.,

NATIONALADVISORYCOMMI’ITEEFORAERONAUTICS

TECHNICALNOTE2811 +..,,r

ON THE CAL6tiTION OF FLOW ABOUT OBJECTS

TRAVELING AT HIGH SUPERSONIC SPEEDS

By A. 7. Eggers,Jr.

Ames AeronauticalLaboratoryMoffettField,Calif.

Washington

October1952

https://ntrs.nasa.gov/search.jsp?R=19930083552 2020-05-19T18:05:21+00:00Z

TECHLIBRARYKAFB,NM

Iv

I:lillllllilllllllilllllllllOEM’%NATIONALADVISCRYCOMMITTEEFORAERONAUTLUS

‘ZEU3NICALiWTE2811

A.proceduresupersonicflows

ONTHECALCULATIONOFFLOWABOUT

TRAYELINGAT HIGHSUPERSONIC

ByA.

forcalculatingwiththemethod

J.Eggers,Jr.

SUMMARY

OBJECTS

SPEEDS

three-dimensionalsteadyandnonsteadyof characteristicsisdevelop4anddis-

cu&ed. Theflowisassumedtobe adiabaticandInviscid,althoughitmaybe rotationalandthegasmayexhibitboththermalandcalori;imper-fections.Thelatterfeaturesofgeneralityareretainedin theanalysissinceit isknownthatthephenomenaassociatedtherewithmaysQnifi-cantlyinfluenceflowsathighsupersonicairspeeds.A furtherstudyofthecompatibilityequationsholdingalongcharacteristiclinesreveals

●

thatatMachnumberssufficientlylargecomparedto 1, flowintheoscu-latingplanesof streamlinesmay,inregionsfreeof shockwaves,often

9 be ofthegeneralizedFrandtl-Meyertype. Surfacestreamlinesmay,undersuchcircumstances,be approximatedby geodesics.Theseresultsholdfornonsteadyaswellas steadyflow,providedonlyslendershapesarecon-sidered,andprovidedtheinducedcurvatureoftheflowassociatedwiththenonsteadymotionsdoesnotexceedin orderofmagnitudethetotalcurvatureoftheflow. Steadytwo-dimensional-flowequationsmaythusbe applicabletoq widerclassofflows,andhenceshapes,athighsuper-sonicspeedsthanwasheretoforethought.

INTRODUCTION

Thecalculationof flowsaboutobjects,primarilymissiles,travel-ingathighsupersonicspeedsisnowgenerallyacceptedas a matterofmorethanacademicinterest.Thedifficultyof thesecalculationsstemsinlargepartfromthefactthatat suchhighspeedsdisturbanceveloc-itiesarenotnecessarilysmallcomparedtothevelocityof sound,norareentropygradientsnecessarilynegligibleinthedisturbedflowfieldabouta body,eventhoughitmaybe ofnormalslenderness.Thus,forexample,therelativelysimple13neartheory,whichhasprovensovalu-ablein studyingflowsatlowsupersonicspeeds,losesmuchofitsutility

. inthestudyofhigh-supersonic-speedflows.Inthequestformethodsespeciallysuitedto calculatinghigh-supersonic-speedflows,notable

2 NACATN2811

progresshasbeenmadeinthedevelo~entof similarityLiwsrelatingtheflowsaboutslenderthree-dimensionalshapesinbothsteady(seereferences1,2, and3)andnonsteadymotion(seereferences4 and~).Steadytwo-dimensionalflowshavereceivedperhapsthegreatestattentionfiomthestandpointofcalculatingspecificflowfields,anditwouldseemthatwithtoolsrangingfromthecharacteristicsmethod(see,e.g.,references6 and7) tothegeneralizedshock-expansionmethod(refer-ence7)theproblemisreasonablywellinhand,atleastinsofarasinviscid,continuumflowis concerned.A moreorlessanalogoussitua-tionexiststithregardtothenonliftingbodyofrevolution(see,e.g.,references6,8, 9,and10)althoughit seemsthatonlyinthecaseoftheconehasa method(reference10)of simplicitycompaabletothatofthelineartheorybeendevelopedforcalculatingthewholeflowfield.

Whenonedepartsfromtheserelativelysimpleflows,thenumberof -toolsforcarryingoutpracticalcalculationsdecreasessharply.Thus,forexample,inthecategoryof inclinedbodiesofrevolution,itappearsthatonlybodiesat smallanglesofattackhavebeenhandledadequately,usuallyby eitherthemethodofcharacteristicsor someotherstep-by-stepcalculativeprocedure(see,e.g.,reference-s6,andXlthrough14-).Inthecaseof steadyflowaboutgeneralthree-dimensionalshayes,asidefromNewtonianflowconcepts,whicharestrictlyapplicableatMachnum-bersexceedingalllimits,onlythecharacteristicsmethodhasapparentlythusfarreceivedseriousattention(references15,16,17,and18).Sauer’streatmentofthismethod(reference18)isespeciallyneat,entailingonlytheassumptionof ideal(i.e.,thermally”andcaloricallyperfect)gasflowandyieldingcompatibilityequationsina relativelysimpleform.Ap@icationofthemethod,althoughitwouldundoubtedlyprovetediousandtimeconsuming,is,aspointedoutby Sauer,certainlyfeasiblewiththeaidofpresentdayhigh-speedcomputingmachines.Itisclear,ofcourse,thattherelativelyexactsolutionsobtainablewiththemethodof characteristicsprovidean invaluablecheckagainstthepre-dictionsofmoreapproximatebutperhapssimplertheories.Indeed,asdemonstratedinreference10,a studyofthecompatibilityequationsof ““thecharacteristicsmethodcanproveusefulindeterminingsimplifiedmethodsforcalculatingmorecomplexflowfields.

Withthesepointsinmind,itisundertakenasthefirstprincipa3objectiveofthepresentreporttoreconsiderthecharacteristicsmethod,particularattentionbeinggivento itsapplicationtoh&h-supersonic-speedflowsinwhich,as isoftenthecase,airdoesnotbehaveasanidealgas. Theassumption,then,ofidealgasflowisrelaxed,andtopreserveinsofaras ispossibletheelementof simplicityinthemethod,pressureandflowinclinationanglesareemployedasdependentvariables(seereference7)ratherthan,forexample,velocitycomponentsaswereusedby Sauer.Extensionofthemethodto thestudyofnonsteadyflowsisalsoconsidered;however,theremainingprincipalobjectiveofthispaperisto showhowresultsofthecharacteristicstheorycanbe exploitedtodeducea simplifiedprocedureforcalculatinghigh-supersonic-speedflowsofboththesteadyandnonsteadytypes.

—

-.

.

—

.—.—

. .

.

NACATN 281.1

AXXLYSISANDDISCUSSION

Compatibilityrelationsdescribingthebehavioroffluidpropertiesalongcharacteristiclinesin supersonicflowmay,of course,be obtainedby proceedingformdlywiththetheoryofcharacteristicsforthequsi-linearpartialdifferentialeqyationswhichdepicttheflow. Intheinterestsofsimplifyingboththederivationoftheserelationsandtheirresultantforms,however,it seemsdesirableto proceedb a moreintuitivemanner,assuminga priorithatthecharacteristiclinesareMachlinesandstresailines(insteadyflow),andutilizingthehplicationfromtwo-dimensional-flowstudiesthatperhapsthemostconvenientdependentvari-ablesarepressureandflowinclinationangles.Withthisapproachinmind,we employtheEulermomentumequations,

2MJ:UZ+VZ+WZ=-QJ (1)at ax by az

(2)

thecontinuityequation,

+ + a(pm+a(pv)+a(pw)_~at ax ay az

theeqyationof state,

P = P(P)S)

andtheenergyequation,

as+uas+vas+was=oat ax ay az

(4)

(5)

where U, V, sndW are. the x, y, andz sxes,

.

thecomponentsofvelocityattime t alongrespectively,ofan elementofthefluidof

(6)

4

densityp,staticpressurep, andentropysl (see

NACATN 281J.

appendixforlist ●

ofsymbols).Toputtheseexpressionsina moretractableform,itisconvenienttoalinethe x axisattime t withthedirectionoftheresultantvelocityat theoriginofthecoordinatesystem.Thusequa- V

tions(1)through(4)andequation(6)simplify,respectively,intheregionofthe.origin,to

(au av aw)*+U*.+P &+&+~ ‘oat

and

asz +U

‘whichrelationsarebasicto thesubsequent

SteadyFlow

—.-

(7)

(8).—

(9) “- -

(lo) - m

(u)

analysis.

..

Characteristicsmethod.-It isclearthatinthiscaseallderivativeswithrespecttotimedisappearfromtheaboverelations.Thus,assumingthereareno shockwavespresentintheregionoftheorigin,2wemaywrite,withtheaidofeqmtions(5)and(n),

lForcertaincalculationsitmaybe desirabletoproceedfrommoregen-eralequationswhichincludeeffectsofheatandmassadditionto (orsubtractionfrom)theflowaswellaseffectsoftipressedforces(e.g., ‘gravitationalormagnetic).Sucha proceduremayeasilybe developed . .fromthatpresentedhereby followingthemethod ofGuderley(refer-encd6)fortwo-dimensionalflow.

-.

21fshockwavesarepresent,theappropriateobliqueshockequationssxe * ;employed.

NACATN28u 5.

where a isthelocalspeedof soundinthefluid.combiningequtions(7),(10),and(12),thereisthenobtainedtherelation

2=s[:(:+5)1

(I-2)

(13)

or,definingA astheamglebetweenthe x axisandthetangentto theprojectionofa stresml.ineinthe x - Y plane>and,inananalogousmanner,theangle.5 inthe x - z plane(seesketch),we have

Projectionof

r

streamhhein

/

L Projectionof streomlhein

Y

23X-X(%+3ap -pu’

X – 2 phne

X-Y plane

(14)

Transformingthederivativeswithrespectto x andz to derivativesinthecharacteristicor ClzandC2Z directionsinthe‘x - z plane(Clzispositivelyinclinedwithrespect,to x, thus a()/ax= [M/(2~=)][ao/aClz+aofiC=]andao/az= (M/2)[ao/aClz-ao/aC2z] ),thereresultsfromthisequation

A+-k=+_&- *- Z-2 *aclzaC= [ ( )1& aC= aC1zM by

(15)

.Inananalogousmanner,thereis obtainedfromequation(9)therelation

● ap ap_ -Pti. aclz (mz-m= ~+~ )(16)

6 NACAm 2811

Addingthesetwoexpressionsthenyields

whilesubtractingyields

S&=&[$-i”(e)]

(17)

(18)

Equations(17)and (18)arecompatibilityequationsforcharacteristicorMachlinesinthe x - z plane.gIndeed,ifitisfurtherreqtiredthatthe x - z planebe theosculatingplaneofthestreamlinepassingthroughtheorigin,thatis,theplanecontainingtheprincipalradiusofcurvatureandtangenttothisstreadine(attheorigin),thentheseequationsaretheessentialrelationsfordeterminingpressureandflowinclinationthroughouta flowfield.Thispointbecomesevidentwhenitis observedthatwiththeimposedrequirement(viz.,~A/bx= O),the -additionalinformationderivedfromstudyingflowinthe x - y planeissimplythatdeducedfromequation(8), or, as wouldbe expected,

?E=Oby (19)

Inproceedingto constructa flowfield,however,itisnecessarytoknowthemannerinwhichtheosculatingplanerotatesand,correspondingly,howtheprincipalcurvaturevariesalonga streamline.Thisinformationiseasilyobtainedfromequations(2)and (3) by Ufferentiatingwithrespectto x, thusyielding

and

a% I. azF=———N2ax

respectively.

Withtheaidofthese

ng..andthe

(20)

(21)”

previouslyderivedexpressions,wenwconsiderhowthecharacteristicsmethodforsteadythree-dimensionalfluWs

“

%t isnotedthattheseexpressionscontainnotonlyderivativesinthecharacteristicdirectionsbutalsoderivativeswithre~pecttotheindependentvariableY. Thistsmeofresultistobe exoectedas

—

4

F

—

poin;edoutby Coburn.’(reference-~9)..

NACATN 2811 7

canbe applied.k particular,onlytheinitialvalueproblemwillbe. consideredhere. Thusit isfirstassumedthatflowconditionsare

known,orin somemannermaybe determinedona surface4inthedisturbedflowfieldtobe investigated.We thenconsiderflowintheregionof3theosculatingplae of oneofthestreamlinespassingthroughthissur-face(seesketchbelow).

sin-’

LTfffc* of surfoce h osculathgplmea, UsingknownconditionsatpointsA andB, pressureandflowinclination

at C srecalculatedemployingequations(17) and (18)as differenceequations.Orientationoftheosculatingplaneandcurvatureofthe.streamlineat C arethendeterminedwiththeaidof equations.(20)and(21)andtheknowledgeof x and&@x. Otherproperties,suchasvelocity,temperature,andMachnumberaredeterminedatpointC inamanneranalogoustothatfortwo-dimensionalflow(see>e.g.jrefer-ence7). Oncethesecalculationshavebeencarriedoutat a numberofpoints,likeC, closetotheinitialsurface,thenthewholeprocedureisrepeated,progressingfromthesenewpoints(orpointsinterpolatedtherefrom)ofknownflowconditionsto otherpointsfartherremovedfromthesurface.~us theflowfieldisconstructedmovingdownstream(orupstream)fromtheinitialsurface.

Variationsonthesecalculationsarefrequentlyrequiredinpracticalapplications;farexample,Ifa shockwaveis encounteredintheconstruc-tion;itisusuallynecessaryto solvesimultaneouslytheequationsfortheshockandthecompatibilityequationforthecharacteristiclineintersectingtheshockinorderto determineflowconditionsat a pointadjacentto theshock.Withan analogousprocedure,pointsonthesur-faceofa bodymaybe treated.Inanycase,it”wouldappearthat

41fno otherinformationontheflowfieldisavailable,thissurfacecannot,as isknownfromthetheoryof characteristics,be a character-isticsurface.. However,as isfrequentlythecaseinaerodynamics,thisrestrictionis e~nated sincetheinitialsurfaceintersectsa shockwaveorbodyoflnownshape,orboth,whi@ resultprovidesadditionalboundaryconditionsattheterminaledgesofthesurface,

8 NACATN 2811

complicationof calculationhasbeenreducedbyworkingwithpressureandflowinclinationanglesasprimarydependentvariables,no lossin

—.

generalitybeingsustainedinsofarasrestrictionsonthethermodynamicbehaviorofthegasisconcerned.Thegasmayexhibitboththermalandcaloricimperfectionswithoutinvalidatingthepreviouslydevelopedequations,theextenttowhichtheseimperfectionsaremamifestinfluenc-ingonlytheformoftheequationof stateofthefluidandtherelationsdefiningitsspecificheats.5 Theprovisionforcaloricimperfectionswouldappear(seereference7)tobe especiallydesirablewhenusingthemethodtocalculatehypersonicflows.

.—

—

——.

Applicationofthemethodtothecalculationofa specificflowfieldhingesonthedetermination.offlowconditionsalongtheinitialsurface.Thisisa separateproblem,thesolutionofwhichhasthusfaratleastbeens~ecialtothe~rticul.ertypeofflowunderconsideration.eIndeed,onlytheairfoilhasapparentlybeentreatedrigorouslyinthisconnectionwithouttherestrictionthatairbehavesasan idealgas(seereference22).

Imperfectgasflow downstreamofthethroatofa hypersonicnozzlecould,of course,be analyzedby thecharacteri~ticsmethodofthis

.

paper.Withtherestrictionto idealgasflows,anadditionalapplication A.ofimportancealsosuggests itself,nemely,tothecalculationofflowsaboutinclinedbodiesofrevolutionatangleofattack(notnecessarilysmall).Inthiscaseflowconditionsalonga surfaceclosetothevertex

6

ofthebdy canbe calculatedapproximatelywiththeaidofreferences25and26,theaccuracyoftheapproximationincreasingwiththeclosenessofthesurfacetothevertex.Flowdownstreamofthissurfacecanthenbe determinedafterthemannerdiscussed,usingequations(17)2(18)>(20),and(21)inthereducedforms

.

5Sinceonlyflowsofdenseairaretreatedhere,heatcapacitylageffectsareconsiderednegligible.

%n thisconnectionsee,forexample,theworkofCrocco(reference20),MunkandPrim(reference21),andKraus(reference22)onthetwo-dimensional.airfoilprobla,andShenmd Lib(reference23)and

*

Cabannes(reference24)onaxiallysymmetricflowaboutbodiesofrevolution. .-

.-

W NICATN2811 9

● end

respectively,where y,theratioof specificheatat constantpressureto specificheatat constsntvolume,wouldbe consid~edconstant(equslto approximately1.4forair).

Letus nowturnourattentiontotheconsiderationofa moreapproximate,andby theseinetokenmoresimplified,methodofcalculat-ingthesteadythree-dimensionalflowofa gasathighsupersonicspeeds.

Approximatemethod.-It iswellat theoutsetofthisanalysistoestablish,insofaras ispracticable,thetypeof flowstobe treated.Inthisconnection,it isconvenientto employthehypersonicsimilarityparsmeter(i.e.,theproductoftheflightMachnumbersadsaythethick-nessratioofa body)as a measuringstick.In flowscharacterizedbyvaluesofthehypersonicsimilarityparsmetersmallccmparedto 1, that

. is,flowsinwhichthebodyisextremelyslenderandliesclosetotheaxisoftheMachcone,thereisno apparentreasontobelievethatthelineartheorywillnotbe asusefulan approximatemethodofcalculation

. asatlowsupersonicspeeds.Inflowscharacterizedby valuesoftheparsmeterup toabout1, thesecond-ordertheoryfirstenunciatedbyBusemann(reference27)forairfoilsandmorerecentlygeneralizedtothree-dimensionalflowsby VanDyke(reference9) andMoore(reference28)shouldprovea usefulapproximation.Ontheotherhand,forflowsaboutmoreorlesssrbitraryshapes,thereisapparentlyno approximatemethcdofcalculationgenerallyapplicablewithengineeringaccuracyat valuesof thehypersonicsimilarityparameterappreciablygreaterthan1.

Inthelimitingcaseof indefinitelyhighfree-stresmMachnumber(andhencesimilarityparameter) anda ratioof specificheatsequalto 1,we havetheNewtonianimpacttheory(reference29)anditsrefinedcounterpart,accountingforcentrifugalforcesinthedisturbedflow,developedfirstby Busemann(reference30)andmorerecentlytreatedby Ivey,IUunker,andBowen(reference”31). Theimpacttheoryhasbeenemployedwithsomesuccessby Grinznlnger,Williams,andYoung(refer-ence32)andotherstopredictsurfacepressuresonbodiesofrevolutionatvaluesoftheshilsxityparameterappreciablygreaterthan1, althoughitshouldbe remarkedinpassingthatthissuccessis inpart,atleast,fortuitous,asperhapsisbestevidencedby thefactthatthemoreexacttheory(withintheframeworkoftheunderl@ngassumptionof M+-,T+l)ofBusemannis considerablylessaccurateundercorrespondingcircumstances.

. As showninreference7, neithertheNewtonieuimpactnortheBusemanntheoryapplywithgoodaccuracytoairfoilsexceptatvaluesof thesimi-larityparameterquitelargecomparedto 1, corresponding,for~emple,inthecaseofthinairfoilsto flightspeedsconsiderablyin &cess oftheescapespeed‘atbe”alevel.Perhapstheforemost-shortcomingofthese

10 .NACATN 2811

theoriesis,however,that,irrespectiveof.iheshapetowhichtheyareapplied,theyprovideno informationonthestructure7ofthedisturbedflowfieldwhichis,ofcourse,offiniteextentadjacenttothesurfaceat flightMachnumberspresentlyof interest(sayMachnumberslessthantheescapeMachnumberat sealevel).Suchinformationis,forexample,importanttothedeterminationoftheflowaboutcontrolsurfacesandthelikewhichmaybe locatedinthisfield.

Inviewoftheprecedingdiscussion,itseemsclearthatinthehigh-supersonic-speedflightregime,theneedforanapproximatemethodofanalysisliesintherealmofflowscharacterizedby valuesofthehyper-sonicsimilarityparametergreaterthan1. An attemptwillthereforebemadeto obtaina methodmeetingpartofthisneed, attentionbeingfocusedprimarilyonflowscharacterizedby largevaluesofthes~larityparsm-eter.To thisend,itis convenientfirstto employequation(14)rewrit-tenintheform

$=*[g[-)-*($)]

where

(22)

●

K

.

.-

s

.

(23) .

Nowconsiderforthemomenta surfacestreamlinealinedinthe x direc-tion,andimposetherequirementthatthe x - z plae be tangenttothisstreamlineandnormalto thesurfaceatthepointoftangency(theorigin). Thex - Y planeisthen,ofcourse,ttigenttothesurfaceatthispoint.Observingthe“lastterminthebracketsontheright-handsideofequation(22),itisnoted(seesketch)that

Y\\ At

+, 1A xr

0’

7’lhisconsequenceistraceableprimarilytotheassumptionof y = 1whichleadsto thewell-knownresultthatthedisturbedflowfieldIsconfinedto an infinitesimalregionsii~acenttothesurfaceofa body.

,

*

NACATN2811 II —

where r istheradiusofcurvatureof thelinenormalto theprojec-tionsof streamlinesinthe x - Y Plme, md passingthroughtheorigin.At thehighMachnumbersunderconsideration,thedisturbedflowfieldisconfinedtoa regionof small.czxtentnormalto thesurfaceofa body;henceitmaybe expectedthat r willbeprimarilya functionofbodyshapeandattitude.a Thisbeingthecase,it followsthenthattheterm(1/~~1) (l/r)willdecreaseinabsolutemagnitudewithincreasingMachnumberoftheflowaboutthebody. Considernowtheterm@5/& )(l-Dz)/(l+Dz). We notethat a5/& = l/R where R istheradiusofcurvatureoftheprojectionofa streamlineinthe x - z planeand,byreasoninganalogoustothatusedin consideringr, isnot~ectedtovarysignificantlywithMachnumberinthedisturbedflowfield.Letus assumeforthemomentthatthequantity(l-Dz)/(l+Dz)isalsorela-tivelyindependentofMachnumber.Withthisassmnption,itis clesrthatequation(22)approachestheeqwtionfortwo-dhensionalflowasthefree-streamMachnumber,andhencethehypersonicsimilarityparam-eteroftheflowbecomeslargecomparedto1. Thecompatibilityeq=tions(equations(17)and(18)) areaffectedina shilarmsmner;thusit isapparentthattheflowwhenviewedinthe x - z planeapproachesthetwo-dimensionaltype. Inthiscase,however,as showninreference7,solongastheMachnumberandratioof specificheatsofthedisturbedfluidarenottoocloseto 1,Dz is smallcomparedto 1, andhencetheflowapproachesthegeneralizedPrandtl-Meyert~e (i.e.,flowinwhichpressureandinclinationangleareapproximatelyfirst-familyMachlines).Ourflowequationmay

()apw NJ2 ab.=— —ax JM~ ax

whereit isrequiredexplicitlythat

II

*,,1ax J==

or,in effect,thatdisturbancesassociatedwith

constantalongcurvedthenbewritten

(24)

(25)

the@imrgenceof stream-linesintangentplanesmustbe ofsecondaryimportancecomparedto tkseassociatedwiththecurvatureof streamlinesinplanesnormalto thesur-face.

Fromtheseconsiderationsitappesrsthattheconclusionofrefer-ence10 thatinviscidflowalongstreamlinesdownstreamofthenoseof

‘Itisinterestingtonotethatin idealgasflows,r becomesjusta. functionofthesevariablesas thevalueofthehypersonicsimilarity

parameterbecomeslargecanpa@ito 1 (seeworkof Oswatitsch,refer-ence33,notingthathisresultscanreadilybe extendedto three-.tensionalidealgasflowsusingthecharacteristicsmethodofthispaper).

12 NACATN 2811

noninclinedbdies of revoluticm traveling at W@ supersonicsPeeds‘s ~oftenofthePrandtl-Meyertype(inre@ons.freeof shock~ves) mayapplyalsoto othersteadythree-dimensionalflows.It istrue,too,thatinthelattercase,justas intheformercase,thisconclusionis 1consistentwiththepredictionsofthehyq?ersonic.simil.aritylawforsteadyflowaboutslendershapes.

Onequestionremainstobe considered,namely,wheredo thestream-linesgointhedisturbedflow?To clarifythismatter,it isconven-ientto studyfuxthertheimplicationsof equation(25).Forthispurpose‘- ~we combineequation(25)withthetransformationequation —

.—

to obtaintherelation .

14’’*l+%-al(26)#

llromthis relationwe deduceeitherthatto theorderofa number(curva-

ture) smallcomparedto

orthat

and

II2&K at3— .M ax

.

(27)

—

:I>>2J’1*1 (28)

pl>>2*l*

whichtypeofflowcagnotEquation(27)impliesverticalflow,however,be treatedby thepresentanalysissinceequation(25)isviolated.sEquation(28)isthentherequirementconsistentwiththebasicassumptions

sl%isconclusionisparticularlyevidentinthecaseofpureverticalflow, ,or say.vorticalflowwitha superimposed~iformstreeqdirectedalongthesxisofthevortex,inwhichcasesbiax.0,andhenceequationcertainlydoesnotfollowfromequation(22). *

,

NACATN 2811 13

●ofthis@yi3i8. Comparingtherelationsof equation(28)withthetransfonuationequation

* &3M ,( b aA .—=— ——~ 2JiiG %y+%y )

leadsonetotheconclusion,however,that

(29)

or,in effect,thatsurfacestreamHnescm, to theaccuracyofthis&is, be consideredgeodesiclinesofthesurface.Withthisinfor-tionwe areenabledto constructtheflow.fieldabouta body,havingonce,determined,,for~le, theflowintheregionoftheleadingedge(oredges)thereof.Thisresultfollowssincea geodesicline,andhencea streamline,onthesurface isftied,provideditsdirectionatanypointisgiven(see,e.g.,reference34).10Withthis~owledgeofthelocationof surfacestresnlines,flowinthephnes tsngenttheretoand.normaltothesurfacemaybe calculatedapproximatelyintherelativelythinregionbetweenthesurfaceandboundingshockwaves,usingthegen-

. eralizedshock-~ansionmethodafterthemannerdescribedinreference7.

. A partialcheckontheseobservationsisaffordedbystudyingtheflowabouta sweptairfoil.Inthiscaseflowatthesurfacemaybecalculatedwithgoodaccuracy,usingtheshock-expansionmethodincom-binationwithsimple-sweeptheory.Forthinairfoils(onthesurfaceofwhichtheappropriategeodesicshaveessentiallythedirectionofthefreestreem)theextendedshock-expansionmethodofthispaperreducestotheslender-airfoilmethodofreference7. Thus,inthiscase,it isevidentfromtheresultsofreference7 thattheextendedmethd willpredictsurfacepressurecoefficientsinerrorby lessthan10percent,providingthecomponentoffree-streamMachnumbernormalto theleadingedgeisgreaterthanabout3. It,isof interestalsoto considera thickairfoilto ascertaintheaccuracywithwhichthismethodappliesto flowwithappreciablecurvature.To thisend,surfacepressurecoefficientsandstreamlineshavebeencalculatedfora 20-percent-thickbiconvexairfoil(atzeroincidence)swept600andoperatingatMachnumbersof10andinfinity(y= 1.4). Theresultsofthesecalculationsarepresentedinfigure1, anditis observedthatthepressuredistributionsdeterminedwiththeshock-expnsionmethodforsweptairfoilsandtheextendedshock-_sion mefiodareinreasonablygoodagreementatbothMachnumbers.

. 10Ifa suddenchangeof surfaceslopecausesan obliqueshockwaveoraconcentratedFrandtl-Meyertypeexpansionfan,thestreamlinesinthedownstreamdirectionsredefinedonthebasisoftheirflowdirection.immediatelyfollowingthediscontinuityin slope.

14 NACATN 2811

Thestreamlinesarealsoinreasonablygoodagreementovertheforward.-

portionoftheairfoil,although,aswouldbe expected,somewhatpoorer8

resultsareobtainedovertheafterportion.Itisnotsurprising,inviewoftheunderlyingassumptionsoftheextendedshock-qsion method,thatitisgener-allymoreaccurateatthehighestMachnumber.

a

Intheprecedingdiscussioncircumstanceswerededucedunderwhichsteadyflowathighsupersonicspeedsaboutthree-dimensionalshapescouldbe constructedapproximately,usingthebasictoolsoftwo-

—

dimensionalsupersonicflowanalysis,nsmely,theobliqueshockeqpationsandFrandtl-Meyerequations.Severalpossibleexceptionsto thesecir-cumstancestiediatelycometomind. Theseincludeconical-typeflowandflowintheregionofthetipofa wing,oratthediscontinuousjunctureofa wingandbody,tomentiona few. Inmch flowsequation(25)maynotbe satisfied,inwhichcasetwo-dimensionalflawinplanesnormalto a surfacecannotbe expected.11 Itmightbereasoned,therefore,thattheseflowscannot,ingeneral,be treatedby theproposedmethod.Thisobservationmaybe correct;however,intheonecaseinvestigatedthusfarinthisconnection,namely,flowintheregionofthenoseofnon- -liftingbodiesofrevolution(seereference10),itwasfoundthatalthoughequation(25)isnotsatisfied,flowalongstreamlinesisnever-thelessofapproximatelytheFwndtl-Meyertype.Thuswe areledto

L-

expectthatperhapsa lessrestrictiverequirementthanthesatisfyingof equation(25)maybe imposedto insurethatflowalongstreamlinesis

——

ofthistype. Sucha requirementis infacteasilyobtainedby reconsid- “eringequation(22)intheform

thusyielding

(31)

Itisevidentthatequation(31)embracesequation(25)asa specialcaseandthatPrandtl-Meyerflowobtainsalongstreamlinesif

abcw la,a ——= --—aclz (32)

M ay

llOnemaynotethatin somecasesofthisnature,theflowinosculatingplanesofthestreamlinesmaybe ofthetwo-dimensionalor eventhesimplerPrandtl-Meyertype,althoughtheseplaes maynotbe normaltothesurface.

.

NACA~ 28u

* JiFIto theorderofa numbersmall.ccuuparedto —M

impliesthatalthoughflowinclinationanglesarent

15

I~,~S resultaxt necessarilycon-

.? stantalong Clz lines,pressureisapproximatelyconstant(seeequa-tion(17)).

Itis clearthattheMcreasedgeneralityoftheaboveresulthasbeenobtainedat someexpenseinourknowledgeofthestreamlineflowpattern.Forexemple,itisnotnowindicatedthat(withinthefrsmeworkofthisanalysis)surfacestreamlinesmaygenerallybetakenas geo-desics- additionalknowledgeoftheflowmustbe hadinorderto deter-minethesestreamlines.Iftheyareknown,however,thecalculationofthewholeflowfieldismateriallyfacilitatedby theaboveconsiderations.To illustrate,considera nonliftingbodyofrevolution(seesketch)

Body surfaceI

&

forwhichwe assumethatflowat thevertexisknowneithecfromsayreference8 orreference10. Themeridiancurveofthebodyisbrokenup intoshortsegmentsas shown,andflowis constructedalongthefirst-fsmilyMachlinesemanatingfromtheintersectionofthesesegments.Therequirementtobe satisfiedisthatthepressurechangeacrosstheselinesbe constantalongtheirlength.12Theconstructionproceedstheninamanneranalogousto thatforthetwo-dimensionalairfoildiscussedinreference7.

Tlnmfaronlysteadyflowshavebeenconsidered.Theproblemnaturallyarisesof extendingtheseconsiderationsto nonsteadyflows.Some,aspectsofthismatterwillnowbe discussed.

NonsteadyFlow

Themethodsofsmalysisinthis-caseareentirelyanalogoustothoseemployedinthestudyof steadyflow,thesingularcontrastingfeature

.

121nthismannersmallchangesinpressurealong Cl linescanbeaccountedforapproximatelyinthepredominantlyconicalflownearthe. vertexofthebody.

.

16 NACATN 2.811

beingthatderivativeswithrespecttotimeinequations(1)through(11)cannotnowbeneglected.Withthispointinmind,onlypertinentresults ●

arediscussedbelow. .—

Characteristicsmethod.-Thecompatibilityequationsrelatingfluidpropertiesalong~ch linesmaybewrittenasfollows:’

●

(33)and

5&=*[&i(9+iPw%w3’)+iwl](34)

Thedefinitionofthe x - z planeastheosculatingplaneofa pathl.ine(streamlinein steadyflow)remainsasbefore,henceequation(19)still ‘appliesinthe x - y planeintheregionoftheorigin.Therotationoftheosculatingplaneandvariationoftheprincipalcurva@reofapathlinewithmotionalongitsrenow,however,obtainedwiththeaidof ●

therelations —

(35)

and

where

(36)

—

(37)

Theseequationsin combinationwiththeenergyandstateequationsareemployedinthesamemanneras inthe.caseof steadyflowto constructaflowfield,proceedingfroman initial.valuesurface.It isclear>how- . _ever,thatbecauseoftheunsteadyfit~e”oftheflow,thissurfaceisnotnecessarilyfixedin space,nor.arefluidproyert+es,xyi$si$$i,ily. .—constantonit. Thus,inordertoconstructtheflow,itwifi,ingeneral, .“=

.—1

—

v ~MATN2811 17

be necessaryto stsrtnewosculatingplanesfromthissurfaceat shortrintervalsoftime,eachplanebeingattachedto a psxticularelementoffluidas itmovesthroughthefield.By wayof comparison,then,we

. recti thatingoingfromsteadytwo-dimensionaltothree-dimensionalflowwiththecharacteristicsmethod,itwasnecessaryto constructtheflowina familyof surfaces(locatedadjacentlyin saythespanwisedirection)ratherthanjusta singlesurface.Analogously,ingoingfromthree-dimensionalsteadyto three-dimensionalnonsteadyflow,itisneces-saryto constructtheflowina fsmilyof spaceslocatedadjacentlyinthelftime’ldirection,ratherthanin justonespace.Quiteobviouslysuchaseriesof calculationsposessoformidableandtimeconsuminga problemastobe questiomblyfeasibleat present;hencetheywillbe consideredinno gxeaterdetailhere. Rather,letus turnourattentiontotheapproximatemethodof calculatingnonsteadyflows.

Approximatemethd.-As inthecaseofthecorrespondingsteady-flowsnalysis,itis convenientto considertheexpressionforpressuregra-dientalonga pathline.Thuswe have

.

wherenowthe x - z planesaretakennormaltothesurfacesweptoutbyelementsoffluidmovingalongthebody. UQoninspectionofthisrelationandthecompatibilityequations,itbecomesclearthatthecriticalrequirementfortwo-dimensionalflowofthegeneralizedRandtl-Meyertypeintheseplanesis,inadditionto theonepreviouslyderivedfromsteadyflowcons~derationa,that

or

where

.

a. beingthevelocityofastheMachnrmiberofthe

(39)

MO : Xo=aot=—

soundintheundisturbedstream.Nowsolongundisturbedstreamandthelocal.Machnumbers

18 NACATN281.1

arelargecomparedto1,MO islargecomparedto1 sincethespeedof...theundisturbedstreamandthespeedofthelocalflowcannotdiffer

s

greatly.Thus,withthisrestriction,therequirementexpressedby .-

equation(39)issimplythattheinducedcurvatureoftheflow1~~/b~l, ,associatedwiththenonsteadymotionsofthefluid,cannotexceedinorderofmagnitudethetotalcurvatureldb/dxloftheflow. Providedthisisthecase,andequation(25)is satisfied,equation(38)reduces,of course,to

Oneobservesthatthisresultisnotapplicableshapesasthatforsteadyfl~w,sincethelocsl

(40)

toaswidea classofMachnumberofthedis-

turbedflowisnowrequiredtobe everywherelargecomparedto1.13Thisadditionalrequirementmanifestsitself,sinceotherwise,nonsteadydisturbancescreatedan appreciabledistanceupstresmand/ordownstreamofa particlecouldsignificantlyinfluenceitsbehaviorinthedisturbedflowfield(seesketch,notingthatincaseofthickbody,particlebisinfluencedby disturbancesoriginatinginparticlesa andc).

Fluid part

M >>/

WOV* fronts ofdlsturboncesgenerotedin portlcleso ond c

~>>~ ~

Af>>j-Shock wove~

Thusequation(40)appliesonlyindisturbedflowfieldsaboutthinorslendershapes(i.e.,shapesproducingflowdeflectionanglessmallcom-paredto1). Insuchcases,pathlinesinthesurfacessweptoutbyelementsoffluidadjacentto theshapesareapproximatedby geodesicsor,evensimpler,linesof curvatureofthesesurfaces.Itisnottobeimplied,ofcourse,thatpathlinesmustalwaysbe suchcurvesin orderforfluidpropertiestobehaveas inPrandtl-Meyerflow. Infact,justas inthecaseofsteadyflow,ifequation(31)ratherthanequation(25)issatisfied,pathlinesarenotnecessarilygeodesics(orlinesof curva-ture)althoughequation(39)andhenceequation(kCl)holdalongtheselines.

h“—

.

~sT’henetsimplificationofrequiringonlythatthehypersonicsimilarity . -parameteroftheflowbe largecompared.to1, is,in general,thatflowinosculatingplanesmaybe treatedasnonsteady,two-dimensional.

.

NACATN 2811 19

. Onenotesthatwithintheframeworkofthisapproximateanalysis,thecalculationofnonsteadyflowsat leastat thesurfaceof slenderbodiestravelingathighsupersonicspeedsshouldnotproveundulydiffi-. cult. To illustrate,consideran oscillatingairfoilas showninthesketch:

Airfoil at iime t2

Y

shockwave

M>>1 Fc.

4P

Directionofro?’atlonPothlineof particle $tri~in9

N’hcq.d%of oirfoi!

leadingedgeat time~h-

Airfoll at timetl-

Thepressureat anypointalongthepathlineshownisreadilydeducedbysimplyintegratingequation(40)alongthislinefromtheleadingedgeoftheairfoilto thepointin question.Thewholeflowfieldas a func-tionoftimemaybe calculatedby emplo@ngthegeneralizedshock-expansionmethcdforsteadyflows(seereference7) ina seriesofplaneslocatedsmalldistancesapartintime. Thisexsmpleservesto emphasize

+ thatthethe historyof fluidelementsmustbe known,at leasttothe@ent offixingtheirinitial.flowdirectionandentropy.It isalsoevidentthatagain,asin thecaseof steadyflow,thegeneralresults. oftheanalysisareconsistentwiththepredictionsofthehypersonicsimilaritylawfornonsteadyflowsaboutslenderrelatedshapes(refer-ence5).

CONCLUDINGREMARKS

A methodofcharacteristicsforsolvingsteadythree-dimensionalsupersonicflowproblemshasbeenconsidered.Itwasfoundthatcompat-ibilityequationsrelattigfluidpropertiesalongcharacteristiclinescouldbe obtainedina simpleformby employingpressureandflowincli-nationanglesasdependentvariables.No sigificsutrestrictionswereimposedoneithertheequationof stateobeyedby thefluid,ortherela-tionsdefiningitssyecificheats.Thesefeaturesofgeneralitywereretainedforthespecificpurposeofenablingmoreaccurateapplicationofthemethodtothecalculationofflowfieldsaboutmissilestravelingathighsupersonicairspeeds.Suchapplicationreqtires,of course,apredeterminedknowledgeoffluidpropertiesalongsomesurfaceinthedisturbedflow.Extensionofthemethd to treatnonsteadyflowswasconsideredbriefly.

.Itwasalsoundertakento obtainanapproximatemethodforcalculat-

ingflowsaboutbodiestravelingathighsupersonicspeeds.ItwasfoundthatwhentheflightMachnumberis sufficientlylargecomparedto1, flow-intheosculatingplanesof streamlinesinregionsfreeof shockwavesmayfrequentlybe ofthegeneralizeFrsndtl-Meyertype- surface

20 NACATN28u

streamlinesinthiseventmaybetaken.as geodesics.Inthecaseofslendershapes,theseresultsapplytononsteadyaswellas steadyflows,

.

providedtheinducedcurvatureof streamlinesdoesnotexceedthetotalcurvatureinorderofmagnitude.Itisconcludedfromtheseandother .considerationsthattwo-dhnehsional-flowequationsmaybe applicabletoa relativelywidec~assofflows,andhenceconfigurations,athighsupersonicspeeds.

AmesAeronauticalLaboratoryNationalAdvisoryC.cmmtLtteeforAeronautics

MoffettField,Calif.,Aug.15,1952

—

.

.

NACATN 281J

APmlx

SYMBOLS

21

%

M

MO

P

s.

t

U$V,W

X,y,z

Y

A

P

.0

localspeedof sound

chordofairfoil(measurednormaltoleadingedge)

characteristiccoordinatesin x - z plane(C~z ispositivelyinclinedwithrespect”tox)

pressurecoefficient(*)

I&h number(ratiooflocalvelocityto localspeedof sound)

Machninnber(ratiooflocalvelocityto speedofsoundintheundisturbedstream)

staticpressure

entropy

time

componentsoffluidvelocityalongthe x, y,andz axes,res-pectively

rectamgulsrcoordinates

“ratioofspecificheatat constantpressureto specificheatat constantvolume

angle(or

angle(or

betweenx sxisandtangenttoprojectionofstreamlinepathline)in x - z plane

betweenx axisandtangenttoprojectionof streamlinepathlLne)in x - y @ane

density

Subscript

free-streamconditions -

.

22

REFERENCES

NACATN 2811

1.Tsien,Hsue-Shen:SimilarityLawsofJour.Math.andPhys.,vol.25,no.

2.

3.

4.

6.

7*

8.

9*

HypersonicFlows. t.—3,O&. 1946,~p.252-259. .

Hayes,WallaceD.: OnHypersonicSimilitude.Quart.Appl.Math.,vol.V, no.1,Apr.1947,pp.105-106--- ———— --

Hamaker,FrankM.,Neice,StanfordE.,andEggers,A. J.,Jr.: The -..SimilarityLawforHypersonicFlowAboutSlenderThree-DimensionalShapes.NACATN2~3, 1951. —

Lin,C.C.,ReiBSner,E.,andTsien,H. S.: OnTwo-DimensionalNon- —steadyMotionofa SlenderBodyina CompressibleFluid.

‘Jou. Math.andphYs.,VO1.27, no.3, m. 1948,PP.220-231.

Hmaker,FrankM.,andWong,T!homasJ.: TheSimilarityLawfor .NonsteadyF&personicFlowsandRequirementsfortheDynamical

.—-

SimilarityofRelatedBodiesinFreeT’light.NACATN2631,1952. =

Guderley,G.: FxtensionoftheCharacteristicsMethod.BritishR.A.E.Library‘ITanslationNo.151. (FromLilientha+GesellschaftRep.139, --pt. 2,PP.15-22),mI. 1947. .1

Eggers,A. J.,Jr.,andS~ertson,ClarenceA.: InViscidFlowAboutAirfoilsatHighSupersonicSpeeds.~cATN26~, 1952. —

Mass.Inst.Tech.,Dept.Elec.Engr.,CenterofAnalysis:TablesofSupersonicFlowAroundCones.By thestaffoftheComputingSection,CenterofAnalysis,underthedirectionof ZdenekKopal.

.

Cambridge,1947.Tech.Rep.No.1. “-.—

VsmDyke,MiltonD.: A StudyofSecond-QrderSupersonic-FlowTheory.N/MATN2200,1951.

— .—.-. -.

10. Eggers,A. J.,Jr.,andSavin,RaymondC.: ApproxmteMethodsfor” ““–CalculatingtheFlowAboutNonliftingBodiesofRevolutionatHighSupersonicAirspeeds.NACATN2579,1951.

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13. Ferri,Antonio:SupersonicF1OWAroundCircularConesatAnglesofAttack.NACATN2236,1950.

.

.

.-

*—

=&—

—

.

.

NACATN 2811 23

.

.

.

.

14. Ferri,Antonio:ofSupersonicAttack.NACA

15● Moeckel,W. E.:

TheFlow

MethodofCharacteristicsfortheDeterminationOverBodiesofRevolutionat51.1 Anglesof

TN1809,1949.—

Useof CharacteristicSurfacesforUnsymmetricalSupersonicFlowFroblems.N.ACATN 1849,1949.

16. Coburn,N.,andDolph,C..L.: TheMethodof CharacteristicsinThree-DimensionalStationarySupersonicFlowofa CompressibleGas.Proc.Symp.Appl.Math.,vol.I,pp.55-66.Amer.Math.Sot.,NewYork,1949.

17● Thornhill,C.K.: TheNumericalMethodof CharacteristicsforHyperbolicProblansinThreeIndependentVariables.ArmsmentResearchEstablishment.Rep.29/48,BritishMinistryofSupply,1948.

18. Sauer,Robert:RecentAdvmcesintheTheoryofSupersonicFlow.NAVORDReport1593,Mar.1951.

19. CobUrn,N.-:‘“c’h~acteristicsonicFlows.Amer.Math.

20. Crocco,Luigi:Singol.aritaNell~IntornodiunaProra

DirectionsinThree-DimensionalSuper-SOC.prOC.,I,1950,pp.241-245.

dellsCorrenteGassosaIperacusticala Diedro.L?Aerotecnica,vol.17,

no.6, June1937,pp.519-534.

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22. I&aUs,Samuel:An AnalysisofSupersonicFlowintheRegionoftheLeadingEdgeof CurvedAirfoils,IncludingChartsforDeterminingSurface-PressureGradientandShock-WaveCurvature.NACATN 2729,1952.

23. Shen,S.F.,andLin,C.C.: OntheAttachedCurvedShock~nFkontofa Sharp-NosedlucidlySymmetrical”BodyPlacedina UniformStream.NACATN2505,1951.

24. Cs@mes, Henri:&de deLrOndede ChocAttach~eclansles .EcoulementsdeR&olution.PartI. LaR&hercheA&onautique,November-December1951,pp.17-23.

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26. Young,G.B.W.,andSiska,C.P.: SupersonicFlowAroundConesatLargeYaw. RANDP-198,revisedAug.6, 1951.

24

27. BUS~ ,A.,andWalchner,O.:sonicSpeeds.BritishR.T.P.

NACATN!2811.

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Univ.ofCalif.press,1946.

30. Busemann,A.: Flfisigkeits-undGasbewegung.Handw&terbuchderNaturwissenschaften,ZweiteAuflage,GustavFischer,Jena,1933,

.—

pp.266-279..—

3-. Ivey,H.Reese,Klunker,E.Bernard,andBowen,EdwardN.: A Method‘ forDeterminingtheAerodynamicCharacteristicsofTwo-andThree-DimensionalShapesatHypersonicSpeeds.NACATN1613,1948.

—

32. Grimminger,G.,Williams,E. P.,andYoung,G.B.W.: LiftonInclinedBodiesofRevolutioninHypersonicFlow. Jour.Aero.Sci.,

—

vol.17,no.---~,Nov.1950,pp.675-690.“...

33● Oswatitsch,Klaus: SimilarityLawsforHypersonicFlows. .Kungl.TebiskaH6gskolan,Stockholm,Institutionenf6rFtygteknik,

.

Tech.Note16,July19,1950,pp.4-5.

34. Graustein,WilklsznC.: DifferentialGeometry.MacMillanCo.,Jan.1935,reprintedApril1947,pp.149-156.

. * , I

\

h$=lo.lo~

Sponw” stw!h%yh f2?)Me=tv a=w#M3e Statbn,x/c?

A@X??/.- Ompon%n of surfdce shrnlh?s ovnfpreswre G$W&Itottis cakdn%d tith h exteno%ishock-exemethodomi#h?shock-ex~sbn metbf tbr M9pt &fmYs (A&mvex aktil Sei%49n,tih%ess rath .0.2, smwpW@ =609 G