naca tn 2443 the similarity law for hyper sonic flow about slender three-dimensional shapes

8/8/2019 NACA TN 2443 the Similarity Law for Hyper Sonic Flow About Slender Three-dimensional Shapes

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FORTECHNICALOTE2443

THESIMILARITYLAWFORHYPERSONICLOWABOUTSLENDERTHREE -DIMENSIONALHAPESBy Fran kM. Hamaker,StanfordE. Neice,a ndA. J . E gger s, J r .

Ames Aeronau ticalLabora toryMoffettField, Calif.


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TECHLIBRARY/WE,M

1 /lBllilllulllIIlllIl0Llb5b70NATIONALIYVISCIRYOMMITTEEORAERONAUTICS

mamcm mm 2Q3THEIMILARITYAWFOREYl?ERSONICLOWABOUT \

SLENDERTHREE+DIMENSIONALHAFESByFrankM.Hamaker,tanford. Neice,andA. J.Eggers,Jr.I

Thesimilarityawfor steady,inviscid hypersoniclow aboutslenderthr ee-dimensiona lhapesis derivedin termsof cust oma ryaer odynam icar am eter s.Tohavesimilari tyof flow, the law statestha t the la te ra l dimensionsf the sha~esin ques t ionandthe ir angleswithrespectto theflightdirectionustbe inverselyroportionalotheirflight,achnqbers. A tirectconsequencefthislawisthattheratioofthelocalstaticpressureo thefree-streamtaticpressuresthesameat correspondingointsin similarlowfields.--Thelawisappliedtothedeterminationf simpleexpressionsorcorrelatingheforcesandmomentsactingon relatedshapesoperatingat hypersonicpeeds.Theshapesconsideredrewings,bodies,andwing-bodyombinations.nthespecialcaseof inclinedodiesofrevolution,heseexpressionsreextendedoincludesomesignificanteffectsftheviscouscrossforce.Resultsofa limitedexperimentalnvestigationfthepressures


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..... ...-. ..z _ .. . . .

2 NACATN 2~3

largecomparedo 1. He alsoreasonedhatsimilitudeouldbe obtained ..inhyj?ersoniclowsaboutslenderhree-dimensionalodiesofarbitraryshape;however,heformofthesimilarityawintermsof cmtomaryaerodynamicarametersasnotdetermined.Ehret,Rossow,andStevens(reference) investigatedhehyper-sonicsimilarityawfornonliftingodiesof revolutiony comparingpressureistributionsalculatedy meansofthemethodof character-istics Theyfoundthelawtobe applicablevera widerangeofMach

nunibersndthic.lmessatios.Theirinvestigationidnot,however,includeheeffectsofvorticityrisingfromthec~ture ofthenoseshockwave. Rossaw(reference) continuedhisinvestigationndfoundthatthelawwasequallyalidwhentheeffectsofvorticityereincludednthecalculations.hesefindingsorroborated,npart,theobservationsfHayesandinficatedhatthelawmaybe usedwithconfidenceo investigateheaerodynamicharacteristicsfnonliftingbodiesofrevolutionthypersonicpeeds.Withthesuccessfulp_@icationfthehypersonicimilarityawtononliftingodiesof revolution,ta~eareddesirableo determinetheformofthelaw,intermsof customaryerodynamicarameters,orslenderhree-dimensionalodiesofarbitraryhape.An tnvesti~tionof themoregenersllawpqmr istopresentthe

a speedof sound

wasthereforendertaken.hepurposeofthisresultsofthisstudy.SYMBOLS


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dragparameterrolling-momentrolling-momentyter \

/iftcoefficient. lift \

liftparameterpitching-momentpitching-momentarameter, \

3

yawing-momentoefficient~~zt)yawing-momentarmeterdimensionlesserturbationotentialunctionviscousforceormomentfunctiondimensionlessodyshapefunctionbo~ slr+peunctionunitvectorsalongcoordinatexes x,y,zjrespectively


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mm m 2JA3Rcstu,v,w

vx,y>za$YbE,vsePP

crossReynoldsumberbasedoncomponentfthefree-stream

crossforce~erunitlengthmaximumbod.yiameterndthevelocityormalto thebodyaxis

charact=istichiclmessrdepthofbodycomponentsfvelocity,, inthedirectionfthe x,y,z axes,respectivelyresultantelocity ,(%rtesimcoordinatesangleofattackangleof sideslipratioof specificeatsangleofrolldimensionlessoordinatesorrespondingo x,y,z,respectivelyorificelocationnthetestconesstreamdensityperturbationeloci@potential

subscripts


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IwIC!AN2t i3DEVELOPMENTFTEESIMlLM31TYFORINVISCIDHREE-DIMENSIONAL

Thefollowingssumptionsremadein this

LAwFLOWanalysis:(1)theMachnumberof theuniformfreestreamislargecomparedo1 (i.e.,theflowishypersonic),2)thedisturbanceelocitiesresmalJcomparedto thefree-streamelocity,nd(3)heflowisofthesteadypotentialtype. It is clearfromthefirsttwoassumptionshattheanalysissstrictly~licableonlyto slendershapesinhypersoniclow.Aswaspointedoutintheintroduction,owever,helastassumptionhouldnotrestrictherange,ofapplicabilityftheresultstopotentialflows.Thepurposeofmakingthisassumptionsto shplifytheanalysis.

A slenderbodyis zcmientedn x,y,z spaoeas shownin sketoh(a) Ywiththefree43tieamvelwity V. directed / .@alongthe x sxiso &.Thegeneraldl.ffer- Xentialequation motionfa steadyflowaboutthebodycanbe writtenintheouowingfcum: (a)(a2-u?)ux(a2-#)vy+ (a2-w2)wzuv(uy+ Vx)-

VW(VZ+ Wy)- W-U(WX + Uz)= oI

(1)


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..-. . - ----

6 mm m 2U3

.

Introducingheperturbationotentialxpressionsfequation3),equation4)thenbecomesY-l V02. a=+ 2 + + (V022v& +~x2. 2 + Py=+ Q~2) (5)

If equations2),(3),and(~)thesteady-state,hree-dimensionalobtatieds follows:arenowintroducedntoequation1),potentialquationfmotionis

[ ao2- ~ (2v@x2 + 9X2+ %2 + ~z2) 1V 0 2mo l ? x%2 %x+[

2-a.

[ ao2-

2(v@yForhypersoniclowaboutslendercomyaredo Vo,anda simpleanalysisurthertidicateshat 9X issmallcomparedo ~ and $Z.AccorMngly,theexactpotentialequationssimplifiedy neglecting, general,lJtermsofhigherorderthan %2 and PZ2,andby neglecting,nparticular,lltermsexcept-V02 inthecoefficientf ~. Equation(6)be reducedto theform may-therefore

1% -


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ma m 2kk3 7.,

.

Thisrelations employeds theequationfmotioninthefollowinganalysts Theboundaryonditionsemaintobe determined.

Theshapeofa slenderhree-dimensionalodyis definedin itsreferenceosition=ntheflowfieldby thefunctionalelationG(x,y,z) O (8)

Theunitnormalat a pointonthesurfaceisgivenby thevectorii =37+m~+nE (9)

andtherequirementhatthebodybe slenders satisfiedy therestriction-1


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8 lWICAN 2443hposingtherequirementpecifiedy equation10),equation12)isfurtherreducedto theform

fit= (z+@. nu)7+ (m+n5)~+ (n- M5)E (13)Ifthevector~ inequationn) isreplacedy fitas definedinthisexpression,henthedesiredgeneralizedoundaryonditionnthesurfacefthebody,isgivenby theequation

Vo(%c+ Wy+tiz) +~y(Gy+ Wz) + ?z(Gz~) = o (14)Inthisequationhederivativesf G are,of course,evaluatednthesurfacef thebodyinthereferenceosition,hilethederivativesof q are,evaluatedt correspondingointsonthebodyinitsrotatedposition.Theremainingoundaryonditions,of course,9xQy=gz=O at x=-m (15)

Inordertoobtainthes-ity lawforflowaboutrelatedbodies,it is conveniento expressheequationsfmotionandboundaryconditionsna nondimensionalorm.A dimensionlessoordinateystemisthereforentroducediththeaffinetransformation(16)

.anda nondimensionperturbationotentialunctionsdefinedy the. .relation z)f(g,%~)= q x t 2 (17)a&c ~


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EAcATN 2443 9

Inan analogousanner,equations14)and(15)fortheboundaryconditionsssumethenondimensionalorms

( Kt )E !-g~K% 0 (20)onthesurface,nd

f~=fq=f{= Oat ~=-~ (a)wherethehypersonicimilarityarametersora constantalueof 7aregivenas follows:

-


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10 mcAm 2U3thattheirlateraldimensionsndangleswithrespectto theflowdirectione inverselyqortionalto theMachm.miberftheflow.

Thisstatementfthelawisess~tiallya generalizationfthatoriginallyresentedy Tsien.ThenewsimilaritysxametersKb,~,K~,and IQ defineadditionalestrictionsn theshapesandattitudesofrelatedodi.es;sowever,hesimilarityarameterKt (andtherestrictionmposedy it)isthesameastheoneinreference>obtainedromtheconsiderationsftwo-dimensionalndaxiallysymmetriclows.In regardto thenewshilarityparameters,ttentioniscalledto ~ which,itisnoticed,oesnotcontain~. Therollangleisthesame,then,forrelatedodiesin similarypersoniclows.T& resultcould-clearthatiftherequiredobe inisalsovalidfor

APPLICATIONS

havebeendeducedintuitively,ndit seemsequallyrotationso an~es ofattack,sideslip,ndrollarethesamesequencearbitrarilyargeOFTHESIMILARITY

(see footnote 3), [email protected]

In theyrecedingectionhe~ersonic similarityawwasdevelupedna generalorm. Thelawis enployednthissectionocorrelatehephysicalropertiesf shnilarflowfieldsandtheaero--c cwaCteristicsOfsomerelatedshapesofpracticalnterest.Someeffectsofviscosityreconsideredntheinvesti~tionftheaerodynamicharacteristicsorinclinedodiesofrevolution.he

assumptionf inviscidlowis,however,etainedlsewherenthisstudy


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NACATN 2ti3Siqplif~ngthisequationo includenlytermsof theproperordertransfo-g theresultingxpressiono nondimensionalormyieldsthefollowingelation: -Y

U.and

P= { 1- (y-l)t2f~Po ~ [Kt2( ~~fn2+Kt2fg1} x

Thederivativesf f are,however,unctionsnlyofthesimilarityparametersndthedimensionlessoordinates;herefore,hisexpressionmaybe writtenas(27)

It is clearfromthisrelationhatforsimilarlows,theratioofthelocalto thefree-streamtaticpressureisthesameat correspondingpoints(~,q,~)ntheflowfields.A directconsequencefthisruleisthatthecenterofpressureisat thesame(~,q,~)ocationnrelatedodiesin [email protected] appliedto relateotherphysicalropertiesfsimilarlowfields,suchas tempe=tures,ensitiesndMachnumbers.

CorrelationfoftheAerodynamicharact=isticsSomeRelatedShapes. BodiesofRevolution.-orbodieqofrevolution,quation27)reducesto theforme

. .. .. .


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12. muxm 2U3%& =t =fi(Kt,&)M&@=&I=fi(Kt,L)

1(28)

~~ = @m = &(Kt,@w-Where C!L,D,and im aredesignatedift,drag,ndpitching-momentparameters,espectively.It isapparentromtheserelationshatthecorrespondingorceandmomentprszaetersaveidenticalaluesforrelatedodiesofrevolutionrovidedhecorrespondingimilarityparametersaveidenticalalues.Itwillnowbe shownthatthisconclusionanbe generalizedo includethesignificantffectsoftheviscouscrossforceson relatedinclinedodies.

Theviscouscrossforcearisesfromtheflow(umallypartbllyseparate@oftheboundary-layerransverseo theb@y axis.A methodof estimatinghisforcealongwiththelift,drag,andTitching-momentcoefficientsssociatedithithasbeensuggestedy Alleninrefer-ence5,andispresentedntheappendixfthepresentpaper.Theresultingxpressionsorthesecoefficientsseeequationc)intheappendix)retransformedotherelationsreobtained:

b% =%%v =%% =

Forslenderodiesofrevolution

nondimensionalormandthefollowing$dcF1(Kt>G)$dcF2(Kt>G)~dcFg(Kt>Ku) 129)ofthetypeunderconsideration,d.

isprimarilyfunctionftheMachnuriberndReynoldsumberof -theflowcomponentormaltothebodyaxis. Consequently,heseexpressionsanbe reducedto theform


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mm m 2W+3 13A Wnitedexperhnentallheckof thesimilari@lawforbodiesof

revolutionasbeenmadeintheAmes10-by 14-inchsupersonicindtunnel.Twoconeshavingthibess ratiosof 0.333and0.204weretestedatMachnunibersf2.1~nd4.4-6,espectively;husthevalueof Kt WaS0.91.quipmentormeasmingforcesandmomentswasnotavailablet thethe ofthesetests;therefore,ressuresnlyweremeasurednthecones.Thesemeasurementseremadeat thelocationsshowninfi~e 1 foranglesofattackqp to 5. Overlappingaluesof & up to 14werethusobtained.Therangesof cross-floweynoldsnunibersoveredin thetestsareshowninfigure2, anditis evidentthatidenticalaluesof Rc couldnotbe obtainedorthetwoconesatthesamevaluesof ~.

Experimentallyeterminedressureatiosareshowninfigure3asa functionf ~. Agreementiththepredictionfthesimilarityawisgenerallybserved,nthatthevaluesof p/p. forcorrespondingpointson thetwobodieslieessentiallylongthesamecurve.Theexceptiono thisagreementson theleesidesofthecones(e=1800)whereit isnotedthatsignificantlyifferenturvesaredefined.Thisdifferencesbelievedobe theresultof dissimilarlowsepa-rationfromthetwocones,causedinurnby themarkeddifferencesnthecross-floweynoldsumberpreviouslyentioned.Separationphenomenahouldbe essentiallyimilart identicalross-floweynoldsnumbers,nwhichcasethecorrespondingaluesof p/p. shouldagree.

Wings,Bodies,andWing-Bodyotiinations.-hegeneralormofthesimilarityawmustbe employednthisphaseof [email protected],then,o obtainexpressionsortheforceandmamentparametersofwings,bodies,andwing-bodyonibinations,t isnecessaryo inte-grateequation27)overrelated,utotherwiserbitraryhayes.Theresultingxpressionsre


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completelyrbitraryhape.9Correlationanbe achieved,owever,ftworestrictionsreplacedon theshapesof theseconfigurations.or

.thecaseofpitchingoment,therestrictionsthatthe 2 directioncosinesof theouternormalsto thesurfaceust,in general,e small bcomparedo thecorresponding directionosines.Thus,forexample,verticalins(alone)av5ngsurfaceslopesinthechordwiseirectiongenerallyf thesameorderofmagnitudes theslopesinthedepth-wiseMrectionareeliminatedromconsideration.ucha shapeisshownin sketch(b). In thecaseofyaw$ngmoment,therestriction

z z

(b)

Y

xzA-Y

(c)is that Z must,ingeneral:e smallconpredto m. Thus,forexample,ings,as shownh sketoh(c),havingchordtiselopesgena%llyofthesameorderofmagnitwles thespanwiselopes,re

.


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mm m 2kh3 ljconsidered,heshd.larityarametersKp and ~ vanishandonlythreeoftheaerodynamicoefficientsemain.Thecorrespondingorceandmomentparametersrereducedto theformsll

132),, Theserelationslsoqply, of course,towingsections.In thiscase,b andthereforeKb areinfinitenditis seenfromequa-tions(19)and (20),thatthetermsinvolvingKb vanishyieldinghetwo-dimensionalquationsorhypersoniclow. Thesimilarityara-meter Kb isthuseliminatedromequation32).equivalentothatpresentednreference.~2Ofpracticalmportancestheconclusionodimensionlessquatiorifmotionas itappliestoIt isnoticedintheequationhattheparameter

Thisresultisbe drawnfromthethinwings.Kb alwaysappears()t2te-theform ~ If b isofthesameorderofmagnitudes= ~2then,consistentiththeotherapproximationsadein developinghis

(7Ktequation,hetermsinvolving~ aretobe neglected.erformingthis~eration,however,ieldstheequationfmotionfortwo-dimensionallow. Thusit is indicatedhat,iftheaspectratioisoftheorderofmagnitudef oneor greater,ypersoniclowaboutwings

maybe treatedappro@natelysa two-dimensional-flowroblem.Thelatterproblemis,of course,elativelyimpleto salve. --


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16 NACATN2ti3CONCLUDINGEMARKS

Thesimilarityawforsteady,nviscidypersoniclowaboutslenderhree-dimensionalhapeshasbeenderivedintermsof customaryaerodynamicrameters.To havedmilarityofflow,thelawstatesthatthelateraldimensionsftheshapesin questionndtheirangleswithrespectotheflightdirectionustbe inverselyroportionalotheirflightMachnuuibers.directconsequencefthislawisthattheratioofthelocalstaticpressureo thefree-streamtaticpressuresthesameat correspondingointsin stilarflowfields.Withtheaidofthislaw,simpleexpressionsereobtainedorcorre-latingtheforcesandmomentsactingonrelatedshapesinhypersonicflows Theshapestreatederew%ngs,bodies,andwing-bodyombina-tions Inthecaseof inclinedodiesofrevolution,heseexpressionsweregeneralizedo includehesignificantffectsf theviscouscrossforce.Thelaw,as itappliestobodiesofrevolution,assub-jectedtoa 15mitedexperimentalheckbycomparingressureseasuredontwoinclinedonesinrelatedflows.Theoryandexperimentereingoodagreementxceptontheleesidesoftheconeswherethedissimilarcross-floweynoldsumbersouldbe expectedo@eld dissimilarseparatedlows.

Therangeofapplicabilityfthelawforpracticalbree-dimensionalhqesappearstomeritinvestigation.fthisrangeis .relativelyswideas thecorrespondingangefornon.inclinedodiesofrevolution,helawshouldproveofvaluein correlatingxperimentaldata,andin simplifyingheoreticalalculationsftheaerodynamiccharacteristicsorfamiliesftheseshapes. .


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17APPENDn

FORCESANDMOMENTSDUETO VISCOUSCROSSFLOWSONBODIESOFREVOLUTION

In reference,Prandtldemonstratedhatlaminarviscousflowsoverinfinitelyonginclinedylindersaybe treatedy considering,independently,hecomponentsftheflownormalandparallelotheaxisofthecylinder.Jones,inreference, appliedthisconceptothestudyofboundary-layerlowsoveryawedcylinders.TheworkofPrandtlandJonessuggests,s indicatedy Alleninreference,thatthecrossforceon slenderinclinedodiesof revolutionaybe esti-matedinthefoIlowinganner:Eachcrosssectionofthebodyistreatedasan elementofan infiniteylinderfthesameradius.Thecrossforceperingequation:unitLengthon sucha cylindersgivenby thefollow-

s~= r cdepoVo2sin2a (Al)Th earethenincrementalift,dra~andmomentproducedy thiscrossforcegivenbytherelations

(A2).

Iifi =r C+ ~ovo2sin2aos ~kg = r Cdc ~ o v6ti*~

moment= r x cdc~oVo2sin2a 1Retainingeadingtermsin a andintegratingverthebody,

.--- .- ---


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18 NACATN2~3

lmFERENms .1. Tsien,Hsue-shen:SimilarityawsofHypersoniclows,JournalOfMathmtics~d PhYSiCS,O1.25>o.3, oct . 1946,pp. 252-259.2. Hayes,WallaceD.: OnHypersonicimilitude.uarterlyfAppliedMathematics,ol.V, no.I,April1947,P.105-106.3.E&et , DorrisM.,Rossow,VernonJ.,andStevens,ictorI.: An

AnalysisftheApplicabilityftheHypersonicimilarityawtotheStudyofFlowaboutBodiesofRevolutiontZeroAngleofAttack NACAT!N22m, 1950.4. Rossow,VernonJ.: ApplicabilityftheHypersonicimilarityuletoPressureistributionshichIncludetheEffectsofRotationforBodiesofRevolutiontZeroAngleofAttack.NICATN 2399,1951.5.Allen,H.Julian:PressureistributionndSomeEffectsofVis-cosityonSlenderInclinedodiesofRevolution.ACATN2044,1970.6.Prandtl,.:Ministryflationso.7. Jones,Robert

Separation.

OnBoundaryayersinThree-Dimensionallow.Aircraftroduction,ol&enrodeReportsndTrans-64)May1,1946.-T.: EffectsofNACAReP.884,

SweepbacknBoundary1947. (FormerlyACA

LayerandTN1402. .


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NACATN 2%3 194

./

(Q)t/c.333

II. A J~ ( b )A=.204/80,


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20

40

30

20

/0

ixc =.333MO=2.75

/ / t/c = .204M.=4.46o 0 4 8 * /2 16

Similurtiy poromefer, Ka, degrees.


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Orifice Locotjon/.8

1.6

0g 1.4a

1.0

@, degreeso 00A 45n 90v 1350 180

b El la

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naca tn 2443 the similarity law for hyper sonic flow about slender three-dimensional shapes

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