d- NATIONALAD~SORYCOMMITTEE …2,3,4 usedassubscriptson # todenotethef~urbasic potentialsforthecantedarrangement lA,lEi,lc,2A,2Busedassubscriptson # todenotespecialforms of @l and

in*

1.s-4

“.“ d-

Z●.- H NATIONALAD~SORYCOMMITTEE

FORAERONAUTICS

TECHNICAL NOTE 4120

THEORETICALCALCULATIONSOF WAVE DRAGAT

ZEROLIFT FORA PARTICUMR STOREARRANGEMENT

By KennethMargolis, FrankS. Malvestito, Jr.,andPeter J. Maxie, Jr.

LangleyAeronauticalLaboratoryLangleyField, Va.

Washington

January1958

https://ntrs.nasa.gov/search.jsp?R=19930084830 2018-08-27T21:49:57+00:00Z

TECHLIBRARYKAFB,NM

NATIONALADVIHIRYCOIMHTEEFORAERONAUTICSlllll~l~~fg~g~ll[iii—

TECHNICALNOTE4120

THEORETICALCALCULATIONSOFSUPERSONICWAW DRAGAT

ZEROLEFTFCRA PARTICULARSTCEU3ARRANGEMENT

~ KennethMargolis,l%ankS.Malvestuto,Jr.,andPeterJ.Ma@e, Jr.

suMt@iRY

An analysis,basedonthelinearizedthin-airfoiltheoryforsuper-sonic’speeds,ofthewavedragat zerolifthasbeencarriedoutforasimpletwo-bodyarrangementconsistingof twowedgelikesurfaces,eachwitha rhombiclateralcrosssectionandemanatingfroma ccmmonapex.Suchsnarrangementcouldbe usedas twostores,eitherembeddedwithinormountedbelowa wing,orasauxiliarybodieswhereintheupperhalvescouldbe usedas storesandthelowerhalvesforbombormissilepurposes.Theccnnpleterangeof supersonicMachnmbershasbeenconsideredanditwasfoundthatby orientingtheaxesof thebodiesrelativetoeachothera givenvolumemaybe redistributedina mannerwhichenablesthewavedragtobe reducedwithinthelowersupersonicspeedrange(wheretheleatingedgeis substantiallysubsonic).At thehigherMachnumbers,the4 wavedragisalwaysincreased.If,inadditiontoa constantvolume,agivenmaximumthickness-chordratiois imposed,thencantingthetwosur-

4 facesresultsinhigherwavedragat all.Machnumbers.Forpurposesofcomparison,analogousdragcalculationsforthecaseof twoparallelwir@Lkebodiesfigurationhaveable(dragwise)arrangements.

withthesamecross-sectionalshapesas thecantedcon-beenincluded.Considerationisalsogiventothefavor-interferencepressuresactingonthebluntbasesofboth

INTRODUCTION

Themagnitudeof thesupersonicwavedragof ccmpleteairplanecon-figurationsisknowntobe dependentnotonlyonthedirectdrageffectsgeneratedby eachccmponentpartof theconfigurationbutalsoontheindirectorinterferenceeffectsintroducedby eachcomponentonallothercomponents.Forairplanesthatrequireexternalfueltanks,promi-nentnacelles,or otherauxiliarybodiesforstorageorbombandmissilePurPosesjthel~ationOf such b-es relative to eachotherandto otherairplanecomponentshasbeenshowntobe an importantconsiderationfromthestandpointofobtaininglowdrag. Inthisconnection,reference1

2 NACATN4120

*pointsoutthatjudiciouspositioningofthecomponentsofa multibodyarrangementcangiverisetobeneficialinterferenceeffectsthatareofsufficientmagnitudetoallowa volumeincreaseof25percentandat the tsametimeactuallydecreasethewavedragat zerolift.Althoughthepossibleincreasesinotherformsof drag(e.g.,frictiondrag,basedrag,anddragduetolift)mightpartiallycancelorevencompletelynullifysuchreductionin zero-liftwavedrag,theimportanceofpositioning —

auxiliarybodiesappearsclear.

Inasmuchas interesthasbeenevidencedinconicalstoresorientedrelativetoeachother,to consideralsothedragproblemfora similarallowa rigorousanalyticalsolutionaswellaations,thepresentpapertreatsthecaseoftwo

theliftandsideforceofitwasbelievedworthwhilearrangement.Inordertotosimplifythecalcula-simplewedge-typestores,

cantedrelativetoeachother,eachwitha rhombicprofileinthespan-wisedirectionandbothemanatingfroma singleapex. Thissimpletwo-bodyarrangementcouldconceivablybe usedas twostores,eitherembeddedwithina wingormmurtedbelowa wing,orasauxiliarybodieswhereintheupperhalvescouldbe usedas storesandthelowerhalvesforbombormissilepurposes.Thezero-liftwavedragofthesystemisevaluatedhereinbymeansoflinearizedsupersonicflowtheoryfortheentiresuper-sonicspeedrange.Correspondingcalculationsforthecaseoftwoparallelwinglikesurfaceswiththessnecross-sectionalshapesas thecantedarrangementareincluded.Considerationisalsogiventothepressuresactingonthebluntbasesofbothconfigurations.

LISTOFS-ES

X,y,z

ZB

~)n

v

P

AP

M

Cartesiancoordinates

z-coordinatedefining

Cartesiancoordinates

free-streemorflight

a$rfoilsection

of sourcepoints

velocity

densityofair

pressureincrement

-c pressure,

Machnumber

-..+)V2

NACATN 4E!0 3

A

Aw

0

k

S,A

CD

CDr

R

v~.

Machnumberparsmeter,fM2-~

disturbance-velocitypotentisL

slopeof surface(takeninstreamdirection)

leading-edgesweepback

leading-edgesweepbackisembedded(fig.1)

ofbody(figs.1 snd3)

ofwinginwhichcantedsz’rangement

inclinationof(positive)

slopeofridge

maximumchord

inneredgeforthecantedarrangement

qtan e+ cotA)line,~

msximumthiclmess

plan-formareaandaspectratio,respectively(Forthecantedarrangement,thesreaincludesthecutoutinnerportionbutexcludesanywingareaexteriortothebodyledhg edgesothat S = (cma)2cotA and A = k cotA(fig.1))

plsn-formareasadaspectratio,respectively,ofa deltawinginwhichthecantedarrangementisassumedtobeembedded(~ = (cm)2cotAw and Aw = 4 cotAw (fig.1))

Wavedragwave-dragcoefficient,@

Wavedregwave-dragcoefficient,Q%

regionofintegration

VOhUIEOfbody

distsncebetweena~xesfortheparallelarrangement

semispanofonebodyfortheparallelarrugement

functionsusedforvelocity

inappendixA forsummarizingexpressionspotentials

-—

NACATN 41204

Subscripts:

1,2 usedas subscriptson A todenotispecificslopes

1,2,3,4 usedassubscriptson # to denotethef~urbasicpotentialsforthecantedarrangement

lA,lEi,lc,2A,2Busedas subscriptson # todenotespecialformsof @l and @2

baseint baseinterference

Theanalysisisassumptionsof small

ANALYSIS

basedon supersonicthin-airfoiltheoryandon thedisturbancesanda cohstantvelocityofsound

throughoutthefluid.Theseassu@ionsleadtothelinearizedequationforthevelocitypotential~:

(1 - M2)#=+ @H + @zz= O (1)

where M iswithrespectsystem.Thepotentialin

theMachnumberoftheflowandthederivativesaretaken b’tothevariablesx, y, and z oftheCartesiencoordititegeneralexpressionforthelinearizedperturbation-velocityspaceduetoa distributionofsourceandsinksingularitiesv

inthe z = O planeis (seerefs.2 and3)

@(x,y,z)= -~J/

~(E.,v)L%d?St

R (x- & - pz(y- q)a- paza

where x, y, and z arethecoordinatesofthepointatwhichthepotentialis desired)coordinates(analogousto x and y) ofthe

thefieldpointand ~ and ~singularities.

(2)

(that is,aretheThefunc-

tion A(E,q)representsthepartic~ar&stributionofsingularities. .andis,of course,dependentuponthebofidaryconditionsimposed.ForthecaseofwingthicknessMstributiom”thataresmenabletothin-airfoil-theorycalculations,thesource-sinkdistributionfunctionisrelatedtotheparticularthicknessdistributioninvolvedandisgivenas

NACATN4120 5

~(~jv)=Rz’(g’’!z,.o(3)

l’heintegrationindicatedinequation(2)isperformedovertheregionRthatisenclosedby thetracesinthe z = O planeoftheMachforeconeemanatingfromthepoint(x,y,z)andby thewingplan-formboundaries.Inasmuchas thepresentproblemisconcernedwiththedragandhencethesurfaceconditions,consistentlinearization.alsorequiresthepotentialtobe evaluatedin the z = O plane.Thus,theapplicableformof——equation(2)becomes

dq (4)

Theparameter~ equals(M2 - 1 anditsreciprocal(absolutevalue) of theMachlines.

Calculationof thewave-dragcoefficientinvolves

l/fJ istheslope

an integrationof .thetangentialcomponentsof surfacepressurewhicharein turnrelatedto thex-derivativeofthevelocitypotential;thatis,

(5)

Apwhere S istheplan-formareaandthequantity— = - g W(x Y)q. v +“

Theparticularthicknessdistributioninthepresentproblemallowsan additionalsimplification.fiasmuchas thequantityA(x,y)isaconstantvalueoveranyoneregionofa wedge-typesurface,equation(~)maybe expressedsimplyintermsofthepotentialbyperformingan inte-grationwithrespectto x;

CD=- (6)

6 NACATN4120

wherethesymbolb denotesthatthey-integrationisperformedoverthespan.Itshouldbe noted,however,thattheexpressionfor ?#(x,y)willgenerallybe differentforeachwedgeregion and,therefore,theevaluationbetweentheleadingedgesadtrailingedgeasrequiredin theintegrandofequation(6)m~t be takeninfinitesteyp.Thesolutiontotheproblemconsistsprimarilyofcalculatingthepotentialforthevari-oussurfaceregionsby useofequation(4)andthencalculatingthedragcoefficientbymeansofequation(6).

Considernowthewing-typearrangementk detail(fig.1). Fourbasicpotentials~, !%, @3,-d @4 =ereq~red~ordertodeter-minethewavedragfortheentiresupersonicrangeofMachnumbers;theareasofintegrationforeachoftheseareindicatedinfigure2. Inaddition,@l and @2 assumedifferentformsforvariousconditions.Acompletesunmaryofthecasestreated,mathematicalconditionsinvolved,andtheappropriateformsofthepotentialsIsgivenintableI. Beforethepotentialsareevaluatedas indicatedby equation(4),appropriateexpressionsfortheslopes~ z (~,~)mu$tbe obtained.

*BFortheout-

boardpanels,theslopeisfoundtobe

and,fortheinboardpanels,

(tanA)(tane)&/%xaz(Ej7)=h2=-ZB 1- (tanA)(tan(3)

(7)

(8)

Thelimitsofintegrationimposedby regionR arereadilyobtainedfromtheinformationgiveninfigure2. Expressionsforeachof thepotentialsobtainedasa resultofperformingthemathematicaloperationsindicatedinequation(4)arepresentedinappendixA. Inaccordancewithequation(6)thewave-dragcoefficientmaynowbe calculated;theresultantexpressionsareratherlengthyandarepresentedinappendixB.As a conveniencetothereaderandasanaidinperformingothercalcula-tionsinvolvingsimilarintegrals,evaluationsofsomespecificindefiniteintegralsthatappearfrequentlyintheanalysishereinaregiveninappendixC.

b—

—.

.

—

v

8

NACATN4120 7

Wave-dragcalculationsanalogoustothoseforthecantedarrangementwerecarriedoutforthecaseoftwoparallelbodiesse@aratedfromeachotherby anerbitrarydistanceZ (seefig.3). Themathematicaldetailsofobtainingthepotentialsendsoforthareomittedherein;onlytherequiredpotentialexpressionsandfinaldragequationsarepresen~dinappendixD. Itmightbe no@d,however,thattheproceduresusedtoobtaintheseexpressionsareidenticaltothoseutilizedfortheapex-adjoiningarr~ement,althoughmuchlesseffortisinvolved.Forexsmple,onlytwobasicpotentialsarerequiredandtheslopesofboththeinboardandoutboardpanelssrethessme.Furthermore,theworkofreference3 canbe usedto determinethepotentialsandwavedragat allsupersonicMachnumbersforthecasewhereno interferenceeffectsarepresent.

RESULTSANDDISCUSSION

As indicatedintheanalysis,closed-fomnformulasarepresentedintheappendixesforthecompleteMachnumberrange.Theseformulasenablethedragcoefficientforanygivenplsn-formarrangewnttobeobtainedreadilyby straightforwardnumericalcomputation.Therefore,ratherthsm.attemptingtosummarizeallpossiblecasesbymeansofplottingseriesofdesigncharts,thepresentpaperemphasizesonlythosesalientpointsthatsreborneoutby thedetailedcalculations.

Theobviousquestioniswheth~thereisanypossibleadvantagedrag-wiseincantingthestoresortinglikesurfacesasifiicated.Theintro-ductionofa finitevalueof e, thatis,openingthetwopanelsof thetwin-wedgearrangementrelativetooneanother,givesrisetotwoeffectswhichinfluencetheresultantwavedrag.Frompurelygeometricconsidera-tions,itis seenthatfora givenleading-edgesweepbackanda cons-tthickness-chordratioan increaseintheanglee resultsinanincreaseofsurfaceslopesinthestresmdirection.Thetrendtowardhigherslopesisevenmorepronouncedif,instead,thethickness-chordratioisallowedtovaryandtheconditionofconstantvolumeisimposed.Inasmuchas thesupersonicwavedragat zeroliftis dependentonthesquareofthemagni-tudeofsuchslopes,theeffectonthedragisadverse;thatis,thedragisincreasedatallMachnumbers.Cantingthetwopanels,however,alsochangesthepressuredistributionand,sincebothpositiveandnegativepressuresareinducedinthefield,an assessmentofthisinterferenceeffectisrequiredto determinethenetchangeinthewave-tiagcoefficient.

Considerthecaseof e = O whereinthearrsngemmtbecomessimplyadeltawingwitha lateralcrosssectioncomposedoftwodiamonds.Forthislimitingcase,theimportanceandmagnitudeoftheincreased-slopeeffectmaybeassessedtirectlybymeansof comparingthedragwiththatobtainedforthemoreconventionalwingwitha lateralcrosssectioncomposedofasingledismond(equationsgiveninappendixB). Thewave-dragcoefficients

8

forbothcasesareplottedagainsttheaspect-ratio—Machnumberparsm-*

eter A@ infigure4. Thetwoconfigurationshave,fora givenvalueofmaximumthickness-chordratio,thesamevoluieandlongitudinaldlstribu- &tionofcross-sectionalareasmdthesamenmd.mumthicknessateachcrosssection.Themaindifferenceis,of course,inthedistributionof thick-ness. Thecomparisonindicatestworesultswhicharetobe expectedinviewofpreviouslypublishedwave-draganalyses:First,higherdragvaluesareobtainedforthe“twin-wedge”case”throughoutthesupersonicspeedrange(theeffectiveslopesin thestreamdirectionaremuchhigherforthiscaseandhencethedragiscorrespondinglyhigher)and,second,thisincreased-drageffectismorepronouncedat thehigherMachnumbers(wheretheleadingedgeissupersonic).Asan interestingsidelight,itmightbenotedthat,inasmuchas thelongitudinaldistributionsofcross-sectionalareasarethesameforbothconfigurations,applicationof the“transonicarearule”wouldyieldthessmedragcoefficientforbothcases.Thus,thedifferencesintheordinatesofthetwocurvesgiveadirectmeasurementofthedegreeofinaccuracyinvolvedwhenthisruleisappliedtothesupersonicspeedrange.

Asmightbe suggestedfromconsiderationoffigure4,thelowersupersonicrange(wherethelea~ngedgeissubstantiallysubsonic)appearstobe thelogicalrangeinwhichtoanticipatenetdragreductionasa resultofcantingthetwosurfaces.IktailedcalculationsbasedontheequationsgiveninappendixB andcoveringthecompletesupersonicspeedrangesubstantiatethise~ectation- onlyinthelowersupersonicspeedrangeandforrelativelysmallopeninganglesistheinterferenceeffectsufficientlylargetooverbalancetheadversedrageffectdueto ethehigherlocalstresmwiseslopes.At thehigherMachnumberswhere‘theleadingedgeapproachesthesonicconditionoris supersonic,thewavedragforthecantedarrangementisalwayshigher.Figure5 presentssome

?

ofthecalculationswhichillustratetheseresultsforthelowersuper-sonicspeedrangeandincludesforcomparisonpurposestheuncantedarrangement(thatis, ptane = O)previouslyshownin figure4.

Theresultsoffigure~ aredir~ctlyapplicableto casesforwhichthemaximumthickness-chordratioisconstant.Thecurvesmaybereplottedintermsofconstantvolumeorforanyothergeometriccon-siderationby introducingtheappropriatemultiplicativefactorsforeachpoint.Consider,forexsmple,theconditionofconstantvolume;thevol~e VB enclosedbybothsurfacesisgivenby theexpression.

or (9)

Y NACATN 4120 9

+andthemsximumthickness-chordratioby

(lo)

Thewave-dragcoefficientgiveninfigure5 isbasedonthepl=-fonnsxea S (whichincludesthecutoutportion)andisrepresentedas

pc~ = ()%6X2= f(f! cotA,ptane) (u)

Itisreadilyapparentfromequation(9)that,fora fixedoveralllength,thevolumemaybemaintainedconstent~byettheradjustingenytwoorall

threeoftheparametersA, 8,and ~. Thus,fora givenvolme

condition,it isdesirabletobasethewave-dragcoefficientonanareawhichis completelyindependentofthecanted-bodygeometrysothatatrueindicationofthedrsgitselfmaybe obtained.A corresponding&ag c~fficientCD’ basedonthearea ~ msythenbe writtenas

() ()& 2(4j3cot A)PCD$& pc~’ = (12)

%where ~ and & aretheareaandaspectratio,respectively,ofadeltawinginwhichthecantedbodyisassumedtobe embedded.Forgiven

v valuesoftheparametersj3tane end ~ cotA, equation(10)givesthespecificthicknessratiorequiredtomaintainthedesiredvolume.Equa-tion(1.1)in conjunctionwithfigure5 deteminesthedragcoeffi-cient ~CD,andtheneqwtion(12)maybe usedtoplotresultsfortheconstant-volumeconditton.

Figure6 presentsresultsobtainedfora specificvalueofthe3~VB

volumeparemeter—(C=)3 = 0“02”

Aspreviouslydiscussed,theordi-

nate (B%)(PCD’) offigure6 givesa directmeasurementofthedr~itselfsince~ and ~ (areausedfornondtinsionalizingCD’)areindependentofthecanted-bodygeometryand,therefore,anyplottedpointinfigure6 maybe legitimately&omparedwithanyotherplottedpointtodeterminewhetherthedregisdecreasedorincreasedwhengeometryparam-etersarechsngedin

Theconclusionsw givenoveralllength

such-a wayastomaintainconstantvolume.- -

tobe drawnfrcmfigure6 areasfollows:Foraanda givenMachnumber(constant~),it ispossible

10 NACIJTN 4120

*

to redistributea givenvolumeina mannerpracticalforstorepurposesandobtaindragreductionby cantingthebodiesslightly.(Compare,forexample,thevaluesindicatedbythefilled-incircles.)If,in tadditiontoa constantvolumea constantmaximumthickness-chordratioisdesired,thencantingthebodieswillalwaysresultinincreaseddrag(followthedashedlines). Similarly,ifinadditionto constantvolumea’constantsweepbackangleA isdesired,thencantingthebodieswillresultinincreaseddrag(followverticallines).Figure6 is,of

course,directlyapplicabletothevolumeparameter~=()*()2, but

()%aXtheresultsareindicativeofthosefoundforothervaluesofthevolumeparameter.Figure5 in conjunctionwithequations(lO)j(11),and(12)maybe usedas justoutlinedto obtaindetailedcurvesforothervolumeconditionsandthiclmessratios.

Calculationsofthesupersonicwavedragfortheparallel-bodyarrangementconsistingoftwosurfacesofdeltaplanform,eachwithasimplewedgeprofileandwithparallelaxesofsymmetry(seefig.3),arepresentedinfigure7. Theinterestingpointtobe notedisthatwhenthetwosurfacesarein theinterferencefieldsofeachothertheresultinginterferencedragisadditive;thatis,thedragofthearrangementis leastwhenthecombinationofMachnmberandlateraldisplacementofthetwoapexesissuchthatthedisturbancefieldof onesurfacedoesnotinfluencethedragoftheothersurface.Thisresultisexpectedinasmuchas thepressureduetoa singlewedgeisthesameinsignoverthewedgesurfaceandinthefieldbeyondthewedge;there- wfore,theintroductionofa similarwedgeintothefield(suchasin thepresentcase)willresultinadditionaldragofthesamesignasthe apressuredragoftheoriginalbody.

Inasmuchasboththecantedandparallelarrangementstreatedin thepresentpaperhavebluntbases,itisadtisabletopointoutthatthedragcalculationsdiscussedthusfardonottakeintoaccounteitherthebasicbasedragortheinterference-dragcontributionresultingfromthepres-surefieldgeneratedby onepanelactingonthebluntbaseoftheoppositepanel.Actually,thisinterferenceeffectgivesrisetoa negativedrag,orthrust,andcouldconceivablybe ofthesameorderofmagnitudeas theinterferencecontributionpreviouslyconsidered.

Inordertoassesstheimportanceofthebase-drag-interferencecon-tribution,theinterferencepressuresactingonthebasehavebeenderivedforbotharrangementsofbodiesandarepresentedinappendixE. Specifi-cally,theformulasgivetheinterference-pressurecoefficient

()~ baseintactingalongthebaselineinthepl~.rofsymmetry,thatis, x = ~ and

NACATN4120 11

*z = o. Theformulaswereobtainedby utilizingequation(4)tofindtheinterferencevelocitypotential(changingtheregionR toincludeonly

+ thatportionboundedby theWch traceandplan-formboundaryoftheoppo-siteorinterferingbody),differentiatingtofindthepressurecoefficient

(4

+)2 a (X,y)—=- .

q Vx, andthenevaluatingtheresultalongtheline x = ~.

Numericalcalculati&sforthisbaseinterferenceeffectwerecarriedoutfortheparallelarrangementofbodiestoascertainwhetherthefavor-abledragincrementwouldcounterbalancetheincreasein dragpreviouslyfoundandindicatedinfigure7. A roughestimateofthedecrementindragwasobtainedby firstplottingthevariationinpressureactingonthebase[illustratedinfig.8 fora distanceparameter&3.O and

severallkchnumber-sweepbacksx%angements)obtainedby useoftheformulasinappendixE, andthenessentiallyintegratingthispressuredistributionoverthebaseareaaffectedby theinterferenceflow.ThedecrementaldragcoefficientMD as foundby thiscrudeapproachisbelievedto givea reasonableapproximationtothemagnitudeofthebase

interferenceeffect.Valuesofthedecrement

bytheabove-outlinedprocedure.forthecaseswere0.13,0.14,and0.013fortheparameterand0.40,respectively.Subtractionofthese

& correspondingordinatesoffigure7 resultin

presentedinfigure8~cotA= 0.10,0.25,dragdecrementsfromthevaluesthatfallbeneath

the“nointerference,~ +w” curvepresentedtherein.Additionalcal-G

culationscoveringtherangeof sweepback,?@chnumber,anddistance-between-bodiesparameterindicatedthesameresult,namely,thatthebaseinterferenceeffectwasgenerallyof sufficientmagnitudeto over-balancetheadverseinterferenceeffectpreviouslyfound(seefig.7)and,thus,theoverallinterferenceeffectonthedragwasfavorable.Anal-ogouscalculationsforthecanted-bodyarrangementcanbe carriedouttiththeuseoftheappropriateformulasofappendixE andtheprocedurepreci-ouslyindicatedfortheparallelarrangementofbodies.Itisapparentthatthemagnitudeof theopeningangleisa criticalparameterwithregardtothenetinterference-dragcontribution.

CONCLUDINGREMARKS

An analysis,basedonthelinearizedthin-airfoiltheoryforsuper-sonicspeeds,ofthewavedragat zerolifthasbeencarriedoutforasimpletwo-bodyarrangementconsistingoftwowedgelikesurfaces,each

G witha rhombiclateralcrosssectionandemanatingfroma conmonapex.

L2 NACATN4120

Suchanarrangementcouldbe usedas twostores,eitherembeddedwithin*

ormountedbelowa wing,orasauxiliarybodieswhereintheupperhalvescouldbe usedas storesandthelowerhalvesforbombormissilepurposes. GThecompleterangeofsupersonicMachnumbershasbeenconsideredanditwasfoundthatby cantingororientingtheaxesofthebodiesrelativetoeachothera givenvolumemaybe redistributedina mannerwhichenablesthewavedragtobereducedat thelowersupe~sonicspeeds.Forpurposesofcomparison,analogousdragcalculationsforthecaseoftwoparallelwinglikebodieswiththesamecross-sectionalshapesas thecantedarrangementhavebeenincluded.Someconsiderationhasalsobeengiventotheproblemofestimatingthefavorabl~(dragwise)interferencepres-suresactingonthebluntbasesofbothconfigurations.Inthecaseoftheparallelbodiesforwhichcalculationsweremade,thisbaseeffect —

-,.seemedmorethansufficienttocanceltheunfavorableinterferenceon

.L

theforwardpartof theconfiguration. —

LangleyAeronauticalLaboratory, —NationalAdvisoryCommitteeforAeronautics,

LangleyField,Vs.,July18, 1957.

.

NACATN 4120 13

Therequiredbodies(seetable

where

F1 =

F2 =

APPENDIXA

SUMMARYOFVHXK!lTYPOWNTIMS

FORCANTEDARRANGEMENT

velocitypotentialsforthecantedarrangementofI)maybe convenientlyexpressedinthefollowingform:

#LA= Fl+F2+F4

@lB=Fl+F3+F4

~C=F5+F2+F~

@~= F6+F2+F4

@2B=F6+F3+F4

$3= F7

fi4=F7+F8

thefunctionsF1 to F8 aredefinedasfollows:

%$%+(kx+ y)cosh-lx + P ~1&

14 NACATN 4120

—

+

+

[

ml - M (~ - ~)co’-l x - $%‘J-= im=YT+ (kc + y)cos-1x+~2@

@(kc+ y)1

% (y- x cotA)

~’-

APPENDIXB

S-Y OFEQUATK$?SFOR~ WAVE-DRAGCOE!?FICJINT

l?orthecasewhereinwave-&agcoefficientis

.

OF!EIECANTEOARRANGEME!W

alltheedgesaresubsonic,that1s, B cotA < 1,theformulaforthe

Forthecondl.l&msonic,thatis, ~ cot

is

inwhichtheleadingedgeissupersoniclmttheremainingedgesaresub-A>l,

)~(tane:cotA <1, theequationforthewave-dragcoefficient

E*=

I I 1!

I

(B3)

WhentheridgelinebecomesEOIIIC,thatiB, P(W 0 j cotA) . 1,equation(B3)reducesto

(%42+ {’ [ 2-P cotA ~om~ P ootA-l+.-— $ CotA-2

“:tA&l- ~”OtA+’mtA-l 2

l+2~mt A-~~= -1

1

2 ~t2 A

(BootA - 1)2 2

(2-P cotA)k 4BcotA + ~2 cet2A~Bh-l ) -2(2 - ~ @t A)

1

+ 2(2- ~ cotA)COB4(2 - 6 netA) (24)(~oot A-1)2 -3+ b#cot A-~2cot2 A s

,’

FortheconditioninwblchbothhUle13110St~dg~iSsubsauic,thatiS,

wave-drag coefficientis

—

the leading edge and ridge linesare

( )~ti9+COtA>l, Btane <l,

2

supersonicbutthe 1P,0)the formula f’orthe :

I

* * ,, & v PW

!, ,! 1, 1

, ●

When the innermostedge

~cD 1(%3@Inax)’=‘p‘O’A/-+ (p cot ‘ (;cot”+ ‘)’-l)f3cot ‘(~cot’+l)’-h 1

Forthespeedconditionwherealledgesaresupersonic,thatis, ~ tan0 >1, thedragcoefficient Is gLven by

1

i-]~’ cot’ A -1

(B7)

ItIstobenotedthatthedragcoefficientgivenbyequations(~) -LO(~) isbasedonanareawhichincludeiithe“cutout”portion;thatis, the referenceareaisthatofa deltawinghatingthesaw leading-edgesweepback.

l?orthespecialcaseof 13= O, considerablesimplificationisintroducedintotheequations,andtheformulasforwave-dragcoeffictentforthis case EM well as thoseforthesingle-wedgedeltaUSedforccmpariBonpurposesinfigure4 we i3~Zed inthefolkwillgtable:

NACATN ku?o 21

Sam

.

.

APmIx c

SPECIFICINTEGRKLSOF INTERESTOCCURRING

INTHEDRAGANALYSIS

Considertheindefiniteintegrals

11 =J

~o~.lCy+ d ~ay+b

12 =s~o~h-lCy+ d ~

ay+b

where a, b, c,and d areconstantssubjecttotheconditionthatay+b so. (Notethattermcanalwayabemadedenominator.)

For a2> C2:

thisconditionisnotrestrictiveinasmuchas thepositiveby reversingsignsinbothnumsratorand

11 .-COS-lCH+ ad ‘bc co&y(c2 ‘a2) + cd -ah*a ay+b4== Ibc- adl

12 = ay+b cosh-lcy+d—+a ay+b

a2# - b2 co~-lCy+ d%2 ay+b

- bc Cos-lY(C2- a2)+ cd - ab

:P22 Ibc- ad]-c

ad - bcc-Y+2a(a2- c2)

c(a2d2- b2c2)- ~%(ad - bc)2a2(a2-

cosh-ly(c2- a2)+ (cd- ah)*c2)3/2 Ibc- ad!

22

14= a~2 - b2 ~o~h-lCy+ d ad - bc—- rY+2~2 ay+b 2a(a2- c2)

NACATN4120

c(a2d2- b2c2)- 2a2b(ad- bc)~o~-lY(C2- a2)+ (cd- ab)z 3/2

2a2(a2- c )

For a2< C2:

ay+b co~-lcy+d+ ad-be11=— — Cosa

ay’b am ‘1

Ibc-ad

.

y(cz- a2)+ (cd- ah”)tbc- adl

ay+b ~o~h-lcy+ d + ad - bc co~h-ly(c2-a2) + (cd-ah)*12=—a

ay+b a-Ibc-a~

13. azyz- bzCOB r-lcy+d: ad-be -y-

282 ay+b 2a(c2- a2)

c(a2d2- b2c2)- 2a%(ad- bc)COB-12a2(c2- a2)3/2

14=a2Y2-b2Cosh-lcy+d+ ad-be ~-r%2 ay+b 2a(c2- a2)

y(cz- az)+ cd - ab

Ibc - adl

●

.

●

.

c(a2d2- b2c2)- 2a2b(ad- bc)cosh-l‘(C2~b~) + cd - ab*~2(c2 - a2)3/2 ad

wherethequantityY = (C2- a2)y2+2(cd- ab)y+ d2 - b2 andtheasteriskindicatesthatiftheinversehyperbokl.ccosineshouldberequiredofa negativenumber-N thenthefollowingrelationshipmustbe used:

cosh-l(-N)= -cosh-lN●

.

NACATN4120 23

APPENDIXD

SUM&$RYOFFORMULASFORVIXOCI’I!YPO’I!ENTI&SANDWAVE

DRAGFORTHEPARALLELARRANGEMENT

Thebasicpotentialsrequiredtoevaluatethewavedragoftheparallelarrangementofsurfaces(seefig.3)maybe obtainedforthecaseofsubsonicleadingedgesinvolvinginterferenceeffectsby theprocedureoutlinedinthebodyofthepaper.At allsupersonicMachnumbersforwhichtheparticularregionof onesurfaceunderconsidera-tionis outsidethedo~in ofinterferencefromtheoppositesurface,thepotentialexpressionsmaybereadl.lyreducedfromreference3. Forconvenience,alltheapplicablepotentialexpressionsaresummarizedinthefollowingformulas:

Subsonicleadingedge;regionoutsideinterferencefield:

Subsonicleatthgedge;regionwithininterferencefield:

(Y+ x cotA)cosh-lx,+‘P2cot‘.+.. $(x

(y+z - X cotA)cosh-l

(y+ Z + X cotA)cosh-l

cotA + y)

x-(y+2)~2 cotA-~(Y+Z-xcotA)

1x+ (y+ 2)~2cotA~(Y+z+xcotA)

(Dl)

(D2)

24 NACATN4120

Supersonicleadingedge;regionoutsideMachtracesfromapex:

@ . ‘(%@X/%X)(y - x cotA)

p~

(D3)

Supersonicleadingedge;regionwithinMachtracesfromapex:

$=[

v(-%-laxbmax) (y - x cotA)cos-lx - y$2cotA

271/-P(XcotA -y) -

(y+ x cot

Theformulasforareaofbothsurfaces

2A)cos-lx + ‘P cOtA1B(X cotA+ y)

wave-dragcoefficientareandaregivenasfollows:

For O<pcotA~~:L+ls

(D4)

basedOritheplan-form

~ 4(1-B’cot’A) + 31 + ~’”cot2A) -ll+(&-@ot’*

4s cash _1 - ,B2cot2A ~jcotA

s

(m)

●

�

b

—

“

—

—_--

,- .=

●

4Y

%=l-’sw-’pco”s-fs-f’-+cotl

25

(D6)

~o~h-l j32cot’A + 1

}

+ sin-l-~ cotA‘~ cotA

(D7)

(M)

26

APPENDIXE .

EXPRESSIONSFOR~ERENCE

NACATN 4120

PRESS~S ACTING

Theformulasinthisappendixgivecoefficientfoz%i)actingalongthebasethatis, x = ~andz=O.

ONEASE

thetiterf erencepressures(inline in the planeof symmetry,

CantedArrangement

theForthe.caseformulais

wherealltheedgesaresubsonic,thatis~ ~cot ACl,

Whention (El)

theleadingedgebecomessonic,thatis, ~ cotA = 1,equa-reducesto

.

b

“

—

—

\ -, .-.

NACATN 4120

.Whenthe

. equation(E3)

ridgeHne becomessonic,thatis, B(~)reducesto

27

= 1

(24)

Fortbe conditionin whichboththe leadingedgeandridge linesare supersonicbut the innermostedgeis subsonic,that is,

tan e + cot A > 1 and ~ t~ e < 1, the pressureformulaisP( * )

At thehigherMachnumbers(wheretheinneredgebecomessonicorsupersonic) theinterferencefielddoesnotaffectthebaseregionof

. theoppositepanel.

* ParallelArrangement

Fortheparallelarrangementofbodiestheexpressionfortheinterferencepressureis

(26)

28 NACATN 41.20

.

REFERENCES.

1.Friedman,MorrisD.,andCohen,Doris:ArrangementofFusiformEudlesToReducetheWaveDragatSupersonicSpeeds.NACARep.1236,1955.(SupersedesNACARMA51120byFriedmanmdTN 3345byFriedmanandCohen.)

2.Eward,JohnC.: DistributionofWaveDragandLiftintheVicinityofWingTipsat SupersonicSpeeds.NACATN1382,1947.

3. Puckett,AllenE.: SupersonicWaveDragofThinAirfoils.Jour.Aero.Sci ., vol. 13, no. 9, Sept. 1946, pp. 475-484.

u

*

.

.

29

“

.

TABLEI.-SUMMARYOF CASES‘lREATED,MATHEMATICALCONDITIONS,

ANDAPPLICABLEPOTENTIALEXPRESSIONS

ApplicableCase Description Mathematicalconditions potential

expressions

I Alledgessubsonic ~cot A<l @m, @2B

II Leadingedgesupersonic ~tanf3+pc0t A<2butotheredgessubsonic ~cot A>l flu,fi~)$5

III Onlyinneredgesubsonic ~-e<l $lc,fi~)fikptane+pcot A>2

Iv Alledgessupersonic j3tane>l fl~,64

.

.

30 NACATN4120

IvMach

Figure1.-Sketchindicatingwingandpertinent

—

cantedarrangement embeddedina deltaparameters_gsedinanalysis. .. ..

.

.

NACATN klm 31

..*(Y-

0 Y-

Y=-xcotli

,tan8+ cot A)

.-Y- x tan6

x

cotA)

y)= t-x

/m7-Y)=x-t

Figure2.-Areasofintegrationandmathematicaldatarequiredtoobtainthevariousvelocitypotentials.

h!\/\//\/ \

,Mach line

T’1%lax

1Ih

Figure3.-Pertinentgeometryforparallelarrangementofsurfacessndsketchesindicating k?ivarious combinations possible for arbitr=y values of Machnumber,aspect ratio, andfistance ~

* .

,,

3.2

2.8

2.4

2.0

1.6

L2

,8

.4

0

I . 1

Ing edge ! supersamc leading edge,

/, -’/

o I 2 3 4 5

Figure 4.- Variationofvave-dra.gcoefficientulthtwo-aurface6 of delta

x

x

.—T

— ——

E6 7 8 9 10 11

A~

aspect-ratio--.hlachnumber pamme ter forplan form.

34 NACATN4120

.

.’-. . . . :, .-: -.2.2

2.0

1.8

1.6

1.4

/3cD 1.2

(tm/cfmx)2

1.0

.8

.6

.4

.2

0

Figure5,-

0 .1’ .2 .3 .4 .5 .6 7 .8 9 Lo/3cotA

Variationofwave-dragcoefficientwithMachnumber-sweepbackparameterforcantedarrangementofsurfaces.

.

.

, t

,014

.ao

.008

(Bq(fq.006

.004

002

n

\

I

“//

I/

\ //

/ \.

4 .

I~ .(32Gmu

!ulQx=Ql—

r-\ //

// // //

1 *

“o .1 .2 .3 .4 .5 .6 .7 .8 .9 10

~cotA

~gure 6.- Vsxlation of wave-drag coefficient with Mach nomber-sweepback parameter for a given3pv~

volumecondition = 0.02 andseveral.thicknessratios.(cD‘(%X)3

basedonareaofa

deltawingwithaspectratio~.)

-.

1.0

.9

.8

.7

.6

BCD -5

(t=pm)~

.4

3

.2

.1

.4 .5 .6 .7 ..9 .9 1.0-” ~00 .1 2 .3

Fip2.de7.- Vaxiation of uave-dr%

e

Interferewe

~dA

cc-efficient with several parameters for the P=dlelarrangement of surSaces.

h ● ,

-.

.24

.20

.16

,12

.08

,04-

0-1.0 -.8 -~

pcotl’1

I \ I I I I I I

i. -.4 -.2 0YF

mgure8.-Illustrativevariationsoftheinterferenceone panel due to the presence of the opposite panelbodies. l/s= 3.0; z.o; x=&.

.2 .4

presmresactingfortheparallel

,6 .8

along the basemmmgemnt of

I.c

or