NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICAL NOTE 3168

A NEW HCIDOGRAPHFOR FREE-STREAMLINETHEORY

By AnatolRoshko

Californiahstitute of Technology

WashingtonJuly1954

,

TECHLIBRARYKAFB,NM

NATIONALADVISORYCONKEFJ?EE

.

u

TECHNICALNOTE3168

A W HODOGRAPHFORFREE-STREAMLINETHEORY

ByAnat01 ROShkO

SUMMARY

InthemethodofHelmholtz-Kirchhoffforseparatedflowpasta flatplate(normalto thestream)theseparationvelocityandthe“basepres-sure”arefixedatthefree-streamvalues.Inthepresenttreatmentamodtiicationisintreducedto allowarbitraryseparationvelocityandbasepressure,sothatvaluesmoreincotiormlty withexperimentmaybechosen.Thesolutiondependsthenonthesingle(base-pressure)parsm-eter k. When k is suitablychosen,thedragandthedetailsofthepotentialflowneartheplateagreewellwithexperiment.Thecompu-tationsdependona particularchoiceoffree-streamlinehodograph,whichhasthefeaturethatitgivesa definitewakewidthforeveryvalueof k.Inthiswaythewakewidthiscorrelatedwiththedrag.

Thesameideasareappliedto workoutthefree-streamlineflowsfora circularcylinderW a wedgeof 90°vertexangle.

INTRODUCI!IOI?

It isremarkablethattheproblemofflowpastbluffbodies,oneoftheearliestto receiveattention,isnotyetunderstood.Theearlyinvestigatorsalreadyhadverygoodinsighttito theproblem;and,althoughconsiderableexperimentalandsometheoreticalinformationonthematterhasbeencollectedsincethen,therehasbeenlittleessen-tid progresstowarda theoreticalformulation. Oneofthemethodsusedto attacktheproblemwasthatofthefree-streamlinetheory,introducedby Helmholtz andexLendedby firchhoff andmanyothers.firchhoff~sexampleofflowpasta flatplate,normaltothestream,iswell-known.Thetheoryisbasedontheobservationthatfortheconfigurateionsinquestiontheflowseparatesfromthebody,leavingbehindita wakeandcrestinga pressuredragquitedistinctfromthatdueto shearingforcesonthesurfaces.Themainaimofthetheoryisto findthe“freestream-lines’*definingthewake,outsidewhichtheflowispotential,andtoccmptietheresultingpressuredrag.

IntheKirchhofftheorythereisa basicassumptionwhichresultsina considerablelossofreality.Thisistheassumptionthatthe

2 NACATN3168

●

velocityonthefreestreamlineat separationSsequaltothefree-streamvelocityUm. Thepressureattheseparationpoints,andonthe“base”ofthebody,behindtheseparationpoints,isthenequaltothefree- Gstreampressure.Thisisnotinagreementwithexperience,whichshowsthatthebasepressureisactuallyalwayslowerthanthefree-streamvalueandthatthedragishigherthanthatcalculatedby theKirchhofftheory.However,theHelmholtz-firchhoffassumptionisattractive,inthatthevelocityallalongthefreestreamlinemaybe consideredcon-stant (U.) upto infinity,whichleadsto a simpleformulationinthehodographplane.Thetheoryhasbeenappliedto manyshapesotherthanthenormalflatplate;infact,thereisa ratheretiensiveliteratureonthesubject.However,inallcases(withthepossibleexceptionofcavityflows)thecomputedresultsfailto agreewithexperience,the Idiscrepancybeingmainlyduetothebasicassumptionabouttheseparation

I

velocity.

Clearly,ifanyprogressistobemadewiththefree-streamlinetheory,itisnecessarythattheseparationvelocitybe allowedto assumevaluesdifferentfrom Um. Themodificationrequiredinthetheorymaybe summedup ina singleparameterk,whichdefinestheseparationvelocityU5 = kU~ andgivesa base-pressurecoefficientCps=1- k2.Thebase-pressurecoefficientisalwayslessthanzero,correspondingtok>l. Ofcourse,itisnotlmownwhatvalueof k shouldbe assumed,butthisisa problathatcannotbe determinedbythefree-streamlinetheory.Itmustcomefromotherconsiderations,principallyofthedynamicsofthewake.

Onemightdoubtthatthefree-streamlinetheoryisapplicableat&

all,particularlyinviewofthelastremark.To sumuptheevidenceforandagainstit,thefolJawingexperimentalobservationsmaybe useful:(1)Thediscontinuitysurfaces,orfreestreamlines,idealizedinthe

c

theory,arewellapproxhatedby theactualshearlayers thatexistinarealfluid,forscmedistancedownstreamoftheseparationpoints(ref.1,p. 553). (2)Onthebackoftheobstacle,downstreamoftheseparationpoints,thedistributionofpressureisremarkablyconstantforalmostanyformofbluffbody,eventheextremeexampleofa flat@ate inclinedatsmallangleofattack(ref.2 orref.3, p. 6’79). Thismeansthatthevelocityatthetwoseparationpointsisthesame,a factwhichisratheressentialtothetheory.(3) T-heshearlayers donotcentinuefardown-streamas assumed,hut“roll.up”to formvortices,alternatelyoneachside.Thisvortexformationoccursbehindallbluffbodies,providedthereisno interferingbarrierbetweentheseparatedshearlayers,atafrequencywhichischaracteristicforeachbodyshape.FageandJohansen(ref.4) notedthatthevortexfrequencyforclifferentbluffbodiescouldbe correlatedby expressingitas a &Lmensiorilessfrequencybasednotonthebodydimensionbutonthedistanced’ betweentheshearlayers,measuredatthesectionwheretheybecomeparallel,before“rollingup.”Whetherornotthevorticesareformed,theideaoffreestreamlines

NACATN3168 3

extendingto infinityisunrealistic,fortheshearlayersdiffuserapidly.Therefore,itappearsratherfruitlesstobe tooconcernedwiththedetailsofthefreestreamlineat infinity; themainaimshouldbe to obtainasolutionwhichisvalidnearthebody. A morerealistlcwayto formulatetheproblemisas follows:

Theflowpasta bluffbodyisconsideredintwoparts.Nearthebodyitmaybe describedby thefree-stresailinetheory,providedthattheparameterk isproperlychosen.Thedescriptionofthewakefartherdownstreammustcomefromotherconsiderations;if itcanalsobe obtainedintermsoftheparameterk,thena completesoltiionmaybe foundbyJoiningthetwoparts.

Evenifit isnotpossibleto completethesolukioninthisway,ontheoreticalgrounds,theresultsofthefree-stresnilinetheoryshouldbequiteuseful,particularlyforcorrelatingvariousbluffshapesexperi-mentally.Forinstance,Iffroma studyof oneortwocasesitcanbedeterminedhow k dependsonwakebreadth,conditionoftheseparatedshearlayers,andsoforth,itmaybepossibleto choosetheappropriatevalueof k foranyotherbluffbody. As anotherexample,therelationsbetween“bluffness,”dragcoefficient,wakebreadth,sheddingfrequency,wakeenergy,andsoforth,mightbe easilyclassified.

TheKirchhoffsolutionforthenomnalflatplateisreviewed.Thenitis shownhowa morerealisticsolutionmaybe obtainedby allowingarbitrarybasepressures.Theexmnplesofa wedgeandcircularcylinder

thes

alsoworkedout.

TheresearchwasconductedatGAL1.XTunderthesponsorshipandwithfinancialassistanceoftheNationalAdvisoryCommitteeforAeronautics.

SYMBOIS

}

A1>A3Y● ● ●AQconstants

alya3J. ● .~an

k2+la=—k2-~

b

*%

*

streamwisedistanceto sectionwherestreamlinesbecomeparallel

dragcoefficient

.

h= k2.12k

k= @Jm

L

n

q

%

R

t

u.

U,v

w

x

Y

z

pressurecoefficient

valueof Cp at separationpoint

averageof CP (overa surface)

averagevalueof ~ onfrontof

NACATN 3168

andonbase

plateorwedge

breadthofanycylindermeasurednormalto stream

distancebetweenparallelfreestreamlines

functionsdefinedforconvenienceof calculation,appendixB

lengthofwedgemeasuredalonga side

wedge-angleparameter,&/11

magnitudeofvelocity(dimensionless)

valueof q at separationpoint

Reynoldsnumber

intermediatemappingfunction

magnitudeoffree-stream

componentsofvelocity

camplexstresmfunction,

streamwisecoordinatein

velocity(dimensionless)

ql+i~

realplane

coordinatenormalto flow

dimensionlessccmq?lexcoordinate,x + iy

.

.

*

NACATN3168

lb

e

P

T

5

half-angleofwedge

angu@?positiononcylindercircumference,measuredfr~stagnationpoint

mar disticeto =ps=tion point oncircularcylinder

1 1 ieinverseofcanplexvelocity,~ = ~ e

directionofflow

velocitypotential

complexvelocity,qe-ie

density

LeviCivitaplane

intermediatemapping

stresmfunction

intermediatemapping

intermediatemapping

‘tshaping”termin Q

function

function,logeg

functionfor k+l

KIRCHKOFIFPROBLEM

Theproblemmadefamousby Kirchhoffwasthatoftheflowpastaflatplatesetnormalto thestream.Itwillbe usefulto reviewhissolution,fo~owingessential.lythenotationofIamb(ref.3, p. 99).In sketch1 thez-planeistheplaneoftheactualflow.Thesolution

@s’1)CI

s

c!

4=--63~cs 1

s’ 1

Sketch1

6 NACATN 3168

Lconsists,asusual,ofmappingz ontothe_~laneoftheccmrplexpoten-tial w = q + i$. Thisisaccomplishedthroughthemediumofthecomplex khodographor v-plane,where v . ~ . u - iv. qe-ie isthecomplexvelocity.It iswiththisplanethattheeleganceoftheHe2mholtz-Kirchhoffmethodisrealized.Theessentialassumptionisthatevery-Were onthefreestreamlinesS1 and S’1 thevelocityisequaltothefree-str.esmvalue Um. Theninthehodographplanethefreestream-

:.

lineis siqlythecircleondly,thetraceofthepla~~l‘UW (=1 ‘er normalization).Sec.Ss‘ inthehodographplaneisknown,sincetheflowdirectionthereisconstant.Thustheboundariesoftheflowinthehodographplaneareknowna prioriandareof simpleform,sothatthemappingtothew-planeiseasilyaccomplished.

Inpracticeit isconvenientto use ~ insteadof V,where

c: leie=g=-=- Thus ~ givesthetrueflowdirectionandthedw“reciproc&ofthevelocitymagnitudeatthecorrespondingpointinthephysicalplane.Oncethemappingsareknown,thesolutionisgivencom-pletelyby

zf 1=$dw= Z(W)

(1)

where q2 determinesthepressure,sinceBernoulli’sequationmaybe

L4

8

usedto evaluatethepressurecoefficient

Cp=l+

Fromrelations(1)thepressureeverywhereccmputed.

intheflowfieldmaybe

Inparticular,at separationandallalongthefreestreamlineCP = Cps’=‘y ‘ince% = 1“ Thepressurecoefficientinthewakeandonthebacksideoftheplateisalsozero.Thedragthenissimplyduetotheexcessofpressureonthefront;itsvalueintheKirchhoffexampleIsCD=

Nowbeingof

0.88.

actual experienceshowsthatthedragisconsiderablylarger, utheorder CD= 2;thetncreaseisduemainlyto suctiononthe

●

an

of

.*

.

NACATN3168

backoftheplate,

freestreamlineat

7

CP5< 0. Correspondingto this,thevelocityonthe

separationishigherthanthefree-streamvalue.Ifqs= k&, k>l, then ~~=1- k2,andthedragcontributionfromthe

backsideis k2 - 1.

Thedifferencebetweentheccmuputedandtheactualdragisa seriousdiscrepancyintheErchhoffsolution,aswellas inthemanysolutionswhichhavebeenworkedoutforvariousbodyshapesusingthesametech-nique.Inotherrespects,theflowintheimmediatevicinityofa flatplatenormalto theflowdoesresembletheconditionsanticipatedinthetheory,asalreadystatedinthe“~troduction,”sothatthefree-streamlinetheoryneednotbe abandoned.Itwouldappearnecessarytomodifythetheoryonlyto theexbentofallowingthevelocityat separationto assume

appropriatevalue qs= kum.

Suchanadjustmentmaybemade,infact,by introducingthehodographsketch2. Here,thevelocityat separationIsal.lowedtobe qs= kUm

q#’s’ @

BUm., ~,I.“ B

s

Sketch2

andto remainatthisvaluealongthefreestreamline(circleinthehodographplane)untilthelatterbecomesparalleltothefreestream(pointB intheholographplane).At infinitytheflowmusthavereturnedto thepoint1, sothatthefreestreamlineinthehodographissimplydrawnby JofningBI,givingthenotchshownin sketch2. Thesingularity(doublet) isstillat I,as intheKIrchhoffhodograph,whichisshownbya dottedlineforcomparison.

This“notchedhodograph”isconvenient,foritmaybe easilymappedontothew-plane;bu%italsoapproximatesexperiments,sincethepressureonthefreestreamlinedoestendto remainconstantforsanedistancedownstreamoftheseparationpoint.Thesolutionforflowpasta normalflatplate,usingthishodograph,isworkedoutinthenextsection.

8 NACATN 3168

NOTCHEDHODOGRAPH

Sketch3 showstheplanesneededtomap

-lipB

theflowfrom thez-to the

Sketch3

w-plane,undertheassumptionthatinthehodographplaneitislikesketch2. Tomaketheradiusofthecircleinthev-and~-planesequalto unity,thevelocityat infinityissetat U@ = l/k;thatis, qs= 1.Thetransformationsare:

whichistheJoukowskytransformation,

●

.

——

th2+ X2.

=h2+l

NACATN 3168

<where

.h

~2-1=—

2k

and

w= l/t2

Solvingfor ( gives

c(

=-ix+ c)X2-1

where.

(2)

k2+l‘= #-l

Then,

d

= *i

[~ ~- + tm-l(~ ++G-j +a t=- ~

1 -wa F] -w

(3)*

theconstantof integrationhavingbeendeterminedfromtheconditionz =Oat w=O. Relations(2)and (3) givetheccxnplete solution..

10 NACATN3168

Ofparticularinterestarepointsontheplateandalongthefreestream-line,whichcorrespondto realvalues of w(+), so thattheexpressionstherearesomewhatsimplified.

has

x=

Y “

TO locatethesepositionsinthephysicalplane z = x + iy,onefromequation(3):

-

at~ k2+,.1Y=~$-1

Equations(4),(5), and (6) locatepositionsontheplate,onthefreestreWine from S to B, andrespectively.At B thefreethedistanced’,whichmay&or (6):

—onthefreestre~inefrom B to m,

streamlinesbeccxneparallel,separatedbyfoundbyputtingq = a2 inequations(5)

d’= nk k2+l (7)kz-~

#

.

NACATN3168

Also,thedownstreamdistanceto B is

()kp+lpb=— + ,Og u@-l ‘k2-~

Thepressurecoefficientmaybe evaluatedfrom

kp&2.1--cP=l -

Icl2

U

(’j’a)

Thevaluesof I(~2 me foundfromequation(3), whichgives,forthepositionscorrespondingto eqmtions(4),(~),and (6) above,theexpressions

To findthedragcoefficient,backoftheplatearecmnputed:

(8)

(9)

theaveragepressuresonthefrontand

whereusehasbeenmadeoftherelation

(lo)

I-2 NACATN 3168

.

ontheplate.Substitutingfromequations(9) givesfortheaveragepressureonthefront

.

[

1 (k2+ 1)2~=x k tan-l— - 1k2~-i 2(k2-1~(u)

Onthebackthepressureisconstant,sotheaveragepressurethereissimply

Finally,thedrag

Cps

coefficientis

= 1- kp (12)

(13)

Thebreadthoftheplate d iseasilyfoundfromequations(4)by setting~=1:

d( )

k2+lfi+ 2k +k2+ltm-1&-1=— -— —k2 =

(14) ~ka+l kp-1

Theexpressionsderivedabovecompleteydescribetheflow,foranyvalue .oftheparwneterk. For k = 1 theyreducetotheKlrchhoffsolution.Inreal.flows,however,k > 1 andtheproblemremainingisto choosethecorrectvalue.Thisproblemwillbe returnedto later (see thesec-tion“Discussion”).Firstitisnecessarytotestthevalidityofthe “-” ‘-notchedhodographby comparingitwithexperimentalresults.Forthis,thereareavailablesomeexcellentmeasurementsbyFageandJohansen(ref.2).

COMPARISONWITHEXPERIMENTSANDWITHANUN3ERHODCGRAPH

InoneofthecasesstudiedbyFageandJohansenthevelocityattheedgeoftheplateandthepressureonthebackcorrespondto k = 1.’34.Usingthisvalueinequations(8) and (9), thepressuredistributiononthefrontoftheplatemaybe computedandcmnparedwiththemeasuredvalues.Figure1 showsthattheagreementisexcellent.

NACATN 3168 13

Nowitisnotat allclearthatsomeotherhodographmightnotgiveequallysatisfactoryagreement,especiald.ysincethe“endpoints”

.ofthepressuredistribution( =land~s=l - k2) arefixedonceCPk hasbeenchosen.AnyotherhodographcurvejoiningthepointsSand I (sketch2)willgivethesamebasepressurecoefficient%$butwillotherwisechangeallthevaluesthathavebeencanputedabove.To investigatethesensitivityof sucha change,S1S’ was chosen tobea sinoothcurve,as shownin sketch4, definedin sucha waythatits

s’ @

kI “Elliptical’f

s Notched

L KZrchhoff

Sketch4

. inverse(inthe~-plane)isan ellipse.Thispermitsan easytrans-formation,thedetailsofwhicharegfveninappendixA. Thepressuredistributionontheplatecalculatedforthe‘telllpticalt’hodographis. alsoshowninfigure1. Itdoesnotagreewiththemeasurementssowellas doesthecaseofthenotchedhodograph.

A muchmoresensitivecmparisonisgivenbythepressuredistribu-tionalongthefreestreamline,as showninfigure2(a)forthetwocases.Thesuperiorityofthenotchedhodographisindicatedby a cmn-parisonwiththeexperimentalcurve(measuredforthispurpose),whichshowsthatthepressuretendsto remainconstantatfirst,asantici-pated.Figure2(b)showsthestreamlinesccmputedforthetwoholographsandforthe~chhoff case.Theshadedregionistheactualshearlayer,measuredbyFageandJohansen,whichthefreestreamlinesareintendedtoapproxhate.

An unusualandveryusefulfeatureofthenotchedhodographisthatthefreestreamlinesbecomeparallelat somesectionB-B’. Inthiswaya definitevalueofthewskewidth d’ isdefinedforeveryvalueof k.

t Thedefinitionofa wakewidthopenssamenewpossibilitieswhichwi~ betakenup later.

14 NACATN3168

Figure3 showshowthedragcoefficientCD andthewakewidth—

d’/d dependonthebase-pressureparameterk; thecalculatedvaluesaregivenintableI.

.—.

Aftertheabovemethodwasworkedout,theauthor’sattentionwasdrawntoanothermethod,whichwasgivenbyRiaboucliinsky(refs.~ and6)andwhichaccomplishesessentiallythesamething.Riabouchins@intro-duces,furtherdownstream,a secondplatewhichisthemirrorimageof -thefirstone. Thetwoplatesandthetwostreamlineswhichjointheircorrespondingedgesenclosea region,orcavity,whichisassignedanarbitrarypressure.Thisisthenthebasepressure,aswellas thecon-stantpressurealongthefreestreamline,andmayagainbe specifiedbytheparameterk. Fora givenk tieouter.potentialflowandtheshapeof thecavityaregivenby thetheory.Themaximumcavitywidth,whichvarieswith k,maybe takenasa measureofthewakewidth d’. Inshort,theRiabouchinskytheory,likethenotched-hodographtheory,givesa flowwhichdependsonthesingleparameterk andofwhichthe“frontpart”maybe usedtoapproximatetheflowneara bluffplate.Foragivenbasepressure,thedragisverynearlythesameasthatfromthenotched-hodographmethod,asmaybe expected-.Thewakewidth,however,issomewhatlarger,andthefree-streamlineshape,ofcourse,issomewhatdifferent.TheRiabouchinskytheoryhasbeenextendedtothecaseofwedgesby PlessetamdShaffer(ref.7),whoalsofounditnecessarytousenumericalmethodstoevaluat-esomeoftheintegralswhichoccur.Itcouldalsobe adaptedtothecaseofa circularcylinder,ashasbeendonehere-forthenotchedhodograph.

Inaddition,anearlypaperbyJoukowsky(ref.8) hasrecentlyccmeA

totheauthor’sattention.Thisgivesa generalmethodforthecasewitharbitraryvelocityspecifiedalonga freestreamline.Thenotched-‘ -hodographresultsforthebluffplateappearthereasa specialexample

.

andareinterpretedastheflowata channelmouthwhichis shieldedby-a flatplateaheadof it.

WEDGES

Theresultsoftheprecedingsectiongivesomeconfidencethatthenotchedhodographwillalsobe suitableforotherbluffbodyshapesandthatmanyofthesolutionsthathavebeenworkedoutforthe?Chchhoffconditionk = 1 canbe generalizedinthesamewayasthecaseofthenormalflatplate.Thegeneralizationisstraightforwardfora wedgeof

NACATN 3168 15

arbitrarynoseangle %, showninsketch5. For k = 1 theproblem

.

Sketch5

QQv

hasbeenworkedoutby Bo’byleff(seeref.3, p. l@).

Theonlyadditionaltransformationrequiredisonewhichtillopen. thesegnentinthe~n-planeontothehalfcircleinthe~-plane,frm

wherethemappingto thew-planeis identicalwiththatworkedoutinthesection“NotchedHolograph.”Theappropriatetransformationis

where

n= 2a/Ye

Also,then

l/n‘1 ‘%

16

(Thesubscripts1 and n,to distinguishflatplateusedtopreventconfusioninthetransformation;theylater.)Themappingfrcm w to <n then,referringis

NACATN

andwedge,aremaybe amittedto-equation(2),

Themappingfrom w to z is

3168

.

(16)

(17)

Sincea generalsolutionof theintegralinequation(17) couldnotbefoundforarbitraryvalues of n, orevenforspecificvalues of n otherthanO or1,onecase(n= 1/2,correspcmdingto a = 45°) wasworkedoutnumerically.Theresultingvaluesfor CD and d’/d,asfunctionsof k,areshownInfigure4 andtabulatedintable11. Thedetailsforthenumericalintegrationae @ven tiappendixB.

CIRCULARCYLINDER

Inattemptingto applythefree-stresmlinecylinder,twonewdifficultiesareencountered.

theorytothecircularOntheonehand,the

traceofthecylindersurfaceinthehodograyhplaneisnotknownapriori.Second,theseparationpointinthephysicalplaneisnotknown,as itwasinthecasesoftheplateandwedge.Thefirstdifficultyisoneonlyofdegree— themappingmayalwaysbe accomplishedinprinciple.Theproblemoftheseparationpoint,however,ismoredifficult;itmaybe appreciatedfromthefollowingdiscussion.

.—.

NACATN 3168 17

Considertheflowpastthecurvedarcshownin sketch6. If the

s

Sketch6

openingangle 2~s isnottoolarge,thenseparationtilloccurat S,justas inthecaseoftheflatplate.Thecurvatureofthefreestream-lineat separationwilldependonthebasepressurecoefficientCps>thatis,onthepsmmeter k. Thelowerthebasepressure,themoresharplywillthestreamlinecurvetowardthecenter.As longasthe“cylinder!’consistsonlyofthecurvedplateshownbytheheavyline,thestreamlinecanadJustitseMto anybasepressure,alwaysseparatingatthepoint S. Butifthecurvedarcisactuallypartofa completecylinder,showndotted,thentheradiusofcurvatureofthefire~inecannotbe smallerthanthatofthecyllnder;otherwisethestreamline

a wouldintersectthecylinder.Itmay,of course,be larger.Foroneparticularvalueof k,thatis,ofthebasepressure,itwillbe @!tequaltotheradiusofthecylinder.Ifthecylinderisactuallythe

. ccmpleteone,then ps isnotknowna priori;butif itbe assumedthatthestreamlineat separationhasthesamecurvatureasthecylinder,thentherewillbe a uniquevalueof ~s foreveryvalueof k. Thus,withthisassumption,a solutionmaybe obtained,dependingas inthepreviouscasesonlyontheparameterk. Ontheotherhand,iftheradiusofcurratureat separationisassumedtobe differentfrom(greaterthan)thatofthecylinder,then 13swillbe saneotherfunctionof k. Thisuncertaintyabouttheconditionsat separationmakesthefree-streamlineproblemofthecylinderconsiderablymoredifficultthanthecasewithfixedseparationpoints.

It seemsworthwhile,asan initialstep,to workoutthecasewherethestreamlinecurvatureisthesameasthatofthecylinder.Thishadalreadybeendonefor k = 1 by Brodets~(ref.9) andlaterby Schmieden(ref.10). Usinga mappingdueto LeviCivita,Brodetskywasableto

. obtainan approximatesolutionby an iterationprocedurewhichconverges,quiterapidly.He foundtheseparationto occurat Ps= 35°,givinga

.

18 NACATN3168.

dragcoefficientCD = 0.5. Itmaybe expectedthatfor k > 1 thevaluesof p~ and CD willbe higher. .

To investigatesucharbitraryvaluesof k,Brodetsky’6methodhasbeenadaptedtothenotchedhodograph.A morecompletediscussionofthemethodisgiveninappendixC. Therestiltsaregiveninfigure5,whichshowshowtheseparationpoint ~s,thedragcoefficient~, andthedistancebetweenstreamlinesd’/d varywith k. Althoughtheiterationwascarriedthroughonlyonest-efiytheresultsupto k = 1.6 ““appeartobe accurateto a fewpercent”(appendixC). At highervaluesof k theaccuracy(withoneIteration)becomesmoreuncertain,sothese “- ~havenotbeenplotted.

Figure6 ccmparesanexperimentalpressuredistributionwithonecomputedonthebasisoftheabovetheoryfor k = 1.4,chosentomatchthebasepressureontheexperimentalcurve.ThereisconsiderableimprovementovertheKirchhoffcasecomputedbyBrodetskyandSchmieden.Onlyinthevicinityoftheseparationpointistherea seriousdis-crepancy,resultinginquitedifferentvaluesfor @s inthetwocases.It isinthisregionthattheuncertaintyabouttheseparationconditionismostnoticeable.Clearlytheassumptionthatthestreamlinehasthesamecurvatureasthecylinderisnotsatisfactory.Withthisassumption,thereisno adversepressuregradient(fig..6),whereasit iswell-knownthatto separatea boundarylayerona continuoussurfaceanadversegradientisnecessary;itexistsintheexperimentalcase.

Onemight,ofcourse,introducemoreplausibleassumptionsaboutthe P

curvatureat separation,butforthisitappearsnecessaryto gobacktoa studyoftheboundary-layerseparation.Thatthenatureoftheboundarylayercannotbe neglectedisclear,sinceathighReynoldsnumbers(above -105)theseparationpointmovesto ~~> 90°. Thetheoryintheaboveformis sui~ableonlyforvaluesless-than

Ineachoftheeverycasetowhichparsmeterk,whichInthisrespectthetheory,inwhichit

DISCUSSION

examplestreatedintheprecedingsections,andinthetheoryapplies,thesolutiondependsonthesinglecannotbe determinedwithoutfurtherconsiderations.theoryisnotclosedas istheclassical(Kirchhoff)issimplyassumedthat k = 1. Thatchoiceof k,

however,isarbitrary,andexperienceshows.~ttobe unrealistic.Onemightjustaswellchoosea valueinmuchbetteragreementwithexperience.Forinstance,ifthevalueisdetermined,experhentally,foronebluffbody,thenitmaybe usedfairlyconfidentlyforothershapesatthesame *

Reynoldsnumber;atleasttheresultwillbemuchbetterthanthatwithh

NACATN3168 19

k=l. However,thisisa roughobsenationbasedonempirical.informa-tion,notonan essentialunderstandi~oftheproblem.A theoretical

. orsemitheoreticalbasis(e.g.,dimensionlessanalysis)isneededtochoosek,butthefree-streamlinetheorycantakeoneno furtherinthisdirection.Someessentiallynewinformationmustbe added;andthis,itappears,willnotbe obtainedwithouta considerationofthewake,themechanicsofwhichplaysa partin settingthebasepressure.

Duetothefactthatthewakesofdifferentbluffbodiesdevelopinthesameway,frcsnthetwoseparatedshearlayers,theyhavemanyfeaturesincommon.k fact,a wakemaybe discussedindependentlyofthebodyifitsgeometricalandvelocityscalesareknowm.ThisisbroughtoutinK&n&n’sanalysisofthevortexstreet,inwhichthetwoparametersneededto closetheproblemarea dimensionanda velocity(relativeto thebody). Theonlyfunctionofthebcdyisto determinethesetwoparame-ters,or scales;itcanhardlyhaveanyfurtherinfluenceon subsequentdevelopmentsinthewake(exceptforReynoldsnunbereffects).Thatis,thewakeiscompletelydeterminedby a specificationofthegeometricalandvelocityscalesearlyin itsdevelopment,inthetransitionfromthebodyregimetothewakeregime..Thevelocityscalemayveryappropriatelybe characterizedby thevelocityalongtheedgesofthefreeshearlayers,whilethegeometricalscalemaybe specifiedby thedistancebetweenthefreeshearlayers.Thesecorrespondto theparametersk and d’/d. Inthefree-streamlinetheorybasedonthenotchedholograph,therelationbetweenk and d’/d isdetermined,fora givenbodyshape,sothatthereisonlyoneindependentparameter,a resultwhichshouldproveveryuseful.

.Whileitmaybepossibleto obtainsameresults,havingonlythe

geometricalandvelocityscalesofthewake,itwillbe necessaryeven-. tual.lyto considertheReynoldsnumbereffects.Thesearerelatedprin-cipallytothestateofthefreeshearlayers,or oftheboundarylayerbeforeseparation.Insteadofa Reynoldsnumberbasedonthebodydimen-sion,itwillprobablybemoreappropriateto introducethethicknessoftheshearlayeranditsratiowithrespectto d’. Thisproblemhasreceivedmuchmoreattentioninthesupersonic“base-pressureproblem”thanintheolderproblemof incompressibleflowpastbluffbodies.

I.

Anotherwayinwhichtheresultsofthenotched-haiagraphtheorymaybe usefulisrelatedto theobservationsmadebyFageandJohansen(ref.2),alreadymentionedinthe“Introduction.”Theyobservedthatthefrequencyofvortexsheddingfroma bluffbodydependednotonthedimensionsofthebodybutonthedistancebetweenthefreeshearlayers,andtheywereableto geta goodcorrelationbetweenbodiesofdifferentshapesby usingthisdistsmceinthetiensionlessfrequency.A singleparameterlikethismightbe used,in conjunctionwiththefree-streamlinetheorysmdwithmeasurementsofthesheddingfrequency,to determinethedrag,forthesewouldgive d’/d andthus k and CD. Hereagainthere

.

20 NACATN3168

.

issomedependenceonshear-layerthickness

Reynoldsnumber,orratherontheratiooftheto d’.

.

Althoughthereseemstobe littlepossibilityatpresentofobtaininga theoreticaldescriptionofthemechanicsofthewake,especiallyintheregionwhereitdevelopsfromthefreeshearlayers,thereis stillthepossibilityoffindinga correlationbetweenbodiesofdifferentshapesona semiempiricalbasis.Sincethecompletionofthiswork,a study,basedonexperiment,hasbeenmadeofthedependenceof k ontheshearlayersandonthedistancebetweenthem.Thefree-streamlinetheoryiscombinedwithsme experimentalresultsto..obtaina correlationbetweenbluffbodiesofdifferentshapes,aswellas someoftherelationsbetweenwakeandbodydiscussedabove.Theresultsofthissemiempirlcalstudyarepresentedinreference11.

CaliforniaInstituteofTechnology,Pasadena,Calif.,August3,1953.

.

.

.

.

NACATN3168 2L

APPENDIXA

EIJXCPI!ICALHODOGRAPH

Ifthefreestreamlineinthe[-planeisan ellipse(sketchAl),

>

0cs

I

s’

kJ )QEs

I

s’

11SketchAl

themappingto thew-planeiseasilyfound. (AlthoughthecorrespondingtraceofS1S’inthev-plane(sketch4) isnotan ellipse,itwil.lbeconvenientlyreferredto astheelMpticalholograph.)

Theellipsein ~ isfirstmappedontotheunitcircle,ina newplane cl,frcmwherethemappingto thew-planeisthesameasthatititheKirchhoffcaseoffigure1. Thefirstmappingisaccomplishedby aJoukowskytransfo~tion,

whilethesecondis (cf.eq.(2))

Thesegive

22 . . NACATN 31.68

.Ontheplate

.

x = o

whileonthefreestreamline

Thebreadthoftheplateisclearly

d =kl’c+4

To evaluatethepressurecoefficientgivenby eq~tion(8),

sothat

ontheplateand

—

NACATN 3168 23

onthefreestreszuline.Theaveragepressure,c~uted fr~eq~tion (lo))onthefrontfaceis

whichmaythenbe usedto findthedrag

.

(j)= ~+k2-1

24 NACATN 3168

APPENDIXB.

.

MAPPINGTHEWEDGE.

Thegeneralmappingforthewedgeisgivenby sketch5 andrela-tions(16)and(17).Onlythewedgesurfaceandtheconstant-pressureportionofthefreestreamlinewillbe ofparticularinterest.Onthesew = q isrealandequation(16)reducesto

where

e =a-nm

and

withTheaveragepressureequation(10),is

normalto oneofthe

-!

(Bl)

(B2)

frontfaces,ccnnparing

(B3)

wherel/q = Icl inequation(Bl).(Thesubscripthasbeenomittedon‘n butretaineon kl.) Thecomponentinthestreemdirectioncontrib-”utestothedragtheamount~ sinm. Butequation(B3)isaveraged

on L,whereasitismoreconvenientto computethedragwithreferencetothebasedimensiond = 2L sina. Onthisbasisthecontributionfromthefrontbeccmes~, andthedragcoefficientis

.

—

.

NACATN 3168 .25

●

=~+l?-1

ThelengthL & a frontfaceis

ThepositionofthepointB is

% J’=Leia+a2eiGdcp1

Inparticular,thedistanced’ betweenfreestreamlines,giventig-y partof zB,iS

d’ =2Lsina+2J

sine @1

Ja2

I

82=2Lsina,+2sincc cosmdp-2cosa sinm

1 1

andthestresmwisedistanceto B, measuredfromthebase,is

b =2cosaJ

cosm&p+2sinaJ

sinm dq1 1

Theabovefollowing

expressionshavebeenreducedto theoperationslistedsummary:

bythe

dq

inthe

.

26 NACATN 3168

++) =~a2CCJSm dq

~2

JI(n,kj= ~ SiIl UJ.)dq)

where

() 2

a2 .k12+ 1

k12- 1

n = 24Tt

.

.

b

.

(D= -tEmr-1 ~-l

1- cp/a2

NACA~ 3168 27

.Then

.

.

()2 nL= ‘1+1 ~

2k~

Thesolutionforgivenvaluesof kl and n correspondsto flowovera

wedgeofhalf-anglea = M/2 andbase-pressureparameterk = kin.

28 NACATN 3168

Brodetskycurvedarccan

APPENDIXC

MAPPINGTHECIRCULARCYLINDER

(ref.9) showshowtheKirchhoffflow(k= 1)pastabe computedbymeansofthemappingsshowninsketchCl.

s’(ie

sEoQI loge;s’

owSI

I&S’ I

sDoT

IC

s’

SketchCl

First,~ ismappedontotheQ-plane,oftenusedinfree-stremlinetheoryanddefinedby

.

.

.

“

i2=lo&c=loge;+i0 (cl)

Therealpartof Q dependsonlyonthemagnitudeofthevelocity,whiletheimaginarypartistheflowdirection.Thevaluesof e onthelineCSgivetheflowdirectionalongthesurface.ForthenormalflatplateCSinthefl-plsaewouldbe straight,simply(3. ilc/2, Now Q istobemappedontothew-plane,andtheideausedisthatthemappingforthearcmaybe obtainedby addinga “correction”tothemappingfortheflatplate,whichisknown.

NACATN 3168 29

‘loaccomplishthis,itishelpfulto replacew bytheIeviCivitalplane T, definedby

(C2)

IntheT-planetheflowismappedontotheinteriorandboundaryofthesemicircle,whichhasthedoubletat 1. Forthenormalplatethemappingis

0 l+T= lo& — (C3)1 ‘T

It isassumedthatforthecurvedarc

Q= loge* + Q’(T) (C4)

where slr isto be determined.Thesingularityoccursinthefirsttermsothat Q’(T) maybe e~ressedasa powerseries,

~’(T) 1A T3 -t-...‘A~’+y 3 (W)

Theconditionthatthestreamlinecurvatureat separationshouldbethesameasthatofthesurface(seethesection“CircularCylinder”)isshownbyBrodetskyto imply

‘1 =-1+ al

A3 =al+a3

1A rathercompletediscussionoftheLeviCivitatransformationandoftheS2-plane,aswellasapplicationsto themapp~gOfcurvedsurfaces)isgivenbyBrillouininreferenceI-2.

30 NACATN 3168

that is,

.

Q = loge&-(’ -a’)T+@’+a)’3+””” (c6)

To evaluatethe ~’s, theradiusofcurvatureofthesurfaceisexpressedintermsofthem.Fora circulararcthey-arethendete~inedtomaketheradiusofcurvatureconstantonthearc. Thesolutionproceedsby iter-ation,startingwith al # 0, a3 = a5= . . . = 0. Br~ets@ showedt~tal . 0.0574givesanarcwhoseradiusisconstantwithin~percent,while al = 0.0585ja3= -0.0083reducesthemaximumdiscrepancyto0.05percent.Thecorrespondingvaluesoftheseparationangle B8 are55.1°and55.0°,whilethedragcoefficientsare0.49and0.500,respec-tively.Thisindicatesthattheconvergenceisquiterapid,al already ““4givinga fairlyaccurateresult.Therefore,inadaptingthemethodtoflowswhere k > 1, itwasconsideredsufficientto useonlythisone

—

termoftheiteration.

For k > 1,the ~k-pbneappearsas ShowninsketchC2,“notched”

SketchC2

likethecorresponding~-plane;thatis~alongSBthevelocitYiscon-stant(q= 1),whilealong131andB’Ithestreamlinesareparallel.Now

—

~k canbemappedontotheQ-plane(sketchCl)bythetransformation

SiIlh2 ~k =(Wp”+(-fl (cl’) ‘

Thephysicalplanecorrespondingto ok willbe calledzk,whilethatcorrespondingto,Q is z.

NACATN 3168 31

.

Nowtheprocedureis simplyasfollows:A valueischosenfor al.

. Thisdeterminesa certaincylindershapeinthez-planewhichcorrespondsto anothershapeintheZk-pkne.Then k iscomputedtogivea con-stantradiusofthecylinderinthezk-phe> orasnearlyas ispossiblewithonlythesingleterm al. Thusforeachvalueof al thereisavalueof k whichgivesa circulararcinthezk-p~e. Thevaluescalculatedaregivenintable111. Eacharchasa definiteopeningangle 2~~ whichdeterminesthecorrespondingseparationangle(sketchCl). Theresults, referredtoalreadyinthesection“CircularCylinder,’!areplottedinfigure5. Ofcourse,usingonly al inseries(c6),itisnotpossibletomakethearcperfectlycircular.Allthatcanbe doneisto checktheaccuracyobtainedat eachvalueof k.Itwasfoundthatat k = 1.6 theradiusisconstantwithin2 percentandcomparabletothisat lowervalues.At highervaluesof k theaccuracybecomeslower;thereforethecurvesoffigure5 havenotbeenextendedbeyondk = 1.6.

Detailsoftherathercumbersomecalculationsforthevariousquan-titiesarenotshownhere.TheyareanalogoustothoseinBrodetsky’spaper,withonlytheadditionalintroductionoftransformation(C7)togivethecorrespondencebetweentheQ- and~k-phes.

Itmightbe we~ topointoutwhyitwasnecessaryto introducethistransformationfrom ~ to Q insteadofworkingdirectlybetween~and T. Withthelatterprocedurethenotchwouldhavebeendistorted,.sincetheapproximatesolutionattemptsonlyto obtaina fitonthesur-faceSC. Intheprocedureused,ontheotherhand,the adjustmentsare

. madeintheQ-planeinsucha waythattheimaginaryvaluesonSCareconstantintheok-plane.Thereisnoneedthento worryaboutBIB’,forequation(C7)mapsitexactlyontothenotchinthe~k-pl.ane.

32 NACATN 3168

1.

2.

3*

4.

59

6.

7=

8.

9*

10.

11●

E.

REFERENCES

FluidMotionPaneloftheAeronauticalResearchCommitteeandOthers,(S.Goldstein,ed.): ModernDevelopmentsinFluidDynamics.Vol.II. TheClarendonPress(Oxford),1938.

Fage,A.,andJohansen,F. C.: OntheFlowofAirBehindanFlatPlateofInfiniteSpan.R.& M. No.1104,BritishA.1927;alsoProc.Roy.Sot.(London),ser.A, vol.116,no.Sept.1,1927,pp.170-197.

InclinedR. C.,773,

Iamb,Horace:1932.

Fage.A.,andR. &M. No.vol. 5, no.

Riabouchinsky,LondonMath.

Riabouchinsky,Proc.Iondon

Hydrodynamics.Sixthcd.,TheUniv.Press(Cambridge),

Johansen,F. C.: TheStructureofVortexSheets.1143,BritishA. R. C.,1927;alsoPhil.Msg.,ser.7,28,Feb.1928,pp.417-441._

D- OnSteadyFluidMotionWithFreeSurfaces.Proc.s;;.,VO1.19, lg21, pp. 206-a5.

D- On SomeCasesofTwo-DimensionalFluidlbtion.ti~h.Soca, ser.2, vol.25,pt. 3, 1926,pp. 187-194.

Plesset,M. S.,andShaffer,P.A.,Jr.:CavityDraginTwoandThreeDtiensions.Jour.Appl.Phys.,vol.19,no.10,Oct.1*,PP.934-939.

Joukowsky,N.E.: I - A ModificationofKirchoffIsMethodofDetermininga TwoDimensionalMotionofa FluidGivena ConstantVelocityAlonganUnknownstreamLine. II- DeterminationoftheMotionof a FluidforAnyConditionGivenona StreamLine.WorksofN.E.Joukowsky,Vol.II,Issue3, Trans.CAHI,No.41,1930. (Originallypublishedin 1890.)

Brodetsky,S.: DiscontinuousCylinders.Proc.Roy.Sot.Feb.1,1923,pp.542-553.

Schmieden,C.: DieUnstetigeIng.-Archiv,Ed.I,Heft1,

FluidMotionPastCircularandElliptic(Iondon),ser.A, vol.102,no.718,

Str6mmgm einenI@eiszylinder.Dec.1929,ppi 104-109.

Roshko,Anatol:OntheDragandSheddingFrequencyofTwo-DimensionalBluffBodies.NACATN3169,1954.

Brillouin,M.: SW lessurfacesdeglissementdeHel.mholtz.Ann.chimieetphys.,ser. 8, vol. 23, 1911,pp. 145-230.

*

.

b

I’UCATN3168 33

.

.

.

k

1.001.051.101.151.201.301.401.501.601.802.00

0.880.868.855.843A%

:g

●735.690.650

TABLEI

FIATPLATE

-4-w-1.0001.1061.178I. 2361.3601.452

.--.- -----

1.7420.6331.733 .6391.723 .6431●710 .6491.696 .660

CD

0.880.970

1.@51.1661.2711.4$%1.7422.0082.2952.9303.650

TABLEII

90°WEIX2E

J

d’/d

9.205.0733.6152.8882.1681.8141.6061.4711.3081.217

------

19.8707.3704●3902.0101.31.1

I

------14.5105.0902.79010IIU.683

b/d

59:40015.goo7.6m4.6102.3$101.5701.170.936.692.569

qlc~ I d’/d b/d

0.637.562.515.475.411.361

0.637.786.905

1.0031.2611.471

4.30 19:5402.280 7.8cm1.889 3.9801.438 1.5281.328 I. 036

TABLEIII

CIRCULARCYLINDER

I al I k

I 0.0574 I 1.000 1IO& l.1~

1.263●0200 1.387

0. 1.470-.0400 1.6cm

3k

1.2

.8

.4

0

Cp

-.4

-.8

-1.2

-1.6

NACATN3168

.

.,

NOTCHED— HODOGRAPHCD= 2.130’

ELLIPTICAL---HODOGRAPH CD=2.165

,

THEORY

w0 FAGE &

JOHANSENCD= 2.13

}E)(P.

‘1.38—~

Figurel.-Pressureonflatplate.

NACATN3168 35

0 I I

\ELLIPTICALHODOGRAPH

(k=l.sql

NOTCHED-HODOGRAPH

(k=l.sq)

-.5

Cp

-1.0

EXPERIMENTAL

-1.5 .5 1.0 I, I

2.0 2.5 3.0X/d

(a) Pressureonfreestreamline.

k=l

u \

ELLIPTICALHODOGRAPH

(k= 1.54)

.

y/d.6

NOi’CHEDHODOGRAPH

(k=l.54).4

i

.2

.

.

0 1.2 1.4x/d

(b) Freestreadines.

Figure2.-C!omparisonofellipticalandnotched

36 NACATN3168

I

1.2

+

1.4 .k

Figure3.- Normal

/

CD

Id/d

1.6

flatplate.

1.8

5

CD

4

3

2

I

o2.0

.

.

.

NACATN3168 37

3.8

Id/d

3.4

3.0

2.e

2.2

1.8

1.4

I.C

1.8

1.6

CD

1.4

1.2

I.0

.8

.6

.41.0 1.1 t.2 1.3 1.4 1.5

k

Figure4.-$20°wedge.

2*2

dyd

I .8

I .4

I .0

8(

6(

Ps

4(

2C1.0 1.2

k1.4

F@me 5.-Cirailarcylinder.

# .

I.7

CD

.3

.9

.51.6

L3

,

.

r

MEA IRED

k k(THEORy) (ACTUAL) I

‘REE-STREAMLINE THEORY

= (k=[.d)

+W ;=(R = 14,500)

0

)?igure6.-Fressuredistributionon circularcylinder.

140 160 180