journal oj:sound and vibration (1981) 78(1), 69-117 ... · trailing edge noise prediction from...

Journal oj:Sound and Vibration (1981) 78(1), 69-117

T R A I L I N G E D G E N O I S E P R E D I C T I O N F R O M M E A S U R E D

S U R F A C E P R E S S U R E S

T. F. BROOKS

NASA Langley Research Center, Hampton, Virginia 23665, U.S.A.

AND

T. H. HODGSON

North Carolina State University, Raleigh, North Carolina 27650, U.S.A.

(Received 30 June 1980, and hz revised[onn 2 March 1981)

A comprehensive experimental investigation of trailing edge noise is reported for the case of a two-dimensional airfoil embedded in a uniform low Mach number flow. The Reynolds number is high and the boundary layer is fully turbulent. Parameters include angle of attack, fl0w velocity, and trailing edge bluntness. By using a coherent output power method, the trailing edge noise spectra and directivity (including forward speed effects) are quantitatively determined. Statistics of the pressure field beneath the turbulent boundary layer are defined in detail. The scattered pressure field (primarily incompressive) very near the trailing edge is measured and successfully modeled by extending existing theory. This helps establish the edge condition (of pertinence to the Kutta condition) which in turn determines the solution for the sound field in this study. By using a statistical model of the turbulent boundary layer pressure field, trailing edge noise is well predicted.

1. INTRODUCTION

Trailing edge noise has received much attention in recent years due to its importance to airframe noise, broadband rotor and propeller noise and other problems such as edge noise of powered lift devices. A most pert inent topic here is that of two-dimensional low Mach number flow over an airfoil with a chord dimension which is large relative to the acoustic wavelength. The Reynolds number of the flow is such that the boundary layer is turbulent over both surfaces of the airfoil (see Figure 1). As convecting turbulence from within the turbulent boundary layer (TBL) passes the trailing edge (TE) into the wake, noise is produced.

Turbulent . ~boundary layer ~,~,,. _ j_~"~

�9 roumg eage Wake

�9

Noise

Figure I. Air foi l in uniform flow of velocity Uo generating noise due to the convection of turbulence from its boundary layer past the trailing edge.

69

7 0 T . F . B R O O K S A N D T. H. H O D G S O N

Numerous theoretical models of trailing edge noise have been developed over the last decade. Howe [I] has reviewed these theories and divided them into three groups: namely, those based on (i) the Lighthill [2] acoustic analogy, e.g., that of Ffowcs Williams and Hall [3]; (ii) linearized hydroacoustic methods, e.g., those of Chase [4] and Chan- diramani [5]: and (iii) " a d h o c " approaches. He has shown that, when appropriately interpreted, all relevant theories produce essentially identical TE noise predictions for vanishingly small Mach numbers. Significantly, Howe [I] has reproduced and generalized the primary conclusions of the diverse theories by using a relatively uniform theoretical approach. The solutions for the noise field include the effect of forward flight and require knowledge of either the turbulent velocity field near the edge or the pressure fluctuations near the edge. However, due to an absence of information as to the precise conditions to apply at the edge, dual solutions are given corresponding to the Kutta condition and no-Kutta condition cases. This represents a weakness in the present state of the theory that produces an uncertainty in the predicted noise levels. The question of whether, or even how, the Kutta condition should be applied in the theoretical modeling of TE noise has heretofore remained unresolved by experiment.

Most edge noise data have been provided by turbulent (one-sided) wall-jet studies [6, 7]. Here the properties of the turbulence near the edge may be considered as given a priori, rather than by Reynolds number and surface profile effects. Notably, Chase [4] (who did not directly address the Kutta condition question) employed the data of Scharton et al. [6] and used surface pressure measurements and an evanescent wave theory model, to suitably predict wall-jet edge noise spectra.

Experiments involving the pertinent problem of two-dimensional airfoils with turbulent boundary layers (Figure 1) have been more limited. This is due to significant difficulties encountered when testing in open flow facilities. The difficulty [8] is that extraneous noise sources such as tunnel nozzle lips, the open jet turbulent shear layer and the downstream collector can make the airfoil TE noise undetectable. Recently, some of the difficulties have been overcome by using various means. Heller and Dobrzynski [9] avoided flow facilities and employed a clean glider to make flyover noise measurements. By assuming the dominance of two-dimensional TE noise, they could use a simple compact dipole acoustic model to determine a transfer function between measured surface pressure and the sound field. Schlinker [10] Used a directional microphone to extract TE noise in a model study in an open jet tunnel. The two-dimensional airfoil was placed between sideplates and the TBL was allowed to develop naturally rather than by tripping for zero angle of attack cases. Surface pressure data were not taken. It was shown that the TE was the dominant source of noise for the airfoil. One-third octave TE noise spectra for several operating conditions were obtained. There were some resolution problems preventing clear distinction of peak frequencies.

In an extensive study Yu and Joshi [11] used space-time correlation analysis to extract TE noise from extraneous sideplate, nozzle and open jet shear noise sources. Normalized TE spectra were deduced from the correlations. Correlations between surface sensors and microphones (as 'well as the use of flow visualization methods) established the importance of TBL large scale structural features on the noise production at the edge. The noise field was determined to be coherent and dipole-like. Surface pressure measurements were made on both sides of the airfoil. However, the sensors were not close enough to the TE to permit measurement of the edge scattered pressure field predicted in the theory [1, 4, 5]. Evaluation of this scatter phenomenon, which is related to the noise production process, would help establish the previously mentioned edge condition. Hahn [12] in a study of the TBL pressure field near a trailing edge did make measurements very close to the edge. However, the sensors were only on one side and the lack of a

T R A I L I N G E D G E NOISE P R E D I C T I O N 71

model of the edge scatter phenomenon produced an imprecise interpretation of the results. It was suggested acoustic pressure near the edge may have been responsible for peculiar results from the coherence measurements.

Experimental work in trailing edge noise has not kept pace with theory. There are issues which have evolved from the theory which affect their applicability to real flow conditions. Therefore, the purpose of the present study is to relate measurement to theory in sufficient detail to help bridge the gap between theory and its practical application. The case considered is that of a two-dimensional airfoil embedded in a uniform low turbulence flow at high Reynolds numbers where the boundary layer is fully turbulent at the TE.

The following is an overview of the four major sections of this paper. In section 2, details of the experiment are given. Central in the tests is the instrumented airfoil section in which a substantial number of pressure sensors are mounted, over both surfaces and many very close to the TE, to permit the evaluation of the TBL pressure field and edge scatter effects. The significance of boundary layer tripping is discussed. Test conditions and the mean flow TBL characteristics are given.

Section 3 is concerned with establishing the statistics of the incident TBL pressure field which approaches the TE. The measured surface pressure fluctuation at any location on the airfoil appears to be made up of two parts: the hydrodynamic pressure field associated with eddy convection in the TBL (the incident TBL pressure field) and the pressure scattered (considered in section 4) from the TE due to the passage of that incident field. In the case where the edge is not sharp an additional component of pressure occurs in a limited frequency range, due to coherent vortex Shedding at the edge. The scattered pressure is found important only at distances less than a hydrodynamic wavelength from the edge. Therefore, the incident pressure field can be deduced from measurements made close to the airfoil TE but yet still far enough from the edge that the scattered pressure may be neglected. Two empirical solutions describing the incident pressure field are given which represent (i) the cross-spectral function, between measurement locations on the surface of the airfoil, to be employed in section 4, and (ii) the wavenumber spectra from the same data for use in section 5.

In section 4, an expression for the scattered pressure field is deduced with the aid of an analytical solution obtained by Howe [1], based on the analysis of Chase [4], for the case of an incident hydrodynamic pressure field scattered by the edge of a semi-infinite plate. The important effects of TBL coherence decay and non-constant eddy convection velocity are introduced into the theory. Comparisons are made with experimental cross spectral data obtained from sensors very close to the TE. The edge condition is determined and is discussed in terms of its physical interpretation in connection with the Kutta condition.

In section 5, a method is described which permits the quantitative measurement of the TE noise spectra. Shear layer refraction and forward speed effects are taken into account in the presentation of directivity data. The TE noise data are synthesized with scaling laws from theories based on the Lighthill analogy. With the edge condition from section 4 the noise field is predicted by using the incident TBL wavenumber spectra, for comparison with that measured.

2. EXPERIMENTAL DESCRIPTION

The experiment was conducted in the anechoic quiet-flow facility at NASA Langley Research Center. The anechoic chamber surrounding the free jet has dimensions of 9.1 m • 6.1 m • 7.6 m high as measured from wedge tiP to wedge tip. A working surface

72 T.F. BROOKS AND T. H. HODGSON

for model changes is attained by removing the 0.6 m deep acoustic wedges from the chamber 's grating floor. The free jet employed in the tests was provided by a vertically mounted nozzle which has rectangular exit dimensions of 0.3 m • 0.46 m and an overall contraction ratio of 34. The nozzle exit is at a height of 2-2 m above the working surface. The free jet exhausts vertically through an acoustically treated exhaust port in the chamber ceiling.

2.1. MECHANICAL APPARATUS

The airfoil used in the test was a NACA 0012 symmetrical airfoil section (thickness to chord ratio of 0.12). The airfoil, with a chord of 60.96 cm and a span of 46.0 cm, was machined from aluminum stock. Slots were milled on the surface on both sides of the airfoil to allow flush mounting of surface pressure sensing devices. Figure 2 shows the airfoil with the sensors mounted and the sensor wiring is seen to be routed from the sides of the airfoil.

Trailing edge, /

Pressure- sensors i / . /

NACA C~). !2 oirfoil I

Figure 2. Test airfoil section, instrumented with flush-mounted pressure transducers.

As shown in Figure 3 the airfoil was supported in the test rig by two heavily reinforced sideplates (152.4 cm x 30.0 cm x 1 cm), which were flush mounted to the nozzle lips. The leading edge of the airfoil was placed 15.2 cm from the nozzle exit. Different geometric angles of attack of the airfoil mean chord line to the incidence flow, ag = 0 ~ 5 ~ and 10 ~ were attained by bolting at different locations through the sideplates into the airfoil section. Porous material was attached at strategic locations on the aluminum sideplates in order to reduce extraneous sideplate edge noise.

The blunt trailing edge thickness of the airfoil was t = 2.5 ram. Different values of t were obtained by gluing hardwood extensions on the edge (see Figure 4). The increase

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TRAILING EDGE NOISE PREDICTION 73

Figure 3. Airfoil test assembly in anechoic quiet-flow facility.

t : 2 . 5 mm , : , . g r a m

" - ~ ' ~ i _ _ . . _ . J. t =0 (sharp .... "~ '~ ~ ' . - . . . . . . _ 1 trailing

Sensors ailing edge extensions

Figure 4. Illustration of trailing edge extensions which were alternately attached to airfoil.

in chord length as measured along the surface was 12.7 mm for t = 0. In addition to the extensions shown, a series of tests was made with sharp-edge metal plate extensions which were held in place at the trailing edge by bolting the extensions between the sideplates. These extended the chord of the airfoil by 15.24 cm and 30.48 cm, respectively. Also tested was a 2.54 cm sharp edge "flap" extension placed at 17.5 ~ off the chord mean axis at the trailing edge. Glossy Teflon tape of 0.08 mm thickness was used to provide a Smooth transition surface for all extensions.

2.2. SURFACE PRESSURE SENSORS

In order to sense the fluctuating pressures in the immediate vicinity of the relatively thin trailing edge of the airfoil, a very small pressure transducer was used. The design incorporated a miniature diaphragm and a monolithic integrated circuit Wheatstone bridge manufactured by Kulite. A more complete description of the design has been

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74 T. F. BROOKS AND T. H. HODGSON

given by Hodgson and Manley [13]. The sensor has a pinhole diameter of 0.34 mm with overall external dimensions of 9.53 mm x 3.18 mm x 0.14 mm depth. An illustration of the sensors after mounting is shown in Figure 5. The Helmholtz resonance frequency of the sensor cavity volume was approximately 62 kHz. Prior to mounting, a calibration of each sensor was obtained by using a special pressure coupler of very small internal dimensions. The sensors were shown to have excellent uniform pressure and phase response up to a frequency of order 30 kHz.

m

"--Sensor pin hole

~ - Sensor device

Figure 5. Section view of trailing edge pressure sensor t~luster.

A bridge differential signal amplification configuration was used in the initial signal conditioning to avoid the DC offset drift signals usually associated with this type of sensor. A low noise differential amplifier with high common-mode rejection operating at a voltage gain of 50 was connected to the transducer bridge output. This was then followed by an AC coupled amplifier to be subsequently describe d.

It is well known that the high frequency response of pressure sensors used in measuring boundary layer pressure fluctuations is dependent on the sensor diameter d and the boundary layer wall similarity parameters, namely the length scale v/u* where v is kinematic viscosity and u* is the friction velocity (a list of symbols is given in the Appendix). The behavior of pinhole sensors has been summarized by Bull [14] for a zero pressure gradient TBL. From Bull's results with d(u*/v)= 32 and tov/u . 2= 0.55, with to being the radian frequency, the sensor would read 2 dB low at a frequency of 20 kHz. For 2 kHz the amplitude error would be negligible. Thus, the sensors used had excellent amplitude and phase response with small resolution errors for the frequency range of interest which was 100 Hz to 10 kHz.

2.3. CO-ORDINATE SYSTEM AND TRANSDUCER PLACEMENT

The co-ordinate system used is shown in Figure 6. The rectangular co-ordinates (xl, x2, x3) are centered at midspan on the trailing edge (TE). The airfoil surface of chord C and span L occupies the region - C < x l <0 , x2 = 0 and -L/2 <x3 <L/2. The sound field observer location is defined as shown in spherical co-ordinates. The 36 surface pressure sensors were flush mounted in symmetrical pairs on opposing surfaces of the airfoil, chordwise along the midspan and spanwise as close as 2-54 mm (0.42% chord) from the trailing edge. Figure 5 shows the trailing edge sensor cluster at midspan. The co-ordinates of the center of the sensing areas are as follows: for x3 = 0, xl = - 2 . 5 4 ,

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~ t u r convecled bulence . ~

. ~ / ' ~ ~ ' ~ . . x 2 .~--Trailing edge

~ E _ ~ ' , , ~ . ~ L / - ~ Sound field / g ..P.o.or.or v l 2 ' e ~ " . / l loco,o ~

dE/ / . / / / / / / / . / I ~1 / . / I ./~ Surface ~ I.~' / . / i

;"//////////~ ~ / " i .<7"7. ~" ', 'n

Iocotio~ / # ' / : ~e I l _ J _ __El

/ / / / / / / / / / / / / / / / / / / / / / / / / / ~ ' / / / / / / . 1

/ x 3

Figure 6. Co-ordinate system showing the convection of eddies within the TBL approaching the TE. The noise field observer is defined in spherical co-ordinates.

-5 .84 , -9 .35, -16.28, -31.0 , -64 .5 , -128"5 and -206 .2 mm; for xl = - 2 " 5 4 mm, xa = 3.25, 8.64, 25.2, 63.5,127.0 and 203.2 mm; also forxl = -2 .54 , x3 = -63 .5 , -127.0 , -177-8 and -221 .0 mm (7.6 mm from sideplate). The use of the trailing edge extensions increased all sensor to trailing edge distances (thereby moving the xl axis) by 5.6 mm for degree of bluntness t = 1.9 mm, by 10.6 mm for t = 1.1 mm and by 12.7 for t = 0.

To measure the radiated noise, eight half-inch Briiel and Kjaer 4133 free field response condensor microphones were placed in the plane perpendicular to the airfoil midspan. The microphone positions employed in the tests are illustrated in Figure 7. Table 1, along with the microphone identification numbers of Figure 7, gives the measured

" ~ M6

t M7

M5 emk~r" ~ . ~ - Side plates \ - -

\ l &,=-90~

~Airfoil

Uo Nozzle

8 m = 90 ~ MI

= ~ 4 ~ M 3 M2

~M8 (microphone no. 8]

Figure 7. Microphone positions defined with respect to the airfoil trailing edge.

76 T. F. BROOKS A N D T. It. H O D G S O N

T A B L E 1

Alicrophone conditions

Microphone number M1 M2 M3 M4 M5 M6 M7 M8

R,,(cm) 122.2 119.2 358.3 122.4 121.9 242.9 172.4 121.9 0,,(deg) 90 -90 -90 120 60 45 135 -135

Acoustic field calibration (point source) Total• 0• 0• -0 .5• - - - - -1 .0 • 0.5+1.0 1-5• 1/3 Oct. (dB)

Shear layer corrections R, (cm) 122.2 119.2 358.3 124.4 120.1 240.0 177.8 124.9 Or (deg) 91.4 -91.4 =91.0 118.4 58.7 37.6 129.5 130.0 AdB 0.1 0.1 -0 .1 1.8 -1.7 -3.5 3.2 3.0

distances, R,,, from the microphone diaphragms to the airfoil trailing edge and also the measured angular positions, On, aft of the trailing edge from the mean chord line (for a~ = 0~

2.4. DATA ACQUISITION AND CALIBRATIONS

All measured signals were recorded on a 14 channel Honeywell 9600 FM analog tape recorder. Initial signal conditioning for the microphones was provided by individually assigned amplifier systems, whereas the conditioning for the surface sensor signals was as previously described. Because of the large number of signals compared with the limited number of channels of the recorder, each test condition required multiple (typically five) recording sequences. Final conditioning for each signal was provided by an Ithaco 455 amplifier/filter assigned to each recording channel. These were used to limit the low frequency noise (I00 Hz high pass normally used) and to optimize the dynamic range.

Prior to testing, amplitude and phase calibrations were performed on the acquisition system including the tape recorder by using a voltage insertion method. In addition an acoustic driver with a press-on adapter was used to cross calibrate the sensors and microphones (grid removed and flush mounted in an adapter). For the flush mounted surface sensors, excellent agreement was found between this calibration and the calibration conducted prior to mounting. With the input signal to the acoustic driver as a common phase reference, it was established that for the frequency range found to be of importance in this study (up to I0 kHz) the sensors were phase matched within a few degrees. The same was found true for the microphones.

The anechoic facility has been found to be of excellent free field quality from recent calibrations. However, it is to be expected that the presence of the sidep]ates and nozzles of the test rig may degrade the free field behavior. To determine the seriousness of this effect, a calibration was made by using an acoustic "point" source which consisted of a tube contraction with a small aperture driven by an acoustic driver. The "point" source was placed at the midspan of the airfoil trailing edge (airfoil removed) and was driven by broadband noise, from 500 Hz to 5 kHz. Analysis of the microphone signals revealed that reflected noise from the test rig did tend to reinforce or cancel (depending on frequency) the directly radiated noise from the source. Results showing deviation from free field conditions for six of the eight microphones are given in Table i. Microphone M8 is seen to be the most affected with it measuring on the average 1.5 dB above that which would be measured in free field. Individual one-third octave levels can vary from

TRAILING E D G E NOISE PREDICTION 7 7

this average value by +1.0 dB. The microphones least affected adversely are M1 and M2 with measured levels very close to those expected for free field conditions (within 0.5 dB). These two microphones provide the primary noise data base for this study. Therefore no significant adverse effect on these noise measurements should be expected, except that the presence of the test rig would influence the smoothness of the spectral data.

2 .5 . D A T A R E D U C T I O N AND STATISTICAL ERRORS

All surface pressure and noise data reported were analyzed from recorded signals. All signals including those from calibrations were analyzed by using a Spectral Dynamics SD-360 processor interfaced with a PDP-11 computer. For the data presented the signal processor employed an average of 256 samples. With data analyzed to a frequency of 10 kHz, the memory period of each sample was 50 ms and the analysis bandwidth was 20 HZ. Pairs of signals were processed to provide, for each pair, the cross spectrum and corresponding autospectra in digital form. From this primary data base the other functional formats, such as coherence function and coherent output power, were computed.

Inherent in the use of Fast Fourier Transform signal processing methods is the necessity of minimizing or correcting statistical variance and bias errors. In this study pairs of signals being processed were coherent but were delayed in time due to the flow convection and/or acoustic propagation between the surface sensors or microphone pairs. Due to the processor's finite memory period of each data sample compared with the signal time delay, a negative bias occurs in the cross spectral density estimate. This bias has been corrected in accordance with accepted equations [15, 16]. This correction, however, had negligible effect on results presented except those involving signals from microphones M3 and M6.

The statistical errors encountered in the data were of the magnitude predicted [15]. These errors are related to the number of averages and the coherence between signals. Its manifestation is a random variance about an otherwise smooth cross spectral density estimate. The method chosen to reduce this variance was simply to increase the data bandwidth from that processed at 20 to 100 Hz (unless otherwise stated). This reduced the variance of the cross spectral density estimates by a factor of x/5. By averaging contiguous frequency cross spectral data (vector quantities) to increase bandwidth, an additional negative bias error results when a time delay exists between the original pair of analyzed signals. This bias was avoided in the averaging of the amplitude by suppressing the phase variation. The phase was separately determined by vectorial summation of the data within the new bandwidth. This procedure is equivalent to that proposed in reference [17], where the nature of this bias error is discussed.

2.6. FLOW FIELD DETAILS

The airflow from the nozzle is uniform across the exit plane and the turbulence intensity is less than 0.4% at all test velocities. The purpose of the sideplates is to keep the airfoil within the uniform potential core of the free jet, maintaining two dimensionality and avoiding interaction with the shear layer of the free jet. The characteristics of the flow field about the airfoil have been determined by the use of tufts, Preston and Pitot tubes, a boundary layer rake, hot wires and surface pressure sensors.

2.6.1. Effect of boundary layer tripping For the airfoil section used in this study natural transition from laminar to turbulent

boundary layer is expected [18] to occur at about 40% chord for the velocities considered, if one has a smooth airfoil of large span within a uniform two-dimensional flow field. This was found to be true only along the midspan (x3 = 0) of the airfoil in the test rig.


Away from the midspan, transition occurs farther downstream. This three dimensional effect is symmetrical about x3 = 0 and is most pronounced at about x3 = +L/4. This results apparently due to the influence, on the potential flow field, of the zero-velocity boundary condition on the sideplates and the sideplates' own resultant boundary layer.

The effect of this spanwise non-uniformity on the fluctuating surface pressure near the trailing edge is shown in Figure 8(a). The figure shows the surface pressure power spectral density, ~ ( f ) (referenced to p o = 2 0 IxPa), for sensors closest to the 2.5 mm blunted edge airfoil. Note that there is a lack of spanwise uniformity..As x3 is increased from x3 = 0.325 cm, which is near the midspan, the level is seen to drop (except for the spectral "hump" near 3 kHz) until the influence of the sideplate TBL is perceived at x3 = 20.32 cm. The drop in level is due to the decrease in T B L thickness away from the midspan.

F __Q

I 0 0

9 0

8 0

T O

I I I I i I i

{Q) | I | I l I I I

" ~ - - ~ . ~ . _ f ~ ~ �9

1 0 0

( b )

9 0 - ~ ,

8 0

7 0 "1 I 1 I I t I I I I I I I I I 0 - 3 I I 0

F r e q u e n c y ( k H z )

Figure 8. Spanwise variation of surface pressure power spectral density (PSD) very close to the 2.5 mm blunted TE for the (a) untripped and (b) tripped boundary layer cases. Uo = 69.5 m/s and -xl = 0.254 cm. Spectra key for x3 (cm): ,0.325; - - - , 6.35; - - - - - - , 12.70; - - - - , 20.32.

To insure two-dimensionality and to obtain a well developed T B L at the trailing edge, artificial transition was employed for all runs of the test. Also, it was felt that it was important to simulate the full scale high Reynolds number case where T B L flow exists on both surfaces over most of the chord. For this purpose, roughness trips of No. 40 carbon random particles were glued to both surfaces of the airfoil. The trips were 2.0 cm wide over the entire span at 15% chord from the leading edge. The effect of the tripping is shown in Figure 8 (b). The power spectra are now seen to be nearly spanwise independent except for lower frequencies for the sensor near the sideplate. The relative influence of tripping the boundary layer on the resultant sound field is illustrated by the cross spectra between microphones M 2 and M 3 shown in Figure 9. The increase in T B L thickness

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T R A I L I N G E D G E NOISE P R E D I C T I O N

I I I I I , I I I I I I I I

l I I

0.3 10

7 9

~ T r i p p e d boundary layer case

,,, / ' , IOdB ,_, ~ i

I ! I I I ! i t I I I I

I

Frequency (kHz)

Figure 9. Cross spec t rum between microphones AI2 and M 3 illustrating the effect on the sound field of tripping the boundary layer on the test airfoil. Uo = 69.5 m / s for the 2-5 m m blunted TE airfoil.

due to tripping results in somewhat higher sound levels with the notable exception of the spectral "hump" near 3 kHz.

2.6.2. Mean characteristics of the turbulent boundary layer Two-dimensionality of the flow field was also confirmed by tufts and spanwise mean

flow measurements with a movable Pitot-rake and 14 Preston tubes (seven on each side and across the span). The local skin friction coefficient cl determined from Preston tube measurements was found to be uniform across the span. This was also found to be true for ag = 5 ~ and 10 ~ for both the pressure and suction sides of the airfoil. The T B L displacement thickness 8", momentum thickness (9, friction velocity u*, cf and mean velocity U1 at the edge of the boundary layer 0.238 cm upstream of the TE are listed in Table 2 for a range of free stream velocities Uo. The values of It*, Cf and U1/Uo are

TABLE 2

Trailing-edge botmdary layer parameters for N A C A 0012 airfoil at 0 ~ angle of attack

8*(cm) Uo(m/s) Ul(m/s) ct u*(m/s) 8*(cm) O(cm) best fit

23.2 21.3 0.00225 0.7064 0.4274 0.2748 0.4125 38.6 35.9 0.00215 1-1623 0.4003 0.2641 0.4082 46.3 43.3 0.00200 1.3533 0.4063 0.2702 0"4063 54.1 50.8 0.00200 1-5855 0.4089 0.2730 0.4030 61.8 58.2 0.00200 1.8169 0.4005 0.2678 0.4013

in agreement with previously measured and predicted values [18, 19] for this NACA 0012 airfoil. This indicates flow similarity to the infinite stream case. The variance seen in the value of 8* with respect to Uo is unexpected and may represent a lack of precision in the measurement of 8*. Therefore values of 8* obtained from a best fit line (see Table 2) have been used for scaling purposes in this paper. Note that one would expect 8 " ~ Uo ~ from the 1/7th power velocity distribution law for flat plates with a zero pressure gradient TBL, whereas the present data are almost invariant with velocity.


2.7. TEST CONDITIONS

The primary test parameters were the free stream velocity Do, geometric angle of attack ag, and trailing edge geometry. The highest value for Uo was 73.4 m/s for a Reynolds number of 3"0 x 106 based on the chord. This corresponds to a value of 1.9 • 104 based on the boundary layer displacement thickness 8* at the trailing edge. The 2.5 mm blunt (t = 2.5 mm) trailing edge airfoil was tested at otg = 0 ~ 5 ~ and 10 ~ The sharp TE was tested at ag = 0 ~ and 5 ~ All other trailing edge geometries were tested at ag = 0 ~ In addition to the airfoil tests, tests employing a 9.5 mm diameter cylindrical rod placed across the span at the trailing edge location (with the airfoil removed) provided a comparison to the acoustic results of the airfoil.

3. FLUCTUATING SURFACE PRESSURE STATISTICS

The statistical properties of the surface pressure field have been defined by using spectral data. The pivotal function used is the cross spectral function G,v between two signals u and o. With G,o being complex and a function of frequency f , G,,o = 16.o(,')1 exp i~o,o(/') where q~,o is the phase angle between coherent components of u and v. When u = v, G , , =10..(s which is the autospectrum of u. When the frequency bandwidth of analysis is 1 Hz, then G, , equals the power spectral density (PSD) , defined here as @(f). In this study, the analysis bandwidth used was 100 Hz but levels were normalized to a 1 Hz bandwidth. The PSI) in terms of radian frequency to is @(~o)= ~(/)/2~.

3.1. POWER SPECTRAL DENSITY AND COHERENCE

Figure 10 shows the chordwise (longitudinal) variation of the P S D for Uo = 69.5 m/s. The results are for the sharp TE case. It shows that the level of the broadband spectrum increases as the trailing edge is approached from upstream. This is expected because the T B L thickness 8 increases in the downstream direction. In general, one expects the PSD to scale in proportion to 8, that is @([)~ 8, and the "characteristic" frequency (corresponding to the spectrum peak) should vary inversely with 8, that is f ~ 8 -1. The latter should be true because the dominant scales of the disturbances within the TBL are of the order of the boundary layer thickness [20]. Pressure fluctuations on the surface should be due primarily to the passage of these disturbances (or eddies) rather than the

[ { X } ~ i i I i i i I i i i i i i i

9O

" ' 4

O

7 0 1 1 l ! I ! I I I I I I I I I t 0 - 3 I I 0

F r e q u e n c y ( k H z )

Figure 10. Chordwise variation of PSI) for sharp TE airfoil. [/0=69.5 m/s. Key for distance -xl (cm) upstream of TE: ----., 1.854; • 2.205; ,2.898; - - - - - , 7-72; . . . . ,14.12; - - - - , 21.89.

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T R A I L I N G E D G E N O I S E P R E D I C T I O N 81

tithe rate of change of the structure of the eddies. Thus, eddies passing with convection velocity V give rise to a characteristic frequency ] '~ V/8.

The PSD's are normalized in Figure 11, where qo is the dynamic pressure of the free stream. The TBL displacement thickness 8*(--8) was measured only in the trailing edge region. Therefore 8* for measurement locations upstream of the edge region was calculated with 1/7th power law for fiat plates. From this, 8* scales with the distance from transition to the 0.8 power. The spectra of Figure 11 are seen to nearly coalesce.

60 r

v .&

o~ O

- - 3 0 I I l l I I I I I I i I I I I I I

-40

- - 50

- 6 0 i i i , I 0 " 0 5 0-1

! I a I l f I I I I t 1

w~'<'Uo Figure 11. Normalized PSD. Data and key from Figure 10.

A better fit of data would be obtained if 8" was scaled with the distance to the 1.2 power. From this one infers that 8 increases more rapidly for this adverse pressure gradient case than for a zero gradient TBL, which is expected. It is noted also that the normalized results are some 5 dB higher in level than those reported for a zero pressure gradient fully developed TBL [21]. The present results are however in agreement with those of Yu and Joshi [11] and of Blake [22], whose measurements were also made in a region of adverse pressure gradient.

Results showing chordwise variation of the PSD for the 2.5 mm blunted TE case are given in Figure 12 for the most downstream sensors. With the sharp edge extension

I00

9O

e

0 80

70

I I I I I I I I I I ! I I I I

: " : = -

, , , I , i l ! t i i 0"3 I I0

F r e q u e n c y (kHz)

Figure 12. Chordwise variation of PSD for 2.5 mm blunted TE airfoil. Uo = 69 .5 m / s . Key for distance - x l (cm) upstream of TE: - - - , 0 .254 ; - - - - , 0 . 5 8 4 ; . . . . , 0 . 9 3 5 ; , 1 .628.


removed the sensors are now very close to the trailing edge. It is seen in Figure 12 that the spectrum for the sensor at xl = 1.628 cm is similar to the spectra for comparable distances upstream for the sharp TE case. However, as the edge is approached substantial changes occur to the spectra. At the closest approach to the edge a spectral "hump" appears in the spectrum (near 3 kHz). This spectral hump is the result of a rather coherent vortex shedding phenomenon occurring in the small separated flow region aft of the blunt TE. In addition to this spectral hump, there is also an overall drop in spectral level at frequencies below the hump. This drop in level will subsequently be shown to be due to destructive interference occurring between the incident hydrodynamic pressure field and scattered pressure from the edge.

~0 N o - - 4 0

3

._~ - 50 0

- 5 0 | i i l I

- 6 0 , ,,,l 0.05 0"I

I I i i I I l i i i i i

" . o . .

t t t t t t i t [ i t t

I 5

wB' lU o

Figure 13. Normalized PSD for a range of free stream velocities for sharp TE airfoil. -xs = 2.898. Key for velocities Uo (m/s): . . . . . . ,30-9; - - - - - , 38.6; - - - , 54.1; ,69.5.

The normalized PSD's for various flow velocities for the sharp T E case are shown in Figure 13. The results are again quite consistent with those of Yu and Joshi [11] and of Blake [22]. The effect of a~ on the PSI) is shown in Figure 14. It is seen that the suction side of the airfoil experiences increases in level for the lower frequency ranges, whereas in this range the pressure side experiences decreased levels. Of course, the suction side develops a thicker boundary layer with increased ~g and the reverse occurs on the pressure side. The trends shown in Figure 14 thus are expected, because, from the previous discussions, of the behavior of the characteristic frequency, 1"-- 8- t .

Further information about the T B L pressure structure is provided by the coherence function .y2 defined between the signals u and o by y 2 = [G,,vI2/G,~G~,, where 0 ~< y ~< 1. The square root of this coherence function, y, is just the normalized cross spectrum. Also Y(D equals the maximum value of the time-based correlation coefficient if the two signals u and v have been passed through narrow band filters centered at jr prior to correlation.

Figure 15 shows the coherence function y2 of signals between two chordwise sensors, 0.351 cm apart. The sensor closest to the 2.5 mm blunted T E is at xt = - 0 . 5 8 4 cm. As expected 3, 2, shown for ~g = 0 ~ and 5 ~ demonstrates that the convecting T B L pressure field is not entirely frozen (unchanging). This is indicated by y2 being less than 1, which shows decay of coherence in the convecting field. The eddies responsible for the lower frequency contributions appear to be changing character (or decaying) downstream, although much less rapidly than those for the higher frequencies. Changing the angle of


I00

90

..<

8O

o o

70

i i i i i i [ i i i i i i i

Suction side

~ - - ~ . . . Suclion side

Pressure side

6 o v v v v i v l , , l n v , , L ~ 0-3 I I0

Frequency (kHz}

Figure 14. Effect of angle of attack cq on P S D for two free stream velocities for 2 . 5 m m blunted airfoil. - x l = 0 . 9 3 5 c m . Key for ag: - - - - , 1 0 ~ suction side; . . . . . , 5 ~ suction side; , 0 ~ - - - , 5 ~ pressure side;

- - - , 1 0 ~ pressure side.

1.0- - - i I i I I I I i I i i i i i Suction side

Pressure side ~ .

�9 ~ 0 . 5 " "

% 0 , , i , I I [ l l l ' ' ' ' ' -

0.3 I I 0

Frequency (kHz)

F i g u r e 15 . E f f e c t on ang le o f a t tack ag o n the c o h e r e n c e f u n c t i o n 72 b e t w e e n c h o r d w i s c (s t rcamwisr a l igned surface pressure sensors on the 2.5 m m b lun ted T E a i r fo i l . Uo = 69.5 m/s . - x l = 0 . 5 8 4 and ~r = 0 . 3 5 1 cm. K e y fo r ar ------, 5 ~ suct ion side; , 0 ~ - - - - - , 5 o pressure side.

attack is seen to affect y 2 primarily in the lower frequency range in a manner consistent with the observations of Figure 14. Coherence should be highest for the characteristic frequency f ~ 8 -1, where t$ is dependent on ag.

For two flow speeds the coherence function is given in Figure 16 for longitudinally (streamwise) and laterally (spanwise) positioned sensors for the sharp TE. It is seen that for a particular flow condition with equivalent sensor spacing (st1 -~ sra) coherence is lower in the lateral direction. In a crude sense the two types of coherence relate to different aspects of the pressure field. Whereas the longitudinal coherence relates more directly to the lifespan (or, inversely, the decay) of the eddies, the lateral coherence relates to

84 T. F. B R O O K S A N D T. H. H O D G S O N

l - O j_ I I I I I I J I I I I t I I

L.

:3,,.. 0.5"-

o . . . .

0-3 I I0

Frequency (kHz)

Figure 16. Effect of free s t ream velocity on coherence funct ion 3, 2 for longitudinally (streamwise) and laterally aligned sensors for the sharp TE airfoil. -x: = 1.524 cm. Key for/do (m/s): ---, 38.6; - - , 69.5.

physical size (or scale) of these eddies. It may be deduced from the shape of the coherence in Figure 16 that the two aspects are interrelated. The eddies with the largest scale have ihe longest lifespan.

3.2. I N C I D E N T I - t Y D R O D Y N A M I C P R E S S U R E F I E L D STATISTICS

As was seen in Figure 10 the sensors closest to the edge for the sharp TE case produced essentially uniform P S D results. Therefore, the edge scatter effect which had an important influence on P S D measurements substantially closer to the edge, as in Figure 12, did not significantly influence the sensor cluster results for the sharp TE case. This conclusion will be further substantiated in the scatter calculations of section 4. Measurements obtained by these sensors can be used to determine the statistics of the incident hydrodynamic pressure field upstream of the TE. The results can in turn be applied to produce the quantitative determination of the edge scatter effect and also to predict the TE noise field.

The co-ordinate system for the flow field used for this discussion is shown in Figure 6. The surface is located in a flow field where the velocity far from the surface is Uo parallel to the positive xl axis. Turbulence is convected above the surface at a velocity V making an angle fl to the xl axis. In non-skewed flow 3 = 0 , so that JvJ= V1 = V. Corresponding in direction to xl and x3 are the distances sr and ~:a denoting relative positions of observation points (pressure sensor locations) on the surface.

The coherence function and corresponding cross spectral phase data can be employed to quantify the T B L coherence decay and to determine convection velocities [21, 12, 23]. The importance of the cross spectral phase angle is that, for a convecting hydrodynamic field, tp,,v equals an important non-dimensional variable introduced by Corcos [23]. For a given separation distance and a given frequency, the phase is

~,.~ = ~ (~ , , et) = ~or V(~,, &), (1)

where V(to, r is the mean convective speed. The phase @,v is a positive quantity when the signal u lags v in time. V(to, ~ ) is positive in the xl direction. The distance is r = (x l ) , -(xO~,, where (xl),, and (xt)v are the co-ordinates of the sensors providing tt and v respectively. A hydrodynamic wavelength can be defined as ,~h = 2~rV(to, ~t)/to. Employing this wavelength concept one can define the wavenumber/.tlo=2~'/,~h. The

i

T R A I L I N G E D G E N O I S E P R E D I C T I O N 8 5

subscript I refers to the x~ direction and the sub-subscript 0 refers to a measured value. Later ~1 will refer to any wavenumber in the xl direction.

Figure 17 shows the normalized cross spectra -/ in the streamwise or longitudinal direction versus the phase q~ = lZ~oSr The results are for the sharp T E case for different values of Uo. Except for values of tz~o~:t corresponding to frequencies below the characteristic frequency, the datum points tend to collapse. Most of the variance is caused by statistical error due to low values of coherence, the effect of this being amplified in this presentation format. Each datum point is the result of matching the coherence value with its corresponding phase. The analysis bandwidth was 200 Hz. In its raw form the phase is restricted to its principal value between +~'. Multiples of zr were added to the principal values by a selection process in which the validity of equation (1) is assumed.

[.0

~. 0.5

l I f i l l l i I l l i l i i i i l i I l l l I i i l l l l l l i t l l l

\ (o) [

>, ! ( k\\ ~'r/=O.ll(flat plate)

-o \ /

- - " ~ \ \ "g : 0 . 1 9

~ \ ,

~ o ~ , ~ ~ =m~ e"=-,~ "" ---- " "

,b,

�9 "r} =0.II ( flat plate) _ -

t:

0 5 10 15 20

Figure 17. Normalized longitudinal cross spectra for (a) Uo=38.6m/s and (b) [/0= 69.5 m/s. - x t= 1.524cm. Curves given for exp(-r/.ulo~t) for different r/. Keys for ~t(cm) values are as follows (with corresponding ~1/8" values in parentheses): (a) i , 0.330 (0.81); O, 0.681 (1.67); A, 1.374 (3.37); ~r, 6"20 (15"20); e , 12"60 (30"88); (b) i , 0"330 (0"83); O, 0"681 (1-71); A, 1"374 (3"46); ~r, 6"20 (15"60); @, 12.60 (31"70).

The fact that data tend to collapse on/-tlos indicates that in general disturbances or "eddies". retain their coherence (or identity) for the time it takes to be convected a distance proport ional to its scale. In Figure 17(a) for the Uo = 38.6 m / s case it is seen that a typical " eddy" of wavelength kh loses approximately one half of its " ident i ty" by the time it takes to be convected a distance of An/2 (/.tlo~ = 7r). A reasonable fit to the


data is found by employing the relationship 7 = exp (-r//Ztosel) where the constant "r/ is set to equal 0.19 for the Uo = 38.6 m/s case and 7/= 0.14 for the 69.5 m/s case. These empirical curves are shown in Figure 17. Additionally shown is the curve for 7/= 0-11 which has been found to fit the data of Corcos [23] for a zero pressure gradient TBL. It is observed that coherence decays more rapidly for the present case, and more so for the lower mean stream velocity. The more rapid decay is an indication of a larger time rate of change of the structure of the turbulence.

In Figure 18 variations of 1' with lateral separation se3 are plotted. Note that se3 has been fiormalized by/Zto(=~o/V) as determined from streamwise coherence measurements. The notation/.t3ose 3 is used with the understanding that #30 =/-/ '1o for set = se3. Where the sensor spacings set and se3 did not match, a method of interpolation was used to make the phase data applicable to the se3 spacing. It should be noted here that the measured lateral phase was zero, as expected, because the flow is normal to the trailing edge: i.e., fl =0 . The data in the figures collapse because coherence over a lateral distance is

1,0

~- 0,5

d i i l i l i l i i i i l l l i l i l i l l i i i I i l l i l l i i l i i

(o)

~ ~ ~ =0,62 �9 , .

"%~_ f ~ =0.714 (fla! plole)

, , I " , # �9 , -

f ,b, C=o.sB

I I I I I I I I I I I I I I | I I I I I - - I F P ' T ~ ' - ~ J I I

0 2 - 5 5 " 0 7 " 5 I 0 - 0

(P = F3o ~3

Figure 18. Normalized lateral cross spectra for (a) Uo = 38.6 m/s and (b) U0 = 69.5 m/s. -x~-- 1.524 cm. Curves given for exp (-~'~3o~3) for different r/. Keys for ~3(cm) values are as follows (with corresponding ~'3/6" values in parentheses): (a) II, 0.325 (0.80); O, 0.540 (1.32); A, 1.652 (4.05)i * , 2.190 (5.37); (b) II, 0.325 (0.82); O, 0.540 (1-36); A, 1.652 (4.16); * , 2.190 (5.51).


proportional t o the scale Ah. By employing the relationship y = exp (--~'/.t3o~3) with the constant s r set to 0.62 for the 38.6 m/s case and to 0.58 for 69.5 m/s a good fit to the data is obtained. This is shown in Figure 18 along with a fit to the data of Corcos with s r = 0.714. Little difference is observed between the present results and those of Corcos.

The empirical curves given are seen to be not generally applicable to frequency ranges below the characteristic frequency (1 kHz for the 69.5 m/s case). Chase [24] has recently examined the limitations of this exponential form and other empirical models. Regardless, below this frequency the same empirical form with different values of ( has been found here to describe individual frequencies: for example, that with s r = 1.6 approximately describes behavior of 750 Hz in Figure 18(b). One can take advantage of this in the prediction of TE noise for the low frequency range (see section 5).

The convection velocity ratio V(oJ, ~l)/Uo as a function of ]" and sr is shown in Figure 19 for the mean stream velocities of 38.6 m/s and 69-5 m/s, respectively. The results

I 'O I I I I I l l l l I I I I I I I I I l l I I I l l [ i i i l l l l I I I I l i l l i i i i l l i l l / / - ~ r ( o )

I ~.,...~L, ~ 15.20

0.5 ," -. .......... ,, ,v,. . I,, ~ ~ i - 0 . 8 1

~o 0 I . . . . I . . . . I . . . . I . . . . I . . . . I , , , , I . . . . I , , I

-~ f.O 1 ~'t/S*=31.70 (b) ~f-'~ .~/15.60

0.5 ~J I F ~ ' c z : : ' : . . . . . . . . . . . . , ' , . . . . ..,/ '__.

_y L'--0"83

, , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I , , , , I 0 5 I0

Frequency (kHz)

Figure 19. Convectionvelocityasafunctionoffrequencyandsensorseparationdistancefor(a) Uo=38.6m/s and (b) Uo = 69.5 m/s.

were obtained from phase data and by using equation (1). Only those data which correspond to frequency ranges showing significant coherence are shown. In both figures it is seen that convection speed increases with increasing separation distance, but, inversely, the frequency range over which the pressure field is coherent is reduced. Coherence measurements between different points amount to a spatial filtering eliminat- ing the contributions from the small eddies which become incoherent in their travel.

88 T. F. B R O O K S A N D T. H. H O D G S O N

With the lifespan of an eddy being proportional to its scale, only the larger scale eddies will remain coherent between two widely separated points. Also the larger scale eddies with their effective center farther from the airfoil surface will have the larger convection velocity. In Figure 19(b), for the largest separation distance, it is seen that only over a frequency range centered around 1 kHz are the pressure signals coherent. Here V/Uo is seen to be about 0.8. At frequencies lower than this characteristic frequency the signals become incoherent. Even at closer separation distances where there is some coherence the convection velocity drops substantially. The TBL is seen to be not supporting the development of eddy scales much larger than 3.

Direct comparison between Figures 19(a) and (b) reveals for most frequency ranges that V/Uo is higher for larger Uo. Also the characteristic frequency is lower for the lower speed, which is explained by the relation ] '~ V/& In Figure 20 the convection velocity data from Figure 19 are plotted with their corresponding phase data. Comparison reveals only insignificant differences between the two sets of results. This shows that the convection velocity measured for a given separation distance depends more on the eddy scale as compared to the distance traveled than on frequency itself.

0.5

o #

] _~TTi i l i t i i I i i l i I l i i l I l l I i l I ~ . j / ~ , = 3 o . 8 8 ( a ) /

O 0 0 O

15 20 . �9 A

I LO.8 I

, , , I , , , , I , , , l l , , , l l , , , , I , , , , I , , , , I , , ,

t .

i (~) / ~,1~*: 31.70 __- O

0 ~ 0 0 0 0 0 _

~, *~15-60

5-46

0.5 ~ ~ ' ~ ~ l'TI -~

I - ~ 0.83

, I I I , , I i I , I , , I I I I I I I I I I I , I I I I , , , I I , , I I

0 5 I0 15 20 ~o=Fio~j

Figure 20. Convection velocity t's. cross spectral phase for (a) /30 = 38.6 m/s and (b) Uo = 69-5 m/s.

i TRAILING EDGE NOISE PREDICTION 89

The empirical curve fits for the coherence decay of Figures 17 and 18 were appropriate for the larger values of ~ o ~ and tx3o~:3. Correspondingly one may obtain an empirical fit to the convection velocity data for large/X~o~q. Figure 21 shows the range of measured values of V/Uo for different measurement distances for the two values of Uo considered.

1.0

~o .~ 0.5

i I I

S -

I I

I ! I I I o Jo is 20 2s so ss

~ , / 8 "

Figure 21. Convection velocity as a function of sensor separation distance for large values of tZlo~l (high requency, tz lo~r > 3.0). Curve shown is fit to data, 0.39(1 + 2~J8") ~ Key for data range is:Z, Uo = 38.6 m/s; ], Uo -= 69.5 m/s.

Fhe data bars contain 90% of the data to Figure 20 for/Zlo~q>3.0. The data are seen o be approximated by the curve

V/Uo = 0.39(1 + 2~q/6") ~ (2a)

~ragmatically the curve should be limited to a maximum of 0.9. The value of 0.39 was :hosen to correspond to the expected lower limit of V/Uo in a TBL as reported by ~'illmarth [20], from consideration of the work of Emmerling [25] and others. In the ower frequency range (small/Xlo~l) V/Uo is not independent of frequency. In this range he model for convection speed for Uo = 69.5 m/s can be chosen as

V / U o -~ [ - 9 x 1 0 - 6 f + 0.5(1 + ~q/8")~ (2b)

~hich roughly fits the data of Figure 19(b) in the lower frequency range (but not much ,elow the "characteristic" frequency). Equation (2b) is used in subsequent discussions ,f the edge scatter phenomenon (see section 4.2).

The foregoing analysis has been concentrated on establishing th6 statistics of the ydrodynamic pressure field in regions which are far enough upstream of the trailing dge for the edge scattering to be unimportant. The primary empirical result may be zritten, by using the similarity hypothesis of Corcos [23], as

Oe, e3 = qb(D exp {-~o1~d-~'~3o1~31 +i~,o~:;}, (3)

,here Ge,r is the cross spectrum between the signals of two sensors separated by a )ngitudinal distance ~:1 and a lateral distance ~r from one another, qb(f) is the PSD of ae incident hydrodynamic pressure field. The constants r/ and ( are given in Figures 7 and 18 for large values of tZto~:3 and tt~o~3. It is found that tXlo=a~/V where V is iven by equation (2a) for large values of t~Xo~ and by equation (2b) for low values of ,1o~r Correspondingly tZ3o is determined by replacing ~:1 by ~r in the equations for V. :,quations (2a) and (3) are appropriate for frequencies above the "characteristic" 'equency of the TBL.

90 T . F . BROOKS AND T. H. HODGSON

3.3. WAVENUMBER SPECTRUM OF INCIDENT PRESSURE FIELD

An alternative representation of the hydrodynamic pressure field can be constructed by modeling the field as a distribution of harmonic evanescent waves. These waves are produced by convecting turbulent elements located above the surface at x2>0. A wavenumber spectrum function H(g l , ~3, co) can be dcfined to satisfy

(p(xl, x3, 4)p(xl +r x3-}" ~3, [ q" T))

= f f f:~o H(/x,, P-3, (o) exp [i(/x 1~1 +/z3~:3 - r dp. 1 d/.t 3 d~. (4)

The left side of this equation is a space-time correlation of the surface pressure p. The integrals are taken over all possible values of wavenumbers/xl and/z3, corresponding to the xx and x3 axis respectively.

By noting that the correlation function of equation (4) is the Fourier transform of the cross spectrum function one has

G"r = f [?oo//( /Xl, / . t3 , ( 0 ) e x p [i(/.t l e l +/x3e3)] d/z1 d/z3

o r

/-/(/z 1,/z3, (o) = (-~)2 Gele~ exp [-i(/z 1~1 +/z3~:3)] d~:l d~:3.

Upon using equation (3) and simplifying one has

/'/'(/d. 1, /t~3, (.0) = r162 1)r ,

where

(5)

(6)

ta3 = 1/~'/~3o = Q/~'oJ, (10)

(/)3(/Z3) = ~" [ 1 .{_ ~2]d.32 ] �9 (11, 12)

These results would equal those of Chandiramani [5] if the constants r/ and ~" were chosen to agree with the data of Corcos [23].

Calculations can now be made for frequencies above the "characteristic" frequency of the TBL. Figure 22 shows the longitudinal and lateral integral scales 61 and C3 as

#1 = 1 / r / t Z l o = "v'/nw, and equations (7) and (8) would give

qh(/xO =--~'1[ 1 ] 7r l+ ta~(p , lo-- /x l ) z '

1 r~o qSl(tx') = ~ J o exp [-ntx,or ] cos [(/x, o - ~1)r de,, (7)

@3(is3) =--1 f~o exp [-~'/X3o~:a] cos (#3~:3) dr (8) 71" J o

One can define the integral scale 81 (and ('3) for the longitudinal (and lateral) direction as

I : /I: 61 = exp [-r/~lo~q]~:l dr exp [-n/Xlor dr (9)

and 63 is similarly defined but with g', t~3o and r replacing r/, ~1o and ~1 respectively. If it could be assumed that convection speed is constant and equal to an appropriate

mean, V, then equation (9) would become

V R A , L , N O EDO . N O , S E I"RED, ,O 9 1

, 0 ' ' ' ' . . . . . . . . . . . .

~ U o = 3 ~ . 6 m/s, 6

xx\ --U0=69'5 m/s,/a I x

@ "Xx

0.1 ~

0 " 0 ] f | I I I I I Be I I I I I 1 1 1 1 I I

04 0'5 I 5 I0 50

Frequency (kHz)

Figure 22. Longitudinal and lateral integral turbulence scale lengths, ~'1 and e3, as a function of frequency for two free stream velocities. -, Scale lengths; - - - , approximate scale lengths using Q = 0.6 Uo.

calculated numerically from equations (9) and (2a) for two free stream velocities. Also shown for comparison are those obtained by using the approximate solution equation (10), in which it was assumed that 17= 0"6Uo. In the calculations the constants r/and are those determined from the present data. The more accurate solutions for t'1 and ~'3 are seen to fall off somewhat more rapidly with increasing frequency than those based on the constant V assumption.

The longitudinal wavenumber function qbl normalized to its integral scale ~'i is presented in Figure 23 for two frequencies and one value of Uo. The numerical results, as obtained by using equations (2a), (7) and (9), are the lower two curves. The figure shows

1,0

0"8

0.6

0 . 4

0-2 e

s

0 . . . . 4-"" -I 0

' I !

i

0

2 3 /.q

f/,74

Figure 23. Longitudinal wavenumber function normalized to integral scale for U o =6 9 .5 m/s. Key for :quencies: �9 , 5 . 0 k H z ; - - - - , 2 .5 kHz; - - - , all frequencies for V = 0 .6 Uo approx.

92 T. F. B R O O K S A N D T. H . H O D G S O N

how the energy is distributed over the /z l wavenumber space. In particular, it shows the distribution of statistically independent wall pressure waves which must be present in order to satisfy the measured cross spectra, as modeled for the hydrodynamic field. The rat io/xl / (1/r /61) may be interpreted as a ratio of an integral scale wavelength (27rr/(~) to the wavelength (2~'//x0 of an evanescent pressure wave. A large value for/z~ means that the wavelength is small (small eddies) and the convection speed is correspondingly low to maintain a constant frequency. The fact that c/5~ is non-zero only for positive values of ~ indicates that, as modeled, all energy is convected downstream and none upstream. Non-zero values of @~ begin at about t.t~r/(~ = 1 which corresponds approximately to the highest convection speed measured (V = 0.9 Uo). The peak values for qb: at about /z~/ (1 = 1.25 correspond to convection velocities of V = 0.68 Uo for the 2.5 kHz case and to V = 0 . 5 8 U 0 for the 5 kHz case. At /z~r /6~=2.5 these velocities, and thus the eddy sizes, are reduced by a factor of 2. It is seen that when a PSI) is measured the perceived signal is due to contributions from a broad range of eddy sizes even when attention is restricted to a narrow frequency range.

For comparative purposes, included in Figure 23 is a presentation of @~ determined from equations (10) and (11) with V assumed constant and equal to 0.6 Uo. This spectrum is unrealistic for an incompressible field because of its non-zero value [26"1 at t-t~ = 0. This function is seen to be highly peaked at tz~r/6~ = 1 as expected. The sharpness of the peak depends on the decay constant r/. If r / w e r e to approach zero, corresponding to no decay, the wavenumber spectrum bandwidth would approach zero.

The measured cross spectral function for the incident pressure field is to be employed presently to explain edge scatter phenomena. The deduced wavenumber spectra will be used to predict the T E noise field.

4. NEAR FIELD EDGE SCA'IffER

The pressure field very close to the T E region is significantly altered from that measured upstream. Very close to the edge an overall drop in spectral level is observed for frequencies below the spectral hump (see Figure 12). The hump is associated with a shedding process due to TE bluntness and this effect is not considered here. Of interest are the broadband spectral alterations which should occur regardless of the precise geometry of the TE, as long as the size of the geometric details are small compared to the hydrodynamic wavelengths, Ah, of the eddies within the TBL.

Solutions will be given for the effect of near field scattering of the hydrodynamic pressure field on surface pressure measurements in the vicinity of the trailing edge. In the analysis use is made of a solution based on evanescent wave theory I-4] for the case where the turbulence is assumed to be non-decaying and constrained to move above the surface on one side of a semi-infinite plate. The present two-sided flow case for an airfoil is treated by a linear superposition of results of the one-sided flow case. The important effect of T B L coherence decay is introduced. The solutions are compared with the measurements for both the sharp and blunt TE. Then the pertinence of these results is discussed with regard to a physical interpretation of the edge condition.

4.1. THEORY

Howe [1], after Chase [4], has given a relation for surface pressure on a plate upstream of a pressure scattering trailing edge, consistent with Chandiramani's results [5]. The form of the relation is appropriate to the ease where the turbulence is frozen as it is


convected downstream. The instantaneous pressure on the plate at position (xl. x3) is

p(x l , xa, ,/)

---21 d/-t i d/.ta d z l l + s g n ( x 2 , E r f ( i x , - . 32 ) '/2 + i t , + to.

X N(/ . t 1,/tL3, Z, toz) exp (i[~tXl -~-/.L3X3-- toz~./]), (13)

where toz = ~ . V(z) =/.t~ Vl(z)+/z3 V3(z). The error function characterizes the edge scatter and will be presently discussed. The symbol sgn (x2) = + or - , depending on which side of the rigid surface is approached. The turbulent flow is constrained to the upper surface (sgn (x2) = +). The term z is the height above the plate where the pressure-causing turbulence is located, z is aligned with the positive x2 co-cordinate.

N is a grouping of factors expressing all the statistical properties of the turbulence. N contains the factor [1] 0 = {)(btt,/x3, z) which is a Fourier transform of dipole sources, induced by the incident turbulence,

Q(xl, x3, z) = to(x1 - Vie, z, x3 - V3e) x V(z),

where to is the vorticity vector whose dependence on time, l , reflects the frozen nature of the incident field. The only time dependence in equation (13) is represented by the -itozr term in the exponent, which comes from the transform of Q. The function N is non-zero and contributes to the integral until z > 6. The integral over z in equation (13) shows that the total pressure field at z = 0 (x2 = +0) is a superposition of contributions from the vorticity at various layers or "s t ra ta" above the plate, represented by the thickness dz at z. Each stratum has its own wavenumber spectral distribution convecting at V(z).

Equation (13) is not in a suitable form for experimental comparison: The experimental data are in the form of auto- and cross spectra of and between pressures measured at various positions on the surface. For the purpose of comparison, the equation will be appropriately simplified for these experimental conditions, and then the effect of T B L coherent decay will be introduced.

For a non-skewed flow field where the mean flow is perpendicular to the edge, N should be a function which is symmetrically peaked about/z3 = 0. In order to eliminate the integration with respect to/.t3, N(/z~,/z3, z, toz) is approximated by N ( ~ t , z, toz)D(pt3), where D is the Dirac delta function. This means the flow is now modeled as two- dimensional with the lines of vorticity extending from - co <x3 < co. This should be a good approximation if subsequent comparison for measurement points in the vicinity of the edge is restricted to the xt axis (x3 = 0).

One can now look at the contribution to the integral from a stratum Az at Zo. Calling this pressure contribution Pzo and noting that the Fourier transform of Pzo is

p~o(Xl, ~') e i'~ d t , (14) P~o(Xl, to, T) =~-g~ r

a~here the integration time T goes to infinity, one obtains, from equation (13),

P ( x l, to, r - - , co) = P~o(X ~, to) = �89 1, Zo, to,o)~ V (zo)}Zlz

x {1 +sgn (x2) Erf (~/ixl(/-Ll + k~o(1 +Mo)))} exp [i/xlxl], (15)

where /-LI =~o~o/Vl(zo)=/xl(Zo) and kzo= to~o/C. This is for one stratum and the total result is obtained from a summation of contributions from all Zo. If V(zo)cou ld be

9 4 T . F . B R O O K S A N D T. H. H O D G S O N

assumed constant over all zo, 0 ~< zo ~< 3, then equation (15) with Az replaced by 8 would be the total, with N taking on an appropriate mean result.

In equation (15) the error function characterizes the diffraction or scatter by the edge. When xl ~ -oo the error function approaches 1. For the upper surface, sgn (x2) = +,

Pzo(X, ~ -oo, to) = Pi exp [i/zlx,],

where Pi is defined as Pi = N(/~I, Zo, tozo) • Az/Vffzo). On the lower surface, sgn (x2) = - ,

P~o(Xl "* -oo, to) = O.

It is convenient to adopt the notation

qxl = �89 - E r f (4ix1[/~1 + Go(1 + Mo)])},

which may he rewritten as

qx, = �89 - ~ exp (-i~'/4)[C(~/(2/rr)lxd[~ ~ + Go(1 -i-/~/o)])

+ i s ( 4 ( 2 / ~ ) l x l l [ ~ + k~o(1 + Mo)])]},

because on the plate the co-ordinate xl = - I x d . C(arg) and S(arg) are Fresnel integrals [27] of the arguments. Equation (15) becomes

P~o(Xl, to) =Pi [1 -qx l ] exp (itzlxl), sgn (x2) = +,

=Piqxl exp (i/zlxl), sgn (x2) = - . (16)

Written in this way, Pi exp (i/zlxa) is the Fourier transform of the incident hydrodynamic pressure field on the plate and would be the whole contributor to P:o if no edge were encountered by the turbulence, qxt effects a contribution to Pzo which is proportional to P~. Piq,, exp (i/-tlxl) is the scattered pressure from the edge. The signs of q~l in equation (16) show that the opposite surfaces scatter equally in magnitude but out of phase by 180 ~ Note that at the trailing edge (x~ = 0) the value of q~ equals 1/2, meaning that P~o(Xl, to)= Pd2 regardless of which side of the plate is approached. Therefore, there is no pressure differential across the trailing edge.

Chandiramani [5] has made a definite distinction between the compressive or freely propagating component of scatter, which becomes the far field TE noise, and the incompressive or near field component of scatter. The function defined here as q~, is representative of the sum of these two components. Close to the edge the incompressive component is expected to dominate whereas the compressive component dominates far upstream. The scatter term q~, is plotted in Figure 24. q~, is a decreasing function of Ixd(t~+Go(l+Mo)) which is maximum at x~ = 0 at the .trailing edge and decreases upstream. The significant drop in amplitude just upstream of the edge is accompanied by a phase shift. In this region the incompressive component of the scattered pressure should dominate. Farther upstream where compressive scatter should dominate the amplitude starts to level off and the phase = constant + ~ dxd + k~o(1 + Mo)lxd. This amplitude and phase behavior would be expected since, based on Chandiramani's type of analysis, the compressive scatter can be shown to vary as

c( ' , / (4/~r)GolXd)-iS( , / (4/rr)GolXd) exp (iGolXd), (ezo)comp . . . . ire-- 2" 'dXd

�9 which is a very slowly varying function for small x~. The above results suggest that the effect of scatter at xl is observed, after the incident field passes xl, by the time required for the field to translate to the trailing edge at velocity V(zo) and radiate back upstream at a velocity c/(1 +Mo). Note that (1 +Mo) -~ is an approximation to (1 -Mo) . In derivin~ equation (13), Howe [1] systematically neglected terms ~ O ( M 2) relative to unity.

0.5

T R A I L I N G E D G E N O I S E P R E D I C T I O N

I Ij..-'- ...e" I I ! I I / I " .I-"

95

0.4

0.3 c~

0-2

0-1

I I I I I. I I I I o J z 3 4 5 6 7 8 9 no

I x, I (~ + kZo(I +Mo))

Figu re 24. P ressure s ca t t e r coeff icient as a func t ion of d i s t ance f rom T E and the h y d r o d y n a m i c and acous t i c w a v e n u m b e r s . R e f e r e n c e phase l ine, ~ ~ , is p h a s e = Ixzl(/xx + k~o(l +A/o)) .

Equation (16) can be used to determine the auto- and cross spectra of the surface pressure on the plate. The cross spectrum between two points xt and .ft on the upper surface (sgn (x2)= +), where the co-ordinates of s coincide with the xi axis, is, using equation (14),

G~,,~, = l i m ( 1 / T ) { P * ( x x , a~, T)P( .~I , a~, T)} . (17)

From equation (16) it is seen that

P ,o (x i, a~ )P,o( ; , , co) = P.*, P,[1 - q**, - q;, + q *~,q;,] exp [- i /z i (xi - ; i )] .

Therefore , from equation (17),

G,~,~, - Gii[1 - q x , - q ~ , + qx,q~,] exp [-i/zl(Xl --'?i)], (18)

where Gi~ is the autospectrum of the incident pressure field. If no edge were present, then of course q~, and qxl would be equal to zero, leaving G , exp [-it~ s(xi - s indicating pure convection of the frozen turbulent pressure field. When the edge is present the autospectrum for a point xl = Xl is Gx,x, = Gii[1 - 2 Re (qx,)+ Iqx,lZ], from which it is seen that a surface pressure measurement near an edge can be significantly influenced by the superposition of the scattered pressure field.

The cross spectrum between locations x~ and Xl, where Xl is located on the lower surface, sgn ( x 2 ) = - , may be determined by using equations (16) and (17). The cross spectrum is

G,,,~,I = G,,[qT, t -q*~q~,] exp [ - i / z t (x i -Xl ) ] , (19)

~r shows that wi thoutscat ter no correlation would exist for points on opposite sides 3f the plate.

For two-sided turbulent flow, the cross spectrum will be modified. From before, the ncident pressure field on the upper side is P~. This may now be called p+O with p;-O Jeing the corresponding field on the lower surface, x2 = - 0 . It is assumed that the incident

9 6 T . F . BROOKS AND T. H. H O D G S O N

fields on opposite sides are non-interactive and independent but are statistically similar: that is (P~~176 = 0 but (p~O),(p;O) = (pTO),(pTO). The incident and scattered fields of each superimpose on the other. Therefore,

p.o (xl ' to)p~o(~l, to) = [p~O +o -o , - P i qx,+Pi q.~] e x p [ - i g l x l ]

x [p~O +o -o - P i qz,+P~ qz,] exp [i/x1s

which leads to

and also

G,,~, = G,[1 - q *, - q~, + 2q ~,q~,] exp [ - i t t 1(xl - s

Gx,~,, = Gii[q*, + qg, - 2q *tqf,,] exp [ - i ~ l ( x l - xl)].

(20)

(21)

4.1.1. Effect of TBL coherent decay Equations (20) and (21) follow from equation (16), which is based on the assumption

that the pressure field is frozen: i.e., Pi is constant in a reference frame moving at velocity V(zo). To include the real effect of coherent decay the factor h, 0<~h ~ 1 , can be introduced. For the present the convection speed is still considered constant for simplicity. To examine hydrodynamic pressures only for the moment, one may take qxl-----q~, = 0: i.e., xl ~ oo. Therefore , for one-sided flow,

* t �9 - P,o(Xl, to)Pzo(s to) = [Pi]* exp [='il.tlxl][hPi + ( ! - h)Pi] exp [1/~1xl], (22)

which leads to

Gx,~ = hGil exp [ - i ~ l ( x t -s (23)

Note that this is in the same form as the empirical result, equation (3), if h = exp and This observation will be employed later. In equation (22), P~ is the transform of the pressure at Xl. The pressure at s is the sum of that which is common to xl and that which is unrelated to that measured at x~. The functional form of equation (22) expresses the assumption that the autospectra of the hydrodynamic pressure field at x~ and s are identical and equal to G~ (homogeneity assumed). The term ( 1 - h)P'i represents turbulence pressure characteristics which have been generated from xl to s

In order to obtain solutions for cross spectra which are equivalent to equations (20) and (21) but include the effect of decay, one can first note that equation (16) is valid for that part of the turbulence which passes the trailing edge. When turbulence characteristics decay before reaching the edge, the equation is valid only when the scatter terms equal zero.

Let Po represent the T B L pressure field at the edge, x, = 0. Equation (16) is the solution for the part of the pressure field at xl related to Po, therefore replacing P~ by hx,Po, where hx, is the decay from xl to xl = 0, one obtains

f P~o(Xl, to) = hxtPo[1 -qx~] exp [ i thxl] for sgn (x2) = +

= h~Poq~t exp [ ig lxl ] for sgn (x2) = - .

Here the superscript I refers to one solution. Another solution, u P,o(Xl, to), is obtained from equation (16) to account for the part decayed, ( 1 - h , , ) P T , prior to reaching the edge. With q~, = 0 one obtains

I I rt P~o(Xl, to) = ( 1 - h~)P~ exp [it t lxl] for sgn (x2) = +,

= 0 for sgn (x2) = - .


Adding the solutions gives

P~o(x~, to) = [ h ~ P o + (1 - h~,)P'~ - h ~ P o q ~ ] exp [i/zlxx]

-- hx~Poqx~ exp [i/z~x~]

97

for sgn (xz) = +,

for sgn (x2) = - .

Note that Poh.~ + Pg ( t - h~,)= Pi, which is the incident pressure at xa, and also note that Po = h,,,P~ + ( 1 - h,,)P~, where ( I - h,,,)P~ is that which was generated from x~ to xx = 0 (as in equation (22)). It is found that

P~o(xt. oJ) 2 , = [Pi - Pih ~ q ~ - P i (1 - hx:)h~q~,] exp [it t lxl] for sgn (x2) = +,

P~o(Xl, t o ) = [ P ~ h ~ t h L q L + P ~ ( 1 - h ~ ) h T , , q ~ , , ] e x p [ i l z l x l ] for sgn (x2) = - , (24)

where xl has been used to indicate terms which are related to the lower surface. The cross spectrum G ~ , for opposing sides for one-sided flow is obtained in a straightforward manner as

,, 2 G ~ , = O . [h~ , (h~ ,q~ , ) - (2h ~, - 2h~, + 1)(h~q*, )(h~q~,)] exp [-i/x x(x~ -.,~1)].

For two-sided flow

G~,~ = G.[h~(hLq~,~) + h~,~(hxtq*~ ) - [2(h ~, + z h~, )

- 2(h~ + h~,) + 2](h~q~*,)(h:~q~)] exp [ - i t t ~ (xt - x~)]. (25)

The cross spectrum G ~ , for measurement locations on the same side can also be obtained for two-sided flow. This is

Oxt;, = Gii{h - hh.~(h.~q; 0 - [ hhx: + (h;~ - h~)(1 - h )]h,,tq* ,

+ [hhx ,h~ + (h;, - h~,) (1 - h ) h ; t + (1 - h;~) 2 + 1]

x h,,~q*~ (h;~q;~)} exp [ - i ~ 1(xl - xl)], (26)

where h as before represents the decay between x~ and xl. Equations (25) and (26) are ~qual to equations (21) and (20) when h = 1 for a non-decaying field.

~.2. C O M P A R I S O N

Equations (25) and (26) represent the predicted behavior of the cross spectra between measurement points on the surface if convection speed may be considered constant, in ~hich case t , i is taken as a function of frequency only. In order to compare the predictions :o experimental data the fact that measured convection speed is also dependent on :listance should be taken into account. To do this one matches the ' terms of equations 125) and (26) to that of equation (3). /xl is replaced by (/Xlo)x, = to /V( to , xx) where the ;ubscript xx means that the quantity must be referenced to the distance Ix,I to the TE. I'he same is true for the /xl terms associated with .rl and xl resulting in (Stlo)~r, and iP-lo)Ir The model for convection speed is chosen as equation (2b) for the lower frequency range where the results are to be compared. Therefore / z l ( x l - x 0 is replaced by ~/-tlo)x,xx-(ttao)~,xa i n the phase terms, and h;,=exp{-rl(Ulo),~,l'2d} and h = ~.xp {-~/(p-lo)~-~olxi-.~d} for the decay terms, etc.

Experimental cross spectral data were obtained for sensors located near the T E region !or both the sharp and 2.5 mm blunted T E for the airfoil at zero angle of attack. Sketches ~f the sensor arrangement are shown in Figure 25 for both edges. The sensor designated is a is taken as representing xx, bl as s and b2, b3 and b4 as xl locations in the :erminology of equations (25) and (26). The sensor a is directly opposite b4. The cross

G * ;pectra Gbi a a r e given in Figure 26(a) for the blunted edge case. Note that b,~ ---- Gxt~ra

9 8 T. F. BROOKS A N D T. H. H O D G S O N

2.5ram blunted trailing edge

b s b2

Sharp trolling edge 0

b=

Figure 25. Geometry of sensor a r rangement near TE where cross spectral data are obtained.

or Gx,, , of equations (25) and (26). The phase tpb,, is given as discrete datum points because of data variance. The correspondences of the datum points and amplitude curves are indicated. The spectrum with the highest level is that of Ga, which is the autospectrum at sensor a. Its phase is identically zero (not shown). Gbl a has a decreased amplitude and its phase is seen to be determined primarily by a time delay effect, i.e., ~bz, /xlola - b t l , of convecting TBL. For the cross spectra between sensors on opposing sides,

~ F ~ ~ "]" ] / / / [ --~ -- ~ / I I / / / ~ ~ n ~

F Y . . Ill ," , Y / i IJ., *7 II " " / I / Y

I : . ~,'o ***o "Pl - ~ L " * ~ i , , , ,,,_1 ,,'; 90 . ~ f r o m measured 4

" - ' ~ " ' - ' " " - - I~ - .~""7 " ~ " ~ C I ' ~ ' ' - L::~",.. -,.-,-v-, ~ s ~a'- t / k , ~, ~ ''~ ; ' ' " ~

. , , , , . ~ . , ~ _ . -~,, I/" /',, _ i \ F A -

60 l i l l l l l l l l i i i l i l l i l i i i l , , , , I I , ' , i I i i l l , I , , ,~,61 , 'D,, i -- 0 I 2 3 4 5 0 I 2 5 4 5

Frequency (kHz) (o) (b)

F i g u r e 26. Cross spectra, (a) measured and (b) predicted, between sensors for the 2.5 mm blunt TE, Uo = 69.5 m/s . Keys for sensor posit ions of Figure 25 are: - - - - , x l = a = - 1 . 6 2 8 cm; I - - - - - , ~l = bl = - 0 . 2 5 4 cm; and for sensors on opposi te side to a : A - - , .~1 = b2 = - 0 . 2 5 4 cm; O , -~1 = ba = - 0 . 5 8 4 cm; -k - - - - , x l = b4 = - 1 - 6 2 8 cm.


i.e., G b:, where b i = b2, b3 and b4, the amplitudes are again lowered and the trend of the phase behavior becomes less explicable unless one takes into account that scatter superimposes on the TBL pressure field.

For the conditions of Figure 26(a), the cross spectra are predicted by using equations (25) and (26), in which the results have been referenced to the measured autospectra at sensor a : i.e., the only experimental data in Figure 26(b) are for G~. Close comparison between Figures 26(a) and (b) reveal that all major characteristics of the cross spectra have been predicted, after allowing for the presence of experimental data variance. Even the theoretical levels for the spectral "humps" are in reasonable agreement.

This comparison between experiment and theory is shown for the sharp TE case in Figure 27, Again suitable agreement is found.

,g 8

/ o - , ,

,~" 01~. , o o . , - I L I

- . L%tt,:" r ,." J 90

80

70

6C

:o "-.-" - V ~ / - . ~ "~C"

F , , , , I , , , , I , , i , l , , , , I , , , , I 0 I 2 3 4 5 0

/ - - f r o m meosured -'~'~..._ ~

~ - , ~ " - - - ' - . . . ,S

t -A

o -_

, , , , I , , ~,'1:7, , I l l A, ~, f , , , ,- I 2 3 4 5

Frequency (kHz)

( a ) (b )

F i g u r e 27 . C r o s s s p e c t r a , (a) m e a s u r e d a n d (b) p r e d i c t e d , b e t w e e n s e n s o r s f o r t h e s h a r p T E . Lro = 6 9 . 5 m / s . K e y s fo r s e n s o r p o s i t i o n s o f F i g u r e 2 5 a r e : - - - - , x l = a = - 2 . 8 9 8 c m ; � 9 -~1 = bi = - 1 . 5 2 4 c m ; a n d fo r s e n s o r s o n o p p o s i t e s ide to a : A - - - , x t = b2 = - 1 . 5 2 4 c m ; O , x] = b3 = - 1 . 8 5 4 c m ; - k - - - , x l = b4 = - 2 . 8 9 8 c m .

A point may be made here with regard to the justification of using pressure sensor results for the sharp TE case to establish the statistics of the incident field. The phase ~b: for a = xl and b i = :71 in Figure 27(a) is dominated by the time delay effect due to convection velocity, as modeled by equation (2b). Calculations with and without neglect 'of the scatter terms in equation (26) have revealed that the expected error due to the �9 presence of scatter effects in the determination of V]Uo in section 3.2 is only a few percent for frequencies above =0.7 kHz. Also the effect on the coherence function


should be small. This would not be true if the sensors were very near the edge such as with the blunted TE case. Here one would expect errors in V/Uo of the order o f l 0 to 20% in the low frequency ranges (above =0 .7 kHz) and the coherence function would be affected. It is suggested that the minimum sensor distance to the edge where scatter can be neglected is =Ah/2, where /~h is the convected hydrodynamic wavelength of interest. Note that if T B L coherence decay had not occurred (i.e., if h = hxt = h~, = 1) then the minimum distance would have to be increased to the order of 2Ah. With regard to the PSD of the incident field, errors less than 1 dB are expected if the distance is above Ah/2 and less than 0.5 dB if the distance is above 3Ah/4.

In Figures 26 and 27 it is seen, for both the blunted and sharp TE cases shown, that when the sensors on the two sides are directly opposite each other then the cross spectra exhibit a series of almost equally spaced spectral humps, the corresponding phases being alternatively zero and +,r(0 ~ and +180~ For the first hump the phase is +Tr and its peak amplitude is =7 to 8 dB down from the autospectra at sensor a, or ba. This has been found to be true for all sensor positions and degrees of T E bluntness. Crudely the first out-of-phase region exists for sensor positions xt, Ah/4 < Ixd <3Ah/4.

T h e phase data for the opposing sensors for different degrees of T E bluntness are summarized in Figure 28 where the measured out-of-phase spectral regions are indicated

!

E

~5

IO

0.5

I I I

-Sensor o

i It_ I / / - Sensor b4.~=

Predicted

Out-of-phase " -

In-phase ~ . - ~ - ( p b 4 o = ' + ' r r ) - ~ - In-phase "

(Pb..=O) ~ .~(Pb.o =0 )

I

I

I / ~ - - Predicted

I I I 0-1 0.5 I 5 I0

- toxi / tJ o

Figure 28. Cross spectral out-of-phase regions for opposite sensors at equal distances upstream of the TE. Key for experimental data is: O O, sharp TE; A A, blunt TE ( t = 2 . 5 m m ) ; [] [], blunt TE (t = 1.9 ram); + +, blunt TE (t = 1.1 mm).

by horizontal lines. The data are given for distances xt from the edge versus the normalized frequency parameter ~xl/Uo. The limits of the in- and out-of-phase regions are predicted by the use of equation (25) for xl =,~1. This is given in the figure by the more vertical lines and is seen to reasonably predict the data for a broad range of xl values. Significantly the measured data and the theory both indicate that regardless of the value of xl the phase is zero up to a finite value of oJxl/Uo(~-0.4). Furthermore, in the limit as x t ->0 then a~ o co. Therefore it is to be expected that the in-phase region would extend over all frequencies for measurements made at the edge (except in a frequency range where coherent shedding occurs due to T E bluntness). In addition, as xl o 0 the two pressure

!fields should become coherent and of equal amplitude. This can be seen from equations ~(25) and (26), or alternatively equations (20) and (21), when x~, s and xl are all allowed to approach zero. The level then becomes ~G. or 3 dB down from that measured upstream. 'For one-sided flow, one would expect from the equations that the autospectrum at the tedge would be �88 G . or 6 dB down from that measured upstream.

The flags on the experimental data of Figure 28 indicate that agreement with this data is fictitious because these sensors were under the dominating influence of the potential !field of coherent shedding phenomena due to TE bluntness. The coherent shedding gives !rise to a fluctuating pressure differential or lift across the finite thickness edge, and thus Ithe • behavior. The scatter phenomenon which has been the subject of this section lgives rise to in- and out-~of-phase spectral regions primarily in a statistical sense. While ~t ~s true that the scatter contributions on opposing sides are 180 ~ out o " " f phase causmg [a fluctuating lift, the effect is second order for the cross spectral function. In equation (21) this fluctuating lift contribution is represented by the q*q~, term. T.._ha ..-..first v.u,..n'4'~" terms given by q* +q~ dominate because the scattered fields are strongly related to the incident pressure fields which are initially measured. Thus the in- and out-of 'phase regions are produced by the superposition of antisymmetric cross spectra caused by independent flow fields on both sides.

The power spectral densities from Figure 12 for positions a and b~, i.e., Gaa and Gblb~, are shown in Figure 29. Also shown is that predicted for Gb,b, by using equation (26) and the measured spectrum Ga~. Agreement is good except in the lowest frequencies where the coherence modeling is weak and, as expected, in the frequency range where the bluntness effect dominates.

/

I 0 0

~o 9O

o ~

o 8O

I I I I I I I I I I i I I l - -"~"

0-5 I 10

Frequency (kHz)

F igure 29. Power spectra] densit ies f o r sensors close to 2.5 m m b lun t T E . D a t a reference to F igure ] 2 . K e y for m e a s u r e d spec t ra : , Gaa ; - - . - - , Gb lb t . T h e s p e c t r u m d e n o t e d as �9 �9 �9 is tha t p red ic ted for Gblb l .

4.3. P E R T I N E N C E T O K U ' I T A C O N D I T I O N F O R T B L F L O W P A S T A N E D G E

Aside from effects due to bluntness, it has been found that the measured pressure field very near the trailing edge is quite predictable in a manner consistent with the evanescent wave theories of Chase [4] and Chandiramani [5]. The evanescent wave model is based on the premise that the turbulent flow responsible for the surface pressure

t is constrained to move above the surface, past the edge and into the wake without altering character or "spilling over" the edge. Implicitly in the theory the assumption is made that the primary flow perturbations causing the pressure are not touching the

102 T. F. BROOKS AND T. H. H O D G S O N

surface or edge, which avoids concerns about possible flow singularities at the edge. Upstream of the edge, with the surface large compared with an eddy size, a condition of specular reflection (or mirror imaging) occurs. This results in the pressure field being double that which occurs when the same eddy is free of the surface in the wake region. Therefore, as the eddy convects past the edge it must give up one half of its pressure amplitude. The transition from a supported pressure field to that which is unsupported is accomplished in the neighborhood of the edge by the pressure scattering process. The scattered field superimposes on the incident field which renders the pressure continuous in the vicinity of the edge and reduces the pressure to its wake value at the edge.

Important general issues [1, 28] have been raised with regard to the theoretical treatment of the boundary condition at the trailing edge. In particular, Howe [1] has questioned the assumption inherent in evanescent wave theory that the turbulent flow perturbations do not wet (or touch) the surface or edge. The contention is that in reality the surface is wetted with the incident turbulence and thus one is concerned with a "Kutta" condition. This Kutta condition is a theoretical representation of the effects of viscosity which insure finiteness of the velocity in the potential flow field near the edge and as a consequence renders the pressure finite at the edge. It is argued that for an incident turbulent vorticity field at z with convection velocity V(z) shed vorticity (of opposite rotation to that incident) should occur near the edge and convect into the wake in the z -- 0 plane with velocity W. Absolute satisfaction of this Kutta condition requires W o V(z) which would enormously relieve the pressure scattering process and correspondingly lower the noise produced.

Attempts were made in the present study to verify directly the existence of this shedding phenomenon but no confirmation was found. In a progress report [29] on this study wake velocity measurements (not given here) were mentioned, based on coherence between pressure sensors and a cross hot wire just downstream of the edge. Recent analysis of the coherent phase data indicates that no measurable portion of that part of the wake vorticity which was coherent with surface pressure was in an opposite sense of rotation to that of the TBL. This is in accordance with observations of flow visualization studies reported by Yu and Joshi [11].

From this evidence and the fact that the measured pressure field, both upstream and very near the edge, is well predicted by evanescent wave theory (without recourse to modifications suggested by Howe [1]) one must question the pertinence of the postulated shedding phenomenon for this problem. Consistency of the results of this section with Howe's models is attained if one takes the velocity W to be vanishingly small in comparison to the TBL convection velocity V. It is noted that this results in identical noise predictions for the Kutta and no-Kutta condition cases. This result is to be subsequently used i n the analysis of the sound field in section 5.

Such a determination, although apparently correct based on the data, leaves significant questions as to the proper role of viscous effects in the modeling of this edge problem. Howe's modification to the surface pressure description (equation (13)) is that the term N, which expresses the incident turbulence characteristics, be replaced by {1-tr(z)}N where o-(z)= W/V(z ) . Although not explicit in the formulation, it would appear that W should be regarded as having a strong xl dependence near xl = 0 with the condition that W-~0 as x1~-co if this is to be accepted as a result due to an applied "edge" condition. Still, however, no such x~ dependence was found in the present study: i.e., G. ~ G.(xt) in the vicinity of the edge. It is suggested that to be constructively viewed here {1-tr(z)}N should be regarded as a redefinition of the incident field where the z dependence of the dissipative influence of viscosity on the formation of and convection of coherent turbulent structures is made explicit. When this is accepted it is seen that

T R A I L I N G E D G E N O I S E P R E D I C T I O N 103 i �9 because of the no slip condition at the surface then for very small z one has o-(z)--> 1, ~and for sufficiently large z one has o'(z)--> 0. In this way, the role of viscosity is different from that of a rapid production of coherent shed vorticity at the edge to match that of ~he incident flow. The role is a more passive one, that of isolating dominant flow ~listurbances from direct contact with the surface such as for example, through the i presence of the T B L sublayer (where o-(z) = 1). This would be consistent with the view 'of evanescent wave theory. I The case where turbulence is confined to one side of the airfoil has not been considered in this study. However, the good prediction and comparison for the noise field of a turbulent wall-jet by Chase [41 indicates a similar edge flow condition to that found here. The case where the airfoil oscillates presents a different physical circumstance from that presented in the present study. Yates [30], using data from Brooks [31], has shown for an oscillating airfoil that real viscous effects near the leading and trailing edges cause a vortical energy dissipation. This results in partitioning of the work done by the moving lirfoil into sound and vorticity modes.

5. TRAILING EDGE NOISE

5.1. M E T H O D O F S O U N D F I E L D D E T E R M I N A T I O N

The unavoidable presence of extraneous noise sources in the test precludes direct measurement of trailing edge noise by the use of a single microphone. Therefore , in order to determine T E noise, cross spectral analysis of pairs of microphone signals was employed in a manner consistent with the coherent output power (COP) method [15, 16].

All microphone locations shown in Figure 7 receive extraneous signals. The autospectrum of the noise at microphones M 1 and M2 , which are at 0,, = 9 0 ~ and - 9 0 ~ respectively, are

GMIMI = Guu q- Gnn, GM2M2 = G~.o + Gram, (27)

where now G,u is taken as the autospectrum of T E noise perceived at M 1 and G~.v is that perceived at M2. Gn, is the sum of extraneous noise spectra at M1 from such sources as the nozzle lips, free jet turbulent shear layer, test rig edge noise, downstream collector and any TBL noise from the airfoil surface. G,~,, is that received at M2. The cross spectrum between M1 and M 2 is

GMIM2 = G,o + G,, , + G,~ + G,,. .

With the knowledge that T E noise depends on the rather transient turbulence statistics which exist at the edge and decay rapidly downstream in the wake, i3,,, and G,o must equal zero. Therefore

G^fIM2 = G,,o + G,,,,,, (28)

where G.o = IG.oI exp (i~o.o). This shows that if Gn., is small compared to G.o then the cross spectrum between M 1 and M 2 would yield the T E noise spectrum. This does not require that Gnn or Gin,, be small but only that the signals producing G,n and G,~,, be mutually incoherent, or nearly so. Now since M1 and M 2 are about equally distant from the TE, IG.oI--IG..I---IGd is the autospectrum of the T E noise perceived by both M1 and M2. The phase q~o depends on the source phase and the small difference in distances between the microphones and the TE. Therefore, in frequency ranges where Gnm can be proven small

! GM1M2 = S ( ~ exp {i[+1r + k ((Rm)M1 -- (R,n)M2)]}, (29)

104 T.F. BROOKS AND T. H. FIODGSON

where k = 2rr]:/c and Rm is the distance from the microphones to the TE. S(]:) is the TE noise spectrum which would be measured at M1 or M2 if no extraneous noise sources were present. In writing this equation one employs the pre-emptive knowledge that TE noise is coherent and antisymmetric on opposing sides of the airfoil. Now, upon assuming that TE noise is given by GMtM2, the TE noise at other microphone locations M] can be determined by using the COP [15, 16] method as

s(f)o, M;= 16M,M I2/IGM1^,21 = ICM2 ,,A2/IG^,I^, I. (30)

G,m in equation (28) is regarded as being a contaminant whose contribution should not be discarded prior to direct evidence that it can be neglected. Necessary evidence can be provided by comparison of the phase of the cross spectra between various microphones with calculated phase relationships, such as in equation (29), for known geometry and source type. GMI~f2 of equation (28) is a vectorial sum of G,,o and G,,, for each frequency; therefore, phase is very sensitive to any contribution of G,,, from likely sources previously mentioned.

An additional concern with regard to the microphone results must be treated. Sound radiated from the TE must pass through the free jet shear layer before reaching the microphones. The sound is refracted in the process. The effect of this refraction is illustrated in the exaggerated sketch of Figure 30. If the velocity U were zero everywhere

/ I - e'ord0d I /

M5 8, 8~- pozilion

t L Y l . . . . . '--{ . . . . . . .

~M4 ~ U=O/ ) W Shear layer

Figure 30. Exaggerated sketch of shear layer refraction of acoustic ray transmission paths.

sound reaching the microphones would follow the "ray path" defined by the measured distance R,, and angle Om. On the present case where U = Uo over a r.egion, a shear layer is present which alters the ray path. This results in both an angle and an amplitude change in the sound field. Several treatments of corrections to these changes have been

reviewed by Amiet [32]. Experimental confirmation of the angle corrections h-as been reportedby Schlinker and Amiet [33].

To correct for shear layer refraction in the cross spectral phase, such as in equation (29), R,, is replaced by Re. R, represents the actual propagation time [33] of the refracted sound times the speed of sound. Values of Re are given in Table 1 (see section 2.3) for

.the different microphone locations when Uo=69.5 m/s. Also in Table 1 are given corrections to the measurement angles 0,, and the amplitude correction zi dB, to be added to the measured levels. The angle 0, is referenced to a retarded source position and a corrected observer position where the distance between the positions is R, = Rm.

T R A I L I N G E D G E N O I S E P R E D I C T I O N 1 0 _ ~ i iAs defined, if there were no shear layer present with flow extending to infinity, the centeI of the wavefront emitted from the edge would be at the retarded source position when the wavefront reaches the corrected observer position. 0, and A dB will be applied to directivity data. It is noted in Table 1 that all corrections for microphones at Or, = +90 ~ (M1, M 2 and M3) are negligible; therefore no distinction will be made regarding e,~ or 8, for these positions.

The applicability of equations (27) through (29) are demonstrated in Figures 31-33. In Figure 31 are shown the autospectra GM1^rl, which are virtually identical to GM2M2,

6(1) I i i i i i i i i i i i i i I i i i i I i i i i

4 0 t

. . . . . . . . . .

GU,MZ w .'~ ." : ' ; : . . . . . L . , . . ~ , .. ~ , ~-~

- . . . . .~, , ',, :"..: ,

20 ~ " " %' " " ": ~ "~ . . _

rfoil V V

I0 t ' , i I ' I 2 I I f , , t I ' , , t I , , , , 0 2 4 6 8 I0

Frequency ( k H z )

Figure 31. Autospectra and cross spectra of microphones on opposite sides of the airfoil illustrating method o extract TE noise. Case is that of Uo = 69.5 m/s , with and without the 2-5 mm blunt TE airfoil in place.

I I I i I I I + l I I I I I I I I I I I I I I I i

+ + .4.'t'4~'++ - + +

J L

+ -I- + + + % + �9 �9 -

,+ +§247 ++~+ + " ~ / ~

~ 4 t " t f - + ~ +

m + -F t t _ . . ", , ~ , , , , + , " ; , . , - , J , , , , i , , , , I , , 0 2 4 6 8 I0

Frequency (kHz)

Figure 32. Cross spectral phase, corresponding to data in Figure 31, illustrating source location. Key for ~ta: L 2.5 mm blunt TE air/oil; + , airfoil removed. Line shown is the predicted phase.

106 T. F. BROOKS AND T. H. H O D G S O N

7 r

o 9-

- - W "

l ' i I I+ I § " I I i ~ f ( ~ l l l 4 - i .~ [p~i i i I"i i i I ' I

+ ++ ~ +m +

+++ + m §

F + + + �9 + + _ + . . . .

+ + + + + �9 + m ! ~ �9 + +

_-Ira + +

- + + 4- 4- 4-

4-

n , �9 t I I , + n t l § im/~-t..~ i n n I i i n i 0 2 4 6 8

w

, , -1 I0

Frequency (kHz)

Figure 33. Cross spectral phase between microphones M 4 and M5 pinpointing the TE as the source location. Key is that of Figure 32. Line -- is that predicted and line - - - is that predicted without shear layer correction.

for the cases where the 2.5 m m blunted T E airfoil at a~ = 0 ~ is placed between the sideplates and also where the airfoil has been r e m o v e d . The airfoi l-removed case is considered a measure of G , , before the airfoil is placed in the test rig. An increased level is Seen in the figure for the airfoil case. A logarithmic subtraction method to extract T E noise, G, , , f rom the two autospectra is inappropriate here. This is because there is little way to determine what part of the additional noise can be attr ibuted to T E noise and what part originated from additional G , , sources such as the presence of the airfoil wake interacting with the edges of the sideplate test rig. To extract T E noise the cross spectra GM1M2 are per formed for the two cases as shown. The case without the airfoil is seen to be much lower than the airfoil case over a large frequency range.

To determine in what part of the frequency range the T E noise spectra is represented by the cross spectra, the phases are compared with that calculated. In Figure 32 the phases ~M*M2, corresponding to G^faM2 of Figure 31, are shown for the two cases. It is seen that for the airfoil case, except for frequencies below = 0 . 6 kHz and above 2 7 kHz, the data tend to coalesce to a straight line originating f rom • at low frequencies. Also shown in the figure is the phase predicted by using equat ion (29). The values for the measured distances are (Rm)M~ = 122.2 cm and (R,,)M2 = 119.2 cm from Table 1. It is noted that calculation reveals that the predicted line deviates f rom a best fit to the data by an amount which would be due to a measurement error of 0.4 cm in one of the distances involved or of 0.2 cm in both distances. Therefore the results are predicted within experimental accuracy in the frequency range specified. The results for the airfoi l-removed case can have various interpretations as to source locations. But certainly only at the lowest and highest frequencies does the noise appear to have a similar origin as the case with the airfoil in place.

Figure 32 establishes that noise due to the airfoil is emitted f rom some location along the midchord line but does not pinpoint the TE. To do this the cross spectra phase q~M4~5 is given in Figure 33. The random nature of the results for the airfoi l-removed case shows that there are no dominant source locations common to both M 4 and M 5 except possibly at the lowest frequencies. However , for the airfoil case the data are seen


to coalesce along a straight line. The predicted line shown is q~^r4M5= O+k((Re)M4-(Re)Ms) where the values for Re are from Table 1. The predicted line deviates from best fit by an amount representative of a 0.8 cm error in the determination of ((R,)M4-(Re)Ms). This is very good agreement in view of the significant effect of the shear layer correction on the prediction. Note that if Rm had been used, instead of Re, the prediction would be altered resulting in a poor comparison as shown in the figure.

The coalescence of G^t4r, f5 data in Figure 33 about that predicted occurs in the same frequency range as that for G^tIM2 in Figure 32. Therefore it is established that the TE is the dominant coherent source of the noise and that in a broad frequency range GM~^t2 represents the TE noise spectra. The terms G , , and G,,,~ are seen to be significant in equation (27) but (7,,, in equation (28) is negligible in all but the lowest and highest frequencies considered.

5.2. RESULTS Trailing edge noise spectra are given in Figure 34 for R, , = 1.2 m and Om = 90 ~ obtained

from GM1M2 measurements. Results are presented for different free stream velocities

5 0 1 - , ' , ~ ~ ~ t , ' , ' ' ' ~ i

/ - / / a ' - - o ~Uo=69'5 m/s /,,,,.__~,/

oL- . . _ / - - L

.~,,..~,,, \ vv,.,:.... \\ ~o:z8 ...... : M_

g. 20 . 9 m / s

I0 Vv I,.

0 w I t I I I I f I ! I I I !

0 - 3 I I0

Frequency (kHz)

F i g u r e 34 . T r a i l i n g e d g e no i se spec t r a l d e n s i t y fo r v a r i o u s f r ee s t r e a m ve loc i t i es a n d d e g r e e s o f T E b l u n t n e s s . ~ , . = 1 .2 m , 0 , . = 9 0 ~ K e y f o r d e g r e e o f b l u n t n e s s : - - , t = 0 ( sha rp ) ; - - - , t = 2 . 5 m m ; - - - - , t = 1 .9 m m ; �9 - - , t --- ! . 1 m m .

rio and degrees of TE bluntness t. These unsmoothed noise data are referenced in level o a 1 Hz bandwidth and po = 20 ttPa as was done previously for the surface pressure lata. For the sharp TE cases it is seen that the peaks of the spectra correspond to the 'characteristic" frequencies noted for the airfoil T B L pressure fields: e.g., see Figure .6. The effect of edge bluntness on the spectra, shown for Uo = 38.6 m/s and 69.5 m/s, s seen to be an additive contribution to that obtained from the sharp edge case. The :ontribution is a spectral hump corresponding to that observed in the pressure measure- nents very close to the TE: e.g., see Figure 12. Note that although thedegree of bluntness s only a fraction of the T B L displacement thickness B* the noise spectra are significantly affected. The coherent shedding occurring at the blunted edge causing this effect appears o have the Strouhal dependence of ft/(-1o = O. 1.


Of primary interest in this study is the TE noise for the sharp edge case. The noise i., that produced due to edge enhancement of aerodynamic sound when turbulence of thc T B L passes into the wake. The theories of trailing edge noise as based on the Lighthill acoustic analogy have provided the fundamental scaling laws. From Howe [1], with ouz edge condition that Howe's parameter W ~ 0, the overall mean square sound pressure in the flyover plane (a = rr/2) should vary as

V 3 [ L~e~ (1 - M o + M~,) sin 2 (0,/2) cos 3/3 (p2> ~p2v2 T k-ff'~] (1 +Mo cos 0,)3[1 + (Mo-Mv,) cos 0,] 2' (31]

where 2 is the mean square turbulent fluctuation velocity and e is a frequency averaged turbulence correlation scale. The far field observer co-ordinates R, and 0, are referenced to the retarded source position in a manner consistent with that illustrated in Figure 30 Mo is the free stream Mach number Uo/c and M~, - V cos/3/c = V/c for our non-skewed case,/3 = 0. The data of Figure 34 are for 0, = 0,, -- 90 ~ To normalize the sharp TE data we have taken v - - V , V - U o and (1 -Mo+M~,) - - -1 . By also taking ~ ' ~ 8 ' the spectra of Figure 35 are obtained. It is seen that one-third octave noise levels of center frequenc)

0

T -Bo "5

-90 0.01

I I I I I I I t I 1 I I I I

I I I I I I I I I I | I I | I I I

0.1

tcS~

Figu re 35. N o r m a l i z e d o n e - t h i r d oc tave T E noise spec t ra f rom da ta of F igure 34 for s h a r p T E case. Or, = 90 ~ Key for ve loc i t i es Uo (m]s) : - - -, 30 .9 ; - - - - , 38 .6 ; - - - , 54 .1 ; - .-, 69 .5 .

fc are employed in the comparison rather than the constant bandwidth levels of Figure 34. The intent is to represent the energy content on an equal basis for each spectrum over the Strouhal number fcS*/Uo. This produces bet ter coalescence of the data. The agreement is fair although the characteristic peak frequencies do not satisfactorily scale as was the case for the normalization of surface pressure PSD in Figure 13. However, the range of Strouhal numbers (0.07 to 0.1) of the characteristic frequency is quite consistent with values discernible from previous studies [9-11] but not with the value assumed by Fink [34] (fS/Uo = 0.1, using 8 rather than 8*).

The overall sound pressure level (SPL) is obtained from integration over the spectra of Figure 34. These levels are given in terms of the free stream velocity in Figure 36. Additional velocity cases are included. Best fit to the sharp T E noise data is seen to follow a USo ~ dependence which is in excellent agreement with the USo predicted [1]. The blunted TE case shown is some 2 dB above the overall levels for the sharp case and follows a U 5"3 dependence. For comparative purposes in the tests, a 9.5 mm diameter cylindrical rod was placed across the span at the TE position with the airfoil removed. By using the same test and analysis procedure a U 6"~ dependence was determined for the rod as expected. The pertinent Strouhal number was found to be f • d iameter /Uo = 0.19.


f l O I I I I

75-- U 0 5 /

70--

~ 65

O .07 6O

55

] I I I 30 40 50 60 70 80

Free stream velocity, U0(m/s)

Figure 36. Overall sound pressure level of TE noise v s . free stream velocity. R,,, = 1.2 m, 0,, = 90 ~ Key for airfoil: A, sharp TE; O, 2.5 mm blunt TE.

The directivity for the sharp and blunted TE cases and that of the rod is given in Figure 37 for Uo = 69.5 m/s. The overall SPL levels have been normalized with regard to microphone distance from the edge and the results for microphones M1 and M2. The data were determined by employing the cross spectral function and the COP result, equation (30). To account for shear layer refraction effects the results are presented in

0e=0~

10 dB o

e,=9o" I / R o d - - , , - t ~ H _ _ t _ . + - " &.=-90" 5 ( ~ ~ ~ - 5 -~Od~-~5 -20 -20 -Z5 -tO -5 5 '

\ I f o da /

Or = 180"

Figure 37. Directivity of airfoil TE noise corrected Ior shear layer refraction effects. Predicted direetivity accounts for the effects of convective amplification. Corresponding results for a small rod in flow is given for comparison. Key for data: O, rod (airfoil removed); O, airfoil sharp TE; O, airfoil 2.5 mm blunted TE (solid symbols for M3). Key for that predicted: , airfoils; - - - , rod.

II0 T.F. BROOKS AND T. H. HODGSON

terms of the retarded angle Or. Also the corresponding amplitude corrections from Table 1 have been applied. Along with the data is given the predicted half-baffled dipole-type directivity as modified by Doppler factors from equation (31) with Uo = 69.5 m/s and V = 0.6 Uo. In addition, the directivity [35] sin 2 0,/(1 +Mo cos 0,) 6 is given for comparison to the rod data.

It is seen in Figure 37 that the sound fields in the forward and aft directions are predicted within 1.5 dB. This represents substantial support for the theoretical directivity behavior as predicted by equation (31). It should be noted that the shear layer amplitude corrections had an important impact on these directivity results. The fact that reasonable agreement is found may be regarded also as a partial confirmation of the amplitude corrections of Amiet [32]. The data deviation which does occur from that predicted cannot be directly explained from the results of the acoustic field calibration summarized in Table 1. This is particularly true for the results for microphone M 3 at 0 r - 9 0 ~ and Rr = Rm = 358.3 cm which are 1-5 to 2.5 dB too low in Figure 37. In the calibration a point source was employed, which did not simulate the dipole-like line sources considered here. A contributing factor to the deviation may be an error buildup associated with our application of the COP method which produced the data of Figure 37. From equation (30) it is seen that a 0.5 dB error, for example, in each of the cross spectra can possibly result in an overall error of 1.5 dB.

5�9 Angle of attack and edge extensions The effect on the noise spectra of an angle of attack change for two free stream

velocities is shown in Figure 38 for the sharp trailing edge case. The data for o~ = 0 ~ are

70

60

I | I I i I I l i I i I

4 0 I t i I I I I I . I I I ! 0-3 I

C e n t e r f r e q u e n c y , ~ ( k H z )

F i g u r e 38 . Ef fec t o f a n g l e o f a t t a c k , ag , o n T E no i se . R,. = 1 .2 m , 0 , . = 9 0 ~ K e y / o r s p e c t r a : - - . - -~ or~ = 5 ~ .

I I I

I ! I

I 0

, ag = 0~

obtained from Figure 34 except now the trailing edge noise is presented in terms of one-third octave spectral levels. The effect of an angle change to otg = 5 ~ is seen to be primarily in the low frequency range. The spectral peaks tend to shift to lower frequencies. Overall noise levels are seen to be not significantly altered�9 As will be explained this behavior should be expected, based on the surface pressure PSD results of Figure 14. �9 In Figure 39 are the results when some significant geometric modifications are made

to the trailing edge region of the airfoil. The sharp T E case (ag = 0 ~ is shown for comparison�9 Three T E extensions were tested, all with sharp edges. The 2.54 cm extension was placed at an angle of 17.5 ~ off the mean chord centerline of the airfoil to simulate

70

50

40 0.3


I I ~ I I I I I I l" I

I I I t I I I I I I

I

Cenler frequency, fc (kHz}

I I

'"!

2

IO

Figure 39. Effect on the noise field of long T E extensions and a T E flap. Key for spectra: - - - , 15.24 crn sha rp T E extension; - - - -

111

, sha rp TE; , 3 0 . 4 8 cm sharp T E extension; . . . . , 2 . 5 4 cm flap at 17.50 on TE.

a flap effect. The 15.24 cm and 30.48 cm extensions were positioned along the chord centerline, which was at a~ = 0 ~ It is seen that, except for some unexplained differences for the lower free stream velocity case, no significant changes in the sound spectra occur. One would have anticipated an increase in level on the order of 1 dB for the extensions due to an expected increase in the T B L thickness 8 at the extended TE. No such effect is clearly observed. This perhaps means that 8 had not substantially changed over the zero pressure gradient extension plates due to some influence of the-upstream adverse pressure gradient. It is remembered that with the extended sur faces the microphones M1 and M 2 which determined the data are no longer at 0,. = 90 ~ but are in the forward quadrant of the TE noise field. The small correction for this fact has not been applied. The primary point of Figure 39 is that TE noise is rather invariant with regard to geometry changes in the edge region as long as the new TE is sharp and the T B L is not drastically modified. The 17.5 ~ flap on the airfoil edge did, of course, turn the flow field in the edge region but did so apparently without causing separation which would be noticed in the sound field. The best agreement between spectra is for the sharp TE and the flap cases. It should be mentioned that for all spectra of Figure 39 and the angle of attack cases of Figure 38 the cross spectral phase data pinpointed the new edge locations as the source positions.

5 . 3 . P R E D I C T I O N S B A S E D O N M E A S U R E D S U R F A C E P R E S S U R E S

From the formulation of the T B L incident pressure wavenumber spectra and the determination of conditions existing at the edge, trailing edge noise can be predicted and compared to that measured. Our edge condition is such that Howe's Kutta and no-Kutta condition noise predictions become identical. Therefore , from Howe [1], the sound field spectrum for an observer at co-ordinates R, 0, a (Figure 2) is

2._~(oJL~ s ina sin2 (0/2) cos fl f~o H,e(#,,o., cos a /c , a,) d/x, S ( ~ c I ( l + M 0 u ) z j _ ~ o l i t - ~ n l - ~ _ ~ / ~ ~ a ), (32)

which is the general form of a result due to Chase [4]. In this co-ordinate system the observer is fixed with respect to the airfoil and both are encountering the same flow field. Correspondence is seen with the "correc ted" observer position and the actual edge location in Figure 30. Here 0 from equation (32) equals 0r and R is the distance between the "correc ted" position and the edge. In equation (32), tt = (/Xl, ~o cos a/c).

1 1 2 T.F. BROOKS AND T. H. HODGSON

For the present exper iment /3 = 0 and ~ ---90 ~ Therefore it =/~, and since n is the unit vector parallel to V one has ~ . n = P-1. Also from the Doppler factors MoR = Uo cos O/c, M~R becomes oJ cos O/clzl and My1 becomes ~/ct.zt.

~Tie is defined as the wavenumber spectrum of the pressure fluctuations that occur in the wake and on the plate or airfoil very close to the trailing edge�9 For the airfoil of this study at ag = 0 ~ /7,e should equal f l /2, according to the discussions of equations (25) and (26). H e r e / 7 is the T B L incident pressure wavenumber spectrum measured upstream of the edge on either side of the airfoil, defined in equation (6) and plotted for several frequencies in Figure 23. For the case where 0 = 0m = 90 ~ equation (32) becomes

s(o,)_ - 2 r r R 2 \ c I J-~o d.t. (33)

Equat ion (33) was used to predict the sound spectra for the sharp T E case for comparison with that measured. Exper imental data for two free s t ream velocities from Figure 34 are shown in Figure 40 along with predictions. The geometry of the experiment

50

40

i 3oi

20 O

I 0

| I I I I I [ I | I I I I i I

�9 , .~

. . . . - " " .~

-""

/ +

k.~.--... + Uo=3S.6 rn/s

0 I I ! I ,,I I I I i I ! I I

0.3 I I0

Frequency (kHz)

Figure 40�9 TE noise spectra for two free stream velocities (from Figure 34) with the spectra predicted based on measured surface pressure. R,n = 1.2 m, 0,, = 90 ~ Key for spectra: , experimental; - - - , prediction; �9 � 9 approx, prediction for low frequencies; + + + +, approx, prediction with V = 0.6 Uo.

and the empirical equations (2a) and (6)-(8), where the constants are given in Figures 17 and 18, were employed. The surface pressure PSD used was that f rom the sensor 2.898 cm upstream of the T E although any of the sensors near this would have given the same results within 0.5 dB. Equation (33) was evaluated numerically. The agreement in both ampli tude and spectral shape is considered excellent (with allowance for experimental variance) for those frequencies higher than the "characterist ic" frequency, for which the empirical equations are applicable.

An approximate closed form solution can be determined f rom equation (32) if it can be assumed that V is effectively constant and equals 17, equation (2a) thus not being used�9 Howe has shown that for this case

oo

-=- 7rR ----~ ( 1 + Mo-~-2(-i - ~ ~ ---M~, sin a ) co

T R A I L I N G E D G E NOISE P R E D I C T I O N 113

where now MeR = 17" cos O/c and Me, = ff"/c. For the case considered above and using equations (6), (10), (11) and (12) one obtains, for 6 = 90 ~

S(~o)----(1/2~r2R2)(f/L/c)C3~(co)/(1- f / c ) , (34)

where as(co) is the PSD upstream of the edge. The predictions from equation (34) are shown in Figure 40 where I7" was taken as 0.6 Uo. These approximate predictions are seen to be somewhat higher in amplitude and of a level falling off less rapidly with frequency than the measured results and those previously predicted. The same trend with frequency was observed in the behavior of (3 in Figure 22.

Shown also in Figure 40 are approximate predictions for the low frequency ranges near and below the characteristic frequencies. General agreement is seen. Each of these curves shown was determined by calculations tailored to particular frequencies. The constants for the empirical forms were found in the manner described previously for Figure 18(b). Employed also were observations as to the behavior of V/Uo in these frequency ranges. By using scaling factors from equation (34) the results were referenced in level to that determined from equation (33). The primary reason for showing these approximate results is to illustrate that the surface pressure measurements can be used to predict particular aspects of the noise spectra. However, these results for low frequen, cies would have little impact on a calculation of overall level.

All of the noise calculations above were simplified because the wavenumber spectra were the same on opposing sides of the airfoil, permitting one to take Hte= 1714 +17/4. When the flow conditions on both sides are not the same, two calculations for the sound field are required and the results must be summed. Calculations have not been made for the non-zero angles of attack. However, good results should be expected, according to the observations of Figures 14 and 38. These suggest little change in the predictions for high frequencies. The characteristic frequency region in the sound field, however, should be somewhat less discernible and lower in level, as suggested in Figure 14 by summing the PSD's of the suction and pressure sides of the airfoil. This expected trend in noise prediction is confirmed for the experimental data of Figure 38.

6. DISCUSSION OF RESULTS

A primary interest in this study has been to examine the usefulness of several trailing edge noise theories by applying them to the measured data. Over the last decade the most used theories have been those based on Lighthill's acoustic analogy, starting with the work of Ffowcs Williams and Hall [3]. Good success in its application has been demonstrated in this paper for the'determination of directivity and for scaling of overall sound pressure level with flow velocity. No attempt has been made to evaluate the more general equations given by Howe [1] which relate the amplitude and spectral shape of TE noise with the statistics of the turbulence near the edge. It was deemed more useful to avoid the complexity of the actual flow field and instead concentrate on the more easily measured surface pressure field which is of course a manifestation of the turbulence flow. This reduced the problem to establishing the statistics in two dimensions rather than three,

The application of evanescent wave theories of Chase [4] and Chandiramani [5], as generalized by Howe [1], to the present results have produced very good agreement, Certainly this indicates that the theories employing surface pressure measurements are very viable in the quantitative prediction of TE noise. Also the techniques and equations given in this paper should prove useful in explaining and quantifying surface pressure measurements in the presence of scattered sound,

114 T. F. B R O O K S A N D T. H . H O D G S O N

The analysis of the near field pressure scatter phenomenon and its agreement with measurement implies the correctness of the basic assumptions employed in the analysis. The TBL flow (in contrast to the pressure field) passes the trailing edge into the wake region apparently unaltered by the presence of the edge. This is not true of course in frequency ranges dominated by any coherent vortex shedding due to TE bluntness, which produces extra noise. Central in the analysis is the assumption that the TBL's existing on opposing sides of the airfoil near the trailing edge are non-interactive and statistically independent. The experimental substantiation of this assumption is important in the light of major questions [4] regarding the application of evanescent wave theory to two-sided flow problems. On the airfoil surface and edge the incident and scattered pressure fields superimpose. This causes the pressure differential at the edge to be zero. This also permits the statistics (wavenumber spectra) of the pressure field at the edge to be determined by measurements made upstream. This is in preference to the procedure proposed by Howe and Chase, who have suggested that all measurements be confined to locations in the wake or on the surface right at the edge, where measurements would be more difficult. In the present case the statistics of the hydrodynamic field were established from measurements made at locations on the order of one-half to one characteristic wavelength from the edge. In the absence of TBL coherence decaY one would have had to use measurements farther upstream because then the influence of the scattered field on the results would not have been as negligible.

As previously described, no hydrodynamic wake shedding activity of the type postulated by Howe has been confirmed for the two-sided TBL flow case of this study. This implies that in the TE noise prediction formulas [1] the convection velocity W of this shed vorticity must be made to approach zero. This results i n there being no distinction between the Kutta and no-Kutta condition solutions. The purpose of the postulated shedding was to remove the velocity singularity (and thus the pressure singularity) at the sharp edge in the potential flow model. It is suggested, for the real flow case where a well-developed TBL approaches the edge, that flow singularities are avoided due to the natural action of viscosity (independent of the distance from the edge) in defining the TBL and not by rapid edge adjustments of the hydrodynamic field such as a shedding activity. The scatter phenomenon provides the pressure adjustments needed in the TBL and wake without significantly affecting flow structure.

In the determination of the sound field in this study a method was employed which incorporated the principles of the coherent output power (COP) method [15, 16]. The use of the method was very successful in quantitatively extracting the TE noise spectra. The COP method is a powerful experimental tool. However, as previously discussed there are restrictions to its use. In the present case, it had to be demonstrated that the coherent noise measure did indeed originate from the trailing edge rather than from some other coherent source. In this regard the method as used is not equivalent to a directional microphone system where locations of interest may be pinpointed to determine local contributions to the sound field./'/either, however, is the method affected by the concerns of directional microphone systems such as frequency and space resolution limitations and associated gain factors [10]. It is felt that in measurement situations where proven feasible the COP method is preferred.

The near field edge scatter model or a modification of it could possibly be employed to help establish optimum sizing of edge treatment (such as porosity) for noise control purposes on edges of rotors, propellers, flaps, powered lift devices, etc. The model provides a means to separate and identify the scattered field (of which far field noise is a component) from the incident hydrodynamic field in terms of the turbulence scale and convection speed.


ACKNOWLEDGMENTS

The assistance of M. B. Manley and R. L. Underwood is gratefully acknowledged. rhis reasearch was funded in part by N A S A Grant NSG 1377. The authors acknowledge helpful discussions with M. S. Howe of Bolt Beranek and Newman, Inc. Also we would like to thank R. H. Schlinker and R. K. Amiet of United Technologies Research Center for providing shear layer refraction correction factors.

REFERENCES

1. M. S. HOWE 1978 Journal of Sound and Vibration 61, 437-465. A review of the theory of trailing edge noise.

2. M. J. LIGHTHILL 1952 Proceedings of the Royal Society London A211, 564-587. On sound generated aerodynamically. I. General theory.

3. J. E. FFOWCS WILLIAMS and L. H. HALL 1970 Journal o:Fluid Mechanics 40, 657--670. Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half-plane.

4. D.M. CHASE 1975 American Institute of Aeronautics and Astronautics Journa113, 1041-1047. Noise radiated from an edge in turbulent flow.

5. K. I. CHANDIRAMANI 1974 Journal of the Acoustical Society of America 55, 19-29. Diffrac- tion of evanescent waves with applications to aerodynamically scattered sound and radiation from unbaffled plates.

6. T. SCHARTON, B. PINKEL and J. WILBY 1973 NASA CR-132270. A study of trailing edge blowing as a means of reducing noise generated by the interaction of flow with a surface.

7. J. C. Yu and C. K. W. TAM 1977 American Institute of Aeronautics and Astronautics Paper No. 77-1291. An experimental investigation of the trailing edge noise mechanism.

8. R. W. PATERSON, P. G. VOGT, M. R. FINK and C. L. MUNCH 1973 Journal of Aircraft 10, 296-302. Vortex noise of isolated airfoils.

9. H. H. HELLER and W. M. DOBRZYNSKI 1978 Journal o:Aircra:t 15, 809-815. Unsteady surface pressure characteristics on aircraft components and far-field radiated airframe noise.

t0. R. n . SCHLINKER 1977 American Institute of Aeronautics and Astronautics Paper No. 77-1269. Airfoil trailing edge noise measurements with a directional microphone.

ll. J. C. Yrd and M. C. JOSHI 1979 American Institute o: Aeronautics and Astronautics Paper No. 79-0603. On sound radiation from the trailing edge of an isolated airfoil in a uniform flow.

[2. M. HAHN 1976 American Institute o: Aeronautics and Astronautics Paper No. 76-335. Turbulent boundary-layer surface-pressure fluctuations near an airfoil trailing edge.

13. T. H. HODGSON and M. B. MANLEY 1980 Center for Acoustical Studies, North Carolina State University. A miniature surface pressure transducer for fluctuating measurements.

t4. M. K. BULL and A. S. W. THOMAS 1976 Journal of the Physics of Fhdds 19, 597-599. High frequency wall-pressure fluctuations in turbulent boundary layers.

15. W. G. HALVORSEN and J. S. BENDAT 1975 Sound and Vibration 9, 15-24. Noise source identification using coherent output power spectra.

t6. A. G. PIERSOL 1978 Journal of Sound and Vibration 56, 215-228. Use of coherence and phase data between two receivers in evaluation of noise environments.

17. A. F. SEYBERT and J. F. HAMILTON 1978 Journal o: Sound and Vibration 60, 1-9. Time delay bias errors in estimating frequency response and coherence functions.

I8. A. E. VON DOENHOFF 1940 NACA Wartime Report L-507. Investigation of the boundary layer about a symmetrical airfoil in a wind tunnel of low turbulence.

19. J.V. BECKER 1940 NACA Warthne ReportL- 682. Boundary-layer transition on the N.A.C.A. 0012 and 23012 airfoils in the 8-foot high-speed wind tunnel.

20. W. W. WILLMARTH 1975 Advances in Applied Mechan&s Journal 15, 159-254. Structure of turbulence in boundary layers.

21. E.J. RICHARDS and D. J. MEAD (Editors) 1968 Noise and Acoustic Fatigue in Aeronautics. London: John Wiley & Sons Limited.

22. W. K. BLAKE 1975 David W. Taylor Naval Ship R&D Center Report No. 4241. A statistical description of pressure and velocity fields at the trailing edges of a flat structure.

23. G. M. CORCOS 1964 Journal o:Fhtid Mechanics 18, 353-378. The structure of the turbulent pressure field in boundary layer flows.

24. D. M. CHASE 1980 Journal of Sound and Vibration 70, 29-67. Modeling the wavevector- I frequency spectrum of turbulent boundary layer wall pressure.

116 T . F . B R O O K S A N D T. H . H O D G S O N

25. R. EMMERLING 1973 Max-Planck-Institut [iir StrSmungs[orschung. The instantaneous structure of the wall pressure under a turbulent boundary layer flow.

26. J. A. B. WILLS 1970 Journal o[ Fhdd Mechanics 45, 65-90. Measurements of the wavenumber/phase velocity spectrum of wall pressure beneath a turbulent boundary layer.

27. M. ABRAMOWITZ and I. A. STEGUM (Editors) 1970 Handbook of MatlwmaticalFunctions. New York: Dover Publishing Company, seventh edition.

28. W. J. MCCROSKEY 1977 Journal of Fhdd Enghwering, Transactions of the American Society of Mechanical Engineers 99, 8-38. Some current research in unsteady fluid dynamics.

29. T. F. BROOKS and T. H. HODGSON 1979 IUTAM/1CA/AIAA-Symposium GStthzgen-- Mechanics of Sound Generation in Flows and NASA Tltl 80134. Investigation of trailing edge noise.

30. J. E. YATES 1980 American hzstitate of Aeronautics and Astronautics Paper No. 80-0971. The importance of viscosity in experimental applications of Kirchhoff-type integral relations.

31. T. F. BROOKS 1977 NASA TP 1048. An experimental evaluation of the application of the Kirchhoff formulation for sound radiation from an oscillating airfoil.

32. R. K. AMIET 1978 Journal of Sound and Vibration 58, 467--482. Refraction of sound by a shear layer.

33. R. H. SCttLINKER and R. K. AMIET 1979 American h~stitute of Aeronautics and Astronautics Paper No. 79-0628. Refraction of sound by a shear layer---experimental assessment.

34. M. R. FINK 1979 Jottnzal of Aircraft 16-10 Art. No. 77-1271R. Noise component method for airframe noise.

35. M. S. HOWE 1975 Journal of Fhdd Mechanics 67, 597--610. The generation of sound by aerodynamic sources in an inhomogeneous steady flow.

C

ct D f Guo Guu h k kz e~ ( i = 1, 3) e L M~ ( / '= 1,2 . . . . . 8) Mo MoR M~.

n N P Po P qo qxt R s(f) t r

T II

Uo

APPENDIX: LIST OF SYMBOLS

speed of sound local skin friction coefficient Dirae delta function frequency cross spectrum between signals tt and v autospectrum of signal it TBL coherent decay factor 2zd'/c r integral turbulent scale in x i direction mean turbulence correlation scale airfoil s p a n microphone identification numbers Uo/c component of Mo in observer direction, Mo(xl/R) component of eddy convection Mach number (ro/c~t. n)n in observer direction (~o/c~t. n) cos/3 unit vector parallel to V term expressing properties of frozen turbulence pressure time history standard pressure reference, 20 I.tPa Fourier transform of p free stream dynamic pressure value of pressure scatter coefficient at xt distance between source and receiver acoustic spectral density trailing edge thickness time integration time time based signal friction velocity free stream velocity

V ( j = 1 , 2 , 3 )

|

i (J = 1, 3) 1 o

( j = l , 3 ) r

t e

)

'(t3 m 1

t 3

z

abscripts

T R A I L I N G E D G E NOISE P R E D I C T I O N

mean velocity at edge of TBL time based signal rms turbulent fluctuation velocity = (V~, 0, V3) TBL convection velocity, lvl = E1 = v for/3 = 0 mean eddy convection velocity shed vorticity velocity (Howe [1]) rectangular co-ordinates height of turbulence above airfoil surface observer sideline angle airfoil geometric angle of attack angle between V and x~ axis coherence function, ]G,,,I2/G,,,,GL.v TBL thickness TBL displacement thickness coherent decay constant in x~ direction coherent decay constant in x3 direction observer polar angle TBL momentum thickness hydrodynamic wavelength wavenumber vector 27r/Ah, measured hydrodynamic wavenumber kinematic viscosity distance between surface sensors in xi direction incident surface pressure wavenumber spectrum wavenumber spectrum at trailing edge mean density in medium delay time cross spectral phase power spectral density (PSD) of incident pressure longitudinal wavenumber function lateral wavenumber function radian frequency l t . V

corrected observer position effective observer position w.r.t, ray travel time incident hydrodynamic field measured quantity; or extraneous noise at M1 extraneous noise at M 2 retarded observer position contribution from turbulence at x2 = Zo

117

journal oj:sound and vibration (1981) 78(1), 69-117 ... · trailing edge noise prediction from...

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