1、NATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICALMEMORANDUM1285LOANCOPY: RETURNTOAFWL TECHNICAL LIBRARYKIRTLAND AFBs NC M./INVESTIGATIONSOF THEWALL-SHEARINGSTRESSINTURBULENTBOUNDARYLAYERSBy H. LudwiegandW. TillmannTranslationof “Untersuchungeniiberdie Wandschubsparumngin turbulentenReibungsschichten”
2、WashingtonMay1950Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICELME240MNIIJM1285 . .INVESTIGATIONSOFTHEWiLL-SHEARINGSTRESSINTURBULENTBOUNDARYLNERS* - -ByH.LudwiegandW.Tillmsnn. .SUMMARYBecauseoftheunsat
3、isfactorystateofknowledgeconcerningthesurfaceshearingstressofboundarylayerswithpressuregradients,theproblemisreexamined.Itisfoundthatforgeneralturbulentboundarylayers.inwallproximity,thatis,inthelsminsrsublayer,inthetransitionzoneandinthepartofthecompleteturbulentzonenearthewall,thesameuniversallawa
4、ppliesasfortheylateflow.Fromthegeneralvalidityofthislawa formulawasdeducedforthelocalhag coeffi-cientcfr,inwhichcf dependsonlyontheReymoldsnumberRe formedwiththemomentumthicknesssndona profilepsrametery. Thisrelationwasconfirmedsatisfactorilybydirectmeasurementswitha newinstrument.Therelatedfriction
5、coefficientcf canthenbedeterminedsimplyfromtheknownvelocityprofile. .-.l?romtheformulafor Cft itfollows,inagreementwiththetests,thatthe cf valuesforboundarylayerswithacceleratingsnddecele-“-ratingpressurearehigherandlower,respectively,thsafortheplate flowatequalReynoldsnumber.Thus.forgreaterReynolds
6、numberssmalllocaldragcoefficientsareattainablenotonlybykeepingtheboundarylayerlaminarbutalsobyappropriatepressurevariationinturbulentboundarylayers.Theriseofthefrictioncoefficienttoamultipleofthatforplateflowinboundarylayerswithpressurerise,asclaimedbyvariousworkers,isheretithdisproved.-.-1.INTRODUC
7、TIONThewall+hesringstressesinlsminarboundsrylayerscanbecom-putedona strictlytheoreticalbasis,sincetherelationshipbetweenvelocityprofilesndshearingstressisknown.But,thisprocedurecannotbeappliedtotheturbulentboundsrylayerssincetherelationship“*“UntersuchungenfiberdieWsmlschubspannunginturbulentenReibu
8、ngsschichten.“ ,.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATM1285fortheshearingstresses,duetotheturbulentexchti-geisstillunknown.Forthisreason,thelawsforturbulentwallfrictionmustbedeterminedbyexperimentalvest-igations.Suchinvestigations“
9、fallintotwoclasses,termedforbrevity,plateflows”andpipeandchanuelflows.”Theapproximateformulasforthefrictiondrag,deducedbythevariousinvesti-gatorsfromthetestdata,areinagreementtosomeextent.Someinvestigationshavebeenmadealsoonboundarylayerswithpressuregradients,bothaccelerating-anddecelerating,butthed
10、ataonwallfrictionareeitherabsentaltogetherorpartlyUnsatisfactory.Theseinvestigationsweremadeina channelofcircularsection(refer-ence1)or,inmostcases.ofrectangularsection(references2,34and5). Forthelatter,onechannelwallwasdesignedasflattestplate,onwhichtheboundarylayertobeexploredwasmeasured.Theopposi
11、tewallwasadjustabletothedesiredpressuredistribution.Itwasspacedfarenoughfromtheexperimentalsurfacetomaintaina corewithpotentialflowbetweenthetwoboundarylayers(freeboundarylayer).Thewall+hesringstreswasdeterminedfromthemeasuredvelocitypro-filebymeansofvonKarmchcancelthetwo-dimensional.ityoftheflowass
12、umedaccordingsmomentumequationfortheinterpretation,andarepr=tovonKarmansumablyresponsiblefortheimprobableresultsoftheaforementionedauthors.Someoftheseauthorshadobservedthat,ingreatlyretardedflows,thelocal.dragcoefficientCf(=Tw/Q,Tw= wall-shesringstress,Q thedynamicpressureoutsidetheboundarylaer)rose
13、abruptlytoamultipleofitsoriginalvalueaftertravelinga certaindistauceinflowdirection,ratherthsmdecreased,asactually52.J”;-$w =momentumthiclmessofboundary52layer,Re=UT =Reynov=kinematicviscosity.Qpsntityg inequation(1)isa fixedfunctionis,naturally,differentforplate,pipe,andchanneldependenceOf on Re is
14、verysmall,thatis,thedifferverylittlefordifferentReynoldsnubers.which,however,“-flow.Thevelocityprofiles(b)Thelocalfrictioncoefficientcf canalmysbereresentedintheformCf=F(Re) (2)(Cff= Tw/: $; TV = wall.+hearingstress;p =density).QpantityF isagaina fixedfunctionforplate,pipe,andchannel” “flow;F canbec
15、cxuputedforplateflowbythemomentmequationwhen gisknown,becausethetheboundarylayer.(c)Forthepartrelationtotalfrictiondragapearsaslossofmomentuminofthevelocityprofilesnearthewall,theu ()fU*Y=*u 7-wasobtained. v(U*=TWP “= shearifitressvelocity)(3)lInthisformula,theReynoldsnumberformedwiththemomentumthic
16、kness52 waschosenascharacteristicquantitybecauseitismoreappropriateforboundarylayerswithvariableoutsideressure. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACA1141285IIis relationholdstruewiththessmefunctionf-forthepart .ofplate,pipe,orchanne
17、lflownexttothewall(ProvidedthattheU*Yfi values arenottoosmallfullyturbulentzonev )50 ,equstion(3) can bereplacedveryaccuratelybyu *=alog* y.+bua and b beinguniversalconstants.TMsapproximatedbya powerlawl/11u() U*Y=*u TtheapproximateI?onmla(3a) -logarithmiclawcanbe .-(Sb).whereC and n areconstantswhi
18、charestillsomewhatdependentonthe u*y/vzoneforwhichtheapproximationistobeespeciallygood.AsalreadystatedinLudwiegtsreport(reference7), it is tobe ,expectedthattheuniversallaw,equation(3) or (Sa),is,asidefromtheplate,pipe,andchsnnelflow,applicablealsotomoregeneralizedboundary-layerflowsinwallproximity.
19、Itnemely,thatwhenthebounary-layerprofilesareplottedinthemanneroflogu/U againstlogy/52(fig,1),parallelstraightlines-areobtainedforsmally/52.Consequently,uisinallcasesproportionaltothesamepowerof y.Fromtheslopeofthestraightlines,thispowerfollowsas 0.13= , whichisin.goodagreementwithequation(Sb)forthe
20、u*y/Vrangeinquestion.However,thisstill.isnocompellingproof-ofthevalidityofequation(Sb)forthereasonthat-thepowerof y caubecheckedbyprofilemeasure-ment,butnottheconstentC,becauseu* Isunknown.Withthevalidityof*equation(3)fortheportionoftheboundarylayernexttothewall,u and,hence,w and cf dependonlyonthev
21、elocityprofileandthematerl.al-constantsoftheflowingmedium;so,whenthevelocityprofileisknown,cfocembecomputed.A COl?re-spndingrelationbetweencf,Re,anda profileparemeteryettobedefinedisderivedinthefollowing.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-
22、,-NACA!lML285 5%2Theterm y= isintroducedasprofileameter;u2 isudefinedasfollows:Ifthelaw,equations(3),(3a),and(Sb),isvalidfor y valuesgreaterthm 52,then u2 issimplythevalueof uatthepointy = .32.But,ifthelaw,uations(3),(3a),and(3b)Yappliesonlyto y valuessmallerthan 52)then U2 isthevaluewhichu wouldass
23、umeifthelaw,Equations(3),(3a),ad (3),wrapplicableuptothepointy =82. Thus,thedoublelogarithmic_plottingof u/U againsty/52(fig.1)givestheprofileparameter7,Ywhentherectilinearpartoftheprofileisextendedasfaras = 182 ,52tsre fromfigure1.andthecorrespondingvalue”ofuThe derivationoftheaboventiohedrelationb
24、etweencf,Re,and y proceedsfromequation(3).Theprofileparsmeterisintroducedbyputtingy = 62 andu2=*uTheequationstatesad U5252/V;therefo(4)thata directconnectionexistsbetweenu*/u52u*()U5252= hl+j v2mustbeapplicable,thefunctionh beingdefinedbyfunctionf._btroductionof2Gruscitz(reference2)definedthequantit
25、yv =1-profileparsmeter;butby U52 thevslueqf u at Y = 52 iSalwaysmeant.-.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6gives,aftersimplerearrangementCf= 2y2h2(ReForabbreviation,thefunctionisNACATM 1285-. .-7)=72H(Re7) (5)written2h2=H. ”So,foralltu
26、rbulentboundary-layerprofileswhosepartnexttothewallisrepresentedbythegenerallqw,equation(3),thefrictioncoefficientisgivenintheformofequation(5).Todefinethefunc- ,tionsh and H inthisequation,equation(4)couldbereplaced,fortheargumentinquestioninaccordancewithequation(3a),byU*andnumericallysolvedfor =h
27、. Butsincetheconstantsa and b .u2inequation(3a)arenotaccuratelyenoughknown,thefollowingmethcdofdefiningH seemstobemoreappropriate.Byequation(1),theprofileparameterfortheprofilesoftheplateflow,designatedYo,iSOY dependenton Rethence“-70 = yo(Re). Oninsertingthisvalueinequation(5),thisequationmustsuppl
28、ythedragcoefficientCf fortheplateflow.Thus,bearinginmindequation(2),thefunctionalequationfor H followsas70%(Reyo)=F(Re) (6)whereF(Re)isthefrictioncoefficientoftheplateflow.This equation, whichmustbefulfllledforall Re,definitelydefinesthe .-functionH forknownfunctionsF(Re)and 70(Re).AbbreviatingRe70(
29、Re)= giterationgivesforRethechainfunctionERe=T-707()70“-.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-7NACATMu85which,insertedineation(6)givesfor Hsnd,whenthefunctionH inequation(5)isthenreplacedbytheprecedingexpression,Since y. varies very little
30、with Re (fig.2), “theofthechainfunctionissogoodthatinthefirstfactorofthefirstdeureeandinthesecondfactor,thetermofhavetobeincl7 -0,7642: Re= 8.00 X“H12= 1.267;y- 0.7263: Re-7.98 x03H12= 1.337; Y“=0.8884: Re= 1.02 x 04*12 = 1.4; 7% (),569 .Figure7.- Severalvelocityprofilesfromtestseriese andfindimensi
31、onless -double-logarithmicrepresentation.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . .20101.0Comtmtprmmmg Re-l,tmxlosconstantpmearre endtmtmlem?id* RE-mlx loga75Comtmt Pra8ann3 aud stripm -1.21 x 104a71 a11a11a71 a11* * #=./15 20 25 35s!#Figure 8.- Severalvelocityprofiles from test serSes e andfwithuniversallaw,equation(34,inwSllproximity. .,1Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
