NATIONAL ADVISORY COMMITTEE FORAERONAUTICS TECHNICALMEMORANDUM1285 LOAN COPY: RETURNTO AFWL TECHNICAL LIBRARY KIRTLAND AFBs NC M. / INVESTIGATIONS OF THEWALL-SHEARING STRESS INTURBULENTBOUNDARYLAYERS By H. LudwiegandW. Tillmann Translationof “Untersuchungen iiber die Wandschubsparumng inturbulentenReibungsschichten” Washington May 1950
26
Embed
NATIONALADVISORYCOMMITTEE FORAERONAUTICS/67531/metadc63021/m2/1/high_res_d/... · 4 NACA’1141285 I I ~is relationholdstruewiththessmefunctionf--forthepart.....
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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‘“
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-
%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,
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 .
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
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. .
.wasused,sinceitwasobtainedonmeasurementsinthesapeexperimentalsetup.As a check,the,yall+hearingstresswasmeasuredwiththecalibratedinstrumentalongtheentiretestlengthat constantspeed.Thefunctionyo = yo(Re}usedforcheckingeqyation(8)[email protected] 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.
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