1、aacEINATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICALNOTENo. 1449THEORETICALSUPERSONICWAVEDRAGOFUNTAPEREDSWEPTBACKANDRECTANGULARWINGSAT ZEROLIFT .BySidneyM.HarmonLangleyMemorialAeronauticalLaboratoryLangleyField, Va.October1947LIBRARYCOPYEJAPR301W3LANGLEYRESEAWCENTERLIBRARYNAMRGltjjA1Provided by IHS
2、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-9.,.,-., NATIONALADVISORYccMMImEE. . .FORTECHNICALNOTENO, ,.- ., .,.AERONAUTICS1449 .T313NlR)iIKNX+I)SUI?EREONICWA”llRAGU?UNTAPE2DSW3ZITBAqm mmuLAR,. BySidneyWINGSATZEROLIFT M.EEUIQOII .,Em4MAIw“”,A theoreticalinvesli
3、fgationofthesupersonicwavedragatzeroliftofa seriesofuntaperedstieptbackwtngshavingthinsymmetricalbiconvexparabolic-arcsectionshasbeenpresentedinNACATNNo.1319.Theinvestigat+mhasbeenextendedto inchzdeMaclInumberswhichbringtheMqchltie behindthewingleadingedgeandalsoto inclpdewingsofrectangularplan.form
4、.Theresultsarepre8eedin.alifiedfozmsothata singlechartpermitsthedirectdetemintionofthewavedragforthisfamilyofwingsoveranextensiverangeofsweebackangle,Machnumber,aspectratio,andthicknessratto.Theresultsobtainedfor thetotalwavedragofthesweptbaokwingsareapplicabletothesamefamilyofwingshavinga correpond
5、ngdegreeofsweeEfomwprd.WhentheMachlineliesbehindthewingleadingedge,thewave-dragcoefficientsofthesweptbackandrectangularwingsareshowntoreachmaximumvaluesa%certainlimitingaspectratiosandconstantforallaspectratiosgreaterthantheselimitingThelitingaspectratio3.sequlto 2CotA1 cotA=; - 1sweptbac”kwing andt
6、o for,therectangularwing,G .isthean+deofsweenbackand M istheMachnber. Theremainvalues.forthewhereAvariationofwingwve+ag cofficientwithMachnumberovertheccmpleterangeofsupersoniclfachnumberisshowntobecomelesspronouncedwithdecreasingaspectratio.Itisalsoshowntluitsweepbaokobtainedbyrotatingthewingpanels
7、rearwardcangiveappreciablereductionsinwingwavrag ooOfficientatallsupersonicsyeeds.“, INTRODUCTION. .Recentdevelopmentsinairfoiltheq?yforsupersonicopeeds(references1to3) indicatepronouncedeffectsofsweepbackandProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IH
8、S-,-,-1,.2 “. ,. _NACATNNO. 1449*aspectratioonthedreg.Inreference,amethodbasedonthin-airfoiltheoryfora frictionlessfluid(reference2)wasappliedtocalculatethesupersonicilavedragatzeroliftfora seriesofwingshavingthinsyetricalbiconvexparabolc-arcsectionswithuntuperedplanformsandvariouanglesofsweepbackan
9、daspectratios.Theresultsinreference3,however,arelj.tedtothosecaesinwhichtheMachlineliesaheadofthewingleadingedge,ortoa raneofMachnumberfra 1.0tothevalueequaltothesecantofthoauleofsweepbtick.Theterm“Mach”lj.ne”asusedhereinreferstotheMaohwavethatoriginatesattheloadingedgeofthecentersection,unlessspeci
10、fied.otherwise.Thepresentpaperextendsthecalculationsofreference3forthesamee,eriesofwingsinorder.topresentwave-dxwgresults forcasesinwhichtheMachlineliesbehjndthewingleadingedge.Thesedataareobtainedforwingu racta”q”randswoptbackplanformsaidcoveranexteniiverangeofMachnumberbeginningwiththvalueatwhicht
11、hoch line”,coinoideswiththewing“leadingcd.Jn ordertoresentamorecompletepictmm,theresults-ofreference3 thatcoverthe”lowerrangeofMach”numiberar$reproduced-hereintogetherwithadditionalcurvesccmputadft$omk-las Civeqinreference3.Theresultsoftheentireinvssti+ptionarepresentedina unifiedformsimilatothaiven
12、in-reference3sothatthewuvedragforthis,familyofwi.fismaybedetgmninsddirect,lyfrcma sjnglechartoveranextensiverenGe”faweepback-aie,Machnumber,aspectratio,andthicknessratio.IntheTenthAnnualWrightBrm. MemorialLecturegivenonDecem%erl?,IQ46,DrvonK6xnindicatedthatatzetioliftthetotalwavedrag”fara qweptforwa
13、rdwing5SidenticalwiththatdbtainedforthesaQewinghavinga corrcmpond.j.fi,amountofewpepbackTheresultsoftheprbsenInvestigatic?fifor thetotaldragofsweptbackwings,therefore,areapplicabletoUiesame?%milyofw$ngshavinga correspondingamountof.sweepfomard.The.dixribu-tionsof.sectiondrag,however,willdifferinthet
14、wocases.Althoughtheculcul.aticnshavebeenmadefor,.thebiconvtixparabolic-arcprofile,thedatamaybeappliedtoj.hdicateCorrespondingresultm for p?ofiiessimilartothebiconvexparabolic-aro.,SYMBOLSprofile.- ,.x,y,z coordinatesofmutuallyperpendlcul.arsytenofaxe6c chordofairfoil section,measu.redinflightdirecti
15、ont/c thidcneesratioofsection,measuredinflight-direction,.,A ”an8 ofsweep,degro”es“.a71a15Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1.;i., ,.- -.:. ,P.y:-d. ;:+ -t, , . . ,*. . .- - -“. . .h. .wingsernispanmeasuredalongy-axissdchordsexceptin.-:
16、.,-.: .-,:.-,-.:L-s *.*.,.,:-2;-.:appendixk .;-K p=aster indicating. ,.s. whga”re+ .s-pa-titi.i”pwlti”ohtiqual:to-y/mJ”sehichords.,. . . .- .,*.-., “,.,. .J. “ . “! 2/% slendernessratiosratiofwingsemispanmee,surefl+.ong,. .-.tiidchordstationtomaxiiaumthicknessofcentersetionv. .velocityinflightdireot
17、ton.,., aemichords ,- -_. . . . . ,. .J%,.” coordinatemeasuredalongy-axfswhih”is.shlftti,to. ,. oPPOsitetipSeCtiOnsemichords” .: .-. Cam, .,sectfonwav+dra.cefficientwithoutti”peffect ., , . . .cd bectionwavrag “coeffiientincludingtipeffect., :. . . .inACd incrementiukectton;waye-dlrqgcofici,”iau5ed,
18、byw-tf.pk.,:. ( , ,AGd Incrementtn.sectionwave-drag coefflc-ientcamedby%ingtiplocatedonsariebald?.ofwingas.sectionProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. ., .c; mCD .ACDaclG“Wtincrementincausedby. .,NACA NO-1449sectionwavedragcoeffcientonon
19、eWg paneltipofoppositewingpanel, ,.wingwave-tiagcoeff”lcient+ithouttipeffectwingwave-dracoefficientincludj.ngtieffectincrementinwave-dragcoefficientcausedbytips,ccmpletewingj.ncremenHnwave-dmagcoefficientononewingpanelcausedbytipofoppcmitewingpanel,completewingcordinateswhichreplacex and y, respecti
20、vely,u“sedto,.indicatOoriqincf,yhih,ZJandm “indicate“ Involvingmultiplioatinb;factorP. .Subticr:iptlotationsforu“a” indicntethoorinintermsofcoordinatesxand y, respectively.;. . .,ANALYSIS trmxifmmatioriof source llne *.Basic4ata.-T.heapalysisisbasedonthin-airfoiltheoryfor .smallpressuredisturbancesr
21、elativetotheamhiiintpressures.Theaxesuse5 arethemutuallypsgpndlcul.arX,y, Z SyStli fi whichthex-axisitakeninthedirectionoffliglxhpositiv6tothe,raar,they-axisisalo thespangcsitiveto-theright,andthez-axithat “jm,thire610nisinfluencedonlybythecomponentofthe,ve3kCitynormal,tpthewing“loadinge,q(reference
22、k). Thepreseures,Werefo%e,areexactlythosethatwouldbe oomputedbytheAckerettheoryofljnsarizotwo-dtiensional:sllper,oticflm.byuse ofthenarroalveiocltycaporletit(reference), Ohoardofthepoint.wherotbMhch wayefrcmthevortexintersectsthowingtzailingedge,thefowLiaentirely%wo-:dimensional.Inthiregj.on,therefo
23、re,thesectionwave-dz%qcoef-ficientbasedonallpwametersmeasurednormalt.othewingleadingedgehast,haconstantAckeretve.luetoraninfinite“z%titaularwing.ThevalueiH:.16 L 2” ,”()?mn=cn (eeaenilixB,equaion(.?4),)measurednormaltothewi leadingedge,TM sectionwave+rameastirednormaltothewingle)ofrefarence3 shawsan
24、importantdifferencebetweentheconflitiona”wheretheMechlineliaheadof()thewingleadiedge m # . Ifm , c% doesnotdeoreasetozerowithincreasingvaluesof y butapproachesa constant:positievalue.ThecontrilmtionsoftheadjacenttipeffecttotheshaleQf,thecd+tiStri.bUtibnLcd aresimilarforbothm.k and mB “ . (Noteiachli
25、nesfromop,ositetipinfig.2(a)infig.:2(b)ofreference3.)Thisccm-Ofthispaperand dll -;. _ “parisoniiicates,therefore,that.i.neneralforagivensweep- backangleahdcomparativelyhighaspectratiosi”wliiohAclliszero( ) orveryqmal.,anincreaseintheMachnumberJ%2:l .,.,Provided by IHSNot for ResaleNo reproduction or
26、 networking permitted without license from IHS-,-,-8whichbringstheMachlinebehindtheleadfngappearscoresult-ina shiftofthecenterofdragintheoutboarddireotion.Theresultsareshowninfgurek(a)forNACATNNo,14k9edgeofthewingpreeaureofthewavetherec%angulxu?wingof infinite aspectratioatseveral Machnumbers.Theser
27、emitscorrespondtotheAckerettheory,whichshowea constantsectionwave-dragcoefficientalongthespan, .Theresultsaregiveninfigurel(b)fortherectangularwingataMachnumberof1.25forseveralaspectratios.Theaspect”ratiosinthisfigure,asinfigure3(b),wereselectedsoastorepresent eachof thedifferenttypesoftpeffects.(Se
28、efig.2.)Effectoftipsonwingwave-dragcdefficienti-Thepresent1 indicated,asnoted In reference3“form ithat theintegratedvalueof Acdl overthewiis zero ifthe2maspectratioisequaltoa greaterthan . InasmuchaaD1s iszero-forA (seefig,a:)thetotalcrementinwave-dragcontributedbythetipis%ro IfA-S.+?” “.Fortherecta
29、ngularwing,m =m and.thetotalincrementin causedbythetipsiazeroiftheaepectratiQA . (SeeappendixB,equation(Bll).)Intherange A;, therefore,thewave4ragCoefficjen.tfortherectangularwingi_sindependentofaspectratioandisequatotheAckeretresultforatwc+dimeneionalwing,IftheaspectratiofortherectangularwingISless
30、thanl/13,theincrementin CD causedbythetipsisfoundtobenegative.Generalizedqurvesforwingwave-dragcoefficien.t.-Fres to8 presentgeeralizedcurvesfordetermmngthewngwave-dragcoefficientoveranexteneiverangeofawoepbackangle, M%chnumber,aspectratio,andthicknessratio.TheresultsareGiveninfiguresand6 fortheswep
31、tbackwings.Asnotedpreviouslyinthe “INTRODUCTION,”thedatainthesefiguresareapplicabletothesamefamilyofwingshavinga correspondingamountofsweepfolward.Tharesultsfortherectangularwingarepresentedtnfigures7and8.ThedatainfigurestoElapplyspecificallytoWtaperedwingsat zero lil?twithbiconvexparabolilc-arcprof
32、ile,sandthe“wingtipscutoffinthedirectionofflight,Theresults, however,maybeappliedtoindicateaproxhatresultsforprofilessimilartor.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-/lfACATNNo.%449 9. .:-.- -,i:. :, :; .:. .-.the-h$edWi pardbb.t;.”.A!;,t.:
33、;f “Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10IftheMachlnedragcoefficientNACATXNo.1449liesbehindthewingleadinCedge,thewingwaye-reachesamaximumvaluewhichisconstantforall.( )aspectratiosgreaterthana certaj.nlimitivaluo A? .f2mE- 1“Figure6 fndic
34、at.estlatingeneralthevatiititionof CD overthecozhpleteranepfMachnumberbecomesleespronouncedastheaspectratioisdecreased.For A“=cotA thevariationof CDwith M iscomparativelysmall,EffectofaspectratioandMachnumberonwi wave-dragcoef-ficientfor=tangularwiIu.-?$ies 7and.8indicatetheeffectsOfaPectra%OandMach
35、nberon CD f#?rectrlarwis.JJFm?e jshowsthevariationof v- withapectratiofor”loo(y “.constantMaahnumbers”.Thewingwqye-dragaogff.icientreachesa.,maximumvalueatanaspectratioequal.to l/ - 1 andremaimconstantforhfghervaluesofaspecbratio.forconstantaspect-ratio8, .-,Whentheaspectratioiseq,tiltoorgreaterthan
36、1, CD isindependentofaspectratioforMachnumbers .equaltoor greaterthan1.41.Thecurvesinfigure!3indicatethatthevariationof CD overthecomplot.eTqngeofMachrminberbeccmmalesepronouncedastheaepectratioisreduced;thuathessmetrendnotedpreviouslyforthesweptbackwings,isshownEffectofwee- . A. 0.2cos2A “The”rsult
37、pinfi.ne%ehind.%hewingleadinedgb:,.,. ,39 Thewave+h?agcoefficientd?”thesweptbackwing”reachesamaximumlvalueatanaspectratioforwhichtheMachlinefrcmtheleading”edgeofthecentersectionintersectsthetrailingedgeatthetip;thismaximumcoefficientreminsconstantforhighervaluesofaspectratio.4. Thewave-dragcoefficie
38、ntoftherectangularwingreachesamaximumvalueatanaspectratio.for,whichtheMachlinefrcmtheleadingedgeofthetipsectionintersectshetrailingedgeofthetipontheoppositewingpanel;“thismaximumcoefficientremainsconstant.athighervaluesofaspectratio.5. Thevave-drag-coefficientiofall.wingpforallaspectratiosdecrease”s
39、withincreasingMachnumber.,-CompleterangeofMachnwnber:.!. -.6.Witha ccmparattvel.yhighd.spectratio,anincreaseintheMaqhnumberwhichmovestheMachllnebehindthewinglead.ingedge .apyears:tomovethecenterof-pressure.ofthewavedragintheout-boarddirection.“ 7. Thewariatf.onofthewi wave-dragcoeffi.cientwithMach .
40、numberoverthecompleterangeof.supersonicMachnumberbecomeslesspronouncedastheaspectratioisdecreased. .-8. SweepbackobtainedbyrotatingthewingpanelsrearwardcangiveappreciablereductionsInthewa-dragcoefficientatallsupersonic.speeds. 9. Sweepbackobtainedbyslidingeachsectionrearward.cangiveappreciablereduct
41、ionsinwave-dragcoefficientonlywhentheMachli?eiswellaheadofthewi leadingedge.When.theMaqhligeapproachesthewingleadingedgeorisbehindt,Hweepbackobtainedbyslidingeachsectionrearwardincreasesthewgwav.ag-coefficient.LangleyMemcrlalAeronauticalLaboratoryNationalAdvisoryCcmmitte6forAeronautics. Langle;Field
42、,Va:,July10,1947. . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,.,NACATNNO.1449.APPENDIXA . ”,FORMULASFORu-RESSZONSANDINTEC+RITIONLIMITSFORINTEGRANDINEQUATION(1)FORtiAJ?Ei?ED_ACKANDtiULAR,WINGSOFBICONVIHPARABOLIC-ARCPROFILEATKERO J“”LIFT m=cot
43、B “:SweptbackWings. .,Thedesiredintegrmdu ibeuation(1)isdeterminedinamannersimilartothatdescribedinappendixA ofreference3,Thesweptbackwingofdesiredprofileshapeamiplanformis-builtupbysuperpositionofthesolutionsobtainedfora sami-infiniteobliquewedge.Onthebauisofthelinearizedtheory,reference2derivesa s
44、olutionrepresentinanobliquesemi-infinite(sweptback)sourceltnemakingtheangle.d ,grweepbackA withthey-axis.The -solutionutilizedforthe”pressefieldorforthedisturbance”velooityis .0,0 6.=R.P.Icoeh-lx-m”2y- “ (Al)- “-”-wherethesubscriptnotationindicatesthatthesourcelinestartsat,theoriginofcoordinates(x=O
45、,y =0). 13quation(ALfsshowninreference2 tosatisfytheboundaryconditionfora thin)$thati, IiIndicalmeaGourcelfiwithareversalinthesignof m.Inequation(A4),the u-expressions-T thatis,Thena71a15.*E,T(X,.y)= ()(Y-ll)fy “-,q ,. ,$C,qix, y)1EL-(y-qf-(Y-n)Provided by IHSNot for ResaleNo reproduction or network
46、ing permitted without license from IHS-,-,-.16. . . . . ,.,” /,.:Fortherectangular. resultfrcmsubstituting*“ NACATNNo.1449.RectanguWings, .:.:L.:.”:,-,;-:“,+. . .tiiwi:.=w arig.t%i“fciilowihg”eeslonam =w inequation(Aj),(A6),and(A7):,.-. ,.,.%-l 1 1(x,y) = : * q - COB-1y_-6):. . . -., , “V dZ:g:;a(”+
47、:y)=.7 cos-i:_-EJ” ,-,- , ,.,., . . .:, . . . .,_ . . ,j+k,q(xsY)=-;(Y- ) Ccih-:*, :,., p,y,=i, .IL.( ) 1+L X-E _!co#.P.kJd. ,. B-Y-1) ._k. ,.LimitsOfIiXte?qtfons, TM limltsofintegrationwithregard.to for-thesectionwavea coefficientsWd wih.regarilto y farthetdtalwiwave-dragcoefficientsarediscussed.T4e.u+onetits caused”byeachoftheelementarysourcelines arezeroatBllpointsoutsideoftherespectiveMachcones.Theexpzwsaionsfortheu-intograndinequation(1)arethereforeevaluatedalongthesectionforvalues-of x beginningattheforwardboundaryoftheMach
