1、I(11,./NATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICALNOTE2597INVESTIGATION OF LAMINAR BOUNDARY INFLUIDS USINGTHE CROCCOMETHODCOMPRESSIBLEBy E. R. Van Driest*North American Aviation, Inc.vWashingtonJanuary1952.-.I,- .-Provided by IHSNot for ResaleNo reproduction or networking permitted without lice
2、nse from IHS-,-,-lG. _TECHL!BW KAFB,NM1, lullllll!luullllllC10L5bbJNATIONALADVISORYCOMMITTEEFORONAUiICS-_,.TECINICALNoml2597INVESTIGATIONOFLAMINARBOUNDARYLAYERINCOMPRESSIBLEFLUIDSUSINGTHECROCCOMETHODByE.R.VanDriestSUMMARYInthepresentinvestigationoftheflowofairina thinlamharboundarylayerona flatplate
3、,theCroccomethodhasbeenusedtosolvethesimultaneousdifferentialequationsofmomentumd energyinvolvedinsuchflow.TheCroccomethodwasusedbecauseitgaveaccurateresultsforarbitraryPrandtlnumbernearunity.TheRrandtlnumberwastakenat0.75,thespecificheatwasheldconstant,andtheSutherlandlawoftiscosim-temperaturevaria
4、tionwasassumedtorepresenttheviscositydatastartingwithaninitialambienttemperatureof-67.60F.Themainresultspresentedherearetheskin-frictionandheat-transfercoefficientsasfunctionsofReynoldsnumber,Machnumber,andwall-to-free-streamtemperatureratio.Variationsofshear,velocity,tempera-ture,andMachnumberacros
5、stheboundarylayerareincluded.TheCroccomethodisdiscussedindetail.moDucTIoNInthepresentinvestigationoftheflowofairina thinlaminarbouhdarylayeronaflatplate,theCroccomethod(reference1)hasbeenusedtosolvethesimultaneousdifferentialequationsofmomentumand “enerinvolvedinsuchflow.TheCroccomethodwasusedbecaus
6、eitgaveaccurateresultsforarbitraryPrandtlnumbernearunity.Thefirstassumptionsforthepresentdiscussionareasfollows:(1)The(2)The(3) The(4)The(5) meboundarylayeristhinpressureadientiszeroplatesurfaceissmoothmotionistwo-dimensionalandsteadyPrandtlnumberisconstanti- - - - !-. .-.- -.-.- - -.Provided by IHS
7、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-I12 NACATN2597-.ThisworkwasdonebyNorthAmericanAviation,Inc.,andhasbeenmadeavailabletotheNationalAdvisoryCommitteeforAeronauticsforpublicationbecauseofitsgeneralinterest.TheauthorgratefullyacknowledgestheassistanceofMe
8、ssrs.WarrenW.Gdlos,JohnN.Hale,andFrederickG.EtheridgeandMrs.PhyllisM.Watte.intheptilicationofthisreport._ SYMBOLSVariables,parameters,”andfunctionsX,yu,vPvkTi%7TM.Pr8*ms5geometricalcoordinatesparallelandplate,respectivelyperpendiculartoflatvelocitycomponentsofbount;)edgeofboundarylayer(incomputation
9、s,at q .R. Reynoldsnuiber(%#/%) CfCflocalskin-frictioncoefficient(%N5)meanskin-frictioncoefficientJx%d?+m%+q rateofheattransferh localheat-transfercoefficient,differencebetween andunityas CH Stantonnumberr recoveryfactor = +m% b=UmP*= P/PmW*= l.+%1%=M/MmTx=T/TmapproachesunityY* ( d-)distanceparsm.et
10、er R.f() =athatis,aipx =O.(This csmbeshown(reference2)tobethecaseforaPrandtlnumberequaltounity,whetherthewallisinsulatedornot.)Henceitfollowsfromequation(6) that a typeofsolutionisT = gl(x)q(u).Whence,sincep and v arefunctionsof,temperature,andthereforeenthalpy,andthereforevelocityu alone,thereresul
11、tsfromequation(5)andboundarycondition(7a)that T = (u)/.Equations(5) and(6) nowbecome .(8)fz#2 + Upp=o(i”+Pr)% + (1-Pr)ig2=O (9)withboundaryconditionsgp=o,i= at.u=O (lOa)=o,i=i%2 at.u=u (10b)m mwheretheprimesignifiesdifferentiationwithrespectto u. Whenthevariablesi, u, p,and p aremadedhens”ionlessbyp
12、uttingi* I I= i/imju*=u umj P*= PPm,and I.L*=v Vm,itisseenthat Imustthen%eequaltoF2 g,2(u). Equations(8) and(9)becomeP+mumfinallyI .-.-.-. _ - . -.- . -.-.,-.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2597withboundaryconditions(12)therefo
13、re,sinceboth u and t3dependuponthefree-streamtemperatureT itfollowsthattheSutherlandlawismostsuitablefobuseinequation(U.).Whenitisassumedthatthespecificheatisconstantand im issetequalto Tm,then canreplaceT* inetableIIIgivesthevaluesof ad l. -Itmaybewelltointerjectherethatthespecialcase P*V*= 1repres
14、entsa gasorstateofa gasforwhichthereisnoeffectofcom-pressibilityonskinfrictionforthinlam.rboundarylayers,althoughthetemperatureofthefluidrisesbecauseofdissipationofthekineticenergy,sndpropertiesofthefluidvaryaccordingly.Nowthattheenthalpydistributionhasbeencalculated,attention,will.berectedtowardthe
15、detailsofsolutionofequation(11).Writingequation(n) inintegralform,takingintoconsiderationtheboundaryconditions(13), oneobtainstheintegralequationin which p*w* isaknownfunctionof u* dependinglawassumed.Themethodofsuccessiveapproximationsutilized.However,asCroccopointedout,thedirect(18)upontheviscosiw
16、illagainhemethodof(-.-.-.- -.- .- . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-tIt1.III1tII10substitution,convergeuponIintegration,resubstitution,a singlesolutionbutyieldsabouttheexactvalue.For,ifaninitialtion h,p;h) -canberepresentedbythestrai
17、ghtline 0.7828+ 0.01781(1-h)!overthepractical-e of Z(1 -h) between-2.2and-4(fg.2).NowtheerrorbetweenZ(1 -h) and g (1-h) willdependuponh theapproximateequa-thesmallnessofthe h,thatis,uponhowwetion(21)representstheexactequation(11).AsCroccopointsoi.rt,i*, JandthereforeP+V+viesraPiYfor pr) = (1-U*) o d
18、%+ u *andcombintigwiththeintegralofequation(11),namely,thereisobtained1,/.(28)(29)(30)I _ . . _.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NAC!ATN2597 13%Similarlyfromequation(21)(u*) = -(1 - u*)(1oneobtainsJ* .-h)+ (l- 2 du+l-h?x)Subtractionoft
19、hisequationfromtheprecedingt(.yJ = (1 - U*)LX-+p*+J”(, -(32)onegiveswhichcanbesolvedinprinciplebyiterationsincef isaknownfunc-tionof . Thatis,startingwithanassumedl, isavail-1ablefromequatio(25),whenceb () iscomputed,whence9However,theaboveiterativeprocesscanberearrangedasfollows:Writeeqution(33)thu
20、s.(34)- . . . . . - - -.-.- . - - .- - .-. . .-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACATN2597(35)Theterm-A cannowbeexpandedintheformg*G(36)8l-arethecoefficientsinaTaylorseriesexpansionof P*P+aboutthepointi*= 1 usingtheSutherlandlaw(equ
21、ation(15)for. . .- - . - .- -. - . .- - .- .- . - -. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 NACATN2597viscosity-temperaturevariation.Theothercoefficientsinequation(39) ,areA. = .13 - (2 - Pr)qA.A1=2- *2 1-Pr)ql+2(l-q)-5- (4-Pr)q1(-=2-(;-p
22、r)ip+F=l-J% 1 1-(4 -pr)q 7-(6 -pr)q24A.3=2- (2-pr)q - (31-Pr)q- 6- (5-2Pr)q12“1 B.%=3_(3-2Pr)ql+2(l-q)- 16-(5 -2Pr)TIco = 15-(4 - m)n( 2“ 1B.cl=- -3pr)TIl+2(l-q)-6 - (5 - 2Pr)q1“1.o -3-n21 DoDl=2 1-Tl+2(l-q) -2+3(1- T).,_. . . - -Provided by IHSNot for ResaleNo reproduction or networking permitted w
23、ithout license from IHS-,-,-3G IWCA!Im2397 17are to be determinedat = 1 -h. Itwillbefoundthatthetermsincludedinequation(39)willgivesufficientaccuracyto -h).Infact,inthecalculationscarriedoutinthepresentreportitwas .-4foundthat 5% wasnegligible,beingoftheorderof10 .1SKIN-FRICTIONANDHEAT-TRANSFERCOEFF
24、ICIENTSSince7=shearstressT.ri%= 2 (u),itfollowsthatthepvus(1) 1 CfCH=Pr1(1)and(49)therefollows,utilizing(50)ItisnowinterestingtoobservefromtableII,that,closely,d2/3Pr 1(1)Prand2PrJ(1)Prl/2sothatpracticallywhenthewhenthea71a55specificheatisvariable,or( l/2Y-lM2 )%=-+cpPumTml+pr m -qspecificheat3.scon
25、sht. (SpecificalQ,for Pr= 0.75,2/3 . 0.8258;2R J(I)= 0.8654Pr I(1)= 0-.8302 and wand Prl/2= O.8660.) -,.,.(51)(52)(53). . . . . - -.- - ._ -. - . - -:. - . . . . r .- . - -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20Supposenowlayer(insulatedth
26、ereisnoheatwall). Then NACATN2597transferintooroutoftheboundary=0 and(variable) (54)or(constant) (55)whichisthewalltemperatureduetoviscousactionalone.However,the“recoveryfactor”isdefinedbyHence,itfollowsqponcomparisonofequations(54)and(56)that,closely,Recoveryfactor=-r=Pr1/2(56)(57)forlaminarboundar
27、ylayers.In-thedeterminationofthefrictionandheattrsnsferattheplate,itwasnotnecesssrytobow theinternalgeometricdistributionofpropertiesofthebouidarylayer.However,ifthegeometricdistri-butionofproperties,suchasvelocityandtemperatureina stabilityanalysis,aredesired,thena suitabletbt isvaluesof and 1911fr
28、omtableIIIwereusedinequation(17).Thefinalsheardistributionwascomputedbyintegrationofequation(11)bythemethodofsuccessiveapproximationsfrom =O to w = 0.98 startingwiththeBlasiussolutionintableI. Intheiterativeprocess,thevalueoftheshearfunctionat = 0.98wasobtainedfromthesolutionofequation(21)(fig.2). T
29、herequiredaccuracyon thethicknesswasarbitrarilytakenatthevelocityratiow = 0.995obtainedbyestimatingthe distributiontothatpoint.Figures7 to12showthedimensionlessshear-functiondistributionforvariousMachnumbersandtemperatureratiosincludingtheinsulated-platecase.Figures13to15givethevariationsoftheveloci
30、ratio, thetemperatureratioT/T andtheMschnumberratioM/figures16to30showthesamedependentvariablesasfunctionsofthesameindependentvari-ablesinthecaseofwall-to-free-streamtemperatureratios1/4,1,2,4,and6. Figures31to47showthesamedependentvariablesasabovebutaslinesof constanttemperatureratioinsteadofasline
31、sofconstsmtfree-streamMachnuriber,thedistanceparameternowbeingtherelativedistancey/8. Forpurposeofcomparisonwithfigures13to15theinternalcharacteristicsoftheboundarylayerareplottedinfigures48.to for Pr= 1. - .- . - .- - - -.- - - - ._-. -, -Provided by IHSNot for ResaleNo reproduction or networking p
32、ermitted without license from IHS-,-,-.,AerophysicsLaboratoryNorthAmericanAviationjInc.Downey,Calif.,January9,195122 NACATN2597Figure51 showsthevariationacrosstheboumlarylayerofthe(2ratioofthetotalenthalpyperunitmass E = i+) atanypointtothatofthefreestreamforaninsulatedplate.For Pr= 1 theratiowouldb
33、econstantatunity.ThefactthatPr islessthunityinthesecalculationsbringsaboutthetransferoftheenergyfromtheinnertotheouterlayersoftheboundarylayer;thisisduetothereadjustmentoftherelativeeffectsofviscosi,heatcapacity,andheatconductiviinfavoroftheheatconductivity. . Provided by IHSNot for ResaleNo reprodu
34、ction or networking permitted without license from IHS-,-,-MACATN2597 23APPKQDIXAw3KKsBEiTFoR cAmxIm3wr AH mKR4!cIcmFoR qThe folMwingheadings mudeqtw.titi am used h mmyingoutan iterationfor *:()k% %2n(%)i*(uJ=T - pe%J+=e (ml- 6). .J -*1/21.m (COlmrl7)1+S2+Qa71P*-A%2%WithC0716tFswckicheat,- (7-l)L.0V
35、ALUESOF IJPr=0.5 Pr=,0.725I J II o 0.1000 .0050.1000.m .0200.2Q01,3008.0452.3cn)4;:2 :g .4014.5034.6135.1849.6073.7267.2$7 .7144,8%5 .3454.8269.9967.4612.9500039.).48$p .9770L.0696-1.0056L.1131.55911.0363L.1671-1.0708-.6369-L,2806.70391.E1OJ).cm50,.0200.0451.m3.E .3941.9159 - -,- .4429 - .4060.9252
36、.4473 .8682 .4(MPr=2.00III,Jo I.Km .W.1s97 .2985.0447.3951.0788.Wl.1213.5750.1709.6531.2246.7105.2787.76782$ .3475-.3527.7863.3532Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-u*).1.2.3.45.6j:;,9).9(.9.930.TABLE mW.mEsmdmrle 11 m3EDuPmErAKms smai F
37、OR ;$ .:?; .7WI.0720 .8QO0.0453 .8538 :0455 .9000.03* .8776 .0393 .- .9,026 - .9400.0235 .9252 .0232 .9630- .9539 - .9M0.0084 .9760 cd% .99001 0 1).04y.O25;: .:3.5026 :1974;$0; . ;95$.8305 :1375.9139 .0EB2.$rplo .0328.9824 .0234.9894 -.9948 .0078.9984 - .99911 .00060 IProvided by IHSNot for ResaleNo
38、 reproduction or networking permitted without license from IHS-,-,- . . . . .- .!s Figure l.- Vi.scosity-lmnperaixre data.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN2597 29.6.s.4.3.2.(o/I 2 3 4 5.88.86/.80Figure2.-ShearfunctionE(1 -h) asa functionof (1 -h).- - -. .-. .- - . - - - -. -. - . . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
