 383-406. On the radiation of sound from an unflanged circular pipe. 22. D. BECHERT, U. MICHEL and E. PFIZENMAIER 1977 American lnstitue of Aeronautics and Astronautics 4th Aeroacoustic Conference AIAA Paper 77-1278. Experiments on the transmission of sound through jets. 23, J. Y. CHUNG and D. A. BLASER 1980 Journal of the AcousticaI Society of America 60, 1570-1577. Transfer function method of measuring acoustic intensity in a duct system with flow. 24. D. W. BECHERT 1980 Journal of Sound and Vibration 70, 389-405. Sound absorption caused by vorticity shedding demonstrated with a jet flow. 25. M. S. HOWE 1980 Journal of Sound and Vibration 70, 407-411. The dissipation of sound at an edge. 26. M. S. HOWE 1979 Journal of Fluid Mechanics 91,209-229. Attenuation ofsound in a low Mach number nozzle flow. 27. L. J. SIVIAN 1935 Journal of the Acoustical Society of America 7, 94-i01. Acoustic impedance of small orifices. 28. 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. 29. U. INGARD 1953 Journal of the Acoustical Society of America 25, 1037-1061. On the theory and design of acoustic resonators.