NUWC-NPT Technical Report 10,69314 November 1994AD-A286 421

Derivation of a Wavenumber Filter Designfor the Measurement of TurbulentWall Pressure Fluctuations

William L. KeithBruce M. AbrahamSubmarine Sonar Department

Q %•94-35902

Naval Undersea Warfare Center DivisionNewport, Rhode Island

Approved for public rclease. distribution is unlimited

PREFACE

This report was prepared under the NU\WC Detachment NCIA London Project entitledNew Lndon Water Tunnel Research. pnncipal investgator W. L. Keith (Code 2141). It v.asfunded by Code 10 under NUWC Division Newport Job Order No 7101B46

The technical revicwer for this report A.as 1 C Chen (Codc 21411

A portion of this research v.as prcsented at the ASME 19Y4 Winter Annual Mceting.Symposiun- on AppicAtIon of Mtro-falrcation to Fluid Mechan.,,%. Chicago. IllIha,,. 6 11November 1 )94

Thc authors• are gateful for the contunued supporn of [I K .M Lama Code 102"

Resiew-ed and Appro,,d: 14 Nostmbe"r 1994

Acling Ilead. Submarinr Sonar lkparinwnt

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1 Ill. L AND SU87ITLL 5 FUNDING NUL41E A~S

Derivation of a Wavenumber Filter Design for theMeasurement of Turbulent Wall Pressure Fluctuations

Wiiam L Keth and Bruce M Abraham

Pf P4 CMOG OAA.42A10%W SAW* ' S£A4N ADDI SS.I t S RI K OR&.WG ORCANVATA CNe RE PORI

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13 ARSTRACY X.-IF tjordiii

Measurernent of the wavenumber. frequency spectrum 0(k ,co) of turbulent wall pressure fluctuationsrequites a liew array (wavenumbter filter) of closely spaced sensors. However, the wide range of wavenumbers ofthe *aIl piressure fluctuations that contribute to the spectrum imposes constraints on sensor size, spacing, andrximber In ti~s study. the Corcos model of the wail pressure field Is used In conjunction with recent experimental

* measuremeris to determine design parameters expressed in terms of boundary layer length and time scales. Forthe case of turbulent boundary layers at moderate to high Reynolds numbers, the resulting wavenumber filterwtsign toquies ml~crolabrication technology.

-**C' tigws 15. NUMBER OF PAGESI ~al oe*F . Design tine Afray. Microlabrication. Pressure Sensor 36

%-v-.w Wal t'.ss.,. Wall Pve~sufe Field. Wavenumbor-Frequency Spectrum M6 PRICE CODE

tA .~~ A~&A(* I 5&CutVTY CLASSIFICA1ION 19 SECURITY CLASSIFICATIONOIF 20. LIMITATION OfABSTRACT

tw* 1 of 11r.S PAGII ABSTRACT

*UNCLASSIFIED I UNCLASSIFIED UNCLASSIF!ED SARPrsalm by1 ANSI Sid Z39- 18~i29&

TABLE OF CONTENTS

Page

LIST 01: ILLUSTRATIONS.......................................... ............ ........ u

LIST OF TABLES..................................................................... I........i

LIST OF SYMBOLS .................. ................ I.......................................ii

INTRODUMtON ...................... ............................. ............ ... II. .........

THlE EFFECT OF SENSOR SIZE ON TInE WALL PRESSUREFREQUENCY SPECTRUM ...........................................................

DERIVATION OF A WAVENUMBER FILTER DESIGN FORMEASL 1EMENTS AT HIGH FREQUENCY ........................................ 3

MEASUREMENTS AT LOW FREQUENCIES............................................ 17

DIMENSIONAL WAVENUMBER FILTER PARAMETERS FORTURBULENT BOUNDARY LAYERS IN AIR AND WATER..................... 20

CONCLUSIONS AND RECOMMENDATIONS .......................................... 20

REFERENCES .................................................................... ............ 23

APPENDIX-SENSOR SIDE EFFECT'S ON MEASURED FREQUENCY

SPECTRA ..................................................................... A-1

ko~lo IvoDT T,~Ii4 4 ~.

DIatu l~ OL

t5 N/

LIST OF ILLUSTRATIONS

Figure Page

1 Dimensional Wavenumber-Frequency Spectra at High and LowFrequencies Estimated From the Cross-Spectral Measurements ofK eith and Barclay (1993) ............................................................................... . 5

2 The Corcos Model of the Nondimensional Wavenumber-FrequencySpectrum at Low and High Frequencies......................................................... 7

3 W avenum ber Filter D etails ............................................................................. 10

4 Signal Processing Flow C hart ......................................................................... 11

5 Spectral Attenuation at the High Frequency Due to the Spanwise orSteam w ise Sensor Length .............................................................................. 13

6 The Product of the Spanwise Sensor Response (for 0 < L2* < 100) andthe True Spectrum for the High-Frequency Case ........................................... 13

7 The Product of the Streamwise Sensor Response (for 0:< LI* < 100) andthe True Spectrum for the High-Frequency Case ........................................... 14

8 The Wavenumber Filter Responses Compared With the True Spectrumfor the High-Frequency Case ........................................................................ . 16

9 Comparison of the Measured Spectrum (for Three Different WavenumberFilters) With the True Spectrum at the High Frequency ................................ 16

10 Nondimensional Wavenumber-Frequency Spectrum for the High- andLow-Frequency Cases From the Corcos Model ...................... . 18

S11 Spectral Attenuation at the Low Frequency Due to the Spanwise orStream w ise Sensor Length ........................................................................... . 18

12 Comparison of the Measured Spectrum (for Three Different Wr -enumberFilters) With the True Spectrum at the Low Frequency ................................. 19

LIST OF TABLES

Table Page

1 Dimensional Wavenumber Filter Parameters for Measurements atHigh Frequencies in Selected Turbulent Boundary Layers ............... 5

144

LIST OF SYMBOLS

kl, k2 Streamwise, Spanwise Wavenumber (radlm)

1% = c0/Uc Convective Wavenumber (radlm)

N Number of Pressure Sensors

p(xI,x2,t) Fluctuating Wall Pressure (piPa)

N. Re UOONv Momentum Thickness Reynolds Number

Ax Streamwise Sensor Spacing (m)

ut= (@/p)1/2 Friction Velocity (in/sec)

8 Boundary Layer Thickness (m)

p Fluid Density (kg/m3 )

Mean Wall Shear Stress (jiPa)

v Fluid Kinematic Viscosity (m2/sec)

((o) Frequency Spectrum (gPa2!(rad/sec))

0(ý,fl,co) Cross Spectrum (LuPa 2/(rad/sec))

c1(kj,k 2,co) Wavenumber-Frequency Spectrum

(IWtPa 2/(rad 3/sec'm2))

I I Magnitude of a Real or Complex Quantity.

iiifivReverse Blank

DERIVATION OF A WAVENUMBER FILTER DESIGN FOR THEME&SUREMENT OF TURBULENT WALL PRESSURE FLUCTUATIONS

INTRODUCTION

The measurement and analysis of turbulent wall pressure fluctuations are important to problems

of flow-induced vibration and noise as well as to szudies of turbulence physics. The space time

field p(x1 ,x2,t) and associated three-dimensional wavenumber-frequency spectrum @(k1 ,k2,ca) are

of primary interest. To date, they have rarely been directly measured in laboratories due to the

extensive number of sensors and signal processing required. A common approach used recently

by Karangelen et al. (1991) and Keith and Barclay (1993) has been to estimate the wavenumber-

frequency spectrum from cross-spectral measurements made with a small number of sensors at

unequal spacings. Although this approach has been useful for estimating the convective ridge

region, uncertainties arise due to the limited number of spatial separations at which data are

obtained. In particular, these uncertainties preclude accurately e imating the lower spectral levels

at wavenumbers outside of the convective ridge region.

Many single point experimental measurements of p(t) and the one-dimensional spectrum 0((o)

have appeared in the literature, as reviewed by Keith et al. (1992). Here, we determine a

wavenumber filter design for the measurement of the two-dimensional spectrum (D(kl,w).

Although arrays of sensors are commonly utilized for acoustic measurements, they are generally

not implemented in incompressible fluid mechanics experiments. The methodology is also

applicable for fluctuating wall shear stress measurements. However, a widely accepted model for

the wall shear stress wavenumber-frequency spectrum required in this analysis does not exist at the

present time. This is largely due to the lack of published research on correlation or cross-spectral

density measurements of the wall shear stress field.

We are primarily concerned with three array parameters: size, spaciing, and number of sensors.

Following Blake and Chase (1971), Smol'yakov and Tkachenko (1983), Farabee and Geib

(1991), and Manoha (1991), we use a linear array of equally spaced sensors. The mechanical and

electrical design of the sensors will not be addressed here. Currently, the size of available sensors

poses limitations on the design of a wavenumber filter. Here we assume that microfabrication

technology will allow a nearly optimum design to be specified at the outset. The design problem

then involves estimating the true field to be measured and determining the effect of the above three

parameters on dte measured field. The sensors are assumed to have a constant frequency response

[7am-_

and uniform sensitivity over the sensing area. We first briefly review.past investigations of sensor

size effects on the wall pressure frequency spectu-m (oD).

THE EFFECT OF SENSOR SIZE ON THEWALL PRESSURE FREQUENCY SPECTRUM

The early work of Corcos (1963) showed that an increase in sensor size leads to a more rapid

roll-off of the measured frequency spectrum O(co) at high frequencies. This effect has been well

established and is supported with the comparisons of published measurements of ((wo) by Keith et

al. (1992). Schewe (1983) measured wall pressure spectra with circular sensors of varying

diameter and concluded that the diameter d+ = dut/v = 19 was adequate to resolve 4)(w) at the

highest energy-containing frequencies in the turbulent field. Schewe did not report measurements

for d+ < 19. The comparisons presented by Keith et al, (1992) showed Schewe's spectral levels

(for d" = 19) scaled with inner variables compared well with the numerical simulation of Choi and

Moin (1990) (for d+- 18). However, Keith et al. (1992) also showed that the Corcos (1963)

correction increased Schewe's spectral levels (for d" = 19) by 2.5 dB at the highest frequencies

measured.

The pressure sensor diameter el required for proper resolution of 4(Co) at all frequencies and

Reynolds numbers remains unkncwn. Schewe's results showed that sensors with d+ > 19 are

inadequate. Head and Bandyopadhyay (1981) and Wei and Willmarth (1989) investigated the

effect of Re on coherent structures in the near-wall region. Wei and Willmarth (1989) concluded

that as Re increases, the hairpin vortices become smaller with increased vorticity in the cores. This

increased stretching of hairpin vortces with increasing Re caused a Reynolds number effect on the

near-wall (y' = 15) normal velocity fluctuations that was not accounted for with an inner variable

scaling. As the Reynolds number (based upon the free-stream velocity and one-half channel

width) increased from 3,000 to 40,000, the outer length scale 8 remained constant and the inner

length scale v/ut decreased by an order of magnitude. Such a Reynolds number effect could cause

the required pressure sensor diameter d' to decrease by an amount greater than simply the decrease

in v/ut. An experiment in which R0 could be varied over a wider range (1,000 <Re < 100,000)

with pressure sensors of varying diameter is required to establish this effect. Current sensor

technology is inadequate for providing sufficiently small sensors for such an investigation.

2

DERIVATION OF A WAVENUMBER FILTER DESIGN FORMEASUREMENTS AT HIGH FREQUENCIES

In order to design a wavenumber filter, we require a model of the true wall pressure field.Although we wish to measure the two dimensional spectrum c(kj,co), we must use a model for the

three-dimensional spectrum (D(k1,k2,c.) in order to account for the two-dimensional spatial

response of the sensors. The Corcos (1963) modcl of the cross-spectrum of the wall pressure field

is given as

4(Nm)= (Dc)(o))A (co.jUc)e'i(AUd)B (,M[Ul/c), (1)

where ý and r1 are the streamwise and spanwise separation variables, respectively. Experimental

data support exponential forms for A(coC-Uc) and B(oyr/Uc) given by

A(wj-JUc) = e~xlt)VUc, B(Ctrl/Uc) = eP k1/Ultu . (2)

Farabee and Casarella (1991) noted that the streamwise decay constant (x reported from various

investigations varies from -0.10 to -0.19, and is a function of various flow parameters. A valuefor cc of -0.125 fits the recent data of Farabee and Casarella (1991) and also Keith and Barclay(1993) at the higher frequencies. A constant of -0.125 implies that a 90-percent reduction inA(o)/Uc) occurs due to the spatial decay of coherent structures as they convect over three

wavelengths in the streamwise direction. At lower values of ow-Uc, the similarity scaling breaks

down, as shown by Farabee and Casarella (1991) and Keith and Barclay (1993). The convectivewavenumbers in this range exhibit a rapid loss of spatial coherence and lower convectionvelocities. The similarity scaling is therefore only valid over a limited range of convective

wavenumbers and related frequencies. Following Willmarth and Roos (1965), the spanwise decay

constant is taken as = -0.7.

T hc wa"enu....fre.ue.cyspectrumn ((ki,k 2,oo) is defined as

(D(kj,k2,o0o) 1 /41t2 T (•loco)et jt+k2lI)dý dil . (3)

Using equations (1) and (2), equation (3) was evaluated with the symbolic manipulation program

MACSYMA (Rand, 1984) to obtain

3

((kl,k2,0)o) Owo)Fi(k¶,kc)F 2(k2,kc), (4)

where

Fj(kj,kc) = { 1/2 kc} {-2or/{ot2 + (kl/kc + 1)2} }

(5)F2(k2,kc) ={ 1127t k1 } {-2p /(D2 + (k2]kc) 2)}

In transforming I(D,ihco), k1 = •oo/Uc is taken as constant, and the variation of Uc with { is

neglected. The Corcos model of the two-dimensional wavenumber-frequency spectrum Cb(kl ,co)

is then

J(kj,(oo)= f D(k1 ,k2,wo)dk2 = '(Oo )FI(kl,kc). (6)

The model of the wavenumber-frequency spectrum given by equation (6) is shown in

dimensional form in figure 1. The low and high frequencies chosen correspond to the recent water

tunnel investigation of Keith and Barclay (1993) (see table 1). There, the lowest measurable

frequency was determined by the acoustic noise in the facility, and the highest frequency by the

electronic noise from the sensor preamplifiers. The cross-spectral measurements of Keith and

Barclay (1993) agreed well with the Corcos (1963) model given by equations (1) and (2), with the

decay constant at = -0.125. The decrease of 53 dB in the convective ridge peak level from the low

to the high frequency reflects the 36-dB decrease in the measured values of I)(cwo), and the 17-dB

decrease in the function FI(kl,kc). The spectrum CI)(klo 0 ) in figure 1 may be viewed as an

estimate of a typical spectrum, based upon cross-spectral measurements zhat could be measured

directly with a wavenumber filter. The high-frequency portion of the spectrum will determine the

minimum sensor size and spacing requirements due to the high wavenumbers involved. The

required length of the wavenumbe filter w•ill incrcase significantly for the low-frequency case in

order to resolve the narrower convective ridge.

We first estimate the highest nondimensional frequency at which measurements of D(kl,o)o)

are to be made that are valid for any turbulent boundary layer. Farabee and Casarella (1991)

concluded that for nondimensional frequencies cov/ut 2 > 0.3, contributions to V(co) are from

velocity fluctuations in the buffer region, and (D (Cw) scales with inner variables. We therefore

define kl*= klv/u1 , k2* k2vlur, *oo' CoY! 2 , Uc* Uc/u, kc*= o** *=v/ut,

4

120 "

11 -'0 wo = 200 rad/s------ .10000 rad/s

S100

"" 90

03a. 70 •

c) 60

50

40--4000 -3000 -2000 -1000 0 1000

k, (rad/m)

Figure 1. Dimensional Wavenumber-Frequency Spectra at High and Low Frequencies EstimatedFrom the Cross-Spectral Measurements of Keith and Barclay (1993)

Table 1. Dimensional Wavenumber Filter Parameters for Measurements at theHigh Frequencies in Selected Turbulent Boundary Layers

UoI g 5v/uc NAx* d+

Invesigation RO Ws) (c) ( ) (cm) (KIs)

('___ ~ (in) (in) JiLL -

Farabee &Casarella 6.0 x 103 28.3 2.8 63.5 1.27

(1991) (air) 92.8 1.1 2.5 x10"3 .50 66

Willnarth & 2.9 x 104 47.6 11.4 53.3 1.07

Wookbidge (1962) 156.0 4.5 2.1 xl0"I .42 383

(air)- - _ -

Keih & Barclay 2,2x 104 6.1 3.3 22.9 .46 444

(1993) (water) 20.0 1.3 9.0 x10 4 .18

Carey et al. (1967) 7.5 x 104 17.0 4.3 9.6 .20 915

(water) .__ 55.7 1.7 3.8 x0"4 .08

Undcrsea Application 2.2 x 105 10.3 21.6 12.7 .25

(sea watl) 33.8 8.5 5.0 x10_4 .10

5

Ax.Axut,/v miad At*" AtUt 2/v. The yoiiieiiii fivqieti 11 A~iI Icvek- ill - tIit 'l v git 1h

tbun1 as

E IkI*,k2*,nk()*) tIbk ~ ()~)Q'(~~)- ~ )I

whicre

1 1 /2-A k,.2 41/((2A' 4 k %1*k 4 1 ))

For E'(kI* ,k2*,n 04U) as detaned jibive,

Theculipmisols f 1) *(WU)o veKSUS Uk)U gi~vefl by Kei'th lIll ut 199.1 ~) h~ln w Ilail Iii Ill,

(11clin.ics ck) * > 3.0) the sixetual levcl~:i ateit last 1() (Ill Ilk-i. Ii.wt i h~lt.i b k ll Stw ii1V

quetucies. We. theltict de cl e t\~4mm 3.0 us the higlicha-st li eqnenhL'v till w~lli-hI (I ,inoth ) \Al IIIi

mneasured. A value fOr U, is itequiv.d it) dote. fine K' Fe n h ioIit .n ' tn i

Spciutt111 d10 MAt eXtenld tO theC lghtglwz hueqnew'ilt.s due it) 111V mnili't-ll (t-,' ay Ill Illti \111'k 111 It %I1..~ 111,1

*occurs Witli typicad finlite Scuasoi seplanttioli.%. Conivectiol VVelocIty Ilwoi~IV'u~IItI I 0ii411e ' litiollhn O1w

phase of the c~ross 'Spectxlnll ale theiexloic ilkil limtited it) lowetIc n1ti '1111 It'plitwst litiqi1 Ilk V

-,for Which lku'ahee and CaIsatella ( 1991) teI to, ted conlvoctioll w Itkiy I wwa~i e cit Il %k.iIl, --

0.34 with TJcj= 1.5. Vot Chlid I~ld MOil's (1990) nollltntik- Alid 1tilitioll, the 11.Ij11k''. lici'pi. lit IiA

coo~ 2.5 Withi U. 11.3. We define 1 Ic' *~i. withl kv*'* v.Z. 'I hit Sliwcia lnihi 1 '. 111'"410 1

evaluated with these pmial'atmwte in slhowa in tiguiwc 2. 'H0111. etuet IS CT110%i iek~' t k -I I

and at k,2* - 0.

Our' waVlveuicullll~ filter' Consists of it line illliy oh N SV~n~IIM. ýe~u , Vill It h.a.'1111! itha

freqaCaIIIcy 1esponse and unil'outti Spatial sensitivity. 'The scaisi ts maek oll Ill~ni~uuu 1111 %1 /

Ij 1u'rITV by 1,2 1 211-C/V, anld 11te eqidll.1y spLCed ait StIM ON 111uie iL It'IIIVIV, A \ *, AS S1%kl Illan

6

----- ------

0 ;

i -o.•v i Iil'lj,,iili~y, t~ I. tiiuLk k2(0.u1

I!,

):O. O0 0 0 0 .8

I-.;,:, II I1

SI I:,

g"".•4,,

'l~ I I' •l ll. . yt. : , I bI'• = .

01 10 l•yj[ 1-(kU ', k,*, (o')]

.~ t.--.'.,.A• ., i'.II

Xl A "4,9)

0.•. 9,' 0.4 , 0,8

Ndytt t

,.LL.S.

L";, A tw 4•

figure 3. We have used rectangular rather than circular sensors to simplify the analysis. The

sensor voltage outputs p(nAx*,t*), n = 0, 1, 2, ..., N-I, are acquired simultaneously in time. The

analog data are first bandpass filtered in the time domain. This may be viewed as a two-step

process. Unwanted acoustic energy inherent to the flow facility (typically at low frequencies) is

removed with frequency filtering. These data are then again filtered at the Nyquist frequency to

avoid temporal aliasing. The sensor outputs are next digitized in the time domain to generate

p(nAx*,mAt*). The discrete temporal Fourier transform of each sensor output is then

M-iPr(nAx ,oao*) = At*/2Tt I wt(m)p(nAx*,mAt*)e-iwo*mAt* . (10)

M=O

Here, wt(m) is the temporal window, and r refers to the temporal record for the ensemble

averaging, r = 1, 2, ..., R. Contemporary analog-to-digital converters and mass storage

capabilities allow sufficient sampling, At*, and record length, M, to minimize errors in

Pr(nAx*,coo*). The discrete spatial Fourier transform of Pr(nAx*,Wo*) is then

N-i - *

Hr(ki*,O)o*) Ax*/27t I ws(n)Pr(nAx*,0)o*)eiklAX* . (11)n=o

The measured wavenumber-frequency spectrum is thus

Sr(kl*,.Oo*) = [1/(NAx*MAt*Cl•C)]IHr(kl *,o * )12 , (12)

where C1 and C2 are corrections for the temporal and spatial windows given by

M-1C1 = (1/M) Y wt 2 (j),

j=0

(13)N-i

= (U/N) Ws2 (j) .j=0

"We choose the Hanning window given by w(q) = 1 - cos 2[7tq/(Q - 1)], q = 0, 1, 2, ... , Q-1, with

side lobes 32 dB below the main lobe, for both the temporal and spatial cases. The ensemble

averaged wavenumber-frequency spectrum is then

8

S (k, ,W I /R Y, Sr*k 1 * . (14

Equations (10) through (14) may be used directly to determine S(kl .(LX). However, existing

fast Fourier transform (FFT) routines arc preferable so as to compute the discrete Fourier

transforms more efficiently. A block diagram showing the sequence of steps in the computation ofS(kl*,wo*) is shown in figure 4.

We now consider the errors that result from the spatial response of the sensors and also from

the wavenumber filter. We note that filtering in the spatial domain is analogous to filtering in the

time domain. However, aliasing in the spatial domain is more difficult to eliminate, as will be

shown. The relationship between the measured and the true spectrum is given by

S(kl*,c0o*) = J J E(il*,1C2*,(o0 *)IQ*(KI*,KC2*)12

(1•NAx*2nC 2)IZ*(IC* - kl*)12dil*dIC2*, (15)

where the normalized sensor response is given by

Q*(KI*,-K2*) = [sin(ic 1*L */2)/(ic *L */2)][sin (K*L2*/2)/( 2*L2*/2)] , (16)

and the wavenumber filter response is

N-IZ*(K=*) = Ax* I ws(q)e-i~l*qAx* . (17)

q=O

We first consider errors resulting from the sensor response Q* (i, 1*X 2 ). The Corcos model

for E*(Kl*,1c 2*,"O0*) given by equation (7) and the sensor response Q*(ii*, C2*) given by

equation (16) are both separable in K1, and K2". We define

- gl*(Kl*,Ll*)g2*(K2*,L2*) (18)

where

9

77 ; ; ii i i I

Kim

10

10•

LmI

Pressure Sensor Array

2 3 4 Pressure Snsors N

Preamplifiers

Band-Pass Frequency Filtering(Analog Filters)

Analog to Digital (A/D) Conversion

Space - Time Fast Fourier Transform* I(Digital Computer)

Ensemble Averaged Magnitude of theFourier Transform Yields e1(ko)

Figure 4. Signal Processing Flow Chart

-• l1

• • , , ,- i , i i i i i i ii•

gI*(KlIdLik) = F*icl,*)Isin(Kl*Ll*/2)/(KI*Ll*/2)12

(19)

g2*(K2*,L 2 *) = F* 2 (K2 *)[sin(c 2 *L 2 */2)/(K2*L 2 */2)]2

and F*l(il*) and F*2(K2") are given by equation (8). From equation (15), the effect of the

spanwise dimension L2* on S(ki*,o)o*) is given by

G 2*(L2*) = f g*2(l•2*,L 2*)dic 2 . (20)

For a particular L2*, the measured spectrum S(kl*, oo*) is reduced by a constant amount at all kl*

and o.'o*. Equation (20) was evaluated using MACSYMA to obtain

G2*(L2*) = 2[e(13 kc*L.*) - kc *L2* - l]/([3kc*L 2 *)2 (21)

The function G2*(L2*) is shown in figure 5. For L2*= 2, a 10-percent (0.5 dB) reduction in the

level of S(kl*,o)o*) occurs, and for L2*= 5, a 20-percent (I dB) reduction occurs. Further insight

into the spatial response of the sensor may be gained by examining the function g2 *(c2 ,Lt2j).

This function is shown in figure 6 for L2-*- 0 (true spectrum), 5, 10, 20, and 100. Here, both the

main lobe of the spanwise sensor response and the dominant energy in the true spectrum are

centered around ic 2*= 0, The spatial filtering of the sensor is significant at higher positive and

negative wavenumbers due to the broad energy content of the true spectrum. The wavenumber

range- 1 < iK2 * < 1 of figure 6 contains approximately 90 percent of the energy existing in the true

spectrum over all wavenumbers. For values of L-* greater than 5, the sensor filtering dominates,

as evident from the side lobe patterns in the figure. We choose L2*-= 5 as an acceptable design

parameter, but note that smaller sensors are desirable.

An acceptable sizc for L1 *ca only b determined by considering the sensor response in

conjunction with the wavenumber filter response, as evident from equation (15). The filtering of

high wavenumbers by the sensor limits the energy that is aliased into the wavenumber range of

interest. We first compare the effect of the sensor response alone on the true spectrum, as thisdetermines the field that the wavenumber filter operates on. Since we are measuring S(k"*,0oO*),

the function g l*(lcl*,Ll*) itself determines the filtering effect of the sensor. This function is

shown in figure 7, for Li = 0 (true spectrum), 5, 10, 20, and 100. The main lobe of the sensor

response is again centered around Kl*= 0, but the dominant energy in the true spectrum is centered

12

1.0

0.9 - GI(L"). Eqn (a4)

"~ . G' 2 (L2 '). Eqn (21), 0.84

0.73

(I 0.6

0.5<i

0.4

000 2;0 4'0 6'0 8'0 100Sensor Length, L, or L

Figure 5. Spectral Attenuation the High Frequency Due to the Spanwisc or

Steamnwise Sensor Length

20-

10- c 3.0

C-J

. L2 0

M -10 .19'

2- , 0

-- 30 '0/

0.401 '. .....

60.0

-1.0 -0.5 0.0 0.5 1.0

Figure 6. The Product of the Spanwise Sensor Response (for 0:• L2* 100) andthe True Spectrum for the High-Frequency Case

- 13

20-,

10 7 .

0-'

-502

-60-

10.~ " "10ao

-40.. ,,

Figure 7. The Product of the Streamnwise Sensor Response (for 0:5 Li' • 100) and

the True Spectrum for the Hiugh-Frequency Case

around K 1 '* = -0.2. For L1 '= 5, the peak in gl '(Kl 4 ,L1 ) occurs at i1' -0.2 , and the peak levelis reduced by 8 percent (or 0.4 dB). For L1 ' = 10, the peak is shifted by 0.5 percent toward theorigin, and the peak level is reduced by 29 percent (or 1.5 dB). For larger values of Ll*, thesensor filtering domiinates the measured spectrum, as shown by the side lobe patterns. We select

L*=5, and determine the effect of the wavenumber filter on the measured spectrum.

With L2* = 5, a 20-percent reduction was shown to occur in S(kl*,Czo ). We express

equation (15) as

S(k 1 ¶,oo*) C3 Jgj (K1 *)(I/NL&x*2x C2 )!Z*(!C1 k,- K~ )1U

where C3 =0.8. In order to determine the effect of the wavenumber filter on the measuredspectrum, we consider the quantity S(kl*,cOo*)1C 3 defined as Sl(kl*,(0o*) and do not include the20-percent reduction, which is constant at all kj '. The filter function given by

V - ki*) =(/NAx*2n7C)IZ*Ocl* - k1')12 (23)

14

is periodic in wavenumber at period 2n/Ax*. The number of sensors N and spacing Ax*

determine the wavenumber bin width Ak* = 2iT/NAx*. We initially define Ax* = 10 (allowing for

finite spacing between the sensors) and N = 100. The filter function V(wl* - ki*) is shown in

figure 8, where the main lobe is centered at k, = -0.2. In comparison, we also show the filter

function for the case Ax* = 5 and N = 100 with the main lobe centered at kI = -0.6 (for clarity).

Decreasing the spacing from 10 to 5 viscous lengths (with the number of sensors held constant)

increases the wavenumbers at which the aliasing lobes occur by a factor of two. In addition, the

main lobe (and aliasing lobes) becomes slightly broader, and the wavenumber bin width increases

by a factor of 2. It is therefore desirable to maximize the number of sensors to achieve a narrow

main lobe and bin width. In addition, minimizing the sensor spacing causes the aliasing lobes to

occur at wavenumbers where the spectral levels are low. We also utilize the filtering provided by

the sensors (previously discussed) to reduce ihe spectral levels at which the aliasing lobes occur,

since the field that the filter function operates on is effectively prefiltered by the sensors.

The measured spectrum S1(kl*,rD3*) was computed numerically for three different cases, as

shown in figure 9. The first case is for 100 infinitesimally small sensors (L1* = 0) and Ax* =10 in

order to illustrate the effect of the filter function V(Kl* - kl*) alone on the measured spectrum.

The measured spectrum is periodic at 2rc/10, and agrees weU with the true spectrum in the

convective ridge region for -0.31 < k, * < -0.09. For k * > -0.09, the measured levels are higher

than the true levels due to aliasing. The second case is for 100 sensors with Lt* = 5 and Ax* =10.

The measured levels are slightly lower than the true levels for kl* < -0.2 due to the filtering from

the sensors (note the measured levels for case 2 and case 3 are identical over this wavenumber

range). For -0.2 < kl* < 0, the measured levels agree well with the true levels, with the aliased

energy reduced due to the sensor filtering. For 0 < kl*, aliased energy is apparent. The third case

involves 200 sensors with Li* = 5 and Ax* = 5. The measured levels agree well with the second

case for kl* < 0. For 0 < kl* the measured levels agree very well with the true spectrum. The

aliased energy apparent for cases I and 2 over this wavenumber range is reduced. This is due to

the smaller sensor spacing, which places the aliasing lobes farther from the main lobe. Here, the

smaller spacing leads to increased accuracy at low values of Iki*I. We select the parameters for the

second case, which result in a good estimate of the true spectrum around the convective ridge and

allow for finite spacing between the sensors, which is desirable for fabricating the array of

sensors.

15

50 . . .... .5040 -- True Spectrum, ow. = 3.0 40

---- Filter Response. N =100. Ax = 10

• 30 Filter Response, N = 100, Ax* = 5 3020 20 -

-10 100)o0 "" 0

S20 ii-2 0-40 ": it ,, : -4

ki

Figure 8. The Wavenumber Filter Responses Compared With the True Spectrumfor the High-Frequency Case

30 ,True Spectrum. u,, -3.0

Measured. Ax* = 10, 1-' =0, N=1100"2 "Measured, Ax' = 10L = 5.N=1 00" 0 a Measured, Ax 5. L = 5. N=200

10-

-0 0 e :,:

.~ 10

o 0

-20

-0.3 -0.2 -. 1 -0.0 0.1 0.2 0.3

kii

DATA POINTS ARE DISPLAYED AT EVERY SECOND WAVENUMBER BIN FOR CLARITY

Figure 9. Comparison of the Measured Spectrum (forThree Different WavenumberFilters) With the True Spectrum at the High Frequency

16

r- _'

MEASUREMENTS AT LOW FREQUENCIES

We now consider the measurement of the wavenumber frequency spectrum at low frequencies.

-The lowest frequency at which measurements may be made is often determined by the background

acoustic noise in the experimental facility. The use of noise cancellation techniques as discussed

by Farabee and Casarella (1991) can allow measurements to be made at frequencies where acousticnoise exists. However, an absolute low-frequency limit valid for any boundary layer cannot be

specified. With regard to the turbulence physics, it is desirable to make measurements at as low a

frequency as is physically possible for a particular facility or test. Here we arbitrarily choose a low

frequency and determine suitable parameters for the measurement of the wavenumber frequency

spectrum.

Farabee and Casarella (1991) defined a frequency region 5 !5 o0/urt < 100, where the

dominant sources for cb(o) are from velocity fluctuations in the outer region of the turbulent

boundary layer. We choose o)o8/ut = 20 which falls within this range. The relationship betweenthe outer and inner frequency scaling is given by wo*Rt = c.0o/urt, where Rt = 8u'T,/v. With Ru

= 2010 (for the highest Reynolds number of Farabee and Casarella (1991)), .o* = .01. The value

for the streamwise decay constant 0t was taken as -0.24, from the results of Farabee and Casarela

(1991) at the lower frequencies. The spanwise decay constant P was taken as -0.7, from theresults of Willmarth and Roos (1965). For convenience, we will retain the inner variable scaling

for displaying the spectra, but note an outer variable scaling is ,:ppropriate at low frequencies, as

discussed by Farabee and Casarella (1991) and Keith et al. (1992).

For measurements at low frequency, sensor size requirements are significantly less restrictive

than for those at high frequency. This is due to the more narrow distribution of energy in both k1 *

and k2* in the true spectrum at low frequencies, as shown in figures 2 and 10. Larger values of

L1 * are in fact desirable for reducing aliased energy, as previously discussed. The reduction in

spectral levels due to the spanwise length is negligible (within 2 percent) for L' < 100, as shown

by the values for G2"(L2*) in figure 11. We define L2* = 100, but note that larger values are

acceptable.

A smaller wavenumber bin width is required to resolve the nanrower distribution of energy

around the convective ridge. The overall length of the wavenumber filter must therefore beincreased. We define the bandwidth Ak over which tde spectrum levels are within 3 dB of the

peak level at the convective ridge. For the high-frequency case, Ak contains eight wavenumber

bins. We choose 100 sensors with length L1 * = 100 and spacing Ax* 1480, which yields eight

17

50-.-True Spectrum, High Frequency. w,* 3.0

40' - --- True Spectrum, Low Frequency, w. 0.01

* 30-0

* 10'

CD

10

o-2Q

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Figure 10. Nondimensional Wavenumber-Frequency Spectrum for the Hfigh- andLow-Frequency Cases From the Corcos Model

1.0<I..

0.9- - G*1 (L*), Eqn (a4)

0 0.8 -- G*2(L2 *), Eqn (21)

S0.7- 0-0*0.01

a,0.6-

< 0.5-~0.4

0.2

0. 1 ----

0 2000 4000 6000 8000 10000

Sensor Length, Lj1 or L2*

, 4-adFigure 11. Spectral Attenuation at tile Low Frequency Due to the Spanwise or

ýý n Streamnwise Sensor Length

. 418

50 I

- True Spectrum, o,", = 0.01

Measured, Ax' =-1480, L1 = 100, N=100

• 40 ,Measured, Ax* = 1480, LI" = 500, N=100

• 0 ]0 Measured, Ax* = 1480, LI = 1000, N=100

_1 30

R 20

0

R 10 R0

01-0.002 -0.001 0.000 0.001 0.002

ki

DATA POINTS ARE DISPLAYED AT EVERY SECOND WAVENUMBER BIN FOR CLARITY

Figure 12. Comparison of the Measured Spectrum (for Three Different WavenumberFilters) With the True Spectrum at the Low Frequency

bins over Ak for the low-frequency case. The measured spectrum for this case is shown in figure12. The convective ridge region is accurately resolved, and the increased levels at higher and

lower wavenumbers occur from aliasing. The overall length of the wavenumber filter for this case

is 148,000 viscous lengths. Also shown in figure 12 are the cases for L1* = 500 and Li* = 1000,

with the same number of sensors and sensor spacing. Increasing the sensor length increases the

wavenumber range over which the convective ridge region is accurately resolved, due to the filter-

ing of wavenumbers that are aliased. For the Lj* = 1000 case, the spectral levels are below the

true levels at the higher negative wavenumbers, but agree well at the higher positive wavenumbers.

The Lj* = 500 length is chosen for the purpose of designing a filter at this low frequency.

Clearly, separate wavenumber filters are required at high and low frequencies. To achieve an

overall filter length of 148,000, required for the low-frequency case (as shown), the number of

sensors needed using a 10 viscous length spacing (corresponding to the high-frequency filter) is

14,800. This is impractical in view of the signal processing requirements as well as the

wavenumber filter fabrication. For a particular application, we may need to design a number of

filters to cover the frequency ranges of interest. The dynamic range required for individual sensors

therefore depends on the frequency range of the wavenumber filter for which they will be used.

19

t .. ,.. . - ..... .. ..;W. .. . n .. .. .. .. ' •.-- •. ...

DIMEINSIO)NAL \VAVI'NUN1)ll34 FiIER~ l'AIANI.I I U%. I i )i1ITUB U LE'NT' BOU1NDARY L AYERUS IN Alit AND) WA I c.1

Wec Consider bOuiildftly hlayCi f runi pai ticulat whid t111111 Iw ild Wie: Itin Ain~ I~i til

ilSO all MxCin13I b0und(ar~y layCl Iinit L-~ in! Xi.N illtn J tiall 011 c~a 11111111 .16111 %Ilk 'i 1% ill,% It% 6 i

bull. The bounidiuy la1yet par-wt tewes anld 1t1C ploilvomsd watvellmili l Il ivi pilfili tc 111 lil lilt Iii 11hfrequency Case are giVell in table 1.Ic~ Regiflld thle two wi iid 1111111tel iilve 'S I titt t 111% W1111111111i1 i41

WoolK~dridge ( 190t2) wborked at a high li orestz canti vehwily. b ut Ilit Icivi~ itIkealmaiutw 'hit 11% 1 I'l slm'I

viscous length scale comparabii~le to thlI ot hi luliceC 1111d Cail.%iu WYMI ~~ I %n ittl 111\ *.I4 lt

usinig wutcr, tile bouiidaiy layer thick nesses weic ecIIIllylklabc Ilil' ilt I ih 14i 1t.1 Ilt, Ci'ill11 viti V. I u I'

of (arey el a!, (1 907) led to a smialler viS tniS Icnth scall NIli11w1 11 iit 1.4-1111i 11111 ~til,, it Ins 1a '14 11

F~or thle undersea tipjliclitioii, tile lbiuiidaly hyt layimjnat iic le well obultailiw ii * itt ( "Ill ., 'I'

ideal silooth Wall, Z.ero ptcs.stne gliadicil, that 11l11t: iase. 'I 1e Ilk-tailioi h11s1t 1 lit I'1s1s,111i I vsls

layer 01 igin Was taken Lis appt~oxiliuately 1(K) it. TIhi: viscouas kIr til;s witir is .t~implai .d 'ill th, toiivi

water ilows, aud thl: hFxmnndwiy layet is thicket,

'llie requtired sensor1 hizes ir these cascs atie 't i leit, idvi '.tit) 10ne 1st lit~ 11 4u111%l iaa llibilli

N~x oft the %Yuveu-cr Uflkliter (bused upol I (X) Solsot s lit 10 (i Vklts h:11111%pt, pilat-) \ li "1 111,111

thle order of 0.1 to 1.0 cmi. Th'le & values 4wiltltlly kused illcii- ehinvsiill IIV'I~IIa ll% 11%11 git cl ' iiI lit-

small viscous length seules ibi thle watrlows ilO d IM 10 ~ d1; aint Vlil-. h it Ill, Wahlas 1111111k Iinvestigatoitof uKeith and Biaiclay, thle actual wiitsu diat tetet was () 0s k-1, wiblli ~, Ist' w~ip. iSI-d 1l.1

444 viscous lengths. The requited wilsmt Sizies, as% well ist'. 11 kestsspll I'itt ig, lilt th4-e &a dwlt

enilphutsiA:, thc ic(ptifelntelt tot' tuiitOtlaicthi~itnio te,.'lulogv10)1Y tillSli'lhIllllllk Aduit11, Ilstit tildiI lit'

Witter.

CONCLJUSIONS AND) I(PAt'ONINIVM)NAI~ONS

I'le analysis proesented here; shows thilt thei iatetln of 11111111 Isle id all ~tStillwavenuunhlel-frlreA ecy spectra at hiigh Itfi trvitce s I ot l j) v'alue s ill pl ati ~ lit-lle11vl -.1 litlilt vil \411114

requires microfabi cation tecciullo~y. lThe %pin iWISe, ias wel i ts tl Ite Vill CIIt1wt m511%l 1st vil'lgit hisSignificant inl reducing tile inleasuied specat levels. I'm tit-1W itiiew.meia CIilt l ilIC ,\' ,.cnllk Itt

frequency specti ut (I)(k1,m)), the stetlCMIMSC SCIIS01i hetigib alON hit1ittsih 111Va Stplln blty 111! 1111

afi`CCtS theU am11ount1 of aiaSed enerIgy.

We have considered two piiticttlarI qulI tit' C Ol imi t~~ wo WMT1VCiik~ i f illvi alt -t s\ Athigh firekilulflcies, it short overall tilter, lenpjh and smititl Nesn iti~ Nitit amlti -wt~ Im tsill lilt ~ It 611i11c.

20

xi.'. I i t' I%%I~ i-%~, liltl~r I ,' Ic lii~Ii I I~~ I I-, tw l ilI I t.1 ol I I g II iIt ItII( greilte I. as is the SIc ISOI spaci IIg.

Il ,' bla' .~ m, w'i~ew-Ii% 111,11 lrcit' l 4 0jq110 N Ol 11Cgltl% icatcr Ill StIM1l iS IC1,thi Me lCL~llired to

v 1t kmilisn~~ut 1'd itlcnIkII w% 11111giu St't luI ., wthi ýllow iiwiIavc- iu1o ilwi fickillency respontses. Ill

pl* litv I, VNV~ II iI,1b1j lilt- 1,11,1 Ii ipllt r~jxill-. tii scli~io s Call 11c miiwasi.ied the wavcn111ibicihv-

IV II $ ~i t V I vy Ill ~j i te 1% 11111% 1h It II k ll M d 1111 ýll i I.i I ml I t% u ccl. \Vhl I ea cc tangt llNe CIIsmsI Siiiay be

1-0*IV$ IVI mIt, bit 11111111l g% hut I~ Il U %%l ' it It$ht u~ I 'Ii I kit~lItk~c llt itci llogy, out &uaal>'sis call also 1w cxtciided

IsItli Ill At PII to 4 We,, hanve kit'lIIV0 I th IN cI I It Io I idv'hci, but I oict ItaKCs wauy be considered,-~~~ ~Nrilh:,: I~ im u t im t11 n uil, lilt%-i ti till, wilkl: Itu'e I sI' %pu 'i 1411 ti ll iiti' t c ic l iL tll wvill slgil~l icaiitiy alTect

op.. Ia te i m I 14t 11% 111rils ~'livi lciti , li t lk ~ ~ h j141 I Ie il 11ctttt1thi 1)SiIliC itjtrs bcu o

4,II Ii. A 'i 0,1111111 fili istiteIctI itIV I" V CI W, Iti Ii iI I;: utI iIg t U011100 IIILis O ijCiICit.:i a el t i t e to b vtise

Jil iti i ii. tees 1 e110 11 w ti litii .stthu y huy)Cl i ll aletiti cocr etdihuuial saiuifllug iL:; w~ell as Jill miuiav c~

ow hri I lit cI;~ iI ~ lit tIil1 jk 'f lit, 111'111% .4 1 otvo Iiln t will 1w ti olipa ticulai. ilitclest, especially for

mm*U 1A i illis w it~i i~ 111 sw,~icical hittlmimiase dhialthileC cotistiailld II' lower Ito values. hIl411.11t1,111, 111C iiill-,141it l biI 1 11I 1i 'ii~ 11IIt I tIIq~l111111 IvJUw ltImuidal y layers will also providt;

111411111l J i$ttil misuijiIth: 1 %i litiejueul 1t1111, tuk liii~ I M 1110chu lugy Will allLAw NULII

Nil

1~vc2-1/22ltvi.eBlank

REFERENCES

Blake, W. K., and D. M. Chase (1971), "Wavenumber-Frequency Spectra of Turbulent-Boundary-Layer Pressure Measured by Microphone Arrays," Journal of the AcousticalSociety of America, vol. 49, no. 3 (pt. 2), pp. 862-877.

Carey, G. F., J. E. Chlupsa, and H. H. Schloemer (1967), "Acoustic Turbulent Water-FlowTunnel," Journal of the Acoustical Society America, vol. 41, no. 2, pp. 373-379.

Choi, H., and P. Moin (1990), "On the Space-Time Characteristics of Wall-PressureFluctuations," Physics of Fluids (A), 2 (8), August, pp. 1450-1460.

Coles, D. (1953), "Measurements in the Boundary Layer on a Smooth Flat Plate in SupersonicFlow, Pt. 1, The Problem of the Turbulent Boundary Layer," Report No. 20-69, Jet

Propulsion Laboratory, California Institute of Technology, Pasadena, CA.

Corcos, G. M. (1963), "Resolution of Pressure in Turbulence," Journal of the Acoustical Societyof America, vol. 35, no. 2, February, pp. 192-199.

Farabee, T. M., and F. E. Geib (1991), "Measurements of Boundary Layer Pressure Fluctuationsat Low Wavenumbers on Sm, ýoth and Rough Walls," Proceedings of the ASME Symposiumon Flow Noise Modeling, Measurement, and Control, NCA vol. 1 l/FED vol. 130, Atlanta,December 1-6.

Farabee, T. M., and M. J. Casarella (1991), "Spectral Features of Wall Pressure FluctuationsBeneath Turbulent Boundary Layers," Physics of Fluids A, 3 (10), October, pp. 2410-2420.

Head, M. R., and P. Bandyopadhyay (1981), "New Aspects of Turbulent Boundary LayerStructure," Journal of Fluid Mechanics, vol. 107, pp. 297-338.

Karangelen, C. C., M. J. Casarella, and T. M. Farabee (1991), "Wavenumber-Frequency Spectraof Turbulent Wall Pressure Fluctuations," Proceedings of the ASME Symposium on FlowNoise Modeling, Measurement, and Control, NCA vol. 1 l/FED vol. 130, Atlanta,December 1-6.

Keith, W. L., and J. J. Barclay (1993), "Effects of a Large Eddy Breakup Device on theFluctuating Wall Pressure Field," Journal of Fluids Engineering, vol. 115, no. 3, September,pp. 389-397.

Kcith, W. L., D. A. Hurdis, and B. M. Abraham (1992), "A Comparison of Turbulent BoundaryLayer Wall Pressure Spectra," Journal of Fluids Engineering, vol. 114, no. 3, September,pp. 338-347.

Manoha, E. (1991), "Wall Pressure Wavenumber-Frequency Spectrum Beneath a TurbulentBoundary Layer Measured With Transducer Arrays Calibrated With an Acoustical Method,"Proceedings of the ASME Symposium on Flow Noise Modeiing, Measurement, and Control,

NCA vol. 1 l/FED vol. 130, Atlanta, December 1-6.

Rand, R. H. (1984), Computer Algebra in Applied Mathematics: An Introduction to MACSYMA,Pitman, MA.

23

Schewe, G. S. (1983), "On the Structure and Resolution of Wall-Pressure FluctuationsAssociated With Turbulent Boundary Layers," Journal of Fluid Mechanics, vol. 134,pp. 311-328.

Smol'yakov, A. V., and V. M. Tkachenko (1983), The Measurement of Turbulent Fluctuations,Springer-Verlag, New York, pp. 178-190.

Wei, T., and W. W. Willmarth (1989), "Reynolds-Number Effects on the Structure of aTurbulent Channel Flow," Journal of Fluid Mechanics, vol. 204, pp. 57-95.

Willmarth, W. W., and F. W. Roos (1965), "Resolution and Structure of the Wall Pressure FieldBeneath a Turbulent Boundary Layer," Journal of Fluid Mechanics, vol. 22, no. 1, pp. 81-94.

Willmarth, W. W., and C. E. Wooldridge (1962), "Measurements of the Fluctuating Pressure atthe Wall Beneath a Thick Turbulent Boundary Layer," Journal of Fluid Mechanics, vol. 14,no. 2, pp. 187-210.

24

• , , i i i I I I i II

APPENDIX

SENSOR SIZE EFFECTS ON MEASURED FREQUENCY SPECTRA

Although we have focused on the measurement of the wavenumber-frequency spectrum,

microfabrication technology is important for single-point measurements of the frequency spectrumOm(coo). For our case, the relationship between the measured and true (one dimensional)

frequency spectrum may be expressed as

omn(coo) =t(wOo) J f gl*(il*L*)g2 *QC2 *L2 *)dKl*dK2 * (A-1)

which reduces to

'rm(O)o) = 'Dt(co0)G 1 * (I *)G2*(L 2*), (A-2)

where G2*(L2*) is given by equation (21) and

00

Gi*(LI*) f g* 1(cl*,L 1*)dl * . (A-3)

Equation (A-3) was evaluated using MACSYMA to obtain

Gl*(Ll*) = 2[e(a kc*Ll*)(2(xsin(kc*L,*) + (cc 2 - 1)cos(kc*Lm*))

- 1+ )kc* Lj*- Q2 + 1]/ [(a 4 + 2(x 2 + 1)kc*2L,*2]. (A-4)

eIThe functions Gl*(L,*) and G2*(L2*) are shown in figure 5 for the high-frequency case. From

equations (21) and (A-4),

dG2*/dL2*I L2*= 0 = 5 dG*/dl*I L* 0'" (A-5)

For small values of L1* and L2*, the spanwise sensor length therefore causes a greater reduction in

spectral level than the same streamwise length. For Lj* = L2* = 10.4, the spanwise and

A-I

streamwise effects are comparable. For larger values of Lj* ard L2*, the streamwise length

dominates the reduction.

The functions GI*(Ll*) and G2*(L2*) are shown in figure 11 for the low-frequency case.These functions are similar to the high-frequency case, with the L2" effect dominant for sensorlengths of less than 2500, and the Li* effect dominant at larger values. However, the actual valuesof Li* and L2* required for a particular reduction are generally two orders of magnitude greaterthan for the high-frequency case. This reflects the change in the energy distribution in the tluespectrum at low frequencies, as shown in figures 2 and 10. The reductions are negligible for Lj*and L2* < 100. For measurements at low frequency, sensor size requirements are thereforesignificantly less restrictive than for those at high frequency.

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L7:;, ; ; i ...i i -1 --i. ..1