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283 683Copy_ Y•',•)

U.-.L Report No. 5591-052-00-00

crc"- WALL PRESSURE CORRELATIONS IN TURBULENT PIPE FILUW

S-nOnl 01 01-0401

". ZO7~ 20 August 196Z

•.

by

Henry P. Bakewell. Jr.

George F. CArey

John J. Libuha

Howard H. Schloemer

and

William A. Volk Winkle

III

1S. , I5 V iUlEUUETER EOlii L581 URTORSFORT T'UMBULL NEW LONDON, CONNECTICUT

-1 Wisi

ABSTRACT

Certain properties of the pressure field induced Ly turbulent airflow at the wall of a cylindrical pipe havebeen investigated over broad frequency bands and in octave fre;-.nacy bands over a limited range of Reynoldsnumbers. The broad-band me4surements Indicate that the pressae field is convected downstream at approxi-mately 0. 7 of the cznterline velocity. The ratio of the root- mean-square z.: -t-re to the dynamic pressure isapp~oximatey equal to 0.006 for the variout keynolds numbers. Longitudinal and lateral space ccrrelations arealso pretened for a limited range of spatial separations. Data obtained in octave frequency bands indicate thatthe ratio of convection velocity to cent---,ne :,elocity decreases as frequency increae, with the ratio varyingfrom approximately unity to 0.64 for the frequency bands investigated. Lonqtudinal and ;ateral correlationdata for various frequencies, spatial separations, and Re)nolds numbers are shown to be functions of Strouhalnumber. Estimates of the longitudinal and lateral scales are given as functions of frequency for the variousReynolds numbers. These data indicate thai the lateral scale is approximately equal to one half of the longi-tudinal scale.

ADMINISTRATIVE INFORMATION

The work described in this report wvs initiated in May 1960 by Dr. William A. Von Winkle and was carriedout under USL Job Orders Noe. 1-612-07-00 and 1-052-00-OC and Navy Project No. S-k011 01 01-0401.

Dr. Von Winkle prepared the program planning specification and supervised the deSign, comtruction, andpreliminary check-oat of the turbulent pipe flow facility nmtil September 1961, when he was appointed USLRepresentative for a special program at Bermuda. He has continued to serve the moject in a consultant capacity.

George F. Carey, who had designed the facility and had guided the project throughouw the construction amcheck-out stages, aune d Dr. Von Winkle's superviscry duties.

Howard H. Sctloemer, Assistant Protessor of Mechanical Engineering at the City College, New York, -.id amembtr of the USL staff for the past thee summers, assisted inthe preliminary design of the tunne and obtainedthe initial wall prtsure correlation data.

Henry P. Bakewe-i, Jr., and John 3. Libuba obtaunedthedata reported 0 this study. In addition, Mr. Bakewellperformedthe mathematical anatyis of the experimental results, and Wr. Libuha ausembled and modified existing

r t sim for the pipe flow facility.

REVIEWED AND APPROVED: 20 AUGUST 1962

Warren Ho r t-Technical Direct- - CoVmm , VTI:~ -cer an d6 Dirca!

TABLE OF CONTENTS

Page

STRC .. . . . . . . .. . . Inside Front Cover

ADWI6NISTRATIVE INFORMATION . . . . . . . .

LIST OF ILLUSTRATIONS ..... ...... . .. ... ......... ii-ill

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

SYMBOLS . . . . . . . . . . . . . . . . IV

INTRODUCTION .... ........... .... 1

GENERAL DESCRIPTION (T THE TUNNE. ........ ......... 1

INSTRUMENTATION ..... . . . . . . .. .............. 3

MEASUREMENTS ............. .............. 5Root-Mean-Square Pressure, Prensse Coefficient, And Correction Factor . . 5Frequency Spectrum .... . . . . . . .. . ............ 6The Convection Velocity ........ . .. . ........... 7Decay Ratio uf the Correlatlon Maxma , I. . ... . .... 9Iongittiih-a1 Space-T.r-. Corzn',i .. ..... ...... .......... 9Longitudinal and Lateral Brozi•-Band Space Correlation . . . . . . "

Longitud!-P.1 and Lateral Correlaton as Ftmc'ions of Strouhal Number. . . 10•-,;,...- i---t----- -- A ------ - ---..--

SUMMARY AND CONC'LUSION ........ ............. 15

DISTRIBUTION LIST ............. . .......... Inside Back Cover

LIST OF ILLUSTRATIONS

Figure Pagc

I Diagram of the Turbulent Pipe Flow Fackity . 172 Block Diagram of Instrumentation System . 17

3 Root-Mean-Square Pressure versus C-6aterline VeW ; . . . . . 18

4 Root- Mean-Square Pressure Coefficient versus Reynolds Number . 195 The Non-Dimeas loallzed Frequency Spectrun . 206 Correction to Power Density for Finite Transducer Size... . .... 21

7 Ratio of Convection Velocity to Centerline Velo,-ty veratr ReynoldsNumber.. ........ . . .. .. . 22

8 Ratio of Convection Velocity to Centerline Velocity venus Frequency . 239 Cross-Correlation versus Time Delay for the Freqtency Band

212-20, 000 cps . ........................ 2410 Cross-Correlation versus Time Delay for the Frequency Band

212-425 cps. . ...................... . 25

11 Cros-Correlation versus Time Delay for the Ftequency Band425-850 cps .......................... 26

12 Cross-Correlation versus Time Delay for the Frequency Band8M5-17 0 0 cpls .............. 27

13 Crcu-Correlation versus Time De.Lay for the Frequency Band1700-3400 cps... . .................... 28

14 C sou-Correlation versus Time Delay for the Frequency Band3400-6800 cp;.. . .................... 29

15 Cross-Correlation versus Time Delay for the Frequency Band6800-13,600 cps. .... .. .. .... ..... ...... . 0

16 Cros-Correjauon versu3 i"oa- Dh.ienionil Time Delay for theFrequency Band 212-20,000 cps .. .... ... . .. 31

17 Cros-Corr-lation versus Non-Di.mensional Time Delay for the

18 C'ors-Couelation versus Non-Dimensional Time Delay for theFrequency Band 425-850 cps ._ . ................ 33

19 Cross-Correlation versus Non-Dimensional Time Delay for theFrequency Band 850-1700 cps . .. ...__. 34

20 Cross-Correlation versus Non-Dimenstonal Time Delay for theFrequency Band 1700-3400 cps ......... . ..... ... . .. 5

21 Cron-Correlation versus Non-Dimensional Tirme Delay for the

Frequency Band 3400-6800 cp.. ....... . . .... .. 3622 Crons-Correlatinn versus Non-Dimensional Time De'-" for the

Frequency Band e800-13, 600 cps. ... . ........ . 3723 Space Correlatiou at a Reynolds Number of 300, 000. ... . . 3824 Space Correlation at a Reynolds Number of 250, 000. . ...... 39

25 Space Correlation at a Reynolds Number of 200, 000 . . 4076 Space Correlation at a Reynolds Number of 150, 000... . . .. 4127 Space unrelation at a P.ynolds Number of 100, 000 . . . ._28 Longitudinal Corelation versus Stroubal Number Based on Ctntedline

Velocity............ . . . 43

-~ _ _ 11 --- ' W - - -' - T _

LIST 01' ILLUSTRATIONS (Cant' d)

Figure

29 Longitudinal Correlation versus Strouhal Number Batod on ConvectionVelocity

30 Lateral CorrLl; lion versus Strouhal Number Based on Centerline

velocity31 Lateral Cc.-elation verst.2-~ obal Number Based on Convection

Velocity32 Estimate of the Longitudinal Scale33 E~stimate of the Longitudinal Scale versus Frequency (Based on Auto-

correlation VDta).

34 Estimate of the Longitudinal Scale versus Frequency (Based otiCorrelation Data as a Ftuictlc-i of Strouh.l Number)

31. Estimate of the Lateral Scale versus Frequzency (Based on CorrelationData as a Function of Strouhtal Number)

LIST OF TABLES

Table F

1 On' txe Btami' UPrilized at Specific Reynolds Numbers. ........2 Values of Constants In Correlation Expressions . . . . . . .

SYMBOLS

d Pipe diameter

f Frequency in cycles per second

p (K. y, t) Fluctuating pressure

t Active radisis of transducer face

R Reynolds nwiaber based on pipe diameter

t Time

T Time delay

UdL Centerline pipe velocity

U0a Convection velocity

31, y Longitudinal and lateral space coordinates

e) ~ Lougi...dinal and lateral separation of transducers

0 Angular frequency

S Strouhal number

ip (f, 17,r Normalized space-time rorre!ation functoni

p Fluid density

Bessel 6inction oL .he first kind of order a

Longitudinal scale

x Lateral scale

Y()2 P where P 2 is the mean square pressure within a third octave band

BANDWIDTH

£ Bounda-y la~yer thickness

iv

WALL PRESSURE CORRELATIONS IN TURBULENT PIPE FLOW

IN'I '.ODUCTION

Certain propeities of the tur:ulent pressure field at the wail of* acyl:ndric-, pipe have been investigated experimentally over a range ofiR -rolds numbers from 100, 000 to 300, 000. In particular, data havi been

e i.ned concerning the convection veloci-. :.,d longitudinal space -timecor.-elation.- both over broad frequency bz.,ds and in octave bands. Longitu-dinal and lateral correlations have also been obtained over broad bands -Asfunctions of spatial separation and in octave bands as functions of Stroulhalnumb r. Finally, measurements nf the root-mean-square pr'essures andthe frequency spectra have been made.

GENERAL DESCRIPTION OF THE TUNNEL

The t',,rbulent pipe flow facility is located in the massive old fort at theY.. Trumbdll site of the Underwater Sound Laboratory. This location was

•lected bncausý. of its low ambient noise level and solid, vibration-freeioundation.

The development of the experimental apparatus, Fig. 1, which was usedin this investigation began with the work of Von Winkle and Weyers.' Know-ledge of the deficiencies of the apparatus used in thei.- investigations was ofconsiderable help in the design of the present system.

Blower noise was attenuated by shock-mounting the blower frame,encasing it in a plywood box lined with six inches nf Riperfine fibcrgaantinsulation, and using a large muffler at the exit ox the blower. This mufflerwas found to act effectively ab a low-pass acoustic filter with a 'utoff fre-quency of 80 cps. In addition, the blower was decoupled from the mufflerby a 3-foot-long section of neoprene tubing wrapped with fiberglass. A shortlength of rubber tubing connected this upstream muffler with the pipe.

A 40-foot-long brass pipe of 3. 5-inch inside diameter and 4. 5-inchoutside diameter is the major component of the experimental system. Thepipe is supported on small, rubber-.lined, wooden blocks which are fastened

I W. A. .ouWinide, Somt Measurements of Longimiuai Spal-.-Time Cosrelatlots .f WaillfPessurerlctuatiotsln i tn~ bt.nt Pipe Flow., USt, Report No. 526, 17 Augus 1961; and P. F. R. Wc 1 brs, Te Vibiarion

ind Acoustic Rllsadon of Thin- Walled .:ylnuezs Caused by internal Turbulent How, Guggenheim AeronaudcaiLaboratory Repor. on Contrac.t NAw-651?, August 1958.

on top of large wooden tables. Except for the test section, this pipe isentirely encased in a 12-inch by 12-inch sand-filled ply,,koud box. Tests usingar accelerorneter mounted on the pipe wall at the test section indicated thatthe pipe was free of resonances. A downstream muffler was alsro provided toattenuate any pressure fluctuations resulting in the mixing of the zxhauststream with ambient air.

The test section itseli was located 96 pipe di ":-.eters downstream of theupstream muffler to insure filly developed turbulent flow. A traversingmechanism which allowed interchangeable use of Pitot tubes, stagnation andstatic pressure probes, and hot wire probes was used to determine thevelocity distribution across the pipe. The resulting 1/8 power law velccitydistribution, obtained from Pitot tube traverses, confirmed other measure-ments and indicated that the turbulent flow was fully developed.'

Static pressure taps were located 9 feet apart along the length of thepipe upstream of the test section to measure differential pressure. Thevelocity of the flow was monitored by observing this pressure differential ona draft gauge. Changes in velocity were accomplished by moving a damperin the inlet of the blower. A remotely controlled drive mechanism was usedto control the damper from the test section location.

The range of mean velocities available for the experiment extended from60 ft/sec to 178 ft/sec. This corresponded to a Reynolds nmnber range,based on pipe diameter, of 100, 000 to 300, 000.

Compressibility eflects were ignored since the highest l3cal Mach number.at a centerline velocity of 214 ft/sec, is 0. 185, well below the nominal

.'-, LaliLUIk•UJCA gtucraily used as the iower limit oi compressibility

phenomena.

The test section was a 2-foot-long section of pipe, which could be removedi-'o-n the pipe run and replaced by other sections (such as an anechoic chamber).For measurements ol pressure fluctuations a 1. 5-inch-diameter hole wasdrilled in the pipe, into which brass plugs could be inserted.

A series of 0.1 525-inch-diameter holes in each plug allowed inter-changeable use of transducers and dummy sections with the result that manydifferent combinations of spacings could be obtained with only two transuucers.

"2Fo: a discimion of full7 developed turbulent p.pe flow, see 11. Schlic: -Ing, Boundar LaXer Theory.McGraw-! "M Book Co., Inc., New York, 1955.

a

Different brass plugs were used to obtain longitudinal and lateral spacecorrelatirp.s. Tolerances of the assembled system were kept close to ensureno protrusions of greater than 0. 001 inches into the air stream. Thisvalue corresponded to approximately half the calculated t-.,ckness of thelaminar stvb-layer.

" S''UAMENTi.=iON

The active face radius of the transducer element was taken as 110 percentof the actual radius of the active element. The resulting active transducerradius was only 0. 0344 inches--51 times smaller than the pfpe radius of1. 75 inches.

The computed longitudinal and lateral scales were 0. 15 and 0. 075 inches,respectively, which were 4. 36 and 2. 18 times larger than the active trans-ducer radius. Since the squares of the recipocals of the last two ratios weredominant in the corrections for finite transducer size, these correctionfactors were very small, as is noted in Figs. 3, 4 and 5.

The transducers were constructed of a lead zirconate ceramic in a mannervery similar to the types used by Franz and by Barger and Von Winkle" intheir investigations of boundary layer pressure fluctuations The electricalcharacteristics of a typical transducer are as follows:

nominal sens.itivity: - 120db//1v/tib, or l±v/pLbar

nominal frequency range: 90 cps to 20 kc

nominal capacitance: 40p}•f

The transducers were essentially free-field calibrated over a irequency rangeof from 90 cps to -0 kc. The open-c-cuit receiving sensitiviftr responsewas reasonably flat and was approximately -120 db/!lv/vb. To minimizeloading losses for this high-impedance source, a one-inch length of Microdotconnecting Lable and a specially modified Enaevco model 2607 amplifier we.,eused to accommodate the high impedarce of the transducer. This transducer-preamplifier combination satisfactorily amplified the low-level signal fromthe transducer to a level sufficient for anall sis and recording.

3 G. 1. Franz, "Turbulent Boundary-Layer Pressure Fluctw,." - on the Dow Dome and Supentructureof the USS ALB ZORE(AGSS 569ý," U. S. Navy Jomurnl of Un.-X-wt,--r Acousics$ vol. !:. no. 1, January1962 (CONIIDENTIAL); and j. A. Bargex and W. A. VonWknkle, "An. Evaluation of a Boidary Layer Stablitza-",. Coating," a paper presented at the. sixty-first meeting of th1e Acoutical Society of America in Philadelpha,ea., on 10-13 May 1961.

3

The measurement system, Fig. Z, consisted of two identical channelsof instrumentatior.. Each channel contained an LDX-107 transducer and

matching Endevco amplifier, a Daven decade attenuator, a Burr-Brown

amplifier, a Gertsch 1/2 octave filter set, and a Ballantine amplier. The

outputs of the two channels were Lsed. as inputs to the polarity coincidence

correlator either directly or through a dual-channel Ampex tape recorder

with a variable time delay. Also incorporated into the system for spectrum

analysis of the noise signal were a Bruel and Kjaer i/3 octave frequency

analyzer and a Br~el and Kjaer proportional band analyzer. All instrumentationexcept the correlator, analyzers, and tape recorder were battery-powered to

minimize electromagnetic radiation interference from external AC power

sources and to minimize grounding problems.

es, A calibration of each channel showed the system response to be linear and

flat from 40 cps to 40 kc and also to be in phase throughout the frequency

range of interest. The correlator used was of the polarity coincidence type

designed by the Naval Ordnance Laboratory as Model WOX-3A. All correlation

measurements were later converted from the polarity ceincidence correidtor

values to true correlation coefficients. Stability over long periods of time anddependence on the phase rathe," than the amplitude of the input signals were

the major advantages of this instrument Since a signal input of 3 volts rms

or greater was required to produce correct results, care was taken to ensure

that the input signal exceeded this value and also that there was an adequate

signal-to-noise ratio. Normally the input signal. was placed at about 10 volts

,rms, which afforded a minimum of 13 db signal-to-zioise ratio. No measure-

mcnts were made in the portion of the spectrum where the ratio was below

this value.

ge For the space correlation measurements, the generated signals were

transmitted directly from the last stage of ampl-fication into the corrcl.lýor.

For the space-time correlation measurements, the generated sigr..tils were

recorded on a dual-channel tape recorder and played back through the

e variable time delay unit into the correlator. A system was then available

which afforded maximurn ease, flexibility, and reliability for making these

measurements.

4

MEASUREMENTS

Root-Mean-Square Pressure. Pressure Coefficient, and

Correction Factors

The pressure fluctuations were detected by using a wall-mounted trans-

ducer and were measured in 1/3 octave bands. The sound pressure levelwas calculated from these measurements by using the bandwidth corrections

and calibration constants.

Root-mean-square pressures at each Reynolds number were obtained byaveraging the sound pressure level in 1/3 octave Lands, i. e.,

where

N = the number of bands

and

Pit = the mean-square 1/3 octave band pressures.

The working frequency range was determined by signal-to-noise ratio ratherthan transducer response.

The root-me. n-square pressure is displayed in Fig. 3 as a function of

centerline velocity. It should be noted that the corrected values are very

near the measured quantities because of the small size of the transducers.

The correction factor4 used is

4w f. ~a

4G. hi, Cocos, k. W. •u'tDe.., and W. A. VonWinkae, On tk. Mekstememn of "ut.ulutt PreureFlictuatloos with a .Tmnsducer of Finite Size, University of Californi? Berkeley, histitute of EngineeringR ,-,rch Report Series 82, Issue No. 12, November 1959, p. 23.

5

where

p2 and p.' actual pressure and measurcd pressure, respectively,

and

A a effective area of transducer face.

Elsewhere in this report it is shown that X ' ̂- 0.S• ..hence, the correction

factor reduces to

1-7. 20.936

25 2

where r = 0. 0344 inches and A . 0.15 inches computed on a broad-band

basis.

A plot of pressrre coefficient versus Reynolds number is presented in

Fig. 4. These values of the pressure -.oefficient are within the range ofvalues reported by other investigators.r The slightly higher values found at

the lower Reynolds numbers are due to the experimental scatter in the

spectral values of the low-frequency components at these Reynolds numbers.

Frequency Spectrun

The non-dimensionalized frequency spectrum is shown in Fig. 5. which

is in general agreement with the results of other irvestigators.' Again, the

correction factor is small for most of the experimental range because of

small values of r/A, . The correction factor' used is of the fornA

sVonWlnlde, op. cit.; W. W. Willmarth, "Space-Time Correistloc" and Spectra of Wall Pressure in aTurbulent Bomndary Layer.* National Aeronautlcs and Spa-. Administration Mhhm-radum No. 3/1.7/59W,March 1959; D. H. Tackt, M. W. Smith, and R. F. Lamibert, "Wal Presure Correlations in Twb.lent AW'..ow,"botirial of the Acourtical Soci2et of Amcrtca, vol. 33, no. 4, AprIl 19•6; and G. M. Corcos .-0d W. A.VonWinkde "Sn.,e Measurements of the Pressure Field at the Boundary of a F0,'1- Developett Pipe Flow," apaper presented at the 1969 'Divisional Meeting of the Division of Fluid Mecwnai; of the Amnicalt PhysicalSc.:..ty, at Ann Arbor, Mic.a., on 23-25 Nomenoer 1959.

GVotWlnlde, op. ci . Wlnmaith, g. cit.; Franz, : aWd M. Harrison, Presure Fct!•.o. onthe Wal A:-teent to a Turbulent Botudary Layer. David Taylor Model si•n Report No. 1260, December 1958.

-Corcos, Cuthbert, and VonWialde, op. cit. p. 21.

6

• ) 4 J, (Y)j, Vy "Y) J.U (y: + j• 2y))Su3 1 (y)* 2Kt' + 6K 4 r 4 2.. +

Y Y

with

K~~) In~Lii

and

oenv

For the data in question only the first two terms are necessary, with the

second term in the order of 10 percent of the first term.

Using the generally accepted definition of scale, i.e., 02./, ._2

all V I

and K 0.5•E along with other constants previously mentioned, the correctionfactor equation reduces to

4 1 . 5 2 (y)

This is displayed in Fig. 6. The corrected spectral values lie within

experimental scatter over a wide range of non-dimensional frequencies, asis seen by inspection of Fig. 5.

The Convection Velocity

Space-time correlation measuremnents of the pressure fluctuations atthe wall have been made in octave bands and ýn broad bands by recording thesigzc.als from two transducers at various longitudinal separations and theaintroducing time delay mechanically between the two channels during playback.

The recorded trace of the cross-correlogram shows a maximum value at afinite value of time delay r (measured fro, .r zero of time delay). Therate at whic* the pressure field at the wall is cor.,,'cted dowur-tearr. is

determined by dividing the longitudinal serjaration of the transducers .y thetime delay necessary to obtain a maximum of correlation.

*1

The measurements of convection velocity were made at the followingReynolds numbers over the broad frequency bands indicated. (The upperlimiting frequency in each case was determined by signal-to-noise ratio.)

Reynolds Number Frequency Band

100, 000 212-4800 cps150,000 212-69f0 cps200,000 212-13,600 cps250, 000 212-20, 000 cps300, 000 212-20,000 cps

Also at each Reynolds number, measurements of convection velocity weremade in as many of the following octave bands as possible:

212-425 cps425-850 cps850-1700 cps1700- 3400 cps3400-6800 cps6800-13, 600 cps

The ratio of the broad-band convection velocity to the centerline velocityhas been determined and plotted against the Reynolds number in Fig. 7. Thisratio has the value of approximately 0,70 and appears to be independent of

veynolds number. Although this value is lower than the 0.80 generallyreported by other investigators,$ it is consistently repeatable. It is possiblethat less averaging of the pressure fluctuations occurs because of the smallratio of the diameter of the active face of the transducer to the boundary

2r 2rlayer thickness; L" - /-!- - 0.039.

X d/2

The ratio of the convection velocity to the centerline velocity has alsobeen determined in octave bands and plotted as a function of the geometricmean center frequency of each band in Fig. 8. The data for the differentRevnolds numbers have been averaged together, as this ratio does not appearto depend on Reynolds number within each octave band. It is of interest thatthe pressure Ructuations, at the wall, of frequencies within the octave bandcentered at 300 cps are couivected d.wnstrearn at approximately the center-line velocity.

8 VonWinkle, Hanrion, Willmarth, and Tack, Smith, and La.mbert; ala M. K. Bull, lnstri-;. atatcon forand Preliminary Measurements of Space-Time 7Ol-elations and Cowvection V'locities of the Prei•a• Field of aTurbrrv'^ doundary Layer University of Southampton (England) AASU Repor No. 149, Augtut 1960.

8

These data substantiate the theo v -hat the ratio of convection velocity tocenterline velocity decreases as the trequency increases with the ratiovarying from approximately unity to 0. 64 for the frequency bands investigated.

Investigations in the lateral airection, as expected, revealed no detectableconvection velocity.

Decay Rate of the Correlation Maxima

A series of space-time correlations was made over a limited range oflongitudinal transducer separations up to a maximum of 0. 924 inches at aReynolds number of 300, 000 in octave bands and over the broad band212-20, 000 cps. The broad-band cross-correlograms are plotted againsttime delay T in Fig. 9, indicating the rate of decay of the correlationmaxima. The octave-band cross-correlograms are presented in a similarmanner in Figs. 10 through 15 and indicate a small rate of decay of the low-frequency pressure fluctuations as compared with that of the higher-frequencycomponents.

Longitudinal Space-Time Correlation

The same series of space-time correlations used to determine decay ratesTru

has been plottea agai.st the p•rramet.: r i octave frequency bands and

over the broad band 212-20, 000 cps at a Reynolds number of 300, 000. Thesedata are presented in Fig. 16 for the broad-band measurements and Figs.17 through 22 for the octave-band measurerant.s. For the various combinationsof transducer separations and frequency bands all the correlation maxima

ruoccur at a value of approximately 1.0 for the parameter - with the

values of the maxima depending on the transducer separation ý . (The highervalues of correlation correspond to the smaller longitudinal separationq )

TUThe slight deviations from the value of 1.0 for the parameter *env at

the correlation maxima are due to the use of the average value of u ,IT with-.n a given frequency band instead. of the individual value of u v dctermircdfor each cross-correlogram.

7

Longitudinal and Lateral Broad-Band Spce Correlation

The normalized space-time correlation function of the fluctuating pre,.sure

is defined as

ap t ,y,t •

The longitudinal and lateral space correlation functions are special casesof the space-time correlationi function •b•y7 and are given respectivelyby b(•,o,o) and •',77,o) .These functions have been obtained experinmentallyover the broadest frequency band available at each Reynolds number for trans-ducer separations of 0. 1515 to 0.9010 inches laterally and 0.2Z76 to 0.924

inches longitudinally. The longitudinal and lateral space correlation data are

plotted against the ratios of transducer separation to pipe radius, •/2and

/2 respectively, in ig.23 truh2-7. As can be seen from the

figures, the smallest available transducer separations are not sufficiently

small to provide detailed informationz ahout the space correlation functions inthe regions of 0<•/d/ 2 <0.I and 0<'7/d/ 2 <0.i .Thus, at the present time,

no estimates of t.he broad-band lateral and longitudinal scales of the turbulencehave been obtained from parabolic. approximations of the space correlationcurve~s. (Other estimates of scale will be discussed later.)

The interesting asl'ect of these data ±,s that at each Reynolds numberinvestigated, for equal values of longitudinal and lateral separation, thelateral space correlation has a higher value than the longitudinal correlation;

•,( ,% )> ,( ,o~,) whet, '! "

khowever, at closer separations than those investigated this ielauonship m-aynot be true. Also, it shouxld be noted that for a given t..ansducer separationthe values of the longitudinal and lateral space correlation functions appearto decrease as the Reynolds number is decreased.

Longitudtnal and Lateral Correlation as Function 3 of Strouh'-l Number

Space correlation data wcre also obtained in octave frequency bands usi,,gthe previously mentioned Reynolds numbers and lo..,4,' dinal and lateral

transducer separ tions. The functions which have be~.s measured _•e srecial

10

cases of a -3rrelation function of the type:

PoAk Y ,y, t)" R.(X + 6,y + 1,Pt +

The subscript f. represents the geometric mean center frequcncy of theoctave band, and the subscript R indicates the Reynolds number for a givenmeasurement. The following table indicates the octave bands used.

Table 1

OCTAVE BANDS UTILIZED AT SPECIFIC REYNOLDS NUMBERS

Af(Cps) f,

212-425 300425-850 600 R = 100,000850-1700 1200) R - 150,000 250,0001700-3400 2400) R 250,000

3400-6800 4800 300,000

6800-13,600 9600

The longitudinal correlation functin q'f.,R(•.o,o) and the lateral

correlation function Ofo.R t, 1,o0 have been deter-nined experimentally

and plotted against the Strouhal nuimber ( -- a 5 iun F s

and 30. The same funcL" -s have also been plotted against the Strouhalnumber with the frequency-dep-ndent convection velocity in the denominatorinstead of the centerline velocity in Figs. 29 and 31.

The long~tudinal correlation data for all R,.Vnolds numbers, frequencies,and separations investigated appear to define a single curve similar to adamped cosine curve:

'At., 0•o~) , COS (AS) e"-b8.

This t;pe of curve can be fitteu, to the data -_lmost op to the third zeroof the funct'on.

The lateral data for all Reynolds nunibt-rs, frequencies, and separations

11

investig4Led comnbine to define a snxgle curve uo th- following nature.

to, d5 I+CS 1 2 -.

g, ,tbO,° C + cS!"' (2 -e's)

The values of the constants used in these mathematical expressions forlongitudinal and lateral correlatien functions are given below for both set8 ofdata plotted against Strouhal number based on either convection velocity orcenterline velocity.

Table 2

VALUES OF CONSTANTS IN CORRELATION EXPRESSIONS

Constants Strouhal Number rigure No.

b-4 a f 28

2 ~ 00

! b -2 292 (75- UC*"V

c = 20 d n 100 I'll, 30uCL

c - iO- 80 17o 31UeOIIV

It should be noted that these empirical expressions for the lateral andlongitudinal correlation functions also satisfy the general requirements that

ai. #a.., (0,o,o• V ,

B~r., (C'o~o) J s0._O .'

o,,,.( (e,0,o 0 " ... (-4,o, 0.

o.. ,,*,c) O kf*, i "t ")

lZ

Estimates of the Longit~uinal and Lateral Scales of Turbulence

The longitudinal space microscale is ebtimated from 9.-e time microscaleand the convection velocity. Thi- is accomplished by apprcxi-nating the

autocorrelation function with a parabola. The interse.-tion of this parabola

with the time delay axis determines a time delay, ° , kiiich when multiplied

by the appropr.ate convection velocity yields an estimate of the longitudinal

scale.

In mathematical terms this relation is expressed as

) •(u,,, 2 L J 'r =o

which is obtained from approximating the auto-correlation function 0(o,o.-j

with a Taylor series about the origin. The first two terms of the series, which

yield a parabola, are used to approximate the function b for small values of

time delays.

The estimates of longitudinal scaie obtained over the various broad-frequency bands are plotted in Fig 32 as a function of Reynolds number.

The average value oi the longitudinal scale k is approximately 0.15 inches.

Estimates of the longitudinal scale were also obtained by the same method

for the various octave fre,:uency bands investigated at each Reynolds number.

These data are presented as functions of fzequency %geometr.c inean center

frequency) for the various Reynolds numbers in Fig., 33. As expected, the

scale becomes smaller as the frequency increases. A definite Reynolds

number dependence, which can not be explained by experimental error alone,

is exhibited by the measurements of longtudinal scale in octave bands.

Estimates of the longitudinal scale in octave frequency bands were also

obtained from the longitudinal correlation coefficient data as represented by

the empirical expression shown in Fig. 29 as a function of Strouhal number

based on convection ve.ocit.. Letting S• be a measure of the distancc between

the origin and the point of intersection on the Stroubal number axis ol the

osculation parabola at the vertex of the correl.' Ain Curve, the following

relation iq obtained:

2 -a' a 2 _1b

13

where

U

and a and b are the constants used in fitting the empirical relation to the

data of Fig. 29. For the case at hand, a = ?r and b = 2, yielding

sec2 I-I../ 4,s 2 ,Aff.-\t w& 2+ 4

orU

A 0.214 -envf,

The values of A f determined from the above relation based on the

Strouhal number representation of the data are plotted versus frequency for

the various Re-ynolds nu:--ers i:n Fig. 34.

A comparison of Fig. 33 with Fig. 34 indicates very close agreement

between the two methods of estimating the longitudinal scale in octave fre-

quency bands.

A similar treatment of the L"teral _orrelation coefficient data shown in

Fig. 31 as a function of Strouhal number based on convection velocity yiel-'sthe following relation:

2. . ... 2(c +d)

UuO

where

and the determined values of the constants are c = 10 and d = 80. Substitu-tion of these values yields

u.- 1 ,

40.105-

14

The values of XA determined from the above relation are shown inFig. 35 c-ý a function of frequency for the various Reynolds numbersinvestigated.

A comparison of the estimates of longitudinal and lateral scales obtalhedfrom the correlation data as functions of Strouhal number indicates that forany given octave frequency band at a specified Reynolds number

A, .05

X k• 0.214

or

X,7 = 0A9Ae

On the basis of this information for octave frequency bands, it hab beenassumed that the broad-band scales are similarly related. Thus, in .ll thecorrection factors on a broad-band basis the lateral scale has been taken ashalf of the longitudinal scale. (It should be noted that if the lateral scale wereactually larger than half of the longitudinal scale, the necessary correctionsfor transducer size tc the root-mean-square pressure and the frequencyspectra would become even smaller than those used in this report.)

SUMMARY AND CONCLUSION

Broad-banQ measurements of the pressure fluctuations are similar tothose reported by other investigators of the turbulent boundary layer. Theratio of the corrected root-mean-square pressure to the dynamic pressurelics within the range of 0. 0062 to 0. 0049. The broad-band piessure fieldat the wall is convected downstream at approximately 0. 70 of the center-lir:e velocity over the range of Reynolds numbers from 100, 000 to 30D, 000.Estimates of the scales of the turbulence indicate that the lateral scale isof the order of half of the longitudinal scale.

The more important aspect of the current investigation is the emphasis

on the octave frequency band measuremente. Convection velocity data ino:ctave frequency bands indicate a aefin~te freq-.ency dependence with theratio of convection velocity to centerline velocity varying from approximately

unity at the low frequencies to 0. 64 at the high frequencies. No dependenceof this ratio on Reynolds number is detecta',..,

Space-time correlations in octave ba.ids indicate that the correlation maximaof the high-frequency components of the pressure field at the wall decay -t a

15

substantially fster rate than the correlation maxima of the low-frequency

components. Longitudinal correlations plotted versus Strouhal number shnw

that the data obtained at various transducer separatione, frecuencies, and

Reynolds numbers define a single function. Lateral correlation data alsodefine a single function of Strouhal number. (These functions are of distinctly

different types, as is readily evident from the data and from the. fitted curves.

Finally, estimates of the scales of the turbulence within octave frequencybands indicate that the lateral scale is approximately half of the longitudinalscale within any specified frequency band.

Since other experimental studies of the turbulent boundary layer have notconcentrated on the frequency-dependent characteristics of the measured

quantities, few comparisons can be made. However, because of the reliabilityof the instrumentation, the consistent repeatability of these data, and the smallsize of the transducers used in the present investigations, it is believed that

these data represent a further step in the direction of an understanding of theturbulent processes.

In conclusion, further investigations are in progress in the turbule.,t flow

facility at the Underwater Sound Laboratory, These investigatioi.s are to becarrieda cut over a wider range of transducer separations and in narrowerfrequency bands. Also an attempt will be made to investigate static pressurefluctuations within the fiow and the relation betw.een pressure and velocityfluctuations.

na

16

LI. 3 VD 0

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won~

I, 4

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17

I .:,~~~~00, -,j,•. .

Cý- JNCORRECTE'D OAT!0-_ _ _DATA CORRECTED ;OR

_ _ _ _ _ ___ ___ _

I .•'C . .. : : ,TRANSDUCER SIZE

I ; I . . .-- . . ._ ___ L_ _ - _

- I" 4-

% 00. ,

_-_____________- i i20• 304 5 78910 0 30 40"0

7C,

I ~~40'

UFT/SEC.

Fig. 3 - Root-Mean--3quare Presure veus Centerline Velocity

18

7 " - . .: . :: :- .• ' : - ;: :--7 :: : : : :-:- , : : :: I - L

010

~P.C ....... _ _ . .. I... ... '..t

002p 0000 '000 000

CNL~ ~ME

M i: 4-Ro-enSurePesr ofiin vtu enisNme

.OC)4 '9

-5

-14- 2.

J- I 4

REYNOLDS NUMB8ER - - tiY~f - ~- ~0 100,000[

2 3 20Q ______

4-COMP"OSITE WF

~TRAN SOUCCR SIZE,

141,*

I -IT

_!d

Fig. 5 - The Non-Dlmensioaalized Frequev'' spectfuni

o. ..... .. .

7t-ttlI I I T,

7f F71 7.1 7

-T .-- I-1 -- 1 -711I ~.;iiii

7- - j t jj

~ ~~i~j1 ~ II t- I I 7

0-7 07 TT0

T.u~

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77 51 !42: A ;;j

luTit:ld Ti!

7.1-7 1 1 lII-+*I

III 'I IIlE~j~j~: :. . REYNOLDS f U161ER FRED BA~rdO1LPS).;,, 0100,000-- 212- 4,800

45~ 1 '25,000-- 212-6,800so1`,oon--- 212-6.800

0; Jj~ 200,000- 212 -13,6C00

~ ~* j~ .j1~i.T 1 '' i* :~ .~~. A 250.COO - 212- 20,000

It .11*. in ti~ I! I ;'*3 300,000- 212- 20 ,000

!it; fi ..ULL IIIm

I.....J...L..L ~ ~... ... .... -, ; r;~ ll. HI 1 I'00,,0~0.0 200,000 300,000

REYNOLDS NUMBER

Fig. 7 -Ratio of Convection Velocity to Cf. iterline Velocity versus Reynolds N'umber

(N) UMBR OFSAMLESCENTER

(64) 4'I

U 671UCNV ... 646 . 6376

DATA FOR O I5U.C()U0[ 1 OREYNOLDS NUM 250.0001 00,00

300,000

7 FI

100 1000 10.000 2,0FREQUENCY IN CPS

Fig. 8 - RLi~o of Convection Velocity to Centerline Velocity veusus Frequency

23

TRANSDUCER SEPARATION-. 000"UT0,CORRELAT.ON

......... 0 -. 738"

REYNOLDS NUMBER- 300,000~.JL... REGLIENCY- 212-20,000 CPS

.6zj

LU 7

LU

S.2

-- -------

0 104 208 312 416 520 624 728r IN MICROSECONDS

Fig. 9 - Cross- Correlation versus Time Delay for the Frequency Rand 21' -20, 000 cps

24

7 ~ 71T>1 ~T~~

iij I.~4-

044

-j 10- 738"

* Eytj()t OD. t .4 f!ER~ 300. 000

2H - 212

: U * %i:j'~INMC0E00 I 2

0~.. ..4 .0 3.. 2 ... 2

rH.1, rs orlto eu ieD~a.frt~FeunyBn )-2ic

02

I I

TRANSEOUCE- SEPARAT UN

nlOO ALTO CORRELA N

---- -- -. 462 :oi

C) - 738 TO

.91 Bs~h4E R-30,OO.U~7

n: iQIE NCr -42 5-850 CPS

.3 ... ..

-1.17

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Fig. 11 _ CO~sICorrelation versus Time Delay for £e Fzrequency BaRQ 42b-850 z p

26

I ij~~ ~T T_ l ~ Ti 7 _____

.7l TRANSDUCER SEPARATIONA- 00") AUTO-CORRELATION

-4-- ]REYNOLDS NUMBER-300,OOO

W6 K~fFREQUENCY-85O-i700CPS]

I4 I V-14

0

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.27

I I ~TRANSDUCER SEPARI IONA- OOO"AUTO-CORRELArIOWNh--t 4ALA.~..0--.462~

f J REYNOLDS NUMBiERS- 3013,000J- FREQUENCY -1700-34OCCPS

T::

4'- i r

.28

.- T TSyR SEPA'PATiON- -A-- CCOQ AUry CORPELATION

0- 462"-..- ,----,----,- 0 - 738"

SV REYNOi.DS NtuY8B;P -30C,000

FREOUENC -34,00 -6800CPS

7- -

__ _ 6

_____ _____I

64_ _

_ _ -A.

LT:V

2 7..~

-2-

C 104 208 312 416 520 e7!r 1 N MICROSECONDS

Fig, 14 - Gross-Correlation versus Time Delay foi th, Frequency Band 3400-68A0 cps

29

T." JUC'ER SEPAR4D NA,

A- coo'U4u-ýcoRELATlON

I9 - 92."SI I~ . .F FR[OUE NCY -68(10-13,600 CPS

0~ ~ fiTTT- 7.

T E 77t: a~~T T~7~~pw ~ : T f ~ 'T j it:II

i u .2leitHLx ;; ; ;,p

... .....Io

I " 1I

-. 2

0 104 208 312 415264 78r IN MICROSECONDS

Fig. 15 -Clow, C' ,rrrldt ion versus Timne D~elay for the Frequency Band 6800-13,600 cps

30

~17 t~T~ 1 . 1 .~4.TRANSDUCER SUrAFATION ~

U-2ý7-1 QL YNOLDS NUMIBER- 300,000

* -~ REQIJENCY-2:2-20,OOOCPS

.I. . . . ..

Z 6t

.4L

t,~ 1 21 [WINj

.2 t Elk

4 6.8 1 2 1.4 .

Pig, 16 - Gross-Correlation versus Non- Dimensional Time DelAy for the Frcquzacy Band 212-20, 000 cps

31

t 77T 77-T- 7

~7~tK

D 0 ' 00;i;crz

C) Ij

-7I1

4 m ITT

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t .

z to

o o 0

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or4

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71 --7

F71 o~.,71... L. .bo

' I*, ~~ I ii.

I D W1NQJA: 1OV780 3n.

LA I 33

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ii- N Cw ;H!

KK......... .. ..

Wi]i

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INJIJIJ3JOJ NIOI IV 3080D~ jn8 I

34

v- 17 TRANSDUCER cr.PARATION

REY*NOLDS NUMBER-300,OOi-REQJE NCY- 1700 -3400 CPS

.. ..j4 T... .......

tLL

0

.4 6.

5.2

'I I ! !' ' " Z-L- L-L-LJJ• 'L'

!1`4A N ;,i')jf P 1 -~A PA II ONi~ I ... 0- 462" r

M"~NI Hs 0I~jtMirF4 300.000. I I'jiuNCY 31(0E0 64C(P,

Z ,! . .. . .7

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ki 44 i• J I

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it.J h .I ; , t t ' I .'1 •

I ;7:J. L7'H Y 7: I: I 1* .1 'I'UI .. ,I

7 . 101 i .

4I1 1.4 1 6

r UCON.

1ig. 21 - Cross-Correlation versus Non-Dimensional Time Delay for the Frequency Band 3400-6800 cps

- ~ ~t TRANSDUCER SE&PARAI ION

01 924"

REYNOLDS NUMBER-3300,000

- -*~. ~FREQUJENCY -680C -t33600C S

______J_ C 60OCPSý

-F7-

4 6 8 I . 2..........

TT

7: 11

2LON GIT UDI NAL A1~1~~t1'~~. . *LATERAL

':REYNOLDS P(IMBER:730C,00

~ *....FREQUENCY- 1 -20,000 CPS 1 it

-4

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Pig 23!Sa4Creaina aRyod ubro 0,00

C38

IN ING'T U N AL IPEY'vOLDS NUMBER -250,000

I FRCQUENCY- 212720,000

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w 17

it it

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Fig 4 baccCorc~as n ata Renols Nmbe of2b0 00

03

[IA .: .l~LONGITjfjb'JIALI *-LATERAL~

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I'ig 2! Spae Cndaton a ~ ey~wA~ Niz~er c 27T-: 0

T., 11 t I0

f rnvtT1II1KT7T T-7 --7777 rTFI1IT1TU LONG;TUI';, AL

'1 Ii 0*LATERAL liRYNOLDS NUMBER -150,000~.

F IREQUEN CY -212-6800 CPS

d:h -

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Fig. 26 - Soare Correllation at a Reynolds Number of 150, 000

41

8LONGITUDINAL

~~ *LATERAL ~ L

I i VLREYNOLDS NUMI5EiR 100,0001j: T7 FREQUENCY -212-4800 CPS

z .6 :1

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u 300,C00 212-20O

2 a ---- ---

.. ... 4. ..... I.2 .... 1....------- ± LI- F:1

0100,OCO 20C,000 300,000REYNOLDS NUMBER

Fig. 32 - EstinutE of the Longitudinal Scale

47

L~liL REYNO( V, NJMIER5

v5.--n-~.-- 0100,000 .

4 0150,000

3~ A 250,000 _

II iII

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FREQUJENCY IN CPSpig. 34 - Estimatf. of the Longitudinal scale versin Frequ.F-ticy(Based on Correlation Data as a Fu'nction of Sirouhal Nwvmbez)

49

4 -.7 RENLD4-NE

3 ff)150,000 IIi? I ~ 200,000

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