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THEJOHNSHOPKINSUNIVERSITYAPPLIEDPHYSICSLABORATORY8621GEORGIAAVE.,ILVERSPRING,M D.Operating underContractNOrd7S86Withthe.Bureauof Ordrience,U.S.Navy 1

AERODYNAMICCHARACTERISTICSOFWINGSATSUPERSONICSPEEDSbyR .M .SnowandE.A.Bonney

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THEJOHNSHOPKINS UNIVERSITYAPPLIEDPHYSICSLABORATORY8621Georgia AvenueSilverSpring,Maryland

Operating UnderContract NOrd7386With theBureauofOrdnance,U.S.Navy

AERODYNAMICCHARACTERISTICS OFWINGSATSUPERSONICSPEEDS

byR .M .Snowand E .A .Bonney

BUMBLEBEEReportNo.55

DISTRIBUTIONOFTHISDOCUMENT I S UNLIMITEDM A R C H 1947


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TABLE OP CONTENTS

Introduction.Reference Tablefor Aerodynamic CoefficientsiI .pplication of Busemann 1oConical Field Method to ThinWings

" b y Robert M ,SnowIntroductionCharacteristic Cones,Boundary ConditionsRectangularWingRaked WingBent Leading Edge 1CaseofIntersecting Envelopes 3Trapezoidal Wing..... 5General SymmetricalQuadrilateral ......6WingWithDihedral 0

I I .AerodynamicCharacteristicsofSolid Rectangular AirfoilsatSupersonic Speeds " b y E .A .Bonney

Summary 23Assumptions 3Nomenclature ...24Discussion 6Derivations. 7Results 4Conclusions4111. The Reverse DeltaPlanform31 7 .Liftand DragCharacteristicsof Delta Wings atSupersonicSpeedsby E .A .BonneyAbstract .44Assumptions 5Nomenclature 5Discussion 6

V .ectional Characteristics ofStandard Airfoilsby E .A.Bonney 8

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V

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I I S B O F F I G U R E SI .p p l i c a t i o n o f Busemann1 s C o n i c a l field Method t o T h i n W i n g s

/ j fi g u r e 1 . Flow f o r R e c t a n g u l a r W i n g ,S e c t i o n " b y P l a n e X j erpendicular toStream3. ^ t o v j F l o w f o r S w e p t - h a c k L e a d i n g E d g e ..3 , P r e s s u r e D i s t r i b u t i o n A l o n g S p a n f o r R e c t a n g u l a r W i n g o r f o r R a k e d - b a c k R e c t a n g u l a r Wing4 .Wing With D i h e d r a l ,S e c t i o n " b y P l a n e Perpendiculart o S t r e a m.*,.. 1 4 5 ;S y m m e t r i c a l T r a p e z o i d a l Wing156 . P r e s s u r e D i s t r i b u t i o n A l o n g a S p a n L i n e f o rT r a p e z o i d 67 . P r e s s u r e D i s t r i b u t i o n A l o n g S p a n L i n e f o r F o r w a r d -p o i n t i n g T r i a n g l e 78 .G e n e r a l Q u a d r i l a t e r a l 79 .Wing WithDihedral.,0

II .Aerodynamic Characteristics ofSolid Rectangular Airfoils atSupersonic Speeds

Figure 1 .Pressure Loss DuetoFlowAround the Tip82 .C3 vs M 83 .AspectRatio Correction to Liftand Wave Drag.384 .Center ofPressureforFinite Aspect R a t i o " .85 . K f c ' s f o r V a r i o u s Wing S e c t i o n s ( s i n g l e t a p e r )....... 3 96 .K^'8 forVariousWingSections(doubletaper)97 .AspectRatio forThin Rectangular WingsSupportedOverEntire Base. 38 .OptimumLiftCoefficient - Entire Base99 .Maximum Lift-Drag Ratio - EntireBase010.Optimum Angleof Attack - Entire Base011.LiftCurveSlcpe- Entire Base.012.Thickness ZUtio - EntireBase01 3 .O p t i m u m A s p e c t R a t i o s f o r T h i n R e c t a n g u l a r W i n g sS u p p o r t e d " b y a H u h 11 4 .O p t i m u m L i f t C o e f f i c i e n t - H u b11 5 . M a x i m u m L i f t - D r a g R a t i o - H b.. 4 11 6 .O p t i m u m A n g l e o f A t t a c k - H u b117.Lift CurveSlope - Huh 218.Thickness Ratio -H uh 2IT

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I V . L i f t and Drag C h a r a c t e r i s t i c s o f D e l t a W i n g s a t S u p e r s o n i c S p e e d sF i g u r e 1 . L i f t C h a r a c t e r i s t i c s o f D e l t a W i n g s02 .F o r m D r a g o f D e l t a A i r f o i l s W h o s e L e a d i n g E d g e s

a r e I n s i d e t h e M a c r . C o n e03 . L i f t - D r a g E a t i o o f D e l t a and R e c t a n g u l a r W i n g s 14 . M a x i m u m L i f t - D r a g R a t i o - D e l t a W i n g s15 . M a x i m u m L i f t - D r a g E a t i o 16 .O p t i m u m A n g l e o f A t t a c k - D e l t a W i n g s17 . R a t i o o f M a x i m u m L i f t - D r a g R a t i o s o f D e l t a andR e v e r s e A r r o w W i n g s...,2 8 .A s p e c t R a t i o o f D e l t a W i n g s .,2 9 . ' Planform S e m i - v e r t e x A n g l e o f D e l t a W i n g s2 1 0 .O p t i m u m L i f t C o e f f i c i e n t - D e l t a W i n g s..... 2 1 1 .O p t i m u m L i f t C o e f f i c i e n t 3 1 2 .S p a n w i s e C e n t e r o f P r e s s u r e L o c a t i o n - D e l t a W i n g s....3 1 3 .( N o m e n c l a t u r e ) . .......3 1 4 .D e l t a Wing Drag v s M a x i m u m T h i c k n e s s P o s i t i o n ,S w e p t T r a i l i n g E d g e ;a = 0 . 53 1 5 . D e l t a Wing Drag v s S w e e p b a c k i L x g l e . , S w e p tT r a i l i n g E d g e ;a = 0 4 1 6 . D e l t a W i n g D r a g v s S w e e p b a c k A n g l e ,S w e p tT r a i l i n g E d g e ;a = 0 . 5 4 1 7 .D e l t a Wing D r a g v s S w e e p b a c k A n g l e ,S w e p tT r a i l i n g E d g e ;b = 0.2 5 1 8 .D e l t a Wing Drag v s Mach N u m b e r ,S w e p t T r a i l i n gE d g e ;b = 0 . 2 5 1 9 . L i f t C u r v s S l o p e - D e l t a a n d A r r o w h e a d P l a n f o r m s .....62 0 . L i f t C u r v e S l o p e - D e l t a a n d A r r o w h e a d P l a n f o r m s .....62 1 .S p a n w i s e C e n t e r o f P r e s s u r e L o c a t i o n - D e l t a and

A r r o w h e a d P l a n f o r m s ,.72 2 .C h o r d w i s e C e n t e r o f P r e s s u r e L o c a t i o n - D e l t a a n dA r r o w h e a d P l a n f o r m s7V .e c t i o n a l C h a r a c t e r i s t i c s o f S t a n d a r d A i r f o i l *F i g u r e 1 .D r a g C o e f f i c i e n t v s S t r e s s ..... 92 .D r a g C o e f f i c i e n t v s T h i c k n e s s R a t i o ...,03 .S t r e s s v s T h i c k n e s s .? ...0j tw^rr? ~tr * - ^ . ,?'-^s^vSiu


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% > * INTRODUCTION

T .

Fivepaperdealing with practicalaspectoftheoreticalworkdoneatthe Applied PhysicsLaboratoryonthesubjectoftheaerodynamicchar-acteristicsof wingsinsupersonicflow arepresentedinthisBUMBLEBEEreport.hesepapers,whichhareappearedasinternalmemorandaoftheApplied PhysicsLaboratory,representthecombinedeffortsofAPL/JHUpersonnelandoutsidecontributorsasnotedinthereferences.

Thefirstpaperderivesliftcoefficientsandcenterof pressurelocationforfiatplatesofpolygonalplanformfromfundamentalconsid-erations,usingBusemann 1sconicalfield method.heotherpapersutilizeBusemann**secondorder approximationformulatodetermineth eaerodynamiccharacteristicsofcertaintypesofwingshavingfinitethickness.on-sequentlytherewill b esomeoverlappingofresults,hutno materialisdeletedinaiimuchasthetw omethodsarequitedifferent.ttentioniscalled,however,todifferencesin nomenclature "betweenthefirstandtheremaining papers. shortdiscussionofth esectionalpropertiesof variousairfoilshapesandtheoptimumtypeisgivenattheendofthisreport.

Itis believed thatthesepaperscovermostoftheunclassifiedengineeringworkontheeffectofwing planformsavailableatthistime.Th ework isbyn omeanscomplete,therebeingmany practicalplanformsforwhichnotheoryhasbeendeveloped,particularlyinthecaseofwingswithfinitethickness.ccordingly thetableonthefollowing pagehasbeenconstructed toindicatethetypesforwhich informationcanbefoundinthisreport,toshowwhereitcanbefound,andtoindicatethemist-ingdata.nthistabletheRomannumeralreferstothepaperinthisreportasindexed intheTableof Contents,whereasthe Arabicnumeralreferstothe pagea whichtheexpression fortheparticularcoefficientmaybefound.n Xindicatesthatth eexactinformationisnotyetknown.

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I .APPLICATION Or BUSXHAHIP S COHICALflELDMETHODTOTHISWINGS" b y Robert M .Snow

IntroductionIn this paperthepressuredistribution,liftcoefficientandcenterof pressurearedeterminedforseveralplanewingsofpolygonalplanforaatsmallanglesofattack," b y th emethodof"conicalfields".heeffectof adihedralbendisalso obtained.Squaresand productsof perturbationvelocitieshavebeenneglected,sincethemethod isbasedonthePrandtl-Glauertlinearised potentialequation.odificationsdu etowing thick-ness,viscouseffects,andinterferenceeffect(withafuselageorwithother wings)arelikewiseneglected.Themethodofconicalfieldsinsupersonicaerodynamicswasdeveloped

by Busemann(Ref.1 ) ,who applied i ' tto'severalimportantproblems.tew-art( 2 )hassolvedthsproblemof adelta wing byessentiallythesamemethod. conicalfieldcorrespondstoalinearhomogeneoussolutionofthelinearised(or Prandtl-Glausrt)potentialequationforsupersonicflow:

( 1 )

HereMistheMach numberofth6mainstream,whichismovingalongthei axis.he perturbationvelocitycomponents( u ,v,w)arealsosolu-tionsofEq .(1)andarehomogeneousofdegreezero,i.e.,u,v,and wareconstantalonganyrayemanatingfromth eorigin.he particularsimplicityofconicalfieldsliesinthefactthatafteratransformationtheperturbation velocitycomponents( u ,v,w)areobtainedassolutionsof Laplace'sequationintwovariables*usemanncreditsthistransfor-mationtoChaplygin, 3 )whomadeuseofitinaformallysimilarproblem.Sincethegeneraltheoryhas baendiscussedrecently by Stewart( 2 )itisonly needed t o .statetheprincipalresultintheformin whichitwillb eMl _*hispapersarevisionofCM-265originallypublishedasnnternalmemorandumofheAppliedPhysicsLaboratory.


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2erodynamicCharacteristicsofWingsatSupersonicSpeedsutilized.et/WetheMachangle( s i n " " 1l/M),and x,y, arectan-gularcoerdiiyitesystem withthez -axispointingdownstream,and

p=tarry/x= gy=tcn iThetransformationr2Ar+ r 2

issuchthathomogeneousfunctionsofdegreezerowhichsatisfyEq . 1 )alsosatisfyLaplace'sequationinthepolarcoordinatesr ,< p ) .Theevaluationofthestreamwisecomponent( w )ofperturbationvelocityisofprimaryimportancesincetheaerodynamicforcesonthewingaredeterminedbywalone.hisfollowsfromth elinearizedBernoulliequation,

p=p-pWuj ( 3 )which,likeEq. 1 ) ,resultsfromneglectingsquaresandproductsofperturbationvelocitiesinthecorrespondingexactequation.nmanyproblemsofthistype,includingthoseconsideredhere,w may b edeter-mined withoutfurtherreferencetothecomponentsuandv.


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A p p l i c a t i o n o f B u s e m a n n ' s C o n i c a l F i e l d M e t h o d t o T h i n W i n g sC h a r a c t e r i s t i c C o n e s , B o u n d a r y C o n d i t i o n

T o i l l u s t r a t e t h e t y p e o f " b o u n d a r y c o n d i t i o n n e e d e d t o d e t e r m i n et h e c o n i c a l f i e l d ,c o n s i d e r t h e s p e c i a l c a s e o f a r e c t a n g u l a r w i n g .T h i s r e c t a n g u l a r wing m a y h e r e g a r d e d a s t h e r e s u l t o f c u t t i n g o f f t h e s n d s o f a t w o - d i m e n s i o n a l a i r f o i l .h i s o p e r a t i o n c a u s e s a m o d i f i c a t i o ni n t h e f l o w ( o r i g i n a l l y t w o - d i m e n s i o n a l ) , w h i c h m a y " b e r e f e r r e d t o a st h e" t i p e f f e c t " .n t h i s c o n n e c t i o n , a f u n d a m e n t a l d i s t i n c t i o n s h o u l db e m a d e " b e t w e e n s u b s o n i c f l o w and s u p e r s o n i c f l o w .or s u b s o n i c f l o w( d i f f e r e n t i a l e q u a t i o n o f e l l i p t i c t y p e )t h e t i p e f f e c t d i e s o f f a s y m -p o t o t i c a l l y with i n c r e a s i n g d i s t a n c e i n b o a r d .or s u p e r s o n i c f l o w ( d i f -f e r e n t i a l e q u a t i o n o f h y p e r b o l i c t y p e ) t h e t i p e f f e c t f a l l s t o z e r o a ta c e r t a i n f i n i t e d i s t a n c e , a n d t h e e n t i r e e f f e c t i s c o n t a i n e d w i t h i n a r e g i o n h o u n d e d " b y r e a l c h a r a c t e r i s t i c s u r f a c e s .or l i n e a r i z e d s u p e r -s o n i c f l o w ,t h e d o m a i n o f i n f l u e n c e o f a n y p o i n t i shounded " b y a " M a c hc o n e " , w h i c h i s o n e n a p p e o f a c o n e o p e n i n g d o w n s t r e a m with s e m i - v e r t e xa n g l e e q u a l t o t h e M a c h a n g l e / *.i g u r e 1 r e p r e s e n t s a s e c t i o n h y ap l a n e p e r p e n d i c u l a r t o t h e m a i n s t r e a m .h e Mach c o n e sf r o m t h e t i p s o f t h e l e a d i n ge d g e d i v i d e t h i s p l a n e i n t o _ _ __t h r e e t y p e s o f r e g i o n s .nt h e c e n t r a l r e g i o n ( I )t h e f l o w i s i n a l l r e s p e c t s t h e s a m e a s i f t h e w i n g w e r e o f i n f i n i t e s p a n , " b e c a u s e n op o i n t o f t h i s c e n t r a l r e g i o nl i e s i n t h e d o m a i n o f i n -f l u e n c e o f a n y p o i n t r e m o v e d GSHX- V f t - , b o u li n t h e m e n t a l p r o c e s s o f o h - n^.^ FW dt a i n i n g t h e r e c t a n g u l a r wingf r o m a n a i r f o i l o f i n f i n i t e s p a n .n t h e o t h e r hand t h e p e r t u r b a t i o n v e l o c i t y c o m p o n -e n t s a r e z e r o i n t h e e x t e r i o rr e g i o n ( I I I ) , b e c a u s e n o I G . p o i n t o f t h i s r e g i o n l i e s i nLO WF O RRECTANGULAR W I N G ,t h e d o m a i n o f i n f l u e n c e o fECTIONBYPU PERPENDICULARa n y p o i n t o n t h e r e c t a n g u l a rw i n g .h e r e q u i r e m e n t o f c o n -t i n u i t y l e a d s t o b o u n d a r y c o n -d i t i o n s w h i c h m u s t b e s a t i s f i e d b y t h e p e r t u r b a t i o n v e l o c i t y c o m p o n e n t su , andw on t h e b o u n d a r y o f e a c h c o n i c a l t r a n s i t i o n r e g i o n ( I I ) .nt h e b o u n d a r y b e t w e e n r e g i o n s ( I I ) a n d ( I I I ) , u , v and w m u s t v a n i s h .O n t h e b o u n d a r y b e t w e e n r e g i o n s( I ) a n d ( I I ) , u , v an d w t a k e o n t h e ( c o n s t a n t ,t w o - d i m e n s i o n a l ) v a l u e s o f r e g i o n ( I ) .

S i m i l a r s t a t e m e n t s a p p l y a l s o t o t h e e x a m p l e o f a s w e p t - b a c k l e a d -i n g e d g e ( F i g .2 ) .h e t w o l i n e s f o r m i n g t h e l e a d i n g e d g e a r e ,o f c o u r s e ,

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AerodynamicCharacteristicsof WingsatSupersonicSpeedsfiniteintheactualcaae,hutth eeffectoftheirfinitenesswillheconfined b y character-isticconepassingthroughthe pointsatwhichtheleadingedgechangesdirection.hepossiMlityoftreating moregeneralpolygonalwings " b ythismethodfollowsfromtheseremarks,orthepolygonsthatmayhetreatedbythismethod,thereareisolatedregionsofuniformflow(identicalwiththeflowforan i n f i n i ' t e wingata certainangleof yaw)sep-aratedbyregionsoftransitionin th eMacheeneswhichstartfromeachvertexofthepolygonandopendownstream.tisconvenientinth efollowingdiscussiontorefertothesespecialMachconessimplya s"theMach cones"or"theMachcone*.

Plorifarmo rdifinl t ion o f y m b o l i . Airflowowardottomfcgt.m

Sect ionylantcrptndlcularotriamI,F l oo m ea*o rnfini tpor.irfoiltnglfawf/2-i,I,Flowarnto rInf in i tepani rfoi ltn g l tfewr/2-StEcnicclFlfldTU FrStream

FIG.2FLOW FOR SWEPT-BACKEAD I N GE06EBoundaryconditionsmustalsobegivenoverthatpartofthe wing whichliesinsidetheMachcone.he velocitycomponentnormaltothewingmusthavethesamevalueasontherestofth ewing,becauseoftheconditionthatthere

benoflowthroughth ewing.he boundaryconditionforwisthatthenormalderivativeof wbeseroonthewing.hisfollowsfromthere-quirementofirrotationality(implicitintheuseofthepotential$) ,8 0thatur=JW _dyandtheconditionthat^isconstantonthewing.hefactthatthe wingdoesnotlieexactlyintheplaneys0isneglected;thisha snoeffectonthe firstorderperturbation.hissimplificationis madethroughout,sothatth eangleofattackentersonlyintheboundaryvaluesand notinthe positionoftheboundaries.RectangularWing

Theproblemof a planerectangularwingwassuccessfullysolvedby Busemann( 1 ) ,byessentiallythemethodemployedhere.chlichtlng( 4 )had previouslyconsideredthesameproDlembyadifterentmethod,andobtainedafalseresultbecauseofananalyticalerror.henthiserroriscorrected,Schlichtlng'smethod becomesconsistentwithth econicalfieldmethod,notonlyforrectangularwings,butalsofortheraked wingsconsideredinthenextsection.Bakedwingswerenottreatedbyeitheroftheseauthors,buttheirmethodsareeasilyextendedtothiscase.)

s:raargarT


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Applicationof Eusemann 8ConicalFieldMethodtoThin WingsAttentionmay b econfinedtoon eendofthewing.heoriginofcoordinatesistakenattheendofth eleadingedge.hepositivex-axisisdirectedspanwiseawayfromthewing.he y -axisisnormaltoth ewing;thedirectionofthepositivey -axismay b eregarded as"upward".hepositives -axieisinthedirectionofflow(theco-

ordinates" b e i n g suchthatthewingisconsidered to b ea t - restwithth eairflowing pastit).hevelocityofthe.incidentflowi s W theangleofattack f i t ,andth eMachangleyusin11/M.or a wingofinfinitespan,theusualtvo-dimensionaltheory*givesVJ-ocWtanjuisur00 abovethewingand

JX)=~VJ.00belowthewing.ntermsofthepolarvariablesRand( p the " b o u n d -aryconditionsforu?are:

=0 fo r


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AerodynamicCharacteristicaofWingsatSupersonicSpeedsThisisnottheconventionalFouriereerie*forafunctionwhichisperiodic with period Z r f T however,itisusefultonoticethattheseriesisperiodicwithperiod ^TT hissuggeststheextensionoftheproblemtoa two-sheeted Riemannsurface " b y analyticcontinuationacrossthecut.tisconvenienttoretainthesymmetry b y consider-ingftorangefrom- gfTto4 - TTthenthe "boundary valuesonthearcsin the"lower"sheetoftheRiemannsurface(thati s ,~2ir


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-

ApplicationfBusemann 'BConicalieldMethodohinWingsOnheowerurfacefhewing(9=- ;w is merely changed insign.he rightmember of Eq.( 5 )or( 5 " )isshown as a function of R/AinFie3 .rom theform(5'),i tisclearthatthecurveissymmetric with respecttothe point(1/2,1/2),sothattheaverage valueof w alonganyspanwiselinefromR =0 to R =A isjust Woo/2.hetrianglecutoffby theMach conecontributesjustonehalfas much totheliftasan equal arealocated in,theregionwheretwo-dimensionalcalculations are valid,it will heseeninthenextsection thatthesamesimple result holds for raked wings with leadingand trailing edges perpendicular totheflow.)ormulaefor liftco-efficientand center of pressurearenow easily obtained.f C j , QO istheliftcoefficientfora planewing o f ' infinitespan,andC^theliftcoefficient fora rectangularwing of totalspan8and chord c t

l-g" t a n j j . ( 6 ) o oThecenter of pressureislocatedata distancehackfrom theleadingedgeexpressed " b y

z= p Q (7)2 l-l/gG/gtanjxTheabovediscussiontacitly

assumesthatthetw o Mach cones fromthetipsd o - notintersectonthewing.owever,thisrestrictionisseen to be unnecessary.f$,sthe(disturbance)potentialinsideoneof thec o n e ' s ,$athe potentialinside theother cone,and$cthepotentialfor a wing ofinfinitespanwithleading edge perpendicu-lar tothestream,thenintheregion common tothetwocones thepotentialis = $,+ $5 >-$t > 0 verify this,i tmay benoted that$ii$j,a(i$oo resolutionsofthe Prandtl-Glauert Eq .( 1 ) ,andsincethatequationi slinear, < i sasolution as well.lso$andit sfirstderivatives( u ,v ,w)arecon-tinuous acrosstheconicalsurfacesboundingtheregionin question.I t maybesaid thatthe tipeffectsfrom thetwotips are additive,sincetheequation defining < |maybe written

'z 0. 8< n

w|{= .6II s0 4

0. 20 0. 2 0.4 0 .6 0 .8 X Vjtan;i 10

Ashe ract ionalistance nboard,spanwise, f romheing ip o heac h cone.FIG.

PRESSURE DISTRIBUTIONLO N GPAN F O R RECTANGULAR WINGR F O R RAKED-BACKECTANGULAR WINGWITH \1\


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&mz^j?i

8erodynamicCharacteristicsof WingsatSupersonicSpeedsItshould b e notedthattheflowintheregioninquestionisnota"conicalfield".hecombinationofthetw oconicalfieldswithdifferentverticesisneveraconicalfield,although i tapproachesa conicalfieldasymptoti-cally downstream.

Thissolutionislimited " b y theconditionthattheMachconefromonetipshouldno tintersecttheotherendofthewing.fthishappensafurther alterationisneededtosatisfy "boundaryconditionsattheedge,andthedifficultyofthe problemisincrsasedenormously.Sinceoverlapping"tipeffects"areadditive,theliftcoefficientand centerof pressureareeasilycalculableaslongastheMach conefromon ewingti pdoesnotcuttheother wing tip.ftercalculation,itisfoundthatEqs. 6 )and ( 7 )also'holdfortheoverlappingcase.tmayhenoticed thatasthequantity5tan u increasesfron0to1(thehighestvalueforwhichthesolution applies;,theratio C ^ / C j , decreasesfrom1to1/2,andthecenterofpressuremovesfromC/2forwardtoC/3.

BakedKingThefollowingtreatmentappliestowingswithapositiverakeangle,definedinsuchawaythatthe leadingedgeisgreaterthanthetrailingedge.f d>t/u theproblemisquitesimple.hepressureon thewingisuniformandistohecalculatedfromtwo-dimensionaltheory.hisisseen " b y consideringthewing ascarved outof aninfin-itespanwing,whichonlyInvolvesremoving portionswhosedomainofinfluencedoesnotcontainanypartofthefinaltrapezoidalwing.If 0


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Application cf Buse'mann'e Conical Field Method to Thin Wings

in=0o rS,>0^9s**rl tanJ U Lcw-0orS,


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/j&HifiiI-

> u

10erodynamic Characteristicsof Wings atSupersonicSpeedsBy eliminating and iti s possibletoobtain6s a functionof S n the planeofthe wing,wheref Oor7T and *A//?e*ftheresultsimplifiesto

de 1 * 'e^+Ro s rs, = 0or TTA A-RoHere 0,c ? 0 arethe polar coordinates oftheedgeofthe wing;p0is0 or 7 7 accordingastherakeis positiveor negative.n thesurfaceof thewings

A~TTR: tt-TrA-RoAs R 1goesfrom 0to A ,R goes fromR0to A,and thedependenceof R eonR islinear.hustheliftfor a raked wingalsovariesaccording

f ( * ) * : s i n " 'VTi iwhereA*f t ' / A isthefraction ofthespanwisedistancefromtheedgeof the wing totheMach cone.hefunction f(X)isshown graphicallyin Pig.S .inceitis now known thatthespanwiseaverageofliftfromtipto coneisJust .one-half of thetwo-dimensional value,i ti seasytoderive formulaeforliftcoefficientand center of pressure.heseformu-laeholdalsoif the Mach conesoverlapon the wingwitnoutcuttingtheopposite wingtip.n other words,theformulaeholdf or ^t f t r t . y a I3where5isthe meanspan,oraverage* ofthespan attheleading andtrailing edges.

c L-sT-1-2 3t a n A*5i ( n )(8>)S j ; and . S i parethespansattheleadingand trailing * j d g e srespect-

ively.hecenter of pressureislocatedatthefollowing distances" b e h i n d theleading edge:

Q j-VsVston^ /jV tanS,+fan S*?- ,* r v (9 )2 I-%% tan> L+V o /s tanS ttanSt


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ApplicationofBusemann'sConicalField Method t o ThinWings1Itmay b e welltorepeatatthispointthattheseresultsholdonlyforpositiverakeangles( S i , > S i p ; .BentLoadingEdgeA problemofconsiderableimportanceisillustratedinFig.2 .Theangles ,and j , arenotnecessarilyacuteangles,asinthecasedrawn,buteachi sassumedtolieintherange>a


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i\

A t ] 12erodynam icCharacteristicsfWingsatupersonicpeedsTobtainhehordwiaeoiflponentfperturbationvelocityforanair-foiltananglefya wWz-S eplacecC,WandM byt,VJtandM,.Thetreamwls eom ponen tfperturbationvelocityfollowsro mmulti-

\\ licationbyin8 :WCWsinS\'i = m o.J4 MHV^ 2 sin28- (10)ior x > 1*7= rsin/?wh e re ,s before, co s=fanLL/tan 6Theboundaryconditionsforheemainingworcsfheuppersemicirclereand

VJsin^2_'2= *s K2forTf-P


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Applicationof Bueemarm ,BConicalField Method toThin Wings3" U 7 I Tr-rZco$2}-JLxu^. -i ton /g " Tf sin*qnypj-1 2 )

Returning tothe generalcase(5 ,andS-^otnecessarily equal)weseektheaverage valueofw along thesegmentof aspanwiseline(i.e.,per-pendiculartothestream)cutoff b ythe Mach cone.tis necessary toevaluateintegralssuch as

/tan' -TaRAy - r c o s T TSinceRfa-iA/i+A* ntegrationb ypartseadsoA Tf-rC - i rinV ._ Tt _T T - V tan75T?dR 4anr+ -secr> rowhereo


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i&fa.

14erodynamic CharacteristicsofWings atSupersonicSpeeds_i-k.j

vertex ontheleading edge.sthesimplestexample,Pig.4shows asection perpendiculartothe mainstream forthecaseof awing with dihedral and withtheleading edgeperpendicularto the main stream.hesitua-tion for a plane wing withfor-ward sweep would differ only inthatthetrace L M J T of the winginJig.4 wouldhestraight,andoverlapping ofthe plane waveswould occur over the " b o t t o m arcaswell asthetop.

Theenvelopeof allMachcones withvertex onLM consistsof two halfplanes with traces ACand ac .imilarlythe Mach coneswithvertexonMN giveriset otheenveloperepresentedbyFGand fg.tisimportanttonoticethat no arcof thecircleis apartofeither envelope.Within the houndsofthelineartheory,shockwavesor rarefactionwavesintersect withoutmutualinter-ference,and the perturbationscaused " b y eachareadditive.ntheregionG-BCMtheflowis uniform;in theregion ABFELthereisanother uniformflow;in theregion " b e t w e e n PBCand thecircletheflowis also uniform,sincethecomponentsofthe perturbation velocity areobtained " b y additionofthecomponentsof perturbation velocity for theothertwo uniformflows.hi s completesthespecification ofthe boundary conditionsfortheupper partof thecircle.hen dealingwith u,v ,or w,thefunctionsoughtassumesa constantvalueof EF,another constant valueonCD(bpthoftheseconstantsobtainable from two-dimensionaltheory;,and a con-stant valueon thearc FC ,namelythesum oftheothertwoconstants.Theboundaryconditionsfor thelowerpartof Fig.4 presentnothing new;u,v,andwtakeoncalculable(constant)valuesonthearcs Beand fD,and the valuezeroon thearcc f .

InCOMrown Lea di n gdg isPtrpendicularo SlrcomFIG.

WING WITH DIHE DRA L, SECTIONYL A N E P E R P E N D I C U L A R TO S T R E A M

Review ofthe problem illustrated by Fig.2 nowshowsthattheanalysisgiven holds alsoforthecaseof a wingwith forwardsweep,thatis,awing with theangle pointing downstream.heonly differenceis aslightmodification of Fig.2 .


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ApplicationofBusemann'sConicalFieldMethodoThinWjngs 1 5T r a p e z o i d a l Wing

T h e l i f t c o e f f i c i e n t and c e n t e r o f p r e s s u r e f o r t h e s y m m e t r i c a lt r a p e z o i d a l wings h o w n i n P i g .5 m a y n o w T Ks t u d i e d .h e l e a d i n g andt r a i l i n g e d g e s a r ep e r p e n d i c u l a r t o t h e mains t r e a m ,and t h e t i pa n g l e(5 ) i s g r e a t e r t h a n t h eM a c h a n g l e .i n c e t h e l e a d i n ge d g e i s p e r p e n d i c u l a r t o t h es t r e a m , 3zsff/za n d t h e s u b s c r i p t m a y h ed r o p p e d f r o m $, andtI n t h e r e g i o n I ,

I t f = I .

I n t h e r e g i o n I I ,t h e a v e r a g ewi s f o u n d f r o m E q .( 1 3 ) :F I G . 5

-2 2 s i n / ?' O SI n t h e r e g i o n I I I ,

Ont a k i n g t h e a v e r a g e o f t h e s e q u a n t i t i e s , w e i g h t e d a c c o r d i n g t o t h ea r e a i n w h i c h e a c h a p p l i e s ,i t i s f o u n d t h a t ' L

CL I.( 1 4 )

COS i m i l a r l y t h e c e n t e r o f p r e s s u r e i s f o u n d t o l i e b e h i n d t h e l e a d i n g e d g eb y t h ed i s t a n c e

- , G{+TT S} ( 1 5 )T h u s i n t h i s c a s e t h e l i f t c o e f f i c i e n t a n d c e n t e r o f p r e s s u r e a r e t h es a m ea s i f t h e w i n e ; w e r e s u b j e c t ot h e u n i f o r m l i f t d i s t r i b u t i o n o fan i n f i n i t e s p a n a i r f o i l , .h e a c t u a l l i f t i s n o t u n i f o r m ;i n t h e r e g i o n It h el i f t i s t h a t o f an i n f i n i t e s p a n a i r f o i l ;i n t h e r e g i o n I I t h el i f t i s l e s s , a n d i n t h e r e g i o n I I It h el i f t i sg r e a t e r b y j u s t e n o u g ht o c o m p e n s a t e f o r t h e d e c r e a s e d l i f t i n I I .s a n e x a m p l e , F i g .6 s h o w st h e p a n w i s e p r e s s u r e d i s t r i b u t i o n f o r t h e c a s e S345"^u, 3o ( > | s ) .


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1-^-i.

1 6 A e r o d y n a m i c C h a r a c t e r i s t i c s o f W i n g s a t S u p e r s o n i c S p e e d s

.2

.0.8

\ jVJ- c 1 %TANU L

FIG.6P R E S S U R EISTRIBUTION ALONG PANIN EOR TRAPEZOIDIT H S 45M30,M = 2

(SEE FiG.5)g e n e r a l S y a m e t r i c a l Q u a d r i l a t e r a l

T h e c a s e o f a q u a d r i l a t e r a l which i s s y m m e t r i c a l a b o u t a d i a g o n a l ,t h a t d i a g o n a l " b e i n g p a r a l l e l t o t h e s t r e a m i s c o n s i d e r e d h e r e .h e e e m i -y e r t e x a n g l e s a t t h e n o s e and t a i l ,s a y a n dSj r e s p e c t i v e l y , a r e n o tn e c e s s a r i l y a c u t e a n g l e s( s e e F i g .8 f o r t h e v a r i o u s p o s s i b i l i t i e s ) ?i ti s a s s u m e d o n l y t h a t e a c h l i e s i n t h e r a n g e JU < 8 . < * f T * > * * -I t i s ,o f c o u r s e , n e c e s s a r y t h a t 6 - * - 5 , < * T T .

T h e forward pointing t r i a n g l e i s a s p e c i a l c a s e ,,aTT/z,I t i s a l s o a s p e c i a l c a s e o f t h e t r a p e z o i d ; e t t i n gStm O o rS-cttUtSi n E q s .( 1 4 ; and ( 1 5 ) l e a d s t o

Z - 2c/3( 1 6 )

H e r e c i s t h e d i s t a n c e f r o m t h e v e r t e x t o t h e t r a i l i n g e d g e - .i g .7s h o w s t h e p r e s s u r e d i s t r i b u t i o n s p a n w i s e f o r t h e c a s e 'S-^S**,/U.30(h~Z) ig 7 is a p p l i c a b l e t o a n y o t h e r - c a s e f o r whichta.nSs V3"tfttM * W a^ orm chanSe o f s c a l e .For t h e m o r e g e n e r a l q u a d r i l a t e r a l o f F i g .8 ,t h e l i f t d i s t r i b u t i o ni s s h o w n by E q .( 1 2 ) i n c o n n e c t i o n with E q ,( 3 ) ;t h e c a l c u l a t i o n o f l i f t and


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*

Application ofBusemann's Conical Field Method to ThinWings 17

V 3T ANp. Stntl-vtrttxanglt45* M o c hngltu .-0* Machnumb*r

F I G .7P R E S S U R EIS TRIBUTIO N ALONGPANINEOR F OR W A R DOI NT I NGR I A N G L E

Airf lowDirection FIG. 8


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t

II

*..

18 Aerodynamic Characteristicsof WingBatSupersonicSpeedcenter of pressureinvolves a tediousintegration.hemostconvenientprocedure for finding the averagepressureover the wing isto integratealong any lins(ABinFig.8 ) paralleltothetrailing edgeon oneside.Thisgives for theaveragevalueof w," b y Bq.(12)

. Tilt A * * ** / *w=sii%T Pdf-where--r t=x >sec$, A+sec/#i1 l=l ftinfiL+KJlT2~ -J-\= A=l s i n ( j L L + 5 , )Jx=tx? sin(6+S ( > TheintegrationleadseventuallytoC Lx5 2,sin2/?-/ysin2>gi17)CLWW'OO^TTin/9,sin 2 /?-s in ^s in 2 /? ,Theanglend ^ 3 ,are,ofcourse,to" b e measured inradians.hisexpressionfor CL/C^OO8symmetricin thetwoangleen d . andthereforeis unchangad " b y interchanging and8 ,

Thecenterof pressureis at

wheretis the weightedaverageoftalong AB,withweights proportionaltothe pressuretis found that

r HtwJgW? .C O S 0 |=rC O S 0,,\rtanTf rdtU,-dt_ osE/g.+cosfe sin2/g sin2/fl-2flcos2A (19)"2c o s 2/ * , -cos2/* 2sin*/S.'2Asin2-2>Ssin2 /f f ,


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Application ofBusemami 1s Conical Field Method to Thin Wings 1 9These formulaecontaina sspecial casesthe forwardpointing tri-angle ( / 3 (=7r/;5)and the backward pointing triangle(=-rr/z)For thesetriangles,theinvariancs ofliftcoefficient(and inthis

particular case,thecenter ofpressurealso)with respecttoreversalof the flow direction may " b aeasilyverified.t hasalreadyheenshownthatliftcoefficientand c e . a t e r of pressurefortheforward pointingtrianglearethesameasifthe pressure were uniform(whichi tisnot;c f .Fig.7) .or the backward pointing trianglethe pressurei uni-form.eversalo fdirection offlowcauses aradical changein pressuredistribution even thoughitdoes notalter theliftcoefficientforthesequadrilateral wings.

Anotherspecialcaseofinterestisthediamond (S,-$ ,i~)*In this case Eqs.(18;and (19)reduceto

sin 2/3-2/^0052/6*T T sin3/tf ( '20)I 2/Si\x\zZ/S

Zsin2^~2/^cos2/C ?C-cosZ (21)

Using Eq.(20)and(21),theliftcoefficientand centerof pressurefor a diamond have" b e e n evaluated and arepresentedi n thefollowingt a " b l e .ti sseenatoncethatthe property ofinvariance under re-versal of flow direction,which was foundto hold for C^ ,does notingeneral applyto thecenter of pressure.fsuchwerethecasethecenter ofpressurepfthediamond would necessarily " b eatZ = c / a ,FromTahleIitisseen thatthecenterofpressureof a diamondact-ually liesforward ofthe midpoint,though never forwardo f7cJiSas long as ^u.

T a " b l eI . 0 100 2 300 400

l(Ctootaacytan/u

.8488.46671.000.8511.46711.015

.8592.46851.064.8720.47091.155

.8897.47431.305(3 50 60 70 80 900

tanS/twy/.9120.47881.556

.9376.48422.000.9646.49052.924

.9885.49665.7591.000.5000CO


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* -4

80 A.ero


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A p p l i c a t i o no f Busemann 1 8 v O o n l c a lF i e l d Method t o T h i n Hi n g s1/T/YT h e t r a n s f o r m a t i o n = m a p s t h e r e l e v a n t p a r t o f t h eu n i t c i r c l e i n t h e = ^Qi < p p l a n e i n t o t h e u p p e r h a l f " o f t h eu n i t c i r c l e i n the^'^'c* f ' l a n e . I t i s c l e a r t h a t w i s g i v e n b y E q .

( 1 1 ) , withr r e p l a c e d b y /pf/r t a n d e v e r y a n g l e ()4zand,< p )multiplied " b y7 7 7 V ' I t r e m a i n s t o s p e c i f y ,,%)K tJK^ n * t e r m s o f t h eg e o m e t r y o f t h e w i n g .s i n t h e p l a n e c a s e ,

C08=7 t C0S =t a n 6 2w h e r eS , a n d$ a r e t h e a n g l e s f r o m t h em a i n s t r e a m d i r e c t i o n t o t h el e a d i n g e d g e .h e " b o u n d a r y v a l u e s K ^ andKg a r e g i v e n b y t h e s a m e e x -p r e s s i o n a s f o r a p l a n e w i n g ,e x c e p t t h a t t h e a n g l e o f a t t a c k i s n o w ,i n g e n e r a l ,d i f f e r e n t f o r e a c h p l a n e o f t h ew i n g .r i t i n gc i . ,an&C X o f o r t h e s e" l o c a l " a n g l e s o f a t t a c k ,o n e f i n d t h a t

-iWtan^ocgWta n }x K~ s in /? ,2- Sjn2I t i s i n t e r e s t i n g t o n o t e t h a t i n t h ec a s e o f s y m m e t r y (,-5,A,c Ca)t h e r e i s a c e r t a i n d i h e d r a l ,n a m e l y Vs Z f o r w h i c h t h e " b o u n d a r yc o n d i t i o n i s w " k o v e r t h ee n t i r e a r c y s o t h a t t h ef l o w i suniformi n t h e w h o l es e c t o r .

f h e r e s t r i c t i o n %+j * f i s s e e n t o b e n o n - e s s e n t i a l a s i n t h ep r e v i o u s c a s e o f a b e n t l e a d i n g e d g e .l s o i t s h o u l d b e pointed o u t t h a tf o r a wing with u p s w e p t d i h e d r a l [tf


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I.

22erodynamic Characteristicof Wings atSupersonicSpeedsReferences

1 .Busemann,A. ,Infinitesimale kegeligeberschallstromune .Schriftender deutschen Akademieder Luftfahrtforschung,Band 7 ,Heft3 ,(1943),pp .105-121.

2 .Stewart,H .J . ,The Liftofa DeltaWing atSupersonicSpeeds .Quarterly of Applied Mathematics,VolumeI V ,No.3 ,( G c t o f c s r 1946).

3 .Chaplygin,S . ,GatJets .Scientific Memoirs,MoscowUniversity,PartV ,(1902),ranslated from the EussianasMCATechnicalMemorandum No.1063.

4 .Schlicting,H. ,Tragflugeltheorle he i Ueberschallgeschwindlgkelt.Luftfahrtforschung 13(1936),pp .320-335.ranslated asNACATechnicalMemorandum No.897.


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I .ERODXHAMIGCHARACTERISTICS01 SOLIDRECTANGULAR AIRFOILSATSUPERSONICSPEEDS*by E .A .Bonncy

SummaryTh elift,drag,moment,andcenter-rof-pressurecharacteristic ofrectangularairfoilscanreadilyheobtainedforanglesofattack " b e l o wthatatwhichtheshock wavedetachesfromtheleadingedgeoftheair-foil.hisreportpresentsexpressionsfortheabovecharacteristicsand establishesoptimum r a l l i e sofaspectratio,angleof attack,andlift-dragratioforanygivenconditionsof allowablestress,airfoilcross-sectionalshapeandMachnumberat wellaemethodofsupportingthewing,i.e.bytheentirebaseoronahub.

AssumptionsItwasnecessarytomakecertainsimplifyingassumptionstokeepthevariousexpretsionsreasonablysimpleand yetaccurate.heya reasfollows:

1 .hechangeinMachanglewith positiveangleof attackoveranairfoiloffinitethicknessandaspectratiowasignored.heanglewillbehigheronthelowersurfaceandlowerontheuppersur-face,therebyoffsettingeachothertoagreatextent.2 .he possibility ofsecondaryti peffectsoriginatingatthepointof maximumthicknessof adoublewedgeairfoilforexample

wasnotconsidered.3 .Considerationofthephenomenaofseparationnecessarilywasomitteddu etothelack of knowledgeofthiseffectinsupersonicflow.hisfactorcancausethecenterofpressureexpressionsob -tainedhereintobesomewhatinerror,particularlyathighanglesofattack. studyofth eeffectsofseparationiscontainedin reference(12).4 . constantskinfrictiondragcoefficientof C ] )f a.00265p ersquarefootof wettedareais usedthroughout.hisisthevalueobtained byvonKarmanforaReynoldsnumberofabout20,000,000.t

isrecognizedthattheactualvaluemaybeconsiderablydifferentthanthisand,inasmuchasskinfrictionisthepredominantfactorinth edragofthinairfoils,cancauseacorrespondingdifferenceinthevaluesforoptimumconditions.*ThispaperisarevisionofCM-247which wasoriginallypublishedasaninternalmemorandumoftheApplied PhysicsLaboratory.lightre-visionwasnecessarytocompleteth ediscussionofthesubject.

23


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24erodynamicCharacteristicsof WingsatSupersonicSpeeds5 .he expression foraspectratiocorrection toliftcurve

olope for an infinitely thin flat platewas used inderiving optimumconditionsforsimplification.he maximum error thatcould resultfromt f t i s assumptioni sabout3 per centfor practical airfoilsizes,

6 .hroughoutthe analysisthe assumptionis madethatS i n ( =G Ca n dC o s a =

This will cause nosizableerror for anglesofattack u p .to15degrees.

Nomenclature

Thefollowing nomenclatureis used throughoutthis paper:h= wingsemi-spanc-chord

y S f t h spectatiocMMachnu m ber44.= VM2 - 1R A*x4 fM- Mach angle =Sin'1I =T a n * *1IM - M -p =localstatic pressureatanypointon theairfoil

P0=freestream staticpressure/^=freestream density

V= velocityin ft/sec.q " 9 ~Y PoM " freestream dynamic pressure2 TL = l i f tA =2bc =wingarea

CL- aliftcoefficientof finitespan airfoil


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J* -..*,&

A e r o d y n a m i c C h a r a c t e r i s t i c s o f S o l i dR e c t a n g u l a r A i r f o i l s a t S u p e r s o n i c S p e e d s 5C ^ l i f t c o e f f i c i e n t o f a n i n f i n i t es p a n a i r f o i l

D = d r a gCD - H- d r a g c o e f f i c i e n t o f f i n i t e s p a n a i r f o i lq AM ~ m o m e n t a " b o u t l e a d i n g e d g e o f wing^ M =M=m o m e n t c o e f f i c i e n t a b o u t l e a d i n g e d g ec q Ac . p .= c e n t e r o f p r e s s u r e m e a s u r e d f r o m l e a d i n g e d g e

CD- * k i n f r i c t i o n d r a g c o e f f i c i e n t ( 5 D =f o r m d r a g c o e f f i c i e n t C j v . = ya v e d r a g c o e f f i c i e n t d u e t o l i f t K \= c o e f f i c i e n t f o r t a p e r i n t h i c k n e s s & - c o e f f i c i e n t ' f o r wing f o r m d r a gK4 * c o e f f i c i e n t f o r wings t r e n g t hw-L0 =wing l o a d i n gK, KxKxK4w ~2T~S tree*

s = s e c t i o n m o d u l u s t i *wingt h i c k n e s s a t r o o t t g - wingt h i c k n e s s a t t i p < -a n g l e o f a t t a c k

/3 s e m i - v e r t e xa n g l e o f l e a d i n g e d g eQ l o c a l a n g l e " b e t w e e n a n y p o i n t o n t h es u r f a c e o f t h ea i r f o i l and t h ef r e e s t r e a m d i r e c t i o nY = r a t i o o f s p e c i f i c h e a t p ' = p - p 0aApsp r e s s u r e c o e f f i c i e n t

S u b s c r i p t U =u p p e r s u r f a c e o fa i r f o i lL =L o w e r s u r f a c e


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I6erodynamicCharacteristicsfWingstupersonicpeeds

I

F - forward half of douDle wedge airfoilE rear halfo=freestream condition*

Pi8CUBsionA.asicequationsoflift,drag,moment?and center of pressurefor infinitely thin flat plates andf or airfoils,f r o t h with infinite

and finiteaspectratios,Various methods(references 4 ,10,and 11;hayeteen developed for

determining the pressureonan airfoil in supersonic flow wherea stockwave has formed and isattached totheleading edge.owevertheydonotalllend themselvestoconvenienthandling forpurposesofdevelop-ing expressionsforliftand drag oftheairfoils.heoneexceptiontothisistheBusemann second order approximation (seeref.4andlo;which is usuallywrittenas follows:

p/p o= + 2*osece^e+^sec2 cosec22 j L L o ( t f + c o s82>i)e2 (D Transforming " b y useofthefundamental relationships,J L t _sm>o =^a,n0sB,4/ko8= jn< *- .a convenientexpressionf or _Psobtained asfollows:

iL,_2_e+*#+9*-&* cec ,ie*2)Tnis metnod i sfound to he very accuratewhen compared tot h * theo-retically correct method of patching curves ofreference( 4 )twodimen-sional method of characteristics)with theerrorapproaching ahout-2per centin lift,drag,and momentatahout60 percentofthedetachmentangle.bovethistheerrorincreasesroughly in aparabolic manner un-tilatthedetachment angleitmayamountto -10to -1 3 percent,depend-ing on thegeometryofthe airfoilhe angleofattack at whichdetach-mentoccurs(called detachmentangle)decreases with decreasing Machnumber and increasing leading edge wedgeangleand mayh edeterminedfor any condition from reference( 4 ) .heerrorin center ofpressurelocation,neglecting separation,when compared totheaccurate methodis negligibleat anyangleof attack uptodetachment.

From thisexpression,i tis possibletodeterminetheaerodynamiccharacteristicsof any shapeof airfoilofinfiniteaspectratio


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AerodynamicCharacter 8ticeofSolidRectangularAirfoilsatSupersonicSpeedsandifthelossinpressureatthetipsforfiniteaspectratiowingsis alsoknown,thentheycanhefoundforanyairfoilofanyaspectratio.orexample,forany wingofinfiniteaspectratioandsymmet-ricalahoutitschordline,theliftcoefficientcanhedeterminedsimply b y th einspectionofequation(2 )as " b e i n g

4


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^<

2 8

. . ^ A i r s '

Aerodynamic CharacteristicsofWings atSupersonicSpeeds

} v

hVH- K -C4*H( 3 )

e,= -cc+/?

Therefore,th etw odimensionalpressuredifferenceoverthefrontandrear halvesoftheairfoil,

( 4 )

Rememhering thattheliftinregionCis equalto halfofthatinregionA (foralikearea)andlikewisetheliftinregion DishalfofthatofB,than

CL 2bc 5 )


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Aerodynamic CharacteristicsofSolidRectangularAir.foils atSuperaonicSpeedsUaing theexpressions,

2 b CTSa n dR=AR\^2lj"A RtheoefficientfiftbecomesCL=2C|CC

29

(6)

( 7 )

This expression appliestoa double wedge airfoilonly.owever,Buse-mann has pointedoutthattheexpression canhemadegeneralfor all air-foils thataresymmetricalahouttheir chord lineanda lineperpendiculartothechord lineatthemidpoint b ysubstituting the parameterofcross-sectionalareadivided b ythe chord squared inplaceofthehalfwedgeangle.herefore for a douhle-wedgeairfoil,

AcsC2 =

tcic2"" t2c= 2 ~ 2Acs.2 =2A'

( 8 )

Valuesof A *forvarioustypesof airfoils arenoted in TaoleI

Thereforeif p 2CgC,=^andG 3=( 9 )

Inasmuch astheliftcoefficientfor infiniteaspectratioisL > tothenhespectatioorrectiono imensionaliftanddragdun

tolift) C , . . / . (11)CL--I-^U-CBA')0 0Thecenter of pressureis ohtained b ysummatingthecenterof pres-

sureofeachregiontimesthetotalpressurein thatregion fortheen-tire airfoil and dividinghythetotal pressureor liftforce.


37/67

LJ^K^'^

~ n if

V i

30erodynamicCharacteristicsofWingsatSupersonicSpeedsReferring againtothesketchoftheairfoil,

,/2b+ 4b-\ 3R_2 /\ lfbttl_3R-2 vH c )= 6 2 b+ 2b-J:/~I2R-61 2 )lj ^, 4c 9R-I4 2RH8(Hi) 2 . J_ . ' 3X 3

fc-P\ .5. 1 1\c 2 9x2=Areaofachegion f CC=(2b2W)i(2R3)2R ,___. _ b c _ ^~2 W 2 4 M - 2 R (B-ili. 3cf,_ b e .(0)-2M - ~4 M -~32 R

Thereforethecenterofpressurewill b e ,(omittingfactorhefromto pandhottom; E

e ,p , ( 2 C | ^ 4 C ga^|(M^(2R-,)4xixj|t(2C,a-4C 2atffel)(2R-3)4xix3]( 2 C la+4C2a/y)J2RH+^xlJ+ ( 2 C , < r - 4 C 2a/5)[2R-3+^x3l

_-3CaA^R-l)2R-I +C3A' W )Thisexpressionlikewiseisgeneralfor allsymmetricalairfoilswhenexpressed asa functionofA 'ratherthan .

fi


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AerodynamicCharacteristicsof SolidRectangularAirfoilsatSupersonicSpaeda 1B.ora givenvalueof allowablestressattherootsection,therewillheanoptimumaspectratiowhich,fora givenlift,willpro-duceth eleastdrag.tlowaspectratios,thetiplosswillhegreatandhenceth earea foragivenliftwillhelargewhileathighaspect

ratiosthethicknessandhenceth edrag foragivenstresswillhelarge;thereforetheoptimumaspectratiowillhesomewherein"betweenthesetw oconditionsand isobtainedasfollows:

D=^e

(15)

() (17)OOI22a2(CL/C|J(l , K^KIYV,, CD^A,*f /cJ


39/67

3 2erodynamicCharacteristicsofWings atSupersonicSpeedsTheparameter Ki*p2 isplotted in Figs.12 and 18toshow the variationin thicknessratio.

Fromtheexpressionsfor drag coefficientr -0O122g2 | % K 3/lR b c D = ^ |-2R)+~sr+cf

& . u d liftcoefficient~ .Q698QC ,. .

therag-liftatiobecomesD3A R b+ C DfM -r.OI745


40/67

AerodynamicCharacteristicsofSolidRectangularAirfoileatSupersonicSpeeds 3whereK3= , - 2 3 )o

Byamorearduousmeans,itmay b eshownthatforawingsupported" b y ah uhorshaft,theexponentof theaspectratiowill b eapproximately3/2forvaluesofshaftdiameterequalto11/2-2timesthemaximumthicknessattheroot.hisisanapproximation( " b e c a u s ethevaluemustagain he2forhuhdiametersequaltoor greaterthanth echord length),hutitwillheshownthatthe method ofsupportdoesnotaffecttheaero-dynamicvaluestoanygreatextent,hutsimplychangesth eoptimumaspectandthicknessratios.Optimizing equation(22)foraspectratio,

d/KM*(CL/c Lj; = 0Usingth eexpressionf oraspectratiocorrectiontoliftcurveslopeofaflatplatefor valuesofR>1,theexpressionforoptimumaspectratia fora wingsupportedovertheentire " b a s e ," b e c o m e s ,

^ 3= R2( 4 R - 3 )2 4 ) andf oswingsupportedb yahuh,

K3.%#, 5 2 5 )^-^CSR-f)Theseexpressionsareplottedin Figs.7and 13toshowthevaria-tionofoptimumaspectratiowithMachnumber andtheaerodynamic-strengthparameterK3.hecorrespondingexpressionsforvaluesofR


41/67

5 34erodynamicCharacteristics ofWingsatSupersonicSpeedfor " b o t h methodsofsupportand themeanralues areshownplottedinFigs.9 ,10 ,15 ,and16.

Inordertodeterminethevalueofwingloading,wto b e used inevaluatingKg , tIsnecessarytoknoweithertheweightandwingareaortheliftcoefficientanddynamicpressure.hedynamicpressurewilldependon t h e . choiceofaltitudeand Machnumber.nasmuchastheopti-mumaspectratio andangleofattackhave " b e e n established,theoptimumliftcoefficientcanreadilyhefoundanddevelopsto b every nearlyin -dependentof . t h emethodofsupportandtheparameterK3.hemeanvalueis plottedinFigs.8 and 1 4 .Results

Expressionsforlift,dragand momentcoefficientsandcenterofpressureforthinflatplatesandairfoilsofinfiniteandfiniteaspectratioaregivenintahleI .heliftcurveslopecorrection andcenterof pressureexpressionsfor aflatplatewillhefound plottedinFigs.3and 4 .heeffectofthicknessonlifthas notbeenaddeddu etoitssmallmagnitudeandthefactthati tdoesnotlenditselftoageneralexpressionforplotting,howevertheeffectoncenterofpressureisquitesizableandtw osamplecaseshave " b e e n plottedinFig.4 .heexpress-ionsoftahleIareexact(withinlimitsofaccuracyofth eBusemannsec-ondorderapproximation)downtoavalueof R -1 .elowthisvalue,alinearextensionto0will heapproximatelycorrect.

Thelossinliftatth etipsduetoflowaroundthetipsasfoundin references( 2 )and( 8 )andisplottedinFig.1 . straight-linevariationisalsoshowntoillustrateth efactthattheliftintheaf-fected region( i n s 5 . d etheMachanglefromthetip;isone-halfofth eamountfortw odimensionalflowforthesamearea.

" A correctionfactor,K\ fortapered(inthickness)airfoilso A rectangularplanformtoheusedintheexpression forKgi sgiveninFig.5andth etermsexplainedin Fig.6 .hesetw ocurvesweretakendirectlyfromreference(6).Figures7to12and13t o f irepresentth eoptimumaerodynamicchar-acteristicsofsymmetricalwingsasa functionofMachnumberandKg v ltheaerodynamic-strengthparameter,forwingssupportedovertheiren-

tirebaseandfor wingssupported byahub.hesimilarityinmagnitudeoftheaerodynamiccoefficientsforwingssupportedovertheirentirebaseascompared towingssupported bya hu bistobenoted.ptimumaspectratioandthicknessratioaredifferentforthetwocases,butcompensateforeachothersothatthecoefficientsarenearly th esame.Conclusions

The Busemannsecondorderapproximationtheoryforth epressureover atw odimensionalwingandtheknowntheoryforpressurelossnearthetipsofafinitespanairfoil provideaconvenientmethodforfinding


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Aerodynamic CharacteristicsofSolidRectangular Airfoils atSupersonicSpeeds 5the aerodynamiccoefficients for flat plates and airfoilsof finitea n d .infiniteaspectratio atanglesof attack " b e l o wthedetachmentangleinsupersonicflow.hiss e c o n d , ordertheoryisvery accuratefor anglesof attack upto aboutsixtyp ercentofthedetachmentangleafter whichi tdepartsfrom theexacttheory atamore-or-less parabolicrate.

By a method alsosuggested " b y Busemann,itis possioletogeneral-izetheexpressionstoincludeanytypeof airfoilsectionwhich issymmetricalaboutitsc h o r d , lineand a lineperpendiculartothechordlineatits midpoint.

Decreasing theaspectratio willdecreasethelift,wavedrag(here-in definedasthedrag duetolift},and momentcoefficientsand causethecenterof pressuretomoveforward.ncreasing thethickness willincreasetheliftand drag coefficientsof airfoilsoffiniteaspectratio veryslightly," b u t willdecreasethemomentcoefficientand causethecenterof pressuretomoveforward quite markedly for airfoilsoffiniteand infinite aspectratio.ncreasingMach number willeitherincreaseor decreasethecoefficientsdependingonthe magnitudeoftheMachnumbers being considered and theaspectratio.

Theexpressionsfor liftcheckverywell with availabletestdata.Thecenterof pressureexpressions,however,makeno allowanceforsep-arationeffects(and otherminor factorsnoted initems1 and 2underAssumptions )and therefore willbesomewhatinerror,theerrorbeing

a function oftheangleofattack.heexpressions will makeit possibletodeterminethe magnitudeoftheseparation effectsoncenter of pres-surelocation however.

Usingtheinformationdeveloped above,theoptimum conditionsofaspectratio andangleof attackand thecorresponding coefficientsforrectangularwingssupported by theentire base and supportedby ahubcan bedetermined.or a given valueoftheaerodynamic-strength para-meter 3 ,theaerodynamiccoefficientsarenota function ofthemethodofsupport,beingalmostidenticalfor eithermethod.heaspectratio,however,will behigher for awingsupported bya hub,thethicknessratio alsobeinghigher in generalto compensateforthiseffectstress-wise.ther effects may benotedbya study ofFig.7through 1 9 .

Theexpressionsforlift,dragand momentcoefficientsand centerof pressurewill befound in TableI .otethatanincrementforskinfrictionhasbeenadded tothedragequations whichwill notappearinthederivation oftheexpression.


43/67

36erodynamicCharacteristicsofWingsatSupersonicSpeedReferences

1 .Schlichting,E . Airfoil Theory atSupersonicSpeed .NACA897 June1939.2 .Taunt,D .R.and Ward,G .N.,S.R.E./Airflow/29.(Secret

report).3 .Rae,R .W.,Ward,G .N.,and Oman,? .L.,S.R.E./Airflow/28 TC1/247.Secretreport;4 .Edmonson,N.,Murnaghan,P .D . ,and Snow,R .M . TheTheoryand Practiceof TwoDimensionalSupersonic PressureCalcu-

lations .umblebee Report No.2 6 ,Dec.1945.5 .Porter,H =H. ,Practical Airfoil Considerations GF-119,

Dec,19,1945(Confidentialreport).6 *Porter,H .H.,and Bonney,E*A. , Aspect Ratio Correction

to LiftforRectangularWings .M-199,J a n . .23,1946(Confidentialreport).7 .Porter,H .H., Aspect Ratio forMinimum Dragof Rectangu-

larWings CF-143,Feb.11,1946(Confidentialreport).8 .Snow,R .M. , FiniteSpan Correction forPlane Airfoils

atSmall AnglesofAttack .CK-214,Feb.14,1946(Re-stricted report).9 .Lemmon,A .W. , Liftto Drag Ratios for Thin Rectangular

Wingsof FiniteAspect Ratio .F-165,March 1 ,1946(Restricted report).10.Lock,G .N .H . , Examplesofthe ApplicationsofBusemann'sFormulatoEvaluatethe Aerodynamic Coefficientson Super-

sonic Airfoils .ept,4 ,1944 TC3/192(Restricted report).11.Busemann,A . HandbuchderExperimentalphysikWien -Harms .

Vol.4 Pgs.3 43 -460 RTPTranslation 2207.12.Ferri,A , Experimental Resultswith Airfoils Tested in the

High SpeedTunnel atGuidonia .NACA TM-946.


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3 8 iI IC Io s

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83

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t2*Mckntutip t,hlckftetilr o o ti06.8 1 .0 y TyplcolDoubleoperedectionwhere K^ an d KxTonbereadro mF ig.5on d6 tj/tiXT ti/tsF'lO.5

Kx'sORARIOUS WINGE C T I O N S(singleaper) FIG.6K)|s FOR V A R I O U SWINGF C T I O N S(doubleaper )

10 1 .5.050 5M-ac hu m b e rFig.7A S P E C TATIOORHINE C T A N 6 U L A R WINGSU P P O R T E DVE RN T I R EAS E

.25

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.0 5

,K3 .000> S -.0005 Cqj-J0053 W.OOI

s

1. 0.5.0.5.0 .5M-n chu m b e r

Fig.8OPT IMUM LIFT C OE FFIC IE NT -ENTIREAS E

39


47/67

iJi

\

1 .05.0.5.0.5M -o chNumber15.0.S.0 M-Machumber 35F ig .9

MAXIMUM L IF T- DRAG RATIO-ENTIRE BASE

Fig .1 0 OP T I M U MNGLE OF ATTACK-ENTIRE BASE

1 .50.5.05M -Machumber0 0 7 A C Dy = .0053f)0 //A \.005 /N /\ J(S.OIOC.004

K,('/r)2.003Ii\ V .OOIJO

k .0020.002 \ .00100005.001 .0002.6401-"o .OOOOB.0IV .5 2 -M a c h .0 2 Number.5 3 0 3

Fig .ILIFT C UR V E SLOPE -ENTIRE BASE

Fig.2T H I CKNE SSATIO-ENTIRE BASE

40


48/67

4 .0

3.5

3.0

O2.5o0C a s 2. o exV)


49/67

-i

nro o oQ 1

\

/ o7 /? o jol toooo MOOoo o 3y

//into

(M=Ooo

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42


50/67

III.THE REVERSEDELTAPLANFORMA section on the reversedelta typeof wing asdiscussed in

CM-258*wast ohave" b e e nincluded in thisreport,howevertheassumpt-ions madeinthat paper were notexactly correct,andthereforeonlyashortdiscussion ofthi8 planform will heincluded here.

Thereversedeltawing ha stheplanform ofan isocelestrianglewiththe" b a s efacing intotheairstream.heangleoftherearsur-faceis greater thanthe Mach angles othatnotiplosseffects willhefeltoverthesurfaceofthe wing.t was assumed intherefer-encereportthatthiBthereforeconstituted a wingwhereon only two-dimensionalflow existed and thecharacteristicscorresponded tothetwo-dimensionalcase.ctuallyhowever,the pressureonasurfacebehinda yawed corner,such asthe aftportion ofthistypeof wingwhen constructed with a double-wedgesection(region Boffigure)isafunction ofthecomponentofMach number whichisnormaltothecorner,%.urthermore,the pressureinsidetheMach conecreatedatthe pointof maximumthickness(region Coffigure)isoftheconical flowtype and must" b ecomputed " b y theconicalfield method.Only in regionA i stheflow ofpuretwo-dimensionalcharacter.

Becauseofthevarioustypesofflowi twillbedifficultt odevelop a generalexpression for form dragand momentcoefficient.Thedesignshould notheoverlooked however,inasmuch asitha s dis-tinctstructuraladvantages and thedrag mayhe very little worsethanthetwo-dimensionalcase.heliftcoefficientwill he very closetothetwo-dimensional value,beingequaltoitforzerothickness.(Seefigure below)

\\\\\

* "Aerodynam icCharacteristicsfReverseArrowWingstupersonicSpeeds,PL/JHUMay946).43


51/67

r

I V .LIFT ANDDRAGCHARACTERISTICSOFDELTAWINGSATSUPERSONICSPEEDSbyE .A .Bonney

AbstractTheliftcharacteristics ofthedeltatypeofwingwiththeleadingedge " b o t hinsideandoutsideoftheMachconefromth erertexofthewinghaveteendetermined byStewart( 1 ) ,Snow( 2 ) ,andothers,andthedragcharacteristics(fordoublewedgesectiononly)have " b e e ndetermined" b yPuckett(3).singthemethodsofreferences(4 )and(5),thisinforma-tionishereinusedtodetermineoptimum conditionsoflift-dragratio,angleofattack,etc.,forcomparison withotherwing planformshape.Theactualpressuredistributionforwingsof finitethicknessionotknownatthiswriting,andthereforemomentandcenterofpressuredataarenotincluded inthisreport.tisshownthat,except forrelative-lylargethicknessratiosatlowMachnumbers,thedeltatypeof winghaslowervaluesofmaximumlift-drag ratiothanthereverse-arrowtypeforwhichtheflowisentirelytwo-dimensional.herelativelylargerootchordlengthrequiredforagivenliftisanotherdisadvantageofthedeltawing.Theprincipaladvantageofthistypeof airfoilisinitsuseasatailsurfacedu etoth eaftcentero fpressurelocation.Intw oreports( 6 ) 7 )concerningtheliftofdelta wings,ithas

beenmentionedthattheresultantforceon th esurfaceofdeltawingsentirelyinsideth eMachconewillhetilted forward ofthenormaltothechordoft^eairfoilhyanamountwhichdependsontheratioofthecomplementofthesweephackangletotheMachangle,approachingalimitof cC/2aheadofthenormalforasweephackangleof90degrees.Thisisdu etothesubsoniceffectwhereinthenormalMach number con-trolling th epressureisles6than 1 ,and willonlyhe possible wheretheleadingedgefthe w i r . gi sroundedtopermitsuctionpeakssimi-lartothesubsoniccase.Thiseffecthasnotbeenrealizedinanyteststodatehowever,andthefollowinganalysisassumesthattheresultantvectorwasal-

waysatrightanglestothechordofthewing.

4 4


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L i f tand DragCharacteristicsofDeltaWingsatSupersonicSpeeds 5

AssumptionsTheassumptionsnotedin reference(4)concerningseparation,skin

friction drag coefficientandlinearizingtrigonometricfunctionswillalso he applicahlein thisreportand inaddition:

1 .heliftcurveslopecorrection fordelta airfoilsisassumed to heindependentofthicknessratio.osizeableerrorwillresultfrom this assumption " b e c a u s e ofthesmallthicknessratios whichare used.

2 .heslopeoftheliftcurveasderived " b y Stewartis atan angleof attacko fzerodegrees.tis assumed herethattheslopei sconstantwith angleofattack.hisassumptionshould introduce nosizeableerror attheanglesformaximumlift-drag ratio which arecon-sideredherein.

NomenclatureThe nomenclatureofreferences( 4 )and( 5 )isused throughoutwiththefollowing additions(Seealso Fig.13):

0=angle between theleading edge and a normaltotheflowdirection

a = Tan 0= parameter ofleading edgegweepback anglecompared- K - with Machangle

(J 0=planform semi-vertex angle=90-0c= " b a s i cchordwisedimension measured from theapextotheposition ofthetip1=rootchordr= chordwisedistancefrom mostrearward pointto pointof

maximum thicknessdivided " b y c

a =( f i - , 1 1c*(l-r)c

1T= thicknessratio= t/cy =distancefrom wing centerlinetocenterof pressuraof

thesemi-pan.


53/67

46erodynamic Characteristicsof Wings atSupersonicSpeedsDiscussion

Thedelta airfoilconsistsofa pianformshape whichi ssymmetricalaboutitscenterline,pointed atthefront,ha sasweptback leading edgeand straighttrailing edgenormaltothecenter line.

For thecasewheretheleading edgeisoutsideofthe Mach cone,theliftremainsidenticalinvalue,hutnotin distribution(2)tothetwo-dimensionalcase.heform drag,however,isslightly higher,hencethemaximum lift-drag ratio will heslightly lower than forthereversearrow typeforwhich two-dimensional lift and dragapply.

For deltashapes wheretheleading edgeisinsidetheMach cone,theliftcurveslopewilldecreasefrom thetwo-dimensional value b yan amount whichi s a functionoftheratiooftheleading edgesweep-hackangletothecomplementoftheMach angle(defined " b ytheparametern )asshowninPig.1( 1 ) .Notethat when n =1 ,thesweephackangleitjustequaltothecomplementofthe Machangle,and n=0correspondsto astraightleading edge.;he form dragwillalso hea functionofnn aswellas" b e i n g a functionofthethicknessratio,location of pointof maximum thickness,and Mach number.hecoefficientof form drag wasdeter ned in reference(3 )and isshownin Fig.2 .tcanheseenthatfor lagevaluesof n ,theform drag isgreatlyreduced fromtheoptimumtwo-dimensional value,where n =0 .ence,forconditionsofhighwingleadings wheretherequired thicknessratio " b e c o m e s largeand form dragoecomes a " b i g proportion ofthetotaldragforoptimum lift-drag condi-tions,thedeltatypeof wing designwilltheoreticallyshowupto ad-vantageover thereversearrow typeinthelowerrangeof Mach numhers.However,for mostpracticalthicknessratios andMach numbers,there-verse arrow willstillgivehigher lift-drag ratios asshownin Figs.3 ,4 ,and 7 . definitedisadvantageofthedelta wingwheren islargeistherelatively largerootchord lengthrequired for a given lift.hecenter of pressuretravelofthistype will prohahiy helargealsodueto thelargechord length and theodd typeof pressuredistribution( 2 )with theleading edgeeither insideor outsidethe Mach cone.

I tisto henoted thatthisanalysisisforthedouble wedgetypeof airfoilcross-sectionalshapeonly,buttheresultsobtainedhereinwill bequalitatively comparaDlefor anytypeofcross-section.

Figure3showsthe actualthicknessratios at which thedelta typeofairfoil with n "1.3 and 2.5 willbesuperiortothereverse arrowtype while Fig.4 showsthesameeffect when considered from theaero-dynamic-strength combinationstandpoint.imitingvaluesof Kg atn s0 ,1.3,1.7,and 2.5 areohown forpurposesofcomparison.ig.5showsthemaximum lift-drag ratiosfortherangeof practical valuesof Kg forvalueofn1.3 and 2.5.

A comparison ofthemaximum lift-drag ratiosofdelta and reversearrowwingtisshownin Fig.7 .heimproved characteristicsofthedeltatype wing for high thlckneasratios atlowMachnumbersisevident fromthecurves*


54/67

L i f t a n d D r a g C h a r a c t e r i s t i c s o fD e l t a f i n g * a t S u p e r s o n i c S p e e d s 7I n r e f e r e n c e s ( 4 )a n d ( 5 ) , t h ef o l l o w i n g e x p r e s s i o n s w e r ed e r i v e df o r l i f t a n d d r a g c o e f f i c i e n t s :

CL S/L_\ (a i n r a d i a n s )VLeo /

E q u a t i o n ( 3 )a p p l i e s f o r w i n g s s u p p o r t e d o v e r t h e i r e n t i r e h a s e .K3 w i l l h e e r a l u a t e d o e l o w .T h e o p t i m u m a n g l e o f A t t a c k i s ,

*,M8 C J 5 . - M -tfopt28.65 UcJC .andheaxisrumift-dragratiob e c o m e s

0A~D m a x yK3M2+ 0Df*Y a l u e eo f CL/CLOO a a * 1 c a n h er e a d d i r e c t l y f r o m f i g s . 1 a n d 2r e s p e c t i v e l y .T h e s k i n f r i c t i o n d r a g c o e f f i c i e n t i s t a k e n a s . 0 0 5 3 , a si n r e f e r e n c e s 4 a n d 5 .T h e a e r o d y n a m i c - s t r e n g t h p a r a m e t e r S 3 i s d e p e n d -e n t o n t h ea p a n v i s e p r e s s u r ed i s t r i b u t i o n a s s h o w n i n t h e f o l l o w i n g d e -r i v a t i o n :

SiL.fi = 2 * 2 . x0x1x12 4 5 * *w h e r e y / b i s t h e r a t i o o f t h ed i s t a n c e o e t w e e n t h e w i n g c e n t e r l i n e a n dt h ec e n t e r o f p r e s s u r e , t o t h e w i n g s e m i - s p a n .

-B y t h em e t h o d s o f r e f e r e n c e ( 2 ) * , i t m a y h e s h o w n t h a t , f o r w i n g sw h o s e l e a d i n g e d g e s a r e o u t s i d eo f t h e M a c h o o n e ,2 cy * ~_3?

* T h i s r e f e r e n c e c o n s i d e r s t h d i s t r i b u t i o n o v e r a f l a t p l a t e o n l y .H o w -e v e r i t i a s s u m e d h e r et h a t t h es p a n w i s e v a r i a t i o n w i l l h e s i m i l a r f o ra w i n g h a v i n g a f i n i t e t h i c k n e s s .


55/67

*rf

^i

43erodynamicCharacteristicsofWingsatSupersonicSpeedswherei isfoundasthelimitingcaseofaquation(28)ofthatreferenceto b e

t=Itan^ 1^Inthoaboveterminology, isdefinedasfollows:

Cos/2= 1* r Tan w .Therefore

= 2 Cos/3( 1+ 2


56/67

Lift and DragCharacteristicsofDeltaVingsatSupersonicSpeeds 9

References1 .tewart,H .J , She LiftofDeltaWingsatSupersonicSpeeds ,

Journalof Applied Mathematics,Oct.1946,p.246.2 .now,f t .M., Application ofBusemann'eConical FieldMethod to

Thin Wingsof PolygonalPlanform ,CM-265Applied Physics Lab-oratory,May2 3 ,1946.Thispaper isreprinted asSection Iofthisreport^

3 .uckett,A .B., Supersonic Wave Drag ofThin Airfoils ,( j j e c e i n -ber2 6 ,1945).aper presenteda ta meeting oftheInstitateof AeronauticalSciences,January1946.

4 .onney,E .A . , Aerodynamic CharacteristicsofSolid,Rectangu-larAirfoilsatSupersonicSpeeds ,CM-247Applied PhysicsLaboratory,May6 ,1945.Thispaperisreprinted asSectionIIofthisreport.)

5 .onney,E .A., AerodynamicCharacteristicsof Reverse-ArrowWingsat SupersonicSpeeds ,CN-258 Applied Physics Laboratory,May 1 4 ,1946.

6 .ones,R .T., PropertiesofLow-aspect-ratio PointedWings atSpeeds Below andAbovetheSpeedo fSound ,NACA ACR No.L5F13,June,1945.

7 .ayes,W .B. ,Browne,S .H . ,andLew, < R .J . LinearizedTheoryof ConicalSupersonicFlowWith Application toTriangularWings NA-46-818.ept.30,1946.

8 .tewart,H .J .and Puckett,A .E., AerodynamicPerformanceofDeltaWings ,(This paper presented atInstituteofAeronau-ticalScience15th annual meeting,NewYork,January1947).


57/67

,

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58/67

14

1 2

1 0 ( V 0 ) max..

I \ 1 0 2. 8\> \

>- ^ t".02Ai ^ fc ^ ,^~.r^^cey

1 .0.5.0.3.0 M -MACHU M B E RFig.3LIFT-DRAGATIOFE L T AAN DECT AN GULARINGS

3.8

1 .5.0.5.0M - M A C HN U M B E R 3.5 Fig.4MA XIMUMLIFT-DRAGATIO-DELTAI N G S1 .0.5.0.5.0.5M-MACHU M B E R 1 .0.5O.5.05 M -MACHU M B E R

Fig.5MA XIMUM L I F T - D R A GATIO Fig.O P T I M U M A N G L EFTTACK-DELTAI N G S

51U>jnt,


59/67

4

1.40

1.30

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..00?

IX )5.0.5.05 MACH N U M B E R Fig.7RATIOOFAXIMUMIFT-DRAGRATIOSFE L T AN D REVERSEARROW W I NG S

2. 0AR

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Fig.8A S P E C TRATIOOFE L T AI N G S

35

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1 .0.5J0.5.05M-MACH NUMBERFig .9P L A N F O R M SEMI-VERTEXN G L E OFELTAINGS

1.050.5X)5M-MACHU M B E R Fig.0OPTIMUM LIFT COEFFICIENT-DELTAINGS

52


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42

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20 0 0 01 .5.0.5.0 M-MACH N U M B E R 3.5 -i ICO S Mtans0Fig.IIOPTI M UM LIFTOE FFI CI E NT Fig.1 2 SPANWISEE N T E R OFRESSURE LOCATION -D E L T AI N G S

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Fig.22CHORDWiSE C E N T E R OF P R E S S U R EOCATION D E L T A AN D A RRO W HEA D P L A N F O R M S 57


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isL

V oSECTIONALCHARACTERISTICSOFSTANDARD AIRFOILS" b y E .A .Bonney

Inselecting theoptimumcross-sectionalshapeofairfoil for usein supersonicflight,oneistempted tochoosethedouble-wedgetype " b e -causeofitslow form drag per unitthickness.hisisa fallacy " b e e . ' s edrag perunitstressisthe proper criterion fordesign.heincrease,drag ofothertypes musthe balancedagainsttheir increasedstrength(sectionmodulus)todeterminetheoptimumshs^e.

Withthisin mind,thefollowing tableand curveshave " b e e n con-structed toshow therelationbetween drag,stress,and thicknessratio forvarioustypesof airfoilsincluding onedesigned " b y Dr.A .E.Puckettinwhich theoptimumshapetogiveminimum drag perunitstresswasthedesign criterion.tisshownthattheoptimumshapeisonlyveryslightly(0.7 per cent)betterthanthe biconvex(doublecirculararc)section.orpracticalpurposes,therefore,(inasmuch asthePuckett wing wouldbedifficultto machine)the biconvextypeofcross-section istheoptimum shapeofairfoilfor given allowablestress attherootofthewing.

Modified I ModifiedShape B i c o n v e x Optimum (Puckett) Double V*44e DoubleWsdge )oubleWedge

ct'13.125 13.87 _ctf24 12 ct*

JL C3 T2 T213.125 13.87 I?24 I12 T212(2-3a)\]j l I.333TE 3.2531 1.8? X*2 a

cDo\/fc

2-l 17.50t f7.38s 24s 1 8s Sa(2-3o)c3

* c ract ionfh o r de n g t ha v i n g wedgeh a p eeachn d) 0 Cfi o r inf in i te span ratio

58


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DRAGOEFFICIENT VS S T R E S S

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