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I!4AC’A, -

RESEARCHMEMORANDUM-

ESTIMATION OF THE

ON INCLINED

FORCES AND MOMENTS ACTING .

BODIES OF REVOLUTION

OF HIGH FINENESS RATIO _-

By II.JulianAllen -L—.!./’”’>.

Ames Aeronautical

COMMITTEENATIONAL ADVISORYFOR AERONAUTICS

WASHINGTONNovember 14,1949

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https://ntrs.nasa.gov/search.jsp?R=19930086014 2018-05-23T15:12:31+00:00Z

TECHLIBRARYKAFB,NM .

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NACARMA9126

NATIONALADVISORYCOMMITTEEFORAERONAUTICS

cll142757

RESEARCH“MEMORANDUM...-. ...............

ESTIMATIONOFTHEFORCESANDMOMENTSACTING

ON INCLINEDBODIESOFREVOLUTION

OFHIGHFININESSRATIO

By’H. JulianAllen

SUMMARY

Thisreportcontainsa discussionoftheaerodynamicforcesandmomentson inclinedbodiesofrevolution.It isknownthata simplepotentialflowsolutionforsuchbodiesdoesnotgiveresultsingoodagreementwithexperiment.An approximatetheorytoallowfortheeffectsofviscosityforsuchbodiesisdevelopedandapplied.It is shownt~ta simpleallowanceforviscouseffectsyieldsresultsinreasonableagree—mentwithexperimentforbodiesofhighfinenessratiosuchaswouldbeusedonmissilesendsupersonicaircraft.Themethodsdevelopedareapplicableatbothsubsonicandsupersonicspeeds.SomediscussionoftheprobableeffectsofReynoldsnumberandmomentson inclinedbodiesofrevolutionis

INTRODUCTION

~owledgeoftheforcesandmomentson

.-—

.;

Machnumberontheforcesandincluded.

bodiesofrevolutionhaslonEbeenof interestinaeronauticalengineering.Theoriginalinterestper:tainedto thecharacteristicsofairshiphulls,ad oneofthefirstlogi-calattemptsat understandingthenatureoftheflowfieldoftheserela—tivelylongclosedbodieswasmadeby MaxMunk(reference1). Munkdemon-stratedthaton suchclosedbodiesat a constantangleofpitchinstraightflightandina nonviscousfluidtmre occurredelementalforcesalongthehullresultingfromchangesinthedownwardmomentumofthefluid.Overtheforwardportionsofthehullshowninfigure1 thedownwardmomentumofthefluidmustincreaseproceedingdownstreambecausetheapparentmassofthecomponentflownormalto theaxisofrevolutionincreasesdueto theenlargingcrosssectionsofthehull. Overthisportionofthehullthereactionisupwardlydirectedforpositiveanglesofattack.Forbodieswithparallelmidsection,representativeoftheolderairships,no crossforceexistsovertheseelementsofthehullsincethereoccursno changeinmomentumofthefluidas theairprogressesalongthesesectionsofconstantarea. At thesternthecontractingcrosssectionsrequirearemovalofmomentumfromtheairstresmand

,.,

hencedownwardlydirected

- ..2 NWA RM A9126

elementalforcesexistalongthehullforpositivehullanglesofattack.It isshowninMunktsworkthatforbodiesofhighfinenessratiothepotentialcrossforceperunitlengthfp atanystationalongthehullisgivenby

fp= (k2-“kl)q ~ sin2U (1)

where S isthecross+ectionalareaofthehull,x isthedistancealongthehullfromthebow,a istheangleofattack,land ka and k= are,respectively,thetransverseandlongitudinQapparentmasscoefficientsforthebody. Thevariationof k=- kl as a functionoffinenessratioisgiveninfigure2,Thiscrossforceat smallangleso~attackcanbeshownfromtheworkofG.N.Ward,reference2, tobe directed’midwaybetweenthenormaltotheaxisofrevolutionofthehullandthenormaltothedirectionof’motionofthehull(i.e.,atanangle a/2). .

It isevidentthatfora closedbody,smchasanairshiphull,at‘apositiveangleofattacktheupwardlydirectedforcesovertheforwardportionmustbe equalto thedownwardlydirectedforcesovertherea sothata pitchingmomentbutno liftordragresults.In figures3 andkare showna comparisonof calculatedsmdexperimentallydeterminedliftandpitchingmomentsasa functionofangleofattackforthehullsoftheAmericanairshipZR~ (U.S.S.Akron)(fromreference3)andtheBritishairshipR-32(fromreference4). It is seenthat,contrsrytothepredictionoftheory,a significantliftforceexistsatangleof’attack.Thepitchingmomentisreasonablywellgivenby thetheory,althoughtheactualmagnitudeofthepitchingmomentisless,ingeneral,thanthepredicted.

UpsonandKlikoff(reference~)havecomparedthecalculatedandobservedcrossforcesforseveralhullshapes,oneofwhichisshowninfigure~. Thediscrepancyat thebowhasbeenshownbyUpsonandKlikofftobe duetothebluffnessof..thebody. A discrepancyalsoexistsbetweencalculatedandexperimentalcharacteristicsovertheremainderofthehull,theactualcrossforceovertherearwardsurfacesalwaysbeingmorepositiveforpositivepitchthanthepotentialtheorypredicts.Ithaslongbeenrecognizedthatthisdiscrepancyresultsfromtheinfluenceofviscosityofthefluid.

In recentyearsthedeterminationof thecrossforcesonbodiesofrevolutionhasagainbecomeoffirstimportancetothedesignersofsuper-sonicaircraftandmissiles.Thesebodiesingeneraldifferfromtheusualairshiphullsintwoimportantaspects:First,thebodiesareofhigherfinenessratio,and,second,thebodieshavea bluntsternor “base.”

%hroughoutthisreportcross–forcecharacteristicsareconsideredintermso~-theangleofpitch.It isclearthatfora bodyofrevolu- .tiontheargumentspresentedapplyequallywelltoanglesofyaworanycombinationsofpitchandyaw.

NACARMA9126 3

b

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.

.

It hasbeenpointedoutby Tsien(reference6) thattheolderairshiptheoriesforthepotentialcrossforcearestillapplicableto such ,bodieseveninsupersonicflowforslenderbodiesat smallanglesofattack.Forsuchunclosedbodiestheairshiptheorywouldpredict,inadditiontoa pitchingmoment,a liftanddrag,butit isagaintobeexpectedthatthefailureto considereffectsofviscositywillleadtoimportantdiscrepanciesbetweenthesecalculatedcharacteristicsandexperi-mentalones.

It isthetheeffectsof

a

A

Ap

cdc

CL

CM

CD

CDa~

ND

D

DU=O

f

fp

f~

.

purposeofthispaperto presentan approximateanalysisofviscosityonbodiesofhighfinenessratio.

SYMBOIS

speedofsound,feetpersecond

characteristicreferenceareaofcientdefinition,squarefeet

plan-formarea,squarefeet

sectiondragcoefficientofin

body

bodyXm

body

body

bodyforforce

a circul.aicylindertermsof itsdiameter

liftcoefficient

momentcoefficientaboutan arbitraryaxisfromthebow

dragcoefficient

dragcoefficient

increaseinbodydragattack

bodydrag,pounds

at zeroangleofattack

coefficientabovethatat

andmomentcoeffi-

perunitlength

at distance

zero singleof

bodydragat zeroamgleofattack,pounds

crossforceperunitlengthalongthebody,poundsperfoot

potentialcrossforceperunitlengthalongthebody,poundsperfoot

viscouscrossforceperunitlengthalongthebody,pound$perfoot

.

k

f pm

k=

k=

2

10

II

M

M.

Mc

q

r

r.

RO

Rc

s

s~

To

v~

Vol

x

Xp

Xm

x

NACARM A9126

viscouscrossforceperunit~eng%halongthebodyofinfinitelength,poundsperfoot

longitudinalapparentmasscoefficient

transverseapparentmasscoefficient

actualbodylength,feet

equivalentlengthofa bluntbasedbody,feet

lift,pounds

momentaboutanarbitraryaxisat distancexm fromthebow,foo%pounds

free-streamMachnumber

crossMachnumber(i.e.,componentMachnumberoftheflownormaltotheaxisofrevolutionofthebody)

free+treemdynamicpressure,poundspersquarefoot

radiusofthebodyatanystationx fromthebow,feet

maximumbodyradius,feet

free+treamReynoldsnumber

crossReynoldsnumber(i.e.,Reynoldsnumberbasedonthecrossvelocity)

cross+ectionalareaofthebodyatanystationx fromthehow,squarefeet

—

---

.

.

“

—

.==

.

cross+ectionalareaofthebaseofthebody,squarefeet

free-streamvelocity,feetpersecond

crossvelocity(i.e.,componentoftheflowvelocitynormalto theaxis of revolution),feetpersecorid —

-. -..—

._

totalvolumeofthebody,cubicfeet .

longitudinaldistancefromthebow,feet—

.distanceofthecentroidofthepkui-formareafromthebow,feet

stationoftheaxisofmoments,feet

referencelengthusedi,~,’,di:jinnj;on.t.-

*

ofmomentcoefficient,feet-..._ -. #.z-.-

-:

.

.

.

NACARM A9126

a angleofattack,degreesorradiansas indicated

7 ratioofthedragcoefficientofa circularcylinderoffinitelengthto thatfora cylinderofinfinitelength

v free-streamkinematicviscosity,feetsquaredpersecond “

P massdensity,slugspercubicfoot

Inreference7,R.reference8, considered

THEORY

T. Jones,followingearliertheeffectsofviscosityon

itelylong~wed cylindersanddemonstratedtha~in

workofL.Prandtlintheflowoverinfin–thecaseofa laminar

flowtheviscouseffectsmaybe consideredby treatizigtheflowacrossthecylinderaxisindependentlyoftheflowalongthecylinder.Forcir–cularcylindersof infinitelengththeviscousforcealongthecylinderis simplythatdueto surfaceshear.Thecomponentflowacrossthecylin–der,however,introduceslargecrossforcesdueto separationof theflow.Joneshasshownthatthecrossforceona yawedcylinderisaccuratelydeterminedhy consideringthecrosscomponentofdragasmaybe seeninfigure6 takenfromreference7. Althoughthedemonstrationofreference7 appliesto laminarflows,itwillbe assumedtobe applicabletoturbu–lentflowsaswell.

Consider,now,a bodyofrevolutionofhighfinenessratio.It isagaintobe expectedthatthecross—forcecharacteristicscouldbe approxi—matelypredictedby treatingeachcircularcrosssectionas an elementofan infinitelylongcircularcylinderofthesamecross-sectionalarea.lWiththisassumptionthelocalcrossforceperunitlengthdueto viscosityfv= wouldbe givenby

pvc2fvm= 2rcd —

c 2(2)

where r isthebodyradiusat anystationx fromthebow, Vc isthecrossvelocity,p isthe

. ofa circularcylinderatmassder&ity,andtheReynoldsnumber

c% isthedragcoefficient

andtheMachnumber

2rVcRc=—

v (3)

vMc=& (4)

where,inaddition,v isthekinematicviscosity,and a isthespeed

,.

6 IiACA~ A9126., —.. ::&.

of soundintheundisturbedstream.

Sincethecrossvelocity *

v~”= V. sina (5)

it followsthattheviscouscrossforcebecomes

fv ‘ 2rc% ~ sti2~ (6)m .—

where q isthedynamicpressure.Thecrosswag Coefficientcdc iSthatofa circularcylinderat thecrossReynoldsnumber

2rVoRc=— sinu = R. sina

v(7)

.

andthecrossMachnumber

Mc = M. sins (8).

where ~ istheMachnumberof.thefreestream. .—

It islmown(seeappendix)thatthedragcoefficientofa c~c~arcylinderoffinitelengthislessthanthatfora cylinderof infinite

._

length.A similarcharacteristicistobe expectedasregardstheviscouscrossforcefora bodyoffinitelengthinobliquefltisothatthevis–

,.=

couscrossforcewillbe lessthanthatgivenbyequation(6). It isalmostcertainthatthelargestportionofthedragreductiondueto -.

finitelengthoccursat theendsofthecylinder.Itwillbe assumed,however,thatthereductionindragforfinenessratioisthesameforeachelementofa bodyoffinitelengthsothatinthatcasetheviscouscrossforcebecomes.

fv = 2qrc~ q sin2a (9)

where q istheratioofcrossdragcoefficientforthebodyoffinitefinenessratiotothatfora bodyof infinitefinenessratio.

Theintegratedviscouscrossforceisthen

where Z isthebody

In determination

length.

oftheliftanddragcharacteristicsatangleofattack,itshouldbe notedthattherealsoexistsa viscousaxialforce

.,

.-

“

NACA~ A9126 7.

.

.

b

.

whichisapproximatelythetotaldragat zeroanglereducedby thereduc—tioninaxialdynamicpressure.Theviscousaxialforceisthen

cDa~ q A COS2a (lo)

where cDa~ isthedragcoefficientat zeroangleand A istheareauponwhichthiscoefficientisbased.

It isnowassumedthatthepotentialsolutionofMunkandtheviscoussolutionmaybe combinedtodeterminethecross<orcedistributionalongthebaiyandtheintegratedforcesandmomentonthebdy. Thepotentialcrossforceperunitlengthactsat anangle a/2 fromthenormaltothefree~treamdirection,whiletheviscousforceactsnormalto theaxisofrevolutionofthebodysothatthedistributionof crossforceintermsofthedynamicpressureisgiven,fromequations(1)and (9),by

f fp a f.=— COS—+Jqq 2q

or

~= (ka–k=)~sin2a cos~+ 2qrc% sin2a (11)

Theliftcoefficientintermsofthereferencesrea A isgivenby

where L isthetotallift.theviscouscrossforce,and

(kZ-k=)sin2a cosCL =

Considerationofthepotentialcrossforce,theviscousaxialforcethenyields

A Jo dx

c%+ Cos 2asina

Thedragcoefficientisobtainedfrom

cD=.&

where D isthetotaldrag,or

A J rcd dx-c

o

(12) !

thecrossandaxialforcesas

NACARMA9126 .-

,

(kz-kl)sin2a sin$ ‘~ ~ + 2q sin3a Z

f Ircdcdx+ ~a_O COS3a

A Od.x.-

A 0 ..—

Thethe

momenbcoefficientabouta givenstation~ isdependentsolelycross-forcedistributionandisgivenby

cM=J& .-

2? sin2a 2 ‘+

frcdc(~-x)h ., ~

Ax ,:0

(14) ‘

whereX isa characteristiclengthfortheevaluationofmomentcoeffi–cient.

Themethodofthepresentreportisclearlytooapproximateto justify.

theimpliedaccuracyoftheprecedihgequations._.It isconsidered.that,. -in.general,thefollowingsimplificationsarewprranted: .—

1. Cosinesofanglesshouldhereplacedby unityandsinesofangles—-

by theangles. .-

2. Thefactork2-kl shouldbe replaced_byunity. ..

3. The viscous =ial force(third)termofequation(12)maybeneglected,whilethecorrespondingterminequation(13)maybe replacedby CDGO*

.

Moreoverthepotentialtermintegralsmaybe evaluatedas’ —

——.

—

and

JzQ (Xm–X) d-x = Vol- Sb(v–xd.-

0 ax

where Sb istheareaofthebodybaseandvolisthebodyvolume.

,Theviscouscross-forcetermintegralsmayalsobe evaluatedas .

—

NACARMA9126

and

.

f

-Lr(xrx)dx= ~Ap(~-xp)

where~ istheplan-;ormareaanafromthebow.

~ isthedistanceto itscentroid

Withtheindicatedchangesequations(11)“to(14)become.respec-. . . . 4

$ively(withu inradians)~ “

.

.

(9=2+~cd.?f)a’-.-.—

(W)

(12a)

(1*)

8 and

[

TolCM=2 -:(’-%)l~+wc%m%)~“k)

To determinetheforceandmomentcharacteristicsitisnecessarytoevaluatethe,coefficientsq and Cdc. In theappendix,availabledataandsomediscussionofthecoefficientsaregiven.

ComparisonofTheoryandExperiment

Testswererecentlycompletedat theAmesAeronauticalLaboratoryofthehigh-fineness-ratiobtiyshowninfigure7. Thetestswerecon-ductedinthesubscmicspeedrangeinthe12-footpressuretunnel(refer–ence9),andinthesupersonicspeedrangeinthe6-by 6-footwindtunnel(asyetunreported).Thetestresultsaffordeda goodopportunitytocomparethetheoryofthisreportwithexperiment.

InnoneofthetestsdidthecrossMachnumber(givenby equation(8)) exceed0.3sotkt, asMy be seen from theappendix,no sensiblecompressibilityeffectexists,whilethecrossReynoldsnumberremainedintherangefOrwhich cdc isconstantandequalto 1.2. Thecoeffi–cientq wasassumedtobe dictatedby theactualfinenessratio

10

2—= &L&z 9*92ro .

I?ACARM A9126 —..=.

forwhich

7 = 0.68

withthesevaluesandthegecznetricpar~etersob~inedfromthe _shapeequation(seefig.7)theequationsforliftcoefficient,drag–coefficientincrement,andmomentcoefficientmaybe determinedfromequations(12a)to (lb) as (withu indegrees)

CL. L = 0.019 u + 0.0025 ~2qA

&Da=o~D.— = 0.00017U* + 0.000043asqA

MCM.— = 0.018a+ 0.00035a2C@z

-.—.

where A and 1 arethemaximumcross-sectionalareaandtheactualbodylength,respectively,andthepitchi~momentisabouta Point

% = 0.704 z

whichisthepointaboutwhichthemomentswereconsideredintheexperl–mentalInvestigationofreference9. Thecalculatedcharacteristicsoflift,dragincrement,andpitchingmomentareccmparedwiththeexperi-mentalvaluesinfigure8. It isseenthatthetheoryofthisreportwellpredictstheliftanddragvariationwithangleofattack.Themagnitudeoftheexperimentalvaluesofthepi.tchi~momentareIwerjhowever,thanthetheoryofthisreportwouldpredict.

DISCUSSIONANDCONCLUDINGREMARKS

It isevident-fromequations(12a)and(lb) thatthevariationofliftandpitchingmomentofa bodyofrevolutionwithangleofattackmust,inthegeneralcase,be nonlinearsincethepotentialcrossforcesdueto thechangeofmomentumofthefluidvariesas thefirstpowerina whiletheviscouscrossforcevariesas thesecondpoweroftheangleofattack.Thisisa particularlyimportantaspectinthedesignofmis-silessincetheguidanceandcontrolproblemw$.11beaffectedifnonlinearcharacteristicsexist.It isofinteresttonotefromequation(l&) thatthepitching+mmentvariationaboutthecenterofgravityforconstantcross-forcedragcoefficientwillbe linearifthecenterofgravityislocatedat thecentroidoftheplan-formarea.‘-

——-.

.

.- .—

—.

“-

.—

.

—.+

NACARM A9126 11

Frcmthetheoryofthisreport,itappearsyossibletomakethepitching+aentvariationaboutthecenterofgravityofthebcuiyzeroinonespecialcase.Thisisthecaseofthebodyforwhich

thatis,a conicalbody

rxx

ifthecenterofgravityisat thecentroidofthecrossarea.

Severalotherimportantexpectationsmaybe impliedfromthetheoryofthisreport:First,sincethecrossReynoldsnumber,whichdeterminesthecross$dragcoefficient,varieswithangleofattack,it ispossibleforthecrossdragcoefficienttoratherabruptlychange(whenRc varieswithanglefrom2 X lCFto .5X 1P) withangleofattackwhichmightleadtorathererraticvariationsof theforcesandmomentswithangleofattack.However,itistobe expectedthatthethree+limensionaleffectspreviouslymentionedwillreducetheindicatedabruptbehavior.It isalsotobenotedthatwind-tunneltestsat lowerscaleswouldnotnecessarilyshowthesepeculiarities.Second,forbodiesmovingat highspeedsthecrossMachnumberwillincreasefromsubsonicto supersmicvaluesas theangleofattackisincreasedandthecrossdragcoefficientmayvaryinanerraticmanner.Thisvariationmayagainleadto correspondingvariationintheforcesandmomentwithangleofattack.Thusitisclearthat .modeltestsofhigh-peedmissilesshouldbe performedoverthewholespeedrangeexpectedfortheconfigurationifthemodeltestsaretobeindicativeofthetruebehavior.However,thefactthatcrossReynoldsnumberas indicatedin theappendixisnotimportantatMachnumbersabove0.5indicatesthatwind-tunneltestsonsmall~calemodelsat highsuper-sonicspeedsshouldaccuratelypredictthebehaviorofthefull-scaleconfigurationsat andabovetheangleofattackforwhichthecrossMachnumberexceeds0.5.

AmesAeronauticalLaboratory,NationalAdtisoryCommitteeforAeronautics,

MoffettField,Calif.

APmNmX

Thesectiondragcoefficientsof circularcylindershavebeen deter–minedfora widerangeofMachandReynoldsnumbersby A nuniberofexperi—reenters.A fairlycomprehensivediscussionofthedragphenmnenaof cir-cularcylindersisgiveninreference10.

12 NACARMA9126

In figure9 is shownthedragcoefficientCdc asa functionofMachnumberforcircularcylindersofdifferentsizes(correspondingtodifferentRejnoldsnuuibers).ThesevalueswereobtainedbyW. F.Lindsey(reference11),,JohnStack(reference12),T.E. Stanton(reference13),andA. Busemann(reference14),aswellas fromsomeunpublishedtestsperformed‘intheAmesl-by 3–1/2-foothigh-peedwindtunnel.=Itwillbe seenthatReynoldsnumberappearsof significanceonlyat lowMachnumbers,sothatforvaluesofcrossMachnumberhigherthan0.5thecurveoffigure9 maybe expectedtoapplyforallReynoldsnumbershigherthanabout102. ThevariationofthedragcoefficientCdc withReynoldsnumberisshowninfigure10alongwithsomeofthehighsub-sonicdragcharacteristicsshowninfigure9 andwiththecfivesofM.F;Relf(reference15) andC.Wieselsberger(reference16). Betweenfigurbs9 and10,thedragcharacteristicsofcircularcylindersas a functi~ofReynoldsandMachnumberarefairlycompletelyestablished.

Thepositionisnot-sofortunatewithregardto ~, theratioofthedragcoefficientofcircularcylinderoffinitelengthtothatofa cir–”cularcylinderofinfinitelength,inthatthisratio,tothea,uthortsknowledge,hasonlybeendeterminedat oneReynoldsnumber(88,000)andat a negligiblylowMachnumber(reference10). Theseresultsaregiven’in figure11andcorrespondto theReynoldsnumberrangeforwhich1.2isthedragcoefficient-ofthecylinderof Infinitelength.

To obtaina roughestimateofthevalueof q at otherReyuoldsandMachnunibers,thefollowingconjectureisgiven.Theend-relievingeffectfora’cylinderoffinitelengthmustbe primarilyconveyedto othersee-- ‘-tionsthroughthelow-velocityregionsintheyake.Evidentlytheratioofthespanwiselengthofthewaketothewakethicbesswouldbe therat~o”whichshoulddetermineq. Thespanwiselengthofthewakewillhe approxi-matelythelengthofthecylinder,whilethewakethicknesswillbe nearlyproportionalto theprcductofthecyllnderdiameterandthedragcoeffi-cient.Itappears,then,thatthevalueof q atReynoldsandMachnumbersforwhich Cdc isnot1.2wouldbe thevalueof q (fromfig.11)foraneffectivecylinderlength-to~iameterratioequaltotheprciluctoftheactuallength-to-diameterratioandtheratioofthedragcoefficient1.2to thesectiondragcoefficientat theReynoldsandMachnumberinthecaseconsidered.

2Thel-by 3-1/2-foottunnelvalues(mainlyusefulindefiningthetrendat highsubsonicvalues)wereobtainedusinga rakeofunshieldedtotal-headtubeandindicateddragcoefficientsabout15percenthigherthanthoseobtainedby others.It isbelievedthatthiseffectwasduetoexcessiveangularityoftheflowat therakewhichwouldindicateincor-rectlyhighvalues. The valueshavebeenproportionatelyreilucaltoagreewithLindseygsvaluesandthisproportionatereductionhasbeenappliedat allotherMachnumbers.

.,

--—--.s.. . ...,.“

.

NACARMA9126 13 “ -

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.

●

✎

It shouldnotbe consideredthat,becausethesectiondragcoeffi-cientshavebeengivenforsupersonicMachnumbers,theequationsdevel–opedinthereportareapplicableat supersoniccrossMachnumbers.ThepotentialsolutionofMunkisderivedontheassumptionofan incompress–ibleflow.Thismomentumsolution,however,shouldbe reasonablyaccurateup to crossMch numbersoftheorderof0.4. Lighthill(reference17)hastreatedtheproblemoftheinviscidcrossforceonbcdiesforthecaseinwhichthecrossWch numberisnotnecessarilysmall.ThesolutionobtainedIsgiveninincreasingpowersoftheangleofattack.Lighthill~ssolution,althoughnumericallycomplex,mayservetoreplacetheinviscidportionsoftheequationsofthisreport.

REFERENCES

1. Munk,~X M.: TheAerodmic Forces~85,~924.

onAirshipHulls.NACARep.

2. Ward,G.N.: SupersonicFlowPastSlenderPointedBodies.QuarterlyJourn.ofMechanicsandAppliedMathematics,vol.2 PartI,Mar.1949,pp.75-97.

3* Freeman,HughB.: ForceMeasurementsona l/4&ScaleModeloftheU. S.Airship“Almon.”NACARep.432,.1932.

4. Jones,R.,Willi~, D.H.,andBell,A.H.: Experiments onModelofa RigidAirshipofNewDesign.R.& M. No.802,June1922.

5. Upson,RalphH.,andKlikoff,w.A.: ApplicationofPracticalHydrbdyuamicstoAirShipDesign.NACARep.405,1931.

6. Tsien,Hsue~hen:SupersonicFlowoveran InclinedBodyofRevolution.Jour.Aero.

7. Jones,Roberttion.NACA

8. Prandtl,L.:

Scl.voL5,no.12,Oct.1938,pp.480M3.- .-

T- EffectsofSweepbackonBoundaryLayerandSepara-Ri;.884,1947.

OnBoundaryLayersin‘l%ree~imnsionalFlow.M.A.P.VolkenrodeReportsand-!l?r~slationsNO.64,my 1,1946.(Avail-ablefromNavyastrans.CGD-68L)

9* Jones,J.Lloyd,andDenwle,FredA.: AerodynamicStudyofa Wing-FuselageCombinationEmployinga WingSweptBack630.–Character–isticsThroughouttheSubsonicSpeedRangewiththeWingCsmberedandTwistedfora UniformLoadat a LiftCoefficientof0.25.NACARMA9D25,1949.

10. Goldstein,S.: ModernDevelopmentsinFluidDynamics.Oxford,TheCkrendonness,v. 2. Sec.195,1938,pp.439-44o.

.,

14 NACARM A9126-..~w.~-.

.

11. Lindsey,W. F.: DragofCylindersofSimpleShapes.NACARep.619,1938.

.

12. Stack,John:CompressibilityEffectsinAeronauticalEngineering.N.4CAACR,1941.

13● Stanton, T.E.: OntheEffectofAirCompressiononImagandPre6___ ~sureDistributioninCylinders,ofInfiriiteAspectRatio.R.& M.No.1210,Nov.1928.

—.——

14. von&rm&?,Th.: TheProblemofResislxmceinCompressibleFluids.-.

Rome,RealeAccademiaD’Italia,1936–XIQ. T-.-—

15. Relf,E.F.: DiscussionoftheResultsofMeasurementsoftheResistanceofWires,withSomeAdditionalTestsontheResistance —ofWiresofSmallDiameter.R.& M.No.102,BritishA.C.A.,1914,

16. Wieselsberger,C.: NewIhtaontheLaws@fFluidResistance.NACATN 84,1922.

17. Lighthill,M. J.: SupersonicFlowPastSlenderYointedBodiesof ._RevolutionatYaw. QuarterlyJour.Mech.@ndAppliedMath.,vol.I,Part1,March1948,pp.76and89.

. .—

.

* . . *

i2

Figure 1.- Schemo#ic diagram of the potential cross-force distribution on a body of

revolution.

G

1

1’ I

1

PCn

—

II

/0 /2 14 /6 /8

Fineness ratio~S’

Figure 2.- inertia foctors for bodies of revolution. zb

,1,

NACARMA9126 c~ —— 17.

.

.

.

.

.

.20

./6

./2 /

.04

Potential theoryo

-4 04 8 /2 /6 20Angleof attack, a, deg

.4‘ I 1 I 1 IPotentia/ theory

.3

.2 /

./A

o

-I“~4 04 8 /2 /6 20

Angleof attack,a, o’eg

Figure3.- Lift andpitching-moment characteristics of a hull modelof the L/S.S.Akron.

18 NACARM A9126 ———-

.

.

—.

Angle of otiuck, a, o’eq.

.5-

Potentialtheory” { ).4 - /

A

/.3 - /‘

.2 “

n“o 4 8 12 /6 20

Angleof attack, a, deg

Figure4.-Liff and pifching”-momenfcharacteristicsofa hull modelof the R-32.

.

. _ ..__

*

.

.4

P

.2

,

.

I

I.

Figure 5- Cuicu/oied

RS-I.

and experimented cross- force on o mode/ of the semi-rigid uicship,

.

20 g~ NACARMA9126—

0 Experiment1 t t

o /0 20 30 40 50 60 70 80 90Angle betweenflow d(rectlonond the

normalto the wire axis, deg=5=

Figure6.-Voriotionof cross

circulur cross-section.‘

force on m obliquewire of

.

.

.-

.

.

%—

b

Shape equation:

rf

–= I-(1–* 23A% ~ )]

I

Dimensions in feet

# .

c1

1P-+– —-l___ . ---_->’.#-

~ = 0./98 ----- -.-”

x t

Figure Z- Body of revolution employed W the example of fhis reporf.

22

/.4

.8

0

-.2

-.4

-.6

42aflE31mH54L NACARM A9126

0.6 EI

0.8 A.

/.2 a

f.40 o1,53 d

0 //

4- /

/

/

-8 -4 0 4 8 /2 /6 20Angleof attack, Q’, deg

,

.

.

—.

.—— —

—

(a)~ift chorffcteristics.

Figure8.- Ca/culOtedundfor the example body

,

.

experimentalaerodynamicchoructeristics .of revolution.

6mEEm==. -.

NACARMA9126

.

.

.

,

23

..-8 -4 0 4 8 /2 /6 20

Angle of uttack, Q, deg=S=

(b) Drag c))urucferisfics.

Figure 8.- Continued.

24

MO=0.4 00.6 Q0.7 00.8 tA0.89 d/.2 &/.3 o/.40 o1.53 a

NACARM A9126

‘Potential theory

-=s=––

.- .- .--8 -4 0 4

Angle of dtock,

~itclvng-mome~f

8 /z /6 ZuCY, deg

choructeristics.

.

.

.

Figure 8.- Concluded.,...,, .%.,.. . .

, , , , .

3.0

❑ voriousdia. - reference i3 (Stanton)o unknowndia.-reference /4 (Busemann)

2.0

1t

1

\

00 .2 .4 .6 .8 10 k2 L4 [6 L8 2.0 2.2

1Mach number, M’

Figure 9. – Drag coefficients of circular cylinders of various sizes as a function of Mach nwnbez ~

24

22

m

1.8

@- 1.6.

2g L4..kg M

g “o

,8

.6

:4

2

010 I@ 10s KY ICY I(Y

i%ynalds mmber, R,

Figure M.- Gimulor cylinder drag coefficient as o function of Reynolds number.

“1 ‘il, ,

I

I ● 9

●✌ ●

☛

● ●

Circular cylinder Iengfh-to-diarneter ratio

Figure II. - Ratio of the drag coefficient of a circular cylinder of finite length b that of a cylinder of

infinite Iengfh, ~, as o function of the length-to-diameter ratio. (i =88,000)