NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALMEMORANDUM1285

LOAN COPY: RETURNTOAFWL TECHNICAL LIBRARY

KIRTLAND AFBs NC M. /

INVESTIGATIONSOF THEWALL-SHEARINGSTRESS

INTURBULENTBOUNDARYLAYERS

By H. LudwiegandW. Tillmann

Translationof “Untersuchungeniiberdie Wandschubsparumngin turbulentenReibungsschichten”

WashingtonMay1950

NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICELME240MNIIJM1285 . .

INVESTIGATIONSOFTHEWiLL-SHEARINGSTRESS

INTURBULENTBOUNDARYLNERS* - --—

ByH. LudwiegandW. Tillmsnn. .

SUMMARY

Becauseoftheunsatisfactorystateofknowledgeconcerningthesurfaceshearingstressofboundarylayerswithpressuregradients,theproblemisreexamined.It is foundthatforgeneralturbulentboundarylayers.inwallproximity,thatis,inthelsminsrsublayer,in thetransi–tionzoneandinthepartofthecomplete~turbulentzonenearthewall,thesameuniversallawappliesas fortheylateflow. Fromthegeneralvalidityof thislawa formulawasdeducedforthelocalhag coeffi-cient cfr,inwhich cf’ dependsonlyontheReymoldsnumberRe formedwiththemomentumthicknesssndona profilepsrametery. Thisrelationwasconfirmedsatisfactorilyby directmeasurementswitha newinstrument.Therelatedfrictioncoefficientcf’ canthenbe determinedsimplyfromtheknownvelocityprofile. .--——-.

l?romtheformulafor Cft it follows,inagreementwiththetests,thatthe cf’ valuesforboundarylayerswithacceleratingsnddecele-“-ratingpressurearehigherandlower,respectively,thsafortheplate ‘flowatequalReynoldsnumber.Thus.forgreaterReynoldsnumberssmalllocaldragcoefficientsareattainablenotonlybykeepingtheboundarylayerlaminarbutalsoby appropriatepressurevariationinturbulentboundarylayers.Theriseofthefrictioncoefficientto a multipleofthatforplateflowinboundarylayerswithpressurerise,asclaimedbyvariousworkers,isheretithdisproved.

.

--

.-

—

1. INTRODUCTION

Thewall+hesringstressesinlsminarboundsrylayerscanbe com-putedon a strictlytheoreticalbasis,sincetherelationshipbetweenvelocityprofilesndshearingstressisknown.But,thisprocedurecannotbe appliedto theturbulentboundsrylayerssincetherelationship‘“

*“UntersuchungenfiberdieWsmlschubspannunginturbulentenReibungsschichten.“ ,.

.-

2 NACATM 1285

fortheshearingstresses,dueto theturbulentexchti-geisstillunknown.Forthisreason,thelawsforturbulentwallfrictionmustbe determinedby experimental@vest-igations.Suchinvestigations“fallintotwoclasses,termedforbrevity,‘plateflows”and‘pipeandchanuelflows.”Theapproximateformulasforthefrictiondrag,deducedby thevariousinvesti-gatorsfromthetestdata,arein agreementtosomeextent.

Someinvestigationshavebeenmadealsoonboundarylayerswithpressuregradients,bothaccelerating-anddecelerating,butthedataonwallfrictionareeitherabsentaltogetherorpartlyUnsatisfactory.Theseinvestigationsweremadeina channelofcircularsection(refer-ence1) or,inmostcases.}ofrectangularsection(references2, 3>4}and5). Forthelatter,onechannelwallwasdesignedasflattestplate,onwhichtheboundarylayertobe exploredwasmeasured.Theoppositewallwasadjustabletothedesiredpressuredistribution.Itwasspacedfarenoughfromtheexperimentalsurfacetomaintaina corewithpotentialflowbetweenthetwoboundarylayers(freeboundarylayer).Thewall+hesringstre~swasdeterminedfromthemeasuredvelocitypro-filebymeansofvonKarm&tsmomentumequation.Theadvantageofthismethodrestsinthefactthatthickboundarylayers(highReynoldsnumbers)canbe producedwitha comparativelysmalllayout.Onesub-stsmtialdrawbackisthenarrowwidthofthe-experimentalflowcomparedto theboundary-layerthickness.Thisislikelytoproducesecondaryflowswh;chcancelthetwo-dimensional.ityoftheflowassumedaccording

‘ ‘smomentumequationfortheinterpretation,andarepr=tovon Karmansumablyresponsiblefortheimprobableresultsoftheaforementionedauthors.Someoftheseauthorshadobservedthat,ingreatlyretardedflows,thelocal.dragcoefficientCf’(=Tw/Q,Tw = wall-shesringstress,Q thedynamicpressureoutsidetheboundaryla~er)roseabruptlytoa multipleofitsoriginalvalueaftertravelinga certaindistauceinflowdirection,ratherthsmdecreased,as actually&i%fcipatedanalogousto thebehaviorofthelaminfiflow.Sincenovalidreasoncould be foundforit,TilJmann(reference6)madean Investigationinenattemptto findoutwhetherthiseffectwassimulatedby secondaryflOws. Thevelocityin a sectionacross’thetunnelwasmeasuredinmagnitudeandMrection.An app?aisaloftheeffectoftheobservedsecondaryflowsgavea cf’ valuelowerby 40percent.Thisfactmadethetestmethodemployedup to thenquestionable.

Quiterecently,Ludwieg(reference7)developeda simpleinstrumentbymeansofwhichthewall+hesringstresscan%e determinedat anyteststationby a heat--transfermeasurement.Thisdirectmethodisnotaffectedby anysecondaryflows.Theinvestigationofthefrictionaldragofturbulentboundarylayerstithpressuregradients~..bothaccele- .,ratinganddecelerating,repeatedwiththisnewinstrument,formsthesubjectofthepresentreport. ●

.

NACATM 1285

II.THEORETICALCONSIDERATIONSONFRICTIONALDRAG

3

Theresultsofyrevioustivestigationsofplateandpipeflowswereasfollows: .

(a)Thevelocityprofileof theboundarylayer@n be represented--—-

intheform

u

()y Re.=g=u (1)=

(u= velocityh boundarylayer,U = velocityoutsideboundarylayer,

Y= wall distance);52 .J”;~-$w = momentumthiclmessofboundary52

layer,Re = UT = Reyno& numberofboundarylayerfomnedwith

momentum thickness62;v = kinematicviscosity.

Qpsntityg inequation(1)isa fixedfunctionis,naturally,differentforplate,pipe,andchanneldependenceOf ~ on Re isverysmall,thatis,the

differverylittlefordifferentReynoldsnubers.

which,however,“-flow. The

velocityprofiles

(b)Thelocalfrictioncoefficientcf’ canalmys be re~resentedintheform

Cf’= F(Re) (2)

(Cff= Tw/: $; TV = wall.+hearingstress;p = density).

QpantityF isagaina fixedfunctionforplate,pipe,andchannel”””‘ “’flow;F canbe ccxuputedforplateflowby themomentmequationwhen gisknown,becausethetheboundarylayer.

(c)Forthepartrelation

totalfrictiondragap@earsas lossofmomentumin

ofthevelocityprofilesnearthewall,the

u ()f U*Y—=*u 7-

wasobtained. v(U*=TWP “= sheari~fitressvelocity)

(3)

lInthisformula,theReynoldsnumberformedwiththemomentumthickness52 waschosenas characteristicquantitybecauseit ismoreappropriateforboundarylayerswithvariableoutside~ressure. .

4 NACA’1141285

II

~is relationholdstruewiththessmefunctionf--forthepart..

of plate,pipe,or channelflownexttothewall..

(

ProvidedthattheU*Yfi values arenottoosmall fullyturbulentzone ~

v )>50 , equs

tion(3) can be replacedveryaccuratelyby

u *—=alog* y.+bu

a and b beinguniversalconstants.TMsapproximatedby a powerlaw

l/11u

()~ U*Y—=*

u T

theapproximateI?onmla

(3a) -

logarithmiclawcanbe .-

(Sb).

where C and n areconstantswhicharestillsomewhatdependentonthe u*y/v zoneforwhichtheapproximationisto be especiallygood.

AsalreadystatedinLudwiegtsreport(reference7), it is tobe,

expectedthattheuniversallaw,equation(3) or (Sa),is,asidefromtheplate,pipe,andchsnnelflow,applicablealsotomoregeneralizedboundary-layerflowsinwallproximity.It&en shouldholdforvelocityprofilesdiverg~gconsiderablyfrczntheprofilesinplateflowandforflowswithmarkedpressuregradients.

A definiteexperimentalproofofthegeneralvalidityofthelaw,equations(3)and(Sa),isaffordedfromthefactoriginallyestablishedby Wieghardt(reference8);nemely,thatwhentheboun~ary-layerprofilesareplottedin themannerof’logu/U againstlogy/52(fig,1),parallelstraightlines-areobtainedforsmally/52. Consequently,uis inallcasesproportionaltothesamepowerof y. ~Fromtheslopeof thestraightlines,thispowerfollowsas 0.13=~ , whichisin

.goodagreementwithequation(Sb)forthe u*y/V rangeinquestion.However,thisstill.isno compellingproof-ofthevalidityofequation(Sb)forthereasonthat-thepowerof y caube checkedby profilemeasure-ment,butnottheconstentC,becauseu* Isunknown.

Withthevalidityof*equation(3)fortheportionoftheboundarylayernexttothewall,u and,hence,‘w and cf’ dependonlyon thevelocityprofileandthematerl.al-constantsoftheflowingmedium;so,whenthevelocityprofileisknown,cf’ocembe computed.A COl?re-

spndingrelationbetweencf~,Re,anda profileparemeteryettobedefinedisderivedinthefollowing.

.

.

NACA!lML285 5

%2Theterm y=— isintroducedasprofile~ameter; u~2 is

udefinedas follows:If thelaw,equations(3),(3a),and(Sb),isvalidfor y valuesgreaterthm 52,then u~2 is simplythevalueof uat thepoint y = .32.But,ifthelaw,~uations(3),(3a),and(3b)Yappliesonlyto y valuessmallerthan 52)then U~2 isthevaluewhich u wouldassumeifthelaw,Equations(3),(3a),ad (3~),w~r~applicableup to thepoint y = 82. Thus,thedoublelogarithmic_plottingof u/U againsty/52(fig.1) givestheprofileparameter7,

Ywhentherectilinearpartoftheprofileisextendedas faras — = 182 ,’52tsre~ fromfigure1.andthecorrespondingvalue”of

uThe derivationoftheabov~entiohedrelationbetweencf’,Re,

and y proceedsfromequation(3). Theprofileparsmeteris introducedby puttingy = 62 and

u~2—=*

u

Theequationstates

ad U5252/V;therefo~

(4)

thata directconnectionexistsbetweenu*/u52

u*‘()

—U5252—= h—l+j v2

mustbe applicable,thefunctionh beingdefinedby functionf..—__

btroductionof

2Grusc~itz(reference2)definedthequantityv = 1 -

profileparsmeter;butby U52 thevslueqf u at Y = 52 iSalwaysmeant.

—

--

—.—

.-

6

gives,aftersimplerearrangement

Cf’= 2y2h2(Re

Forabbreviation,thefunctionis

NACATM 1285

-. .-

7) = 72H(Re7) (5)

written2h2=H. ”

So,forallturbulentboundary-layerprofileswhosepartnexttothewallisrepresentedby thegenerallqw,equation(3),thefrictioncoefficientisgivenintheformofequation(5). To definethefunc- ,tionsh and H inthisequation,equation(4)couldbe replaced,fortheargumentin questioninaccordancewithequation(3a),by

U*andnumericallysolvedfor —=h. Butsincetheconstantsa and b .

u~2

inequation(3a)arenotaccuratelyenoughknown,thefollowingmethcdofdefiningH seemstobemoreappropriate.

By equation(1),theprofileparameterfortheprofilesoftheplateflow,designatedYo,iSO~Y dependenton Rethence“-70 = yo(Re). Oninsertingthisvalueinequation(5),thisequationmustsupplythedragcoefficientCf’ fortheplateflow.Thus,bearinginmindequation(2),thefunctionalequationfor H followsas

70%(Reyo)=F(Re) (6)

where F(Re) isthefrictioncoefficientoftheplateflow.This ‘equation, whichmustbe fulfllledforall Re,definitelydefinesthe .-functionH forknownfunctionsF(Re)and 70(Re).

Abbreviating

Re 70(Re)= g

iterationgivesf’orRe thechainfunction

ERe =

T–-70

7

()70 “--

-.

.

.

>

7NACATM u85

which,insertedine~ation(6)givesfor H

snd,whenthefunctionH in equation(5)isthenreplacedby theprecedingexpression,

Since y. varies very littlewith Re (fig.2), “theofthechainfunctionis so goodthatinthefirstfactorof thefirstdeureeandinthesecondfactor,thetermofhaveto be incl&led.Therefore

C’=*

(7)

convergenceonlythetermzerodegree

(8)

Thisformulagivesthefrictioncoefficientcf’ forgeneralboundarylayers(forexsmple,withpressureriseor fall)inrelationto theReynoldsnumberRe andtheprofilepsrametery. ItWSSderivedontheassumptionthattheuniversal.law,equation(3),isapplicableinwallproximity.ThefunctionsF and YO canbe takenfromtheexperimentaldataonplateflow. .- ,..

As an approximation,itis sufficientto insertinequation(8)thefunctionsF and y whichfollowfromtheassumptionofthecon—ventional1/7powerlaw!?orthevelocityprofile.Owingto theaffinityoftheprofiles,yO isunaffectedby Re andcanbe computedby asimpleintegrationwhichgivesthevalue Y. = 0.717~Thecorres~on~ngfunctionF withGruschtitz’snumericalconstsnt(reference2)reads

Cf’= F(Re)= 0.0251Re–1/4

8

mom thesevaluesfor 70 fid F, insertedin

Cf’= O.044gy7/!Re-1/4

NACATM 1285

equation(8),fo~ows

-.

Sincethe1/7power.lawforthevelocity~str~butionandthesu&sequent1/4powerlaw”forthefrictioncoefficientarevalidonlyin ‘roughapproximation,thederivedcf’ fognulais comparativelyinaccurate.A betteradaptationto’theactuallyappearingdragcoefficientsisobtainedby a slightchangeinthenumericalconstants,whichresultsintheformula

= O.o%oy1.70~e4.268‘f‘ (9)

,,Thisformulaapproximatesequation(8)quiteclosely,whenthe

functionF isreplacedby theSchultziGmnowplatefrictionlaw(explainedinthenextsection)andtheti_ction70 by thecurverepresentedin figure2. Intherange of ‘1x 103<”Re<4 X 104,thediscrepanciesareless“than3 percent. —.

Fromthesimpleapproximateformula(9),itisseenthatat con-stantRe thedragcoefficientcf’ isproportion&.to71”705. . ~—.Since 7 decreasesalongthete~tlengthforbound- layerswithpressureriseand Re increases,cf’ decreasessharply,whichisentirelycontrarytothefindingsofMangler(reference4)andWieghardt(reference5),whoidentifieda substantial”:ncreasegf the cf$ value;therefore;itwasdecidedtochecktherelation(8)derivedforthefrictioncoefficientcft by experimentswhichwillbe describedinthefollowing. .

III.EEHAVIOROFDRAGCOEFFICIENTINBOUNDARYLAYERS

●

Theboundaryrectangularcross(reference9) and

-.WITHPRESSUREGRADIENTS

—

. ,..-layerswereinvestigated3nthesame_testlengthofsection,as alreadydescribedbySchultz-Grunowwhichhadbeendesignedaccordingtq_theconventional.

principleforboundary-iayermeasurementsexplainedintheintroductionofthepresentrepor’t.But,whiletheiradjustablewallhadbeensetforconstantpressureover thetestlength,thewall,’inthepresentstudy,wasadjustedsothatthedesiredpre”#surevaria~ionresulted.Thevelocityprofilesoftheboundarylayers–weredeterminedat tento’

—

.: —

<

—

.

.—

.

.-—

.

NACA~ 1285

twelvestationsslongthecenterlineoftheThewallshearingstresswaameasuredattheLudwieg’sinstrument(reference7).

9-.

smooth,flatplate(1.4x 6m).samepointsby meansof

Altogether,fourdifferenttestseriesyerecarriedthrough:

(a)At constantpressurein flowdirection(plateflow) “ ..(b)At moderatepressurerise

(c)At strongpressurerise

(d)At’pressuredrop---—-

Theinstrumentwascalibratedpriorto themeasurementsandinthefollowingmanner:Thechannelwassetforplateflowandtheinstrumentmountedattwodifferentstationsontheplate.Eachtestruncoveredtheentirespeedrange,thecorrespondingcalibrationshearingstressbeingdeterminedby thefrictionlawforplateflow. F&omtheavailableapproximationformulasforthedragcoefficientofplateflow,theSchu.lttirunowformula(reference9)

3Cf’ “0.0334(log Re)l-838

.wasused,sinceitwasobtainedonmeasurementsinthesapeexperimentalsetup.As a check,the,yall+hearingstresswasmeasuredwiththecalibratedinstrumentalongtheentiretestlengthat constantspeed.Thefunctionyo = yo(Re}usedforcheckingeqyation(8)wasdeterminedfromthesin@tsaeouslymeasuredvelocityprofiles.It isplotted.againstlogRe in figure2, slongwiththe y. fYomtheSchultz-Grunowmeasurements,forcomparison.we writerlstestpOintsliesomewhatabovetheSchultz-Grunowcurveat smallRe numbers.Theheavysolidcurveisusedasbasisinthesubsequentinterpretations.- -—

In figure(3a),thedragcoefficientcfr isplotteddoublelogarithmicallyagainstthe Re numberforthefourtestseries,&ongwiththeSchultz-Grunowfrictionlawforcomparison.Thetestpointsoftheseriesmadeas a checkat cons,tantpressure(plateflow)coincidewiththeSchultz4runow curve and,thus,provethecorrectness of the

31ntheSchultzArunowreport,Cf’ is Indicatedas tictiOnOftheRe~oldsnumberformedwithlengthx; inthepresentsrticle,it isreducedtotheReynoldsnumberRe formedwiththemomentumt%ickness82.

10 NACATM 1285

—calibrationmeasurement.Thetwotestserieswithmoderatemd strongpressureriseexhibitcf~ valuesattheendofthetestlength,whichareconsiderablybelowtheSchultz-Grunowcurveforplateflow.Thisthereforemeansthattheassumptionofa Cft valuedependingon Reonly,asusedby13uri(reference1),Gruschtitz(reference2),SquiresndYoung(reference10),Kehl(reference3), ‘advon.DoenhoffandTetervin(referenceJJ)aabasisforcomputingturbulentboundarylayerswithpressuregradients,isnotcorrect;Itrefutes,inparticular,thetest”dataofMangler(reference4)andWieghsrdt(reference~),whoclaimeda markedincreasein cf’ onboundarylayersinretsrdedflowaftera certainentrancelengthinflowdirection.Tbetestpointsof “-theseries,withpressuredrop,arelocateda littleabovetheSchultz-Grunowcurve.

Followingthesepreparations,we proceedtothecheckingofeqp~tion(8). To simplifythemodeofwritingofthisequation,thefollowingabbreviationisintroduced:

Afterputtingequation(8)inthefolloidngform

(8a)

thesamefbnctlonal.relationshipNe&ngbetweenRe -and cf? inplate

()-70

flowprevailsalsobetweencf*~ Yand Re —. Thusfigure(3b)

[()]

?’o(Re)2 .-

?0showslog cf’~

()~ alongwiththeSchultz-plottedagainstlogRe ~0 ._

Grunowlaw-forplat=flow. Ifequation(8)is correct,thepointsofalltestseriesmustfallontheplottedcurve.Thepointsof thetestserieswithconstantpressureweresimplytr~sferredfromfigure(3a),sincea recalculationissuperfluous.Thepointsoftheremainingtestserieslie“satisfactorilyontheSchultz-Grunowcurve.Thetestpointsoftheserieswithpressurerise,.whichresultin considerablysmallerCf” valuesthanthecorrespondingplateflow,areinespeciallygood

:

agreementi”with”theSck~tz&unowcurF@j~-dsoprovee“~ation(8)in.

NACATM1285 U —.

thebestconceivablemanner.Onlytherecomputedpointsoftheserieswithpressuredropliea littlebelowtheplotted’curve,butthisdoesnottiplya failureofequation(8)becausetheSchultz<runowlawisratheruncertainat smallRe numbers,as seenfroma comparisonwiththeapproximateformulasof otherauthors(references10,12,13,14,and15),whichdeparttiomtheSchultz-Grunowcurveby thesame”orderofmagnitudeasthepresenttestpoints.

Fromequation(8)(orevenmorereadilyapparentaccording%ot,heapproximateequation(9)),it followsthatwithapproachto thepointofseparationof a turbulentboundarylayer(y+O), thedragcoeffi-cient Cf’ tendstowardzero.So,closeto thezoneof,separation,verysmall cft valuesmustaypear,whichwe haveattemptedto proveinthetestserieswithstrongpressurerise. It resultedin aCf[valueof0.0010insteadof a cf~ of0.0020forplateflowatthesame Re number.No acceptablelowerdragcoefficientscouldbeobtainedwiththeexperimentalsetupbecausetheflowseparatedfirstinthecornersofthetunnelsection.

Forthederivationofequation(8),it,wasassumedthattheuniversallaw,equation(3)forthewall-adjacentpartofthevelocityprofilesintheboundarylayerisapplicablealsoto generalboundarylayers.Therefore,allprofilesintherepresentationof u/u* against

log~ mustcoincideinwallproximity.Thisbehavioris satisfactorily

confirmedon severalprofileschosenatrandomrepresentedin figure4.Theshesringstressvelocitiesu* weredeterminedfromthethermallymeasuredcf’ vslues.Inthissemilogarithmicrepresentationthecoincidentwall+djacentprofileportionis a straightline,by-reasonofequation(3a).Onlythetestpointsnesrestto thewallliea littlebelowthisstrai@tline,sincetheysxeno longerin thecompletelyturbulentrengeofthevelocityprofile,butin thetransitionalzonetothelsminsrsublayer(Relchardt’smeasurements,reference16). Inthisrepresentation,theprofileshapewithpressureriseordropisaffectedonlyin thepsrtawayfromthewell.

IV.SINGLEPARAMETRICCHARACTERISTICOFTURBULENT

IKXJNDARY-IJLYERPROFIIW

Thesingleparametriccharacteristicofturbulentvelocityprofilesat anypressurevariation,foundby &uschwitz(reference2) andrepro-ducedin figure1,hasbeenconsistentlyverifiedbyvariousauthors(references3,8, end11),buthasnotbeenexplained.Thereasonfoq _.

12

— —.—

NACATM 1285 - “““–,

I

theapproximatelysingleparametriccharacteristiciga~parentfromthedataoftheforegoing.

As seenin figure4, allboundary-layerprofile6-nearthewall -.coincideintherepresentationof u/u* againstu*y]V (or )log$ .

FYomit,therepresentationu/U againsty/82 inwhich,accordingto ~Gruschwftz,allprofilessreto forma one-parsmeter~!ly of curves,areobtainedby af$inedistortionoftheordinatewith u*/U andtheabscissawith v/u52. Sincebothquantitiesu*/U ~d V/u*b2areindependentof oneanother,itmightbe expectedthat--eventhewall-adjacentpartofthevelocityprofilewouldbecometw=parsmetricintherepresentationu/U againsty/82.

,.,Butthegeneralvelocitylaw,

equation(3),whichappliestothecoincidentwall-adjacentpartoftheprofilein figure4, canbe approximatedby a simplepowerlaw,(3b),outsideofthelsminarsublayer.A powerlawwithanymutuallyunaffectedaffinedistortionsin ordinateandabscissadirectionalwayschangesagainintopowerlawswiththesamee~onent~whichgivea singleparametricfamilyofprofilesinwallproxiiiity. –

Theadjoining”partofthevelocityprofile,inwhichthevelocityvariesveryslowly,1samplydefinedforgivenwall-adjacentpart.Sincethejoiningmusttakeplacecontinuouslyandwithcontinuousderivatives,

u. 1 asymptotically,andthetheprofilemustapproachthevalue ~ .

integralmust

thickness}.

However,profilerefersonlyto theturbulentpartoftheprofileanddoesnotincludethelaminarsublayer.

be J“;( - :)d(:)=_ 1 (definitionofmomentum

thissingle-parametriccharacteristicforthevelocity

Since,by reasonof,thesingle-pgxmnetriccharacteristic,thevelocityprofilescanbe definitelyidentifiedby snyparameter,theconnectionbetweentwoprofileparametersmustbedefi&te. ThisiSrepresentedinfigure5 for 7 aridquantityH12~H12_beingtheratioofdisplacementthickness451 tomomentumthickness52. Inthis

plot,thetestpointswhichcorrespondto dissimilarpr~filesfromalltestserieslieverynicelyon onecurve(witfiexceptionofthetestseries“constantpressureand strip”;seenextsection),whichconfirmsthesingle–parametriccharacteristicinthebestconceivablemanner.

4Displacement.thickness81.~~.;j&.”‘

t

“

NACATM 1285

Consequently,H12 isa suitableprofilepsrameter,isusedfrequentlyincomputationsoftheturbulentit isnaturaltouseit inequation(9)insteadof

13

becausethisquantityboundarylayers,m“dY. Therelationship

between7 and H12,representedin figure5, canbe approximatedbytheformula

which,introducedin eqyation(9),gives

Cf’= 0.246XloQ”678H12

theapproximate

x ~e-o.268

whichreproducesthedragcoefficientscf’ inrelationparameterH12 sndReynoldsnumberRe satisfactorily.

-.

.

..

formula

(9a)

toprofile

V. SCOPEOFAPPLICATIONOFTHEESTABLISHEDCORRELATIONS

Aftertheresultsobtainedintheioregoinghaveproved-trueforboundarylayerswithpressuregradients,thenextproblemisto checktheextentofvalidityofthederivedrelationsat anydisturbanceoftheboundarylayer.Twoadditionalseriesoftestsweremade,namely,

(e)At constantpressureandwithturbulencegrid‘

(f)At constantpressureandWithsquarestriy “

Forthemeasurementsofthefirsttestseries,a setupde-scribedbyWieghardt(reference17)wasused. A coarsescreenofmetalstripswasplacedbeforethetunnelnozzlewhichincreasedtheturbulenceoftheairflowconsiderably(diminutionbfthecriticalspherecharacter-isticcoefficientUD/V from 3.75x 105to1.3x 105).Therestoftheprocedurewasthessmeas forthetestseriesat constantpressure(a).In figure6a,themeasuredcf~ isplottedagainsttheReynoldsnumber”Reindoublelogarithmicrepresentation.TheincreasedturbulenceoftheouterflowhasIncreasedthefrictioncoefficientonanaverageof

10percentcomperedto theplateflow. ()y. 2 is

In figure6b,cf’~

()7 indoublelogarit~crepresentation,‘--plottedagainagainstRe —70

correspondingto figure3b. Thetestpointsofthis,seriescoincide iwiththetestpointsfortheplate:1ow,thusprtiingtheapplicabilityofequation(8)tosuchintenselyturbulentflows.

14 NACATM1285 “–S

—

In contrastto thistestseries(e),wherethedisturbanceoftheboundarylayerstartedattheouteredge,the“disturbancein series(f)originate~atthewall. To thisendja continuouss@sre strip(13by “13mm)wasfitted”ata distanceof x = m“”from_kheJ.eadingedgeofthetestplate,crosswisetothedirectionofflow.Thethicknessoftheboundarylayeratthispointwasabout@ mm. Velocityprofilesandwall-shearingstressesweremeasuredatdifferent””distancesbehindthestrip.Thevaluesfor Cf’ arereproducedin fi~e 6q. Thetestpointsatshorterdistficebehindthestri~~are, of c&.rse,lowerthanforplateflow.Withincreasingdistance,“thecf~ .yaluesrisesteeplytohighervaluesthanforplateflowanddroptowardtheendofthetestlength,butnotaslowas forplateflow.~hehighercf’ valuesrelativetoplateflowareattributabletotheincre~”edturbulencecausedby theseparationat thestrip.In figure6b,the cf’ values,recomputedaccordingtoequation(8a)jarereproduced.Thefirstpointrightbehindthestripdoesnot fallonthecufieof‘theplatelaw,whilethetestpointsoftheorderofmagnitudeofteriboundary-layer

-,.J ,,—

thicknessalreadyfulfillequation(8)‘6gaigunequivocally.Thereasonforthefailureofequation(8)forthefirsttestpointsliesinthefactthatthestipulatedvalidityofequation(3)inwallproximityisdisturbedby thestrip.Thisfactisborneoutby theprofilesrepre-sentedin figure7 inthemanneroffigurel-.Profile_No.4 isa pro=filemeasuredcloselybehindthe,strip.A comparisonwithfigure1indicatesthathereinwallproximity,thelaw,equation(3),isnolongervalid.Moreover,theprofileisoutsideofthesingle-parameterfamily.ProfileNo.1 was~asuredintherangewhereequation(8) isapplicable”again.Notethestraightlinev~iat$onwiththeSme sb@as infigure1 forthevalidityofeqyat+on(3).Theylateprofileof”figure1 isreproducedforcomparison.Figure7 furt~ershowsa profileofthetestserieswithturbulencescreen,whichagainindicatesthevalidity,,ofequation(3)as anticipatedfromtheforegoing.

—

—

.-

Figure8 representsseveralprofiles fromthelasttestseriesforwhichequation(8)isapplicableagain,plottedInthe~anneroffigure4,alongwitha plateflowprofileforcomparison.Here,also,everyplottedprofilecomplieswitheqyation(3)inwallproximity.

.. —... .Translatedby J.Vanier

—.——NationalAdvisoryCommittee —forAeronaut-its

,.,, . . . .:..

NACATM 1285

FmFERENcEs

15

1. Buri, A.: EineBerechnumgsgrundlagefirdieturbulenceGrenzschichtbeibeschleunigteruhdverz6gerterGrundstr6nmng.Dissert.Ziirich1931. —-

.—2.Gruschwitz,E.:

..—DieturbulenceRelbungsschichtinebenerStr”-g

.-

beiDruckabfallund-anstieg:In$.-Arch.vol.2,no.3, 1931,PP.3aT346. .

3. Kehl,.A.: Untersuchungenfiberkonvergenteunddivergence,turbulenceReibungsschichten.Ing.Ach. vol13,no.5,1943,pp.293-329.

4.Mangler,W.: DasVerhaltenderWandschubspann~ginturbulentenReibungsschichtenmitDruckanstieg.~~ 3052,1943.

5.Wieghardt,K.: fierdleWsndschubspannunginturbulentenReibungs-schichtenbeivertiderlichemAussendruck.ZWBUM 6@3, 1943.

6. ‘I!ilhuann, W.: ~berdieWandschubspannungturbulenterReibungs–schichtenbeiDruckanstieg.DiplomarbeitG6ttingen1947.

7. Ludtieg,.H.: EinGer&tzurMessungderWandschubspannungturbulenterReibungsschichten.Ing.-Arch.(AvailableasNACATM 1284.)

8. Wiegh-adt,K.: “hereinigeUntersuchungensnturbulentenReibungs-schichten.ZAMM25/27,19b7,p. 146.

9. Schultz-Grunow,F.: NeuesReibungswiderstandsgesetzl%rglatte“Platten.Luftfahrtforschungvol.17,no.8, 1940,p. 239. (Avail-ableasNACATM 986.)

10.SquirejH. B. smdYoung,A. D. : TheCalculationoftheProfileDragofAerofoils.R. & M. No.1838,BritishA.R.C.,1938.

11.VonDoenhoff,A. E. andTetervin,N.: DeterminationofGener&lRelationsfortheBehaviourofTurbulentBoundaryLayers.NACARep.772,1943.

12. Prandtl,L.: ZurturbulentenStr6munginRohrenundl-E.ngsPlatten.Ergebnissed.AVAG6ttingen,IV.Lieferung,1932,p. 18.

13. Schoenherr,K. E.: ResistanceofFlatSurfacesMovingthroughaFluid.Trsms.Sot.NavalArchitectsandMarineEng.vol.40,1932,P. 279.

.

16 NACATM 1285.-

14.Nikuradse,J.: TurbulenceReibungsschichtenanderPlatte.ZWB1942. L

15.Falkner,V. M.: A NewLawforCalculatingDrag,–Aircraf%EngineeringVOI. XV, no.169, 1943, p.65. —. —

16. Reichardt,H.:ZAMMvol.20,

17.Wieghardt,K.:einerPlatte.

DieWhe~bertragunginturbulenten’Reib@gsschichten.no.6,194o,p. 297. (Available‘asNACATM 1047.)

.

~berdieTurbulenceStr”- imRohrundl“hgsZAMMvol. 24, 1944, 1?.2940 .

.

.

.

.

NACATM u85 17

5 -LO -05

● Constantpressure

+ MOderskpressure rise —-al

lx Heavypressure rise

A Pressure drop A

/“//’

+

//’

x

x

/lLA . /

2 -a

x//●’ 3 +

/ $+/

/ 4/< x’

*+ -03

,2 x~;/“

/4 /

-04/x

/’,/

I –(25

–06

log +$

) log JL—A- %?

- log 7I

1: F@= 2.86 x 103Iq2 = 1.289; 7 = 0.718

2: Re = 7.98 X103H12= 1.3%; 7 = O.tM~

3: ~ = 1.48 XK)$H12= 1.442; Y = 0.6Z3

4; Re = 2.65x 104H12= 1.511; Y -0.584

5: Re= 3.25X 104%2 - 1.721; 7 = 0.477

Figure1.-Sh@e-parametricfamilyofvelocityprofilesinboundarylayers. withpressuregradientsindimensionless,double-logarithmic.representation.

.

(Ill

0.7

_--’./

I I

——

log E@

&30 34

7“36 3$ 40 42 4.4 46

Figurd 2.- Proffle paradeter 70 of plate flow as fmction of the F@cuolds nu.mkr Re

I

I

.“. ,-$

Za

Z6

Z4

72

10+ log q

Zo -1-3JI 32

Figuxe 3.-

,,

I . lo-percmt dlttel’eme

I

3.4 3%

(a)

Dragcoefficients

%’Cf 1

Sdnlltz-clrmm

x+

●

☞

☎

A

Comtantpressure

Wderak premm rise

m’rlat pi-e99um rim

PTwmra drop

+

i-_~+.

x

++

+

x +

+

x kg R’J

plotted against Re.

in tmuds.ry layers withpressuregradients.

I,, 1

[01To 2lo+ log c#—

,7

A

A AA

Schunzihuww

x

-=--xi=

—

1- lo-pr=d dmterm=e

au 32 3.4 3.6 3!6 4.0 a 44 46

()(b) Cfl - :2 plotted against

Figure 3.-

Re.-$accordingtoequation(8a).

Concluded.

+ . ,. ,, ,.

50

40

30

m

to

TI

Coafmt pressm● Re -7.98 x 10s

Mxlmat9 pn?smreIisO+ Re-2.66 x lf14

Ored plwsum *x Re-1.46*lo4@ R9 -9,25 x Id

PTOasum &op

A m .2.86 x log

--1-.-+&e

./-+”” +

●

e

0x

I x+

[0 E 20 25 30 a5

e*a-

*+++

+

+x

Figure 4.- Several velocityprofilesofboundarylayerswithpressuregradientsanduniversallaw,equation(3a),inwsllproximity.

I

O.a

0.7

(26

Q5

0.41.2

7

f.3

I

●

☞

x

A

El

Co* pressure

Moclerste pressure rise

Grest pressuw rise

Pressure drop

Constaut pressure sl@turbulence grid

Constsnt pressure snd strip

1.5 /.6

gFigure 5.- IwhtioDsbip between profile pmunet.ers7 and H12.

. .,.1

NACA~ 1285 23..__ ___

7.3

Z6

7.4

12

Z6

Z4

10+ logCf’

● Constsmtpressure

~ Comksnt pressure dturbulemegrid

I n Constsntpressuresndstrip

.

Y--- .●

Schultz-Gruuow ●

●●

●

I 1 N10-percentdiftererce I (o)

3.2 3.4 3,6

(a) Cf’plotted

[012lo+l.ogc~$

3a 4.0

--1-Schultz-Grunow

+** “~

7232 3.4

1 -10 -peroentdifference

1 4.0

0(b) Cff . >2 plottedagainstRe~ according70

Figure6.-DragcoefficientsCf’inthebound~md f.

—-

4.2

42

layersoftestseriese-.

24

.

NACATM 1285

.

● Constentpressure

+ Constantpres.he andturbulence grid

•l Constantpressure “

//n

/-

,0

❑

+

/

a‘/0’ ●’,A//

,/e’

*/#

,/

-0.1

-0.4

-a5

10 log g,13k----A

.

1: Re=1.21.X:I04’12 = 1.249;7 -0,764

2: Re= 8.00 X“@H12= 1.267;y- 0.726

3: Re-7.98 x<03H12= 1.337; Y“=0.888

4: Re= 1.02 x ~04*12 = 1.~4; 7% (),569 .

Figure7.- Severalvelocityprofilesfromtestseriese andf indimensionless -double-logarithmicrepresentation.

. . .

20

101.0

Comtmtprmmmg Re-’l,tmxlos

constantpmearre endtm’tmlem?@@’id

* RE-mlx log

❑

Comtmt Pra8ann3 aud stripm -1.21 x 104

●☛☛

● ☛

‘* * #=./

15 20 25 35

s!

#

Figure 8.- Several velocityprofiles from test serSes e andfwithuniversallaw,equation(34,inwSllproximity. .,

1