An experimental and theoretical investigation of the propagation of sound waves through a turbulent boundary layer

Journal ofSound

and Vibration (1988) 127(l),

AN EXPERIMENTAL OF THE

AND

PROPAGATION A TURBULENT M.

SALIKUDDIN,

91-121

THEORETICAL OF SOUND BOUNDARY

C. K. W. TAMS AND

INVESTIGATION

WAVES

THROUGH

LAYER? R. H. BURRIN

Lockhead Aeronautical Systems Company, Georgia Division, Marietta, Georgia 30063, U.S.A. (Received 27 June 1987, and in revised form 17 May 1988)

As the sound waves from aircraft turbopropellers (“propfans”) pass through the turbulent boundary layer on the surface of the fuselage, the weak shock waves associated with the sound pulses and the higher order harmonics are filtered out by the velocity and density gradients as well as by the turbulence of the boundary layer. To assess these effects and to provide a prediction scheme, a theoretical model of the transmission of propfan noise through a turbulent boundary layer was developed. In this model, the effect of refraction due to velocity and density gradients of the turbulent boundary layer has been taken into account. Unlike all the available analytical models, the effect of turbulence damping has been incorporated by adding appropriate eddy viscosity terms in the equations of motion. An experiment was conducted to determine the attenuation of sound crossing a turbulent boundary layer over a flat plate attached on one side of a high speed square-duct jet at Mach numbers as high as 0.8. A point sound source with its opening located inside the potential core of the jet was used to generate sound pulses. The acoustic amplitudes were measured by two microphones, one located outside the boundary layer, and the other located under it, flush with the flat plate. Substantial transmission losses were observed, particularly for high frequency noise in the forward directions. These results are compared with the predicted transmission loss to validate the boundary layer interaction theory developed for a point sound source. Reasonably good agreement between the experiment and the theory is achieved. The theoretical model of the transmission of propfan noise through a turbulent boundary layer which includes the effect of propfan rotation is used to predict the transmission loss for JetStar cruise condition. These predicted results are compared with the JetStar flight data measured on the fuselage surface. Reasonably good agreement between the flight data and the prediction is achieved.

1. INTRODUCTION

Noise measurements carried out under the JetStar Flight Program [l] revealed that the turbulent boundary layer on the fuselage of an aircraft had an important shielding effect against sound waves generated by advanced turbopropellers (“propfans”). Because the flight experiments were not designed specifically to study this shielding effect, the data obtained could not be quantified in a straightforward manner. Nor could the measured data be used easily to provide a basic understanding of the observed phenomenon. To estimate correctly the noise levels at the fuselage surface, the effects of the fuselage t This work was funded by Lockheed’s Independent Research and Development Program. A portion of this paper was presented at the AIAA 10th Aeroacoustics Conference (AIAA-86-1968). 1986; this paper also will be published as a part of a larger article “High-Speed Turbopropeller Noise”, in the Encyclopediaof Fluid Mechanics, Volume 8. $ Present address: Department of Mathematics,

0022460X/88/220091

+31 $03.00/O

Florida State University, Tallahassee, Florida. 91 @ 1988 Academic Press Limited

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boundary layer interaction with the radiated noise from the turboprops need to be understood. A number of theoretical investigations have been carried out to analyze and understand the problem of sound transmission through boundary layers. In earlier work, McAninch [2] and Hanson [3] used highly simplified physical models aiming primarily at obtaining a qualitative explanation and understanding of the phenomenon. Subsequently, a much improved model was adopted by McAninch and Rawls [4]. Most recently, Hanson and Magliozzi [5] employed a model which was very comprehensive in scope. By means of this model, the effect of boundary layer refraction and fuselage scattering on the incident turbopropeller noise were studied. However, in all these models [2-51 the turbulence in the boundary layer was never taken into consideration. As the sound waves from turbopropellers pass through the turbulent boundary layer on the surface of the fuselage, the weak shock waves associated with the sound pulses and the higher order harmonics are filtered out by the velocity and density gradients as well as by the turbulence of the boundary layer. To assess these effects and to provide a prediction scheme, a theoretical model of the transmission of propfan noise through a turbulent boundary layer was developed. In this model, the effect of refraction due to velocity and density gradients of the turbulent boundary layer has been taken into account. In contrast to previously available analytical models, the effect of turbulent damping has been incorporated by adding appropriate eddy viscosity terms in the equations of motion. A computer program capable of predicting the acoustic transmission loss through the turbulent boundary layer has been developed. To evaluate the effect of turbulent damping alone, a companion computer program using an inviscid model has also been developed. By comparing the predicted values of these two programs, it is now possible to assess the effect of turbulent damping quantitatively for different frequencies, Mach numbers, and angles of incidence. To validate the boundary layer interaction theory, an experiment was conducted to determine the attenuation of sound by a tubulent boundary layer. A flat plate was, in essence, an extension of one of the walls of a square duct, through which exhausted a high speed jet. A point sound source was introduced in the potential core of the square jet to generate sound pulses. These acoustic signals were measured by two microphones, one mounted flush on the flat plate and the other one located outside the turbulent boundary layer as shown in Figure 1. The acoustic energy loss due to the transmission through the boundary layer was evaluated from these measurements. The boundary-layer profile and the turbulent intensity were also measured by a hot-wire anemometer.

Flow

> > > >

Smulated propeller n06e

-

??

* > f

u -

Sound

waves

-

Repesentative fuseloge surface

Figure

1. Schematic

for “boundary

layer-propfan

noise”,

interaction

experiments.

SOUND

PROPAGATION

THROUGH

BOUNDARY

LAYER

93

The acoustic signal measured above the boundary layer and the corresponding boundary layer profile were used to predict the acoustic transmission losses due to the boundary layer interaction. Thus the boundary layer interaction theory was validated by comparing the transmission losses derived from measurement with those predicted by the theory. A unique impulse technique developed at the Lockheed-Georgia Company [6,7] was used in the acoustic measurements. The acoustic signal used in this technique was a sharp pulse that contained information over a wide frequency band, (A delta function spectral density, which is an infinitely narrow pulse after inverse Fourier transformation, contains all frequencies.) In addition, using this technique makes it possible to separate the direct pulse generated by the sound source from its reflections from the surroundings simply by ensuring that the width of the pulse is such that the reflected pulses are adequately separated in time from the direct pulse and from each other. In the presence of high velocity flow, however, it becomes extremely difficult to isolate the pulses from possibly dominant background jet noise. Signal averaging has been applied to the acoustic signals to overcome this problem [7]. If a sufficient number of individual records separated by several seconds are averaged, the stochastic contribution from the jet flow-associated fluctuations are averaged out, thus enabling recovery of the pulse time history. It should also be pointed out that if continuous sinusoidal signals were used, it would be very difficult to separate the direct signals from the reflected ones. The impulse and editing technique overcomes the difficulties associated with continuous signals. In this paper the results ‘of the controlled laboratory experiment accompanied by the theoretical analysis of the propagation of sound waves through a turbulent boundary layer are reported. Also, the transmission losses for JetStar cruise condition were predicted and compared with the data measured on the fuselage in flight. The experimental set-up, test procedure, and the data acquisition and analysis procedures are described in section 2. The experimental results of these investigations are preiented in section 3. The theoretical investigation including the formulation of the prediction model and the results are described in section 4. The results in this section include the predictions for the experimental configuration and for the JetStar cruise condition and their comparison with experimental data and JetStar flight data, respectively. Finally, the important conclusions are discussed in section 5.

2. EXPERIMENTAL

SET-UP

AND

PROCEDURE

2.1. EXPERIMENTAL FACILITY Experiments to determine the attenuation of sound by a turbulent boundary layer using a wall jet were conducted at the Lockheed-Georgia Compressible Flow Calibration Facility (CFCF). The calibration facility uses dry compressed air at a stagnation pressure of 300 psig. Three large tanks with a combined capacity of 13 000 cubic feet act as reservoirs for the facility. An on-line compressor maintains the pressure in the tanks. For this study, an extra plenum and a miniature high speed open jet wind tunnel facility were added. It consisted of a series of foam and screens and a 25: 1 contraction. The flow exited from the contraction through a four inch by four inch square jet onto a flat plate, forming a wall jet, as shown in Figure 2. The flow rate was controlled by means of a small pressure regulator which in turn controlled a large 10 inch regulator in the pipe leading up to a wind tunnel. Using this arrangement, fine control of velocities from Mach O-1 to Mach 1 was attainable. A traverse system, mounted in the jet stream, was used to traverse miniature probes for aerodynamic measurements.

94

M. SALIKUDDIN

ET

Sound

AL.

source

Microphone

above the

rmcrophone

MIcrophone

above

t Flush-walled rmcrophone Test sectton

Figure

2. The test set-up

for “boundary

Plenum

layer-propfan

noise”,

section

interaction

experiment>

An Apple IIc personal computer with an ISAAC A/D and D/A converter was used to drive the traverse remotely. The traverse has a step resolution of l/4000 inch. The Apple/ISAAC system was also used to acquire aerodynamic data, which were stored on a floppy disk during the experiment and analyzed later. Velocities were measured using modified DISA type 55P15 subminiature hot-wire probes in conjunction with a DISA 55M anemometer system. The modification to these probes was a simple epoxy bridge mounted between the prongs of the probe, approximately l/ 10 inch downstream of the wire. The bridge was found to increase the life of the probe markedly at high speeds by suppressing the prong vibration. Two pressure taps, one located at the entrance to the contraction, the other at the exit, were used to monitor the Mach number during the course of the experiments. Calibration of the hot wires was accomplished by controlling the flow velocity using the pressure regulator and monitoring the output voltage of the hot wire. At high speeds, the hot wire responds to changes in pu, p and u being the density and velocity of the air [8]. A calibration of volts squared versus the square root of pu was found to approximate a straight line. A typical velocity profile of the square jet, acquired at Mach number

SOUND

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BOUNDARY

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95

M..,,=O-5, shown in Figure 3, indicates that the potential core extends up to about 3.5 inches above the flat plate at about five inches from the jet exit. A point sound source fabricated for this purpose was introduced in the potential core of the square jet by keeping the source exit at 3.2 inches above the flat plate. The vertical position of the sound source was determined based on the potential core height of the square jet shown in Figure 3. This source consisted of an acoustic driver connected to a 1.125 inch diameter long tube which, through a contraction, finally terminated to a 0.25 inch diameter exit. The acoustic driver was mounted on a traversable support, and the support was placed on a traverse rail. Therefore, the point source could be traversed along the flow direction in the potential core maintaining a fixed distance from the flat plate (see Figure 2). To measure the acoustic signals generated by the point source, two microphones were mounted at a distance of five inches from the duct exit, one flush on the flat plate and the other one inch above the flat plate facing the flow (which would be outside the turbulent boundary layer). 2.2.

EXPERIMENTAL

PROCEUDRE

Instead of using a single frequency, or for that matter broadband noise, sharp pulses of extremely short duration were used. This method enabled the direct pulses to be separated from those reflected from the surrounding test set-up surfaces. By suitably editing out the undesired parts of the time history and Fourier transforming the direct signal, the acoustic pressures at a wide range of frequencies were determined. Phase-locked signal averaging of a suitable number of samples was used to educe the pulse buried within the dominant background jet noise. The position of above-the-boundary-layer microphone was very critical, since this microphone measured the direct sound coming from the source and its reflection from the flat plate. An optimum spacing of one inch, between this microphone and the flat plate, was established by conducting a few preliminary tests to find where the direct pulse was clearly separated from its reflection. The basic operational principle was to excite the acoustic driver with a train of narrow rectangular pulses from a pulse generator. A dual channel pulse generator, which had the capability not only to vary the period, pulse width, and pulse intensity, but also to introduce any desired delay between its two channels, A and B, was used in this case. While the output of channel A was used to drive the source, the appropriately delayed output of channel B was used as the trigger for the signal averaging process. The output ,

Figure

3. Mean velocity

,

,

,

,

,

,

,

,

,

profile above the flat plate, at x = 5 inches

from the jet exit; A& = 0.5.

96

M. SALIKUDDIN

ET

AL.

of the source as measured by the microphones mounted flush on the flat plate wall and outside the turbulent boundary layer and the trigger signal of channel B were recorded on a multi-channel tape recorder for subsequent data analysis. The data analysis procedure involved signal averaging and Fourier transformation of the averaged signal using an appropriate FFT analyzer. The long tube of 1.125 inch diameter connected to the acoustic driver and contracted to an exit of 0.25 inch diameter was used to allow the output pulse from the driver to propagate in the tube for as long a period of time as possible. This process of high intensity pulse propagation in a hard-walled tube introduces non-linear propagation effects on the pulse, and so the resultant acoustic pulse becomes steeper compared to that coming out of the driver. The objective of this process was to generate pulses with high frequency content [9], a requirement for the present experiments, using a low frequency acoustic driver. The extent of the frequency-range increase depends on the intensity of the pulse, the propagation time in the tube, and the size of the tube. The higher the pulse intensity, the longer the propagation time; and the smaller the tube diameter, the greater the increase. However, if the tube diameter is reduced excessively, even though the frequency content of the pulse will increase, its acoustic energy will decrease due to dissipation. The application of non-linear propagation to generate high frequency signals from low frequency acoustic drivers was developed at LockheedGeorgia under an IR&D project, and a patent application is filed with the United States Patent Office. During the experiments, the acoustic signals were measured by microphones mounted flush on the flat plate wall and outside the turbulent boundary layer. The boundary layer profile and the turbulence intensity were measured by a hot-wire anemometer while the computer-controlled traverse mechanism positioned the probe. During hot-wire measurements, the microphone outside the boundary layer was removed to avoid boundary layer distortion (see Figure 4). The velocity profile covering the entire potential core was also measured to determine the height of the potential core above the flat plate so that the acoustic source could always be placed inside the potential core. During acoustic measurements, the hot-wire traverse system was removed from the facility to avoid acoustic interference (see Figure 5). The acoustic measurements were made first with the microphone above the boundary layer. This microphone was then removed and the flush-wall

Figure 4. The test set-up for aerodynamic

measurements.

SOUND

Figure

PROPAGATION

5. The test set-up

for acoustic

THROUGH

measurements

BOUNDARY

in the presence

LAYER

97

of the Aat plate.

microphone measurements were made immediately thereafter to avoid acoustic interference and boundary layer distortion. Measurements with both the microphones were made at different angular positions with respect to the acoustic source by traversing the source along the direction of the flow. At some angular positions and at higher flow Mach numbers, the direct pulse, measured by the microphone one inch above the flat plate, was not completely separated from its reflection by broadening of the pulses. In this situation, “above the boundary layer” measurements were taken with the flat plate removed (see Figure 6). 2.3.

DATA

ANALYSIS

measurements made by flush-wall and above the boundary layer microThe acoustic phones at various angles and Mach numbers were analyzed on an FFT analyzer by applying the signal averaging procedure as outlined at the beginning of this section. A typical result indicating the effect of signal averaging is shown in Figure 7. The time

Figure 6. The test set-up

for acoustic

measurements

without

the flat plate.

98

M. SALIKUDDIN

A-1.5

Figure boundary

ET AL.

msF Time

7. Illustration of signal recovery by time domain averaging; acoustic layer; M, = 0.5, x = 90”. (a) Before averaging; (b) after averaging.

signal

measured

above

the

domain signals after averaging were Fourier transformed to derive the spectral distribution. Various corrections should be applied to these spectra to derive the absolute sound pressure levels. Distance corrections were required for both the measurements since their locations with respect to the acoustic source were different. The flush-wall measurements need corrections to account for wall reflections. Above the boundary layer, measurements need free-field corrections for the nose cone mounted on that microphone. The free-field corrections in this case were dependent on the angle of sound emanating from the acoustic source with respect to the flow direction. The available information regarding these corrections is not adequate. To avoid complications and the danger of inaccuracies involved in the limited correction data available, the measured spectra at a given angle and flow Mach number were normalized with the corresponding data derived for the no-flow case. Thus, the derived relative sound pressure level spectra were the measure of sound at the surface of the flat plate and above the boundary layer. The attenuation of sound due to the presence of boundary layer was the difference between these two spectra for a given angle and Mach number. It should be noted that a far-field sound propagation is assumed in this process of normalization. The shortest distance between the source and the measurement location was 2.2 inches for above the boundary layer microphone for X = 90”. Based on the wavelength and this distance, the assumption of far-field sound propagation is valid for frequencies above 8-10 lcHz.

3. EXPERIMENTAL

RESULTS

The boundary layer velocity prohles just above the flat plate at various Mach numbers were measured at the flush-wall microphone location, about five inches from the jet exit, by using a hot-wire anemometer. Figure 8 shows the boundary layer profiles for A& = O-5

SOUND

PROPAGATION

r

THROUGH I

I

I

BOUNDARY

LAYER

99

I

0.6z _ g p 0.4-

Figure

8. Mean velocity

profiles just above the flat plate, at x = 5 inches from the jet exit.

and 0.8. These results indicated that relatively thick turbulent boundary layers of approximately 0.5 inch existed at five inches from the jet exit for all the Mach numbers between 0.3 and O-8. This information was necessary to scale the frequency of a full-scale turbopropeller noise for the experimental configuration. The objective of the acoustic measurements in this program was to estimate the transmission loss as sound, from a simulated turbopropeller, passes through the turbulent boundary layer on a simulated fuselage. Based on the dimensions of the JetStar fuselage, a turbulent boundary layer of four to five inches thickness would be developed on the fuselage in the propeller plane at cruise. However, in the laboratory experiment it was not possible to create such a thick boundary layer at higher Mach numbers. In the present study, the boundary layer thickness achieved for higher Mach numbers was approximately O-5 inch. Therefore, to maintain a constant ratio of acoustic wavelength to boundary layer thickness, the frequency of the simulated propfan noise had to be raised by a factor of 8-10 times the full-scale frequency. Since the blade passage frequency of the SR-3, eight-bladed propfan used in JetStar was of the order of 1 kHz, the comparable frequency for the simulated sound must be 8-10 kHz. Therefore, to evaluate the transmission losses for the blade passage frequency and at least its first three harmonics, the simulated sound must contain frequencies between 10 and 40 kHz. On the basis of this criterion, the acoustic source was built to generate pulses containing frequencies as high as 40 kHz. To evaluate the acoustic transmission loss through the turbulent boundary layer, acoustic measurements only in the presence of flow are required. However, as discussed earlier, the relative sound pressure levels with respect to the no-flow condition were used for transmission loss evaluation. Therefore, acoustic measurements were also made at various angles, x, for the no-flow condition. Typical time histories, measured above and below the boundary layer, for the no-flow condition at various angles, x, are shown in Figure 9. Clearly, the reflected pulses from the plate appear after the direct pulses when measured above the boundary layer. However, the signals measured on the wall surface are single pulses. The magnitude of the pulses at either of the measurement locations are different for different angles. This is due to the varying distances between the measurement point and the sound source for different measurement angles, x. By suitably editing the unwanted portions of the time history, and Fourier transforming the direct signal, the acoustic pressures at a wide range of frequencies were determined. Typical spectral distributions of sound pressure levels above and below the boundary layer at A4, = 0 for various angles are shown in Figure 10. The spectral levels, at either

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ms-b

AL.

I.5 msTime

Figure 9. Typical M,=O.O.

time domain

signals

at various

angles,

Frequency

x, measured

above and under

the boundary

layer;

(kHz)

Figure 10. Typical SPL spectra at various angles, x, measured (a) above the boundary layer and (b) on the surface of the flat plate; M, = 0.0; -, x = 70”; - - -, x = 90”; -A_, x = 100”; _ _ _ _ _, x = 1100; a_ _ _, x = 120”.

SOUND

PROPAGATION

THROUGH

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101

of the measurement locations, first increase with x up to x = 90” and then start to decrease. This is similar to the behavior observed with the time domain signals in Figure 9. It should be noted that the sound pressure levels at the above boundary layer microphone should be higher than those on the wall surface for a given x at M, = 0 due to different propagation path lengths. However, as seen in Figure 10, the sound pressure levels on the wall are slightly higher than those above the boundary layer location. This is due to the doubling effect of the wall, which increases the SPL’s by 6 dB when measured on a rigid surface. This increase due to the wall effect was higher than the decrease of about 3.25 dB due to the position of the two microphones. The shapes of the sound pressure level spectra, at a given x, measured at either of the microphone locations, respectively, should be identical. However, as observed in Figure 10, the above-boundary-layer sound pressure levels drop faster with increasing frequency than those at the wall. This is due to the different responses of the microphones. A nose cone was mounted on the above-boundary-layer microphone as opposed to a grid for the wall microphone. Acoustic measurements were made at various flow Mach numbers, M,. Typical phase averaged time histories, measured above and below the boundary layer, at M, = 0.5, are shown in Figure 11. The time histories, shown in Figure 11, indicate that the pulses are relatively wider than in the no-flow situation (see Figure 9) due to the convection effects. In addition, the broadening of the pulses increases with increasing x. Due to this behavior at higher flow conditions and at larger values of x, the high frequency content of the pulses is reduced tremendously. Therefore, accurate estimation of spectral levels at these situations was not possible. Spectral distributions of sound pressure levels above and below the boundary layer locations for Mm=005 are shown in Figure 12. The decrease of high frequency energy

Above boundary

the layer

On the surface of the flot plate

)C-----l~5ms~~1~5ms~ Time Figure 11. Typical M,=O.S.

time domain signals at various angles, x, measured above and under the boundary

layer;

102

M.

75 0

ET AL.

SALIKUDDIN

I IO

I 20

Frequency

I 30

: 40

(kliz)

Figure 12. Typical SPL spectra at various angles, x, measured surface of the flat plate; M, = 0.5; -, x = 70”; - - -, x = 90”; -

(a) above the boundary layer and (b) on the -. x = 100”; - - - - -, ,y= 110”; - - -, x = 120”.

with increasing flow and x, as observed from the time history data, is clearly seen in this figure. By using the spectral distributions of sound pressure levels at a given Mach number and the corresponding levels at M, = 0, the relative sound pressure level spectra were derived for above and below the boundary layer for various angles x. The relative sound pressure level spectra in the surface of the plate are compared with those above the boundary layer for Mach numbers 0.3, 0.5, and O-8 in Figures 13-15, respectively, to indicate acoustic transmission losses at various angles, x. As observed from these figures, the acoustic transmission losses increase with increasing frequency, angle x, and Mach number. It was not possible in the present study to evaluate transmission losses at higher angles for higher flow conditions due to the strong convection effect. In fact, for M, = 0.8 (see Figure 15) even for angles x = 90” and loo”, results were obtained only up to about 30 kHz. The effect of flow Mach number is further demonstrated in Figure 16 by comparing the relative sound pressure level spectra on the surface of the plate with those above the boundary layer at a fixed angle, x = 1 lo”, for various Mm. Clearly, the acoustic transmission loss increases with M,. It should be noted that the spectral distributions of sound pressure levels measured in the presence of flow (see Figure 12) show some amount of oscillatory behavior. This behavior is carried into the relative sound pressure level spectra (see Figures 13-16) which is the difference of the spectral levels measured with and without the presence of flow. When the transmission loss spectra are evaluated, the oscillations become more prominent since the peaks and valleys of relative sound pressure level spectra at the wall do not necessarily coincide with those for the spectra above the boundary layer. These are spurious oscillations due to the residual flow noise contamination. Complete elimination of background flow noise is possible by increasing the number of time domain averaging substantially, but that is not possible in this case because of experimental and hardware

PROPAGATION

THROUGH

Frequency

Figure 13. Comparison layer; M, = 0.3.

BOUNDARY

LAYER

103

(kHz)

of relative SPL spectra at various angles, x, measured above and under the boundary

limitations. However, the effects of flow noise contamination can be overcome by smoothing the oscillatory spectrum (this is demonstrated in the Appendix). Therefore, the relative sound pressure level spectra are smoothed out by drawing mean lines through the actual oscillatory spectra. Using these spectra, the transmission loss results are evaluated and summarized in Figures 17 and 18.

Sound ottenuo by the boundary I

Frequency

Figure 14. Comparison layer; M, = 0.5.

CkHz)

of relative SPL spectra at various angles, x, measured above and under the’boundary

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ET AL.

Sound

Frequency

Figure 15. Comparison layer; M, = 0%

of relative

SPL spectra

-101

101

by the I

0

CkHz)

at various

angles, x, measured

above and under the boundary

I

I

I

I

I

I

I

I

boundary I

IO

layer 20 Frequency

Figure 16. Comparison of relative the boundary layer; x = 110”.

attenuation

SPL spectra

at various

30

40

(kHz)

math

numbers,

M,,

measured

above and under

SOUND

PROPAGATION

THROUGH

Measurement

angle,

BOUNDARY

LAYER

105

x (degrees)

Figure 17. Measured boundary layer transmission loss directivities at various flow Mach numbers, M,; _ _ ---,M,=O.3;_0_ _-,M,=O.4;3_,M,=O.5;_11_ _-,&=0.6; 0.7; --A--, Mm= 0.8. (a) 10 kHz; (b) 20 kHz; (c) 30 kHz.

--A--,M_,=0.2;-•--•-*----, M, =

Figure 17 shows the transmission loss directivities evaluated for different Mach numbers (Mm) at three frequencies. At each frequency the amount of transmission loss increases with x and M,. For lower values of x, the effect of boundary layer interaction seems to increase the sound pressure level slightly. This increase varied between 0 and 2 dB and data are not plotted in this figure to avoid overcrowding. To emphasize the fact that the boundary layer transmission loss increases with increasing x, Mm and frequency, the experimental data are replotted in Figure 18 with respect to Mach number (Moo) for different frequencies at three measurement angles (x). 4. THEORETICAL

INVESTIGATION

A number of theoretical investigations have been carried out to analyze and understand the problem of sound transmission through boundary layers [2-51. In the model developed by Hanson and Magliozzi [5], the propfan was represented by a rotating source. Unlike in previous works, the cylindrical geometry of the aircraft fuselage was fully taken into account. A realistic turbulent boundary layer velocity profile was also included in the calculation. The acoustic disturbances were assumed to be inviscid and time periodic. By means of this model Hanson and Magliozzi were able to study the effect of boundary layer refraction and fuselage scattering on the incident sound waves generated by an advanced turbopropeller. However, in all these models [2-51 the turbulence in the boundary layer was never taken into consideration.

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M. SALIKUDDIN

0

0.2 Mach

ET AL.

04

06

38

number, Mm

Figure 18. Variation of measured boundary layer transmission at various frequencies; --O--, 10 kHz; - ?? -, 20 kHz; -A- -,

loss with respect to flow Mach number A&, 30 kHz. (a) x = 100”; (b) x = 110”; (c) x = 120”.

As the sound waves from turbopropellers pass through the turbulent boundary layer the surface of the fuselage, the weak shock waves associated with the sound pulses and the higher order harmonics are filtered out by the velocity and density gradients as well as by the turbulence of the boundary layer. To assess these effects and to provide a prediction scheme, a theoretical model of the transmission of propfan noise through a turbulent boundary layer was developed. In this model, the effect of refraction due to velocity and density gradients of the turbulent boundary layer has been taken into account. Unlike in all previously available analytical models, the effect of damping by turbulence has been incorporated by adding appropriate eddy viscosity terms in the equations of motion. Physically, boundary layer turbulence consists of random unsteady motion of the fluid elements of the boundary layer. As sound waves propagate through such a shear layer both damping and scattering occur. Damping arises through dissipation via an eddy viscosity type (turbulent mixing) mechanism. This generally leads to a reduction in the transmitted sound wave amplitude. Scattering arises because of the randomness and non-uniformity of the turbulent fluid motion. It gives rise to the phenomenon of spectral broadening of the sound waves. In the high frequency geometrical acoustics limit, the spectra broadening phenomenon associated with the random scattering by the turbulent interface of a shear layer has been investigated in some detail by Campos [ 10,111. Here, this effect will not be pursued. Our primary objective is to assess quantitatively the effect of turbulence damping on the amplitudes of the sound waves that are transmitted to the wall of the fuselage. on

4.1. THE PHYSICAL MODEL AND SOLUTION Consider a point source of sound located at a height H above a flat plate in a uniform flow with velocity t(, as shown in Figure 19. Let 6 be the thickness of the boundary layer adjacent to the flat plate. To simulate the (damping) effect of the turbulence in the boundary layer on the sound waves passing through it an eddy viscosity term is added

SOUND

Figure

PROPAGATION

19. A point source

of sound

THROUGH

BOUNDARY

above a turbulent

boundary

LAYER

107

layer on a flat plate.

to the momentum equation. In the past, the use of eddy viscosity terms in calculating the mean velocity profile of turbulent boundary layers has been found to be quite successful and satisfactory. In the present sound transmission problem the effect of turbulent mixing is, as in the case of mean flow calculation, to smooth out any imposed velocity gradients. Thus for sound waves with wavelengths comparable to, or longer than, the thickness of the boundary layer the use of eddy viscosity terms to simulate the effect of turbulent mixing appears to be reasonable, at least as a first approximation. The same eddy viscosity coefficient as for mean flow calculations is used. With respect to a Cartesian co-ordinate system centered on the surface of the plate directly below the point source with the x-axis pointing in the direction of the mean flow and the y-axis normal to the surface of the plate passing through the point source, the linearized equations of motion are

$+a -@)+a

- (picx)=QoS(x)6(y-H)6(z)ti”‘,

(1)

where p, v’and p are the density, velocity and pressure perturbations. Quantities with an overbar are mean flow variables. (u, u, w) are the velocity components in the (x, y, z) directions. Q0 is the source strength and w is the angular frequency. 6(x) is the Dirac delta function. y is the specific heat ratio of the gas. Subscript cc denotes flow quantities in the uniform flow state away from the boundary layer. V,is the eddy viscosity coefficient. It is equal to zero outside the boundary layer. On the surface of the plate the no-slip boundary condition of y=o ii=0 at (4) must be satisfied. It is obvious that all the flow variables will have a time dependence in the form of exp [-iwt]. For convenience, this factor is not written out and in what follows; only the spatial part of the dependent variable is explicitly shown. The above boundary value problem can best be solved by first introducing Fourier transforms in the x and z variables. Define the Fourier transform of p and its inverse by P(X, Y, z) e

i(++kzz) dx dz,

(54

00

Ph

Y, z) =

II --oo

p’(y, k,, k,) e-i(k~x+k~r) dk, dk,

(5b)

108

ET AL.

M. SALIKUDDIN

where k,, k, and kz are the wavenumbers On applying the Fourier transforms to the other dependent variables in favor of outside the boundary layer (Y > 6), where 3:+

(w - u,k,)*_

dy

a:.

along the x, Y and z directions, respectively. equations (1) to (3) and after eliminating all 6, it is straightforward to find, in the region Y, = 0, that the governing equation for p* is

k2 _ k2 p’= ei”“m“xhx) 1 z & 00 S(Y -H), 3

(6)

where aas is the ambient speed of sound. On using standard techniques to handle the S-function on the right side of equation (6) it is easy to find that the solution of this equation, satisfying the outgoing wave condition, is

where (w

-

e&x)*

A=

a,

_

k2

_ x

2

k2 z

-f
3

Sg.

I

Thus, at the edge of the boundary layer, the Fourier transform of the pressure incident sound waves, labelled with subscript i and superscript 0, is given by

On inverting

the Fourier

transform

the pressure,

II Oc

Pb,s,z)=~

tw

-

uCOkx)

in physical

co-ordinates,

etAcHm61+iCk,x+kzz)

dk

is

dk_ x

(10)

_

A

--cc

of the

Inside the boundary layer Y, is unequal to zero. On applying the Fourier transforms and eliminating dependent variables as before, equations (1) to (3) can be reduced to the following fourth order system: 1 _i(w - Uk,)v,

a -2

1 dp ----dy

p

-(k:+k:)(

2iv,k, dri di -d* dy dy

(o - Uk,)*

1 [

a

-2

l-i(~-~~kx)~‘)]l-2ik~~~~,

iv,k, +-d2

d*ti dy2 (11)

v,fi=i*+[-i(w-dk,)+v,Ck:+kI)lV. (12) dy2 P dy In deriving these equations the eddy viscosity, v,, has been taken to be a constant although in the numerical computation, carried out later, it is assumed to vary with y in accordance with experimental observation. In the region adjoining the boundary layer the mean Bow is uniform, i.e., independent of y, so that equations (11) and (12) reduce to a set of equations with constant coefficients. These simplified equations can be solved in closed form. The four linearly independent solutions labelled with subscripts i, r, 1 and 2 for easy distinction are as follows: (a) the incident

acoustic

wave solution, 6, =

ia e

-ia(r-6)

p,[v,(k’,+kf+cu2)-i(w-uu,kx]’

(13)

SOUND

PROPAGATION

THROUGH

BOUNDARY

LAYER

109

(b) the reflected acoustic wave solution,

b=e

ia(y-8)

_

-

,

eiu(y-6)

icr

(14)

“=pm[~,[k~+k~+U2)-i(~-uu,k,)]’

(c) the first viscous solution,

A=09

o’, =

e-P(Y--s). ,

(15)

(d) the second viscous solution,

i2=0,

c2 =

eP(Y-s)

(16)

Here (17)

’

1 l/2

kz+kz-i(w-u,k,)/v,

,

-:cargp
(18)

It is to be noted that if equation (13) is multiplied by $“(kx, kz) of equation (9) then at the edge of the boundary layer this solution will match with that of the incident sound wave solution generated by the oscillating point source. Solution (d) becomes unbounded as y + co. Therefore, this solution must be ignored in the present acoustic wave transmission problem. Corresponding to solutions (a), (b) and (c) three linearly independent solutions of equations (11) and (12) can be constructed. The numerical calculation of these solutions is discussed briefly later. Let [;;;

t; ;;I,

[%I;

2; z;],

[!i;

t; :;I,

(19)

be the solutions of equations (11) and (12) which become equations (13), (14) and (15) respectively as y + co. By using these three solutions a general solution inside the boundary layer, which matches the incident sound wave solution at the edge of the boundary layer, is easily found to be i(v,

k, kz) = MY,

kc, k)+4%y,

c(Y, k, k) = [ci(Y, k, k)

k, k) + %(y,

k,

kHi~“‘Wx,W,

+ AC,(y,4x9 k) + Bv’,(yv kc, k)lF!“‘(k,

k),

(20)

where A and B are arbitrary constants. They are independent of y but are functions of k, and k,. At y = 0 the solution must satisfy the no-slip boundary condition of equation (4). Upon using the linearized energy equation this no-slip condition may be replaced by the two conditions v’= 0,

-(iw/jFi*)p’+d6/dy

= 0.

(2I,22)

Substitution of equation (20) into equations (21) and (22) gives two linear algebraic equations for A and B. These equations can readily be solved by Cramer’s rule. With A and B found, the pressure on the wall can easily be calculated by equation (20). Thus d(0, kx, k,) = Nkx, k,, w)$“(kx,

k,),

(23)

where F(kx, kz, w)=k(O,

k, k=)+&(O,

%, k=)+gP’i(O, kx, k=)*

(24)

110

ET AL.

M. SALIKUDDIN

In physical co-ordinates the pressure distribution on the wall with the turbulent boundary layer correction is given by the inverse Fourier transform of equation (23): i.e., F(k,,

k,,

w)

(@

-

umkx)

eiA(H-S)+i(kvx+kzz)

dk

A

*

dk

5.

(25)

Interest will now be confined to the high frequency or the far-field limit. Under this condition the integrals in equations (25) and (10) may be evaluated asymptotically. To facilitate the asymptotic evaluation a change in integration and co-ordinate variables is first made. Let k, = wl;,,

k, = wk;,

R = x/(H

- 6),

i= z/(H

- 6).

(26)

Then equation (25) may be rewritten as

WI&w)(

1-

u&J ei’~O(H-s)‘am dk; dk;, [(1-u,k;)2/a~-~~-~f]1’2

F(wk;,

(27a)

where i=[(1-u,lS,)2-a~(~~+~S)]“2+aoo(~+X;,z).

(27b)

For large w(H - 6)/a, the double integrals of equations (25) and (10) may be evaluated by the method of stationary phase. The stationary point in the k,-k, plane of the double integrals is defined by the conditions a$&& = 0,

a&YE; = 0.

(28a, b)

On solving equations (28a, b) simultaneously it is straightforward to find that there is only one stationary point. After some algebra the stationary values of k, and k,, labelled by a subscript s, are found to be k,,=wk;=-M,u/a,(l-M~)+wx/a,(l-M~)[x2+(1-M~)((H-6)2+z2)]“2, kZ,=wkZ=~z/a,[x2+(1-M~)((H-S)2+z2)]”2,

(29)

where Mm is the uniform flow Mach number (Mm = ~,/a,). With the stationary point determined, the asymptotic value of the integral of equation (27a) may be evaluated according to a standard procedure (see, e.g., reference [12]). Similarly, the double integral of equation (10) can also be calculated. The asymptotic forms of these integrals are not written out here, as primary interest is only in their ratio to each other. Physically, this ratio represents the boundary layer correction to the pressure of the incident sound wave. Following Hanson and Magliozzi [5], one can define the boundary layer correction factor C, as the ratio (in dB) between the acoustic pressure with wall and boundary layer, to the acoustic pressure at the same location in space but without the wall and the boundary layer: i.e., c, = 20 log,,

[p:;:Zsl

dB*

(30a)

Upon accounting for the doubling effect of the wall, the transmission loss can be evaluated as (6.0 - C,) in dB. By means of the asymptotic forms of the integrals of equations (27a) and (10) it is straightforward to find for a point (x,.,,, z,,,) on the flat plate, G = 20 log,, IF(%,, k,,, w)l

dB,

(30b)

where F(ks, k,,, w), k,, and k,, are given by equations (24) and (29) respectively with x = x, and z = z,. x, and z, are related to x,,,.~~ and zwallgeometrically as shown in Figure 19.

SOUND

PROPAGATION

THROUGH

BOUNDARY

111

LAYER

4.2. NUMERICAL METHODS For large turbulent Reynolds number, u~OS/Y,,the fourth order system of equations (11) and (12) is known to be stiff. The problem lies in that, for large turbulent Reynolds numbers, the two eddy viscosity solutions have extremely large growth or decay rate. The result is that any numerical round-off error will also grow at this enormous rate. This prevents an accurate numerical computation of the incident and reflected acoustic wave solutions because, in a few steps of integration, the solution becomes so much contaminated by the rapidly growing error that the numerical values become practically proportional to that of the viscous solution. Calculation of the linearly independent acoustic solutions by conventional methods are, therefore, numerically not feasible. Sophisticated techniques, as described below are needed to overcome this problem, In dealing with a similar problem in hydrodynamic stability theory, a method known as orthonormalization has been developed. This method allows one to construct numerically all linearly independent solutions of a system of stiff differential equations of the kind such as equations (11) and (12). An excellent review of the method and its computer implementation has been given by Scott and Watts [13]. This orthonormalization procedure with a slight modification has been found to be applicable to the present sound transmission problem. All the results based on the eddy viscosity model described below are obtained by using this method. Before the fourth order system of equations (11) and (12) can be integrated, the eddy viscosity, v,, must first be specified. On following Prandtl’s mixing length theory V, is taken to have the form Y,= I*]au/ay].

(31)

Cebeci and Smith [ 141 have discussed the appropriate representation of the mixing length 1. According to one of their suggestions, the boundary layer is divided into two layers. In the outer layer the mixing length, I,, is a constant. In the bottom or inner layer the mixing length, pi, decreases linearly as the flat plate is approached. Thus 10=a,S

for

y>y,,

li=K_V

for

(32)

J'o
where y, is a small distance from the wall and y, is to be obtained from the continuity of 1. The recommended numerical values of K and (Y, are 0.40 and 0.075 respectively. These values have been adopted in these calculations. 4.3. INVISCID MODEL In the limit Y,+ 0 the governing equations in the boundary layer, i.e., equations (11) and (12), reduce to F_2ik

1

pLtic = dy ’

_

-i

u=p(o-Jk&

di

(33934)

Eliminating v’gives (35) The appropriate boundary condition imposed at the wall is that of equation (21) which, together with equation (34), yields Y = 0,

djI/dy = 0.

(36)

The simplified inviscid model problem consisting of equations (35), (36) and the incident sound wave of equation (10) can be solved in a fashion identical to that for the eddy

112

rvt. SALIKUDDIN

m

AL

viscosity model theory. The advantage of the inviscid model is that the second order equation (35) can be integrated numerically in a straightforward manner whereas, for the eddy viscosity model, the corresponding fourth order differential system of equations (11) and (12) is known to be stiff. Special numerical techniques are required to construct the appropriate linearly independent solutions. This is discussed in section 4.2. For some wavenumber k,, the factor (W - tik,) may vanish at some point y, in the boundary layer. When this occurs equation (35) becomes singular along the path of integration and the inviscid model breaks down. This same phenomenon was encountered previously by investigators studying the problem of hydrodynamic stability of shear layers. One way to resolve the difficulty is to use the method of deformed contour integration. This method has been elaborated by Tam and Morris [15]. Following this method the integration of equation (35) is to be performed in the complex y-plane. For a boundary layer velocity profile as shown in Figure 19 the path of integration is to be deformed below the branch cut and the critical point y,, as depicted in Figure 20. The deformed contour integration can be carried out numerically by a marching type integration scheme.

h(y)

T

Figure

4.4.

20.

Deformed

TRANSMISSION LAYER

path of integration

LOSS

PREDICTION

in the complex

FOR

y-plane

POINT

for a boundary

SOURCE-FLAT

layer velocity

PLATE

profile.

BOUNDARY

CONFIGURATION

In this section, the boundary layer transmission losses for acoustic signals, measured experimentally, described in section 3, are compared with those predicted by the analytical model. A computer program was developed for the prediction of transmission loss of acoustic signals. The most important input required for the prediction program is the turbulent boundary layer profile at the measurement location. This was supplied from the experiment in an indirect way. A turbulent boundary layer profile of the following form [16] was assumed in the prediction program: u/u, = tanh”’ (O-082 y/ 0*).

(37)

For small y/8* this gives u/u, = 0.7 (y/e*)“‘,

(38)

where u is the velocity at a vertical distance y from the flat plate, and u, and 0* are the free stream velocity and the momentum thickness of the boundary layer, respectively. The momentum thickness 8* was computed from the measured boundary layer profile and was used in the prediction scheme.

SOUND

PROPAGATION

THROUGH

BOUNDARY

LAYER

113

Before comparing the experimental data with predicted results the effect of viscous damping in the process of sound propagation through the boundary layer is studied for the point source-flat plate configuration. To evaluate the effect of turbulence damping on the transmitted acoustic waves, boundary layer correction factors were numerically computed for both the eddy viscosity boundary layer model and the inviscid model. Figure 21 shows the difference between the boundary layer correction factor based on an eddy viscosity boundary layer model and that of an inviscid model with respect to the Mach number A4, (Figure 21(a)), and with respect to frequencies (Figure 21(b)) as a function of measurement angle, x. It turns out that the effect depends strongly on the frequency of the sound wave, jet Mach number, and the measurement angle x. Generally speaking, the turbulence damping effect is small in the region aft of the source (i.e., x s 900). However, in the region forward of the sound source the damping effect becomes increasingly large. This effect becomes larger for the higher frequencies. This implies that the waveform of the transmitted sound wave would be smooth as the higher order harmonics which are essential to waveforms with sharp spatial changes would be filtered off by the turbulent boundary layer. Directivity comparisons of acoustic transmission losses for different M, and at a number of frequencies are made in Figure 22. The measured and the calculated transmission losses show a relatively constant value in the rearward directions and then show a rapid rise in the forward directions (i.e., with increasing x). This comparison shows qualitative agreement between the experimental results and prediction. The predicted and the measured transmission losses increase with measurement angle, x, Mach number, M,, and frequency. However, this increasing trend is much higher for the predicted losses as compared to those measured. The theoretical model underpredicts the losses at 0,

I

50

I

I

60

I

I

I 70

I 80

I

I 90

I

I

I loo

, 110

I

‘\( 120

Observotlon angle, x (degrees) Figure 21. Effects of eddy viscosity damping on transmitted acoustic waves for the point source-flat plate of 30 kHz; -, M, = O-5; - - -, configuration; (a) for various jet Mach numbers, M,, at a frequency M-=06; - - -, M-=0.7; - - -, M,= 0.8; (b) at various frequencies, M-=0.5; -, 10 kHz; - -, 20 kHz; - - -, 30 kHz.

114

M.

ET AL.

SALIKIJDDIN

‘\

\

I6

, 50

, 60

I 70

I

Measurement

I 80

I

I 90

\’

I

I 100

!\I

I

110

120

angle, x (degrees)

Figure 22. Comparison of boundary layer transmission-loss directivities tion; symbols, measured data; lines, prediction; -O-, h4, = 0.5; --W -A -, M, = 0.8. (a) 10 kHz; (b) 20 kHz; (c) 30 kHz.

measured - -,

IV,

experimentally = 0.6;

-0

with predic- -,

hf,

= 0.7;

lower values of x, M, and frequency and overpredicts at higher x, M, and frequency compared to experimental values. Prandtl’s mixing length theory used to calculate the turbulent Reynolds number in the boundary layer was developed for steady state mean Bow calculation, so it is probably applicable for low frequency sound transmission problems. However, for the estimation of transmission losses for high frequency sound the Reynolds number could well be frequency dependent. To study the dependence of transmission loss on turbulent Reynolds number the transmission losses were predicted by using different values of the K and (or combinations of the Cebeci and Smith model [ 141. Figure 23 shows the directivity plots for transmission losses at Mm = O-5 for a frequency of 30 kHz. The decrease in turbulent Reynolds number, caused by the increase in K and cr, values increases the transmission loss values considerably. The agreement between theory and experiment for lower Mach numbers, shown in Figure 22, can be improved if predictions are made with a lower turbulent Reynolds number by using an appropriate set of K and (Y, values (see Figure 23). It must also be pointed out that at high x values and especially for high frequency sound waves, the computation scheme could incur excessive cumulative and round-off errors. 4.5.

APPLICATION

The

locally

TO

THE

PROPFAN-FUSELAGE

CONFlGURATION

point source-flat plate boundary layer model to the fuselage boundary layer of an aricraft

developed above with an advanced

may be applied turbopropeller.

PROPAGATION

THROUGH

BOUNDARY

LAYER

115

6-

6-

03

101 50

I 60

I 70

I 80

Measurement

I 90 angle,

I 100 x

I 110

1 I20

(degrees)

Figure 23. Effect of turbulent Reynolds number on boundary layer transmission-loss directivities of30 kHz; it4,= 0.5; - - -, CT,= 0.075, K = 0.4; --, a, 0.15, K = 0.8; - - -, (Y,= 0.375, K = 2.0;-, K = 4.0.

at frequency (I, = 0.75,

For the type of aircraft under consideration and for the JetStar flight experiment the radius of the fuselage is much larger than the thickness of the fuselage boundary layer and the wavelength of the acoustic waves at blade passing frequency. Under these conditions the boundary layer may, to a first approximation, be considered as locally identical to that of a flat plate. A propfan is a rotating noise source. Therefore, unless the propfan is located very far away, it would be advantageous for the purpose of calculating the incident sound waves on the turbulent boundary layer of the fuselage to retain the rotating noise source characteristics. Let (r, OS,x) be a cylindrical co-ordinate system centered at the propfan with the x-axis pointed in the direction of flow as shown in Figure 24. In terms of these co-ordinates the pressure field associated with the thickness and loading noise of a propfan rotating with angular velocity w can be written in the form (see reference [ 171)

Figure 24. Diagram looking downstream at the propfan and the co-ordinate system

in the x-direction (p, 4, x) centered

showing the co-ordinate at the fuselage.

system (r, 0,. x) centered

116

M.

SALIKUDDIN

AL.

ET

where KnB( ) is the modified Bessel function and ~,,a( k) depends on the blade geometry and the loading distribution. B is the number of blades. uo3 is the freestream or cruise velocity and aao is the ambient speed of sound. For the nth blade passing harmonic, the pressure field is OT

P,(C &,x,

t) =

I ((

Jk2 - (ku, -

K,B

wnB)2/a&

p,B( k)

eicnBos+kx-nBw’ dk. ) (40)

-Lx

For a point sufficiently far away, i.e., R = ( r2 + x~)“~ is large, the above integral can be evaluated by the method of stationary phase. It is straightforward to find that the stationary wavenumber k, is given by wnB cos ,y ks =

a,( 1 - M&)Jl

M,wnB

- ML sin*x-a,(

(41)

1 - ML)

where cos x =x/(x’+ r2)“2. In evaluating the integral (40) by the method of stationary phase the modified Bessel function KnS is first replaced by its asymptotic form. After the asymptotic evaluation is completed, the asymptotic form of KnB is replaced by the original function. Then, for large R, it is easy to find that P,(C 6, x, t) -where H’,‘i(

pnB( k,)sin x Ra,

7r3’2i cnmi/2H(,~(p,Br) ei(nB~s+k5x-nB4, 21’2

(1 - M& sin2x)3’4

) is the Hankel

function

of the first kind and

Pne = wnB sin x/a=41 Now the pressure field of expression respect to the fuselage co-ordinates reference [ 181)

(42)

- Mk sin2x.

(43)

(42) can be decomposed into cylindrical waves with (p, 4, x) by means of the addition theorem (see

Here RL = H-R, is the distance from the center of the propfan to the center of the fuselage. R, is the radius of the fuselage and H is the height of the center of the propfan above the surface of the fuselage. On the surface of the fuselage the wave numbers of the mth incident cylindrical wave are approximately equal to k, = k,,

k, = m/ Rp

(45)

Therefore, by regarding the boundary layer as locally plane, the transmitted wave at the bottom of the boundary layer corresponding to the mth cylindrical wave of equation (44) is given by the product of the transfer function of F(k,, k,, co) of equation (24), with k, and k, given by expressions (45) and the complex amplitude of the incident wave. In this way, the sound pressure at the surface of the fuselage under the turbulent boundary layer can easily be found. By using this expression for the transmitted sound field, the following boundary layer co ection factor CF, can readily be established: f

CF

at

(R,, 4, x) = 20 log,,

H”’ n~+m(Pn~W

m=-‘x

J,UL&)F(k,

I.

ml&, 0) e’“@’

(4)

SOUND

4.6.

TRANSMISSION

PROPAGATION

LOSS PREDICTED

THROUGH

FOR JETSTAR

BOUNDARY

117

LAYER

PROPFAN-FUSELAGE

CONFIGURATION

The acoustic data measured on the fuselage surface for the JetStar flight test are described in reference [l]. These results include in-flight noise data for an eight-bladed SR-3 propfan measured at several locations on the fuselage at a Mach number of 0.8 and at an altitude of 30 000 feet. These are the propfan noise data propagated through the fuselage boundary layer. In this section, these data are compared with the predicted results. The noise generated by the SR-3 propfan was predicted by using the weakly non-linear theory for advanced turbopropeller noise developed by Tam and Salikuddin [ 171. The boundary layer transmission losses predicted by the theory, described in section 4.5, was then incorporated with the predicted propfan noise data to evaluate the noise levels on the fuselage surface. Figure 25 shows the comparison of JetStar flight data with the predicted results for the blade passing frequency and its harmonic at various measurement angles x. The agreement between the theory and measurement is reasonably good for the blade passing frequency at all the measurement angles. However, this agreement deteriorates for the harmonic frequency at higher measurement angles (i.e., above 90”).

harmonic

60

00

70

Measurement

no. 2

so angle,

100

110

120

x (degrees)

Figure 25. Comparison of measured and predicted sound pressure level directivities of the SR-3 eight-bladed propfan on the JetStar fuselage surface at the cruise condition at 30 000 feet altitude; M, = 0.8; 0, measurements; -, predictions.

6.

CONCLUSIONS

An experiment was conducted to determine the attenuation of sound crossing a turbulent boundary layer over a flat plate attached on one side of a high speed square jet at Mach numbers as high as O-8. Substantial transmission losses were measured, particularly for high frequency noise in the forward direction. The transmission loss increases with Mach number, frequency, and forward direction. It was not possible to estimate the transmission losses at higher Mach numbers for very forward directions. In these cases, the impulsive signals generated by the acoustic source were drastically reduced and broadened by severe convection effects. This caused a very low signal-to-noise ratio that prevented accurate signal recovery. However, this problem can be overcome by using an acoustic source that can generate pulses of much

118

M. SALIKUDDIN ET AL.

higher intensity. Thus, signal recovery will be possible, and the transmission losses for these severe conditions can be estimated. To assess the acoustic transmission loss as the sound waves from turbopropellers pass through a turbulent boundary layer on the surface of the fuselage, a theoretical model was developed. In this model, the effect of refraction due to velocity and density gradients of the turbulent boundary layer has been taken into account. The effect of eddy viscosity on acoustic transmission loss was evaluated for the point source-flat plate experiment. The effect depends strongly on the frequency of the sound wave, the measurement angle, and the jet Mach number. The predicted transmission losses for the point source-flat plate tests were compared with the experimental results. The agreement between the experimental results and the predictions are reasonable for some cases. In the point source experiments, the acoustic energy scattered by turbulence is automatically discarded in the measured signal by the process of time domain averaging. However, it is expected that the amount of scattering is very small compared to the attenuation and refraction of sound due to other mechanisms. The analytical model is consistent with the experiments, since it does not take turbulence scattering into consideration. The theoretical model was also used to predict the transmission losses for the SR-3 eight-bladed propfan for the cruise condition of the JetStar flight experiment. These results are compared with the measured noise data. Reasonable agreement is achieved between measured and predicted data.

ACKNOWLEDGMENTS The authors are particularly thankful to Dr K. K. Ahuja for his deep involvement in this study. Special thanks are due to Mr J. A. Gallagher who helped in conducting tests for aerodynamic measurements. REFERENCES 1. B. M. BROOKS 1983 HSER 8882. Analysis of JetStar prop-fan acoustic flight test data. 2. G. L. MCANINCH 1983 Journal of Sound and Vibration 88, 271-274. A note on propagation through a realistic boundary layer. 3. D. B. HANSON 1984 Journal of Sound and Vibration 92, 591-598. Shielding of propfan cabin noise by the boundary layer. 4. G. L. MCANINCH and J. W. RAWLS 1984 AIAA Paper 84-0249. Effects of boundary layer refraction and fuselage scattering on fuselage surface noise from advanced turboprop propellers. 5. D. B. HANSON and B. MAGLIOZZI1985 Journal ofAircraft 22,63-70. Propagation of propeller tone noise through a fuselage boundary layer. and K.K.AHuJA~~~O JournalofSound 6. M. SALIKUDDIN,P.D.DEAN,H.E.PLUMBLEE,JR. and Vibration 70,487-501. An impulse test technique with application to acoustic measurements. 7. M. SALIKUDDIN, K. K. AHUJA and W. H. BROWN 1984 Journal ofSound and Vibration 94, 33-61. An improved impulse method for studies on acoustic transmission in flow ducts with use of signal synthesis and averaging of acoustic pulses. 8. B. NORMAN 1967 DISA Information Publications. Hot-wire anemometer calibration at high subsonic speeds. 9. M. SALIKUDDIN and K. K. AHUJA 1987 AZAA Paper 87-2688. A device to generate high frequency noise from commercially available low frequency drivers. Also 1988 Journal of Sound and Vibration 123, 261-280.

10. L. M. B. C. CAMPOS 1978 Journal of Fluid Mechanics 89, 723-749. The spectral broadening of sound by turbulent shear layers. Part 1. The transmission of sound through turbulence and shear layers. 11. L. M. B. C. CAMPOS 1978 Journal of Fluid Mechanics 89, 751-783. The spectral broadening of sound by turbulent shear layers. Part 2. The spectral broadening of sound and aircraft noise. 12. N. BLEISTEINand R. A. HANDELSMAN 1975 Asymptotic Expansions of Integrals. New York: Holt, Rinehart and Winston.

SOUND

PROPAGATION

THROUGH

BOUNDARY

LAYER

119

SCOIT and H. A. W~r-rs 1977 SIAM Journal of Numerical Analysis 141, 40-70. Computational solutions of linear two-point boundary value problems via orthonormalization. T. CEBECI and A. H. 0. SMITH 1974 Analysis of Turbulent Boundary Layers. New York: Academic Press. C. K. W. TAM and P. J. MORRIS 1980 Journal of Fluid Mechanics 9f$ 349-381. The radiation of sound by the instability waves of a compressible plane turbulent shear layer. H. SCHLICHTING 1962 Boundary Layer Theory. New York: McGraw-Hill. C. K. W. TAM and M. SALIKUDDIN 1986 Journal of Fluid Mechanics 164, 127-154. Weakly nonlinear acoustic and shock-wave theory of the noise of advanced high-speed turbopropellers. M. ABRAMOWITZ and Z. A. STEGUN 1964 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Washington, D.C. National Bureau of Standards.

13. M. R.

14. 15. 16. 17. 18.

APPENDIX:

EFFECT

OF RESIDUAL

BACKGROUND

NOISE

ON

IMPULSIVE

NOISE

This appendix contains a brief discussion of the effect of residual background noise on impulsive noise. When a periodic impulsive signal is contaminated with random background noise, like jet noise, the impulsive signal can be recovered by using time domain signal averaging [7]. Complete recovery of the impulsive signal is possible if the number of averages is sufficiently high. The number of averages depends on the signal-tonoise ratio which reflects the degree of contamination. For some cases, experimental and hardware limitations may make it impossible to perform the required number of averages. To achieve a large number of signal averages, the contaminated signal must be acquired over a long period of time. For a high speed jet flow experiment, it is very difficult to maintain constant flow speed and temperature over a long period of time. In addition, when the impulsive noise generating device (i.e., the acoustic driver) is run continuously for a long time its temperature increases. The medium in the duct assembly between the driver and the flow is then heated up. The change in jet speed and the temperature increase within the duct assembly alter the propagation speed of the signal. When individual records are superimposed in the signal averaging process, the impulsive signals must be precisely aligned in time. The change in the pulse propagation speed slightly shifts the signal in time. This reduces the sharpness of the pulse peak and, thereby, reduces its high frequency content. Therefore, both the signal averaging process and the experimental conditions determine the optimum number of averages needed. Sometimes the hardware used for data processing may impose limitations. The FFT analyzer currently used here for signal averaging limits the maximum number of averages to 4096. When, for whatever reason, signal recovery is not absolutely complete, some contamination error will appear in the results. In most applications, these residual errors are insignificant. But in some situations, as in the evaluation of transmission losses as described in this paper, the error is significant. Some empirical method such as smoothing is then needed to overcome the problem. The use of smoothing in the present study has been validated by experimental means, and the validation is the topic of this appendix. A.l.

EXPERIMENTAL

ARRANGEMENT

In an actual experimental situation similar to the one described in this paper, the impulsive signal is injected into the noise environment (i.e., jet noise). The jet modifies the signal entering the flow stream. Therefore, it is impossible to get the uncontaminated signal to compare with the recovered one. However, this problem can be simulated by contaminating the acoustic impulsive signal with electronically generated random noise. Figure Al shows an arrangement to generate pulses with high frequency content. A periodic pulse with period 50 ms, width 55 ps, and peak amplitude of 50 V was fed to the acoustic driver sketched in Figure Al. The driver output was fed through a 16 : 1 contraction to a 25-inch long, l/Cinch constant diameter tube. The output, a string of

120 sharp tube. This oscillator. compared

very

M.

SALIKUDDIN

ET AL.

pulses [9], was measured by a microphone placed along the axis of the source signal was mixed with random (white) noise of 2 Hz to 200 kHz from an In this validation study, the signal recovered from the contaminated signal is with the signal measured by the microphone before contamination. A.2. EXPERIMENTAL

RESULTS

AND

THE VALIDATION

Figure A2 shows the time domain signal. The signal averaging was carried out for a record length of 10 ms which corresponds to an upper frequency limit of 50 kHz. For clarity, only 5 ms signals from the middle of the records are plotted in Figure A2. Figure A2 (a) shows the impulsive signal as captured from the microphone and its averaged version. This averaging was carried out to eliminate electronically generated random noise associated with the pulse. Figure A2 (b) shows the contaminated signal that was generated by mixing the impulsive signal with the white noise. The peak level of the white noise was the same as the peak level of the pulse. The contaminated signal was recorded on a tape recorder on one channel and the triggering signal was recorded on another. Different numbers of averages of the recorded signal were made, and the corresponding recovered signals are plotted in Figure A2 (c). The recording was used so that the averaging for each case could start with same signal for consistency. Acoustc drwer

Figure Al.

c

Experimental

set-up.

5ms

A

Tme

Figure A2. Time domain signals; dotted lines show the editing process (a) Impulsive signal; (b) contaminated signal; (c) averaged signal.

applied

to the signals

before

FFT.

SOUND

PROPAGATION

THROUGH

BOUNDARY

LAYER

121

The signal before contamination and the signals recovered from the contaminated signals were Fourier transformed and are plotted in Figure A3. In this figure, the spectrum of each of the recovered signals is compared with the spectrum of the signal before contamination, which is called exact data. When only 64 averages were applied, the recovered spectral distribution wa substantially different from the exact data, especially at higher frequencies. The spectrum with 256 averages agrees better with the exact data but has some oscillations above 10 kHz. However, a mean line drawn through the oscillatory spectrum coincides with the exact data.

Frequency

CkHz)

Figure A3. Spectral data showing the effect of time domain signal averaging on contaminated signal; -, exact data; - - - - -, recovered data by time domain signal averaging.

The spectral distribution of the signal with 1024 averages agrees very well with the exact data. However, with 4096 averages, the signal seems to have lost some high frequency energy. Excessive averaging, even for this semi-ideal case, has introduced some loss of high frequency energy due to the temperature change in the driver. Therefore, for the situation of this demonstration, 1024 is the ideal number of averages for accurate signal recovery. In a practical situation, one may not be able to obtain the desired number of averages. In that case, the next lower number of averages (256 in this simulation) might give a reasonably accurate result. Some smoothing may be done to the spectral data if that is needed. For the experiment described in this paper the level of contamination was much higher for higher Mach numbers compared to the amount used in this simulation. For those cases, even 1024 and 4096 averages were not sufficient to recover the signal absolutely. A number of validation tests similar to the one described here were conducted to simulate the different experimental situations. Based on those results, the smoothing process was applied to the transmission loss spectra only in the higher frequency range.