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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 [email protected],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
[email protected]$%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