NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALINOTE3221

STUDYOF TEE SUBSONICFORCESANDMOMENTSONAN

INCLINEDPLATE OF INFINITESPAN

By Bradford H. Wick

Ames Aeronautical LaboratoryMoffett Field, Calif.

Washington

June1954

AFMBC

TECHLIBRARYKAFB,NM

Iilllllllllllullll[lllllNATIONALADVISORYCOMMITTEEFORAERONAUTIC.

STUDYOFTHESUBSONICFORCESANDMOMENTSONAN

INCLINEDPLATEOF INFINITESPAN

ByBradfordH.Wick

SUMMARY

A studyhasbeenmadeof existingexperimentalandtheoreticalresultsforan inclinedflatplateof infinitespan,andof theextenttowhichtheresultsareindicativeofthoseforthinairfoilsections.Thestudyincludedan examinationof theflowaboutan inclinedplate,theforcesontheplate,andtheadequacyof theoryinpredictingtheforces.Theoriesconsideredwere thewell-knownthin-airfoiltheory,andthetheoryofdiscontinuouspotentialflowandmodificationsthereof.Theeffectsofcompressibilitywereexsmined.

s Theresultsof thestudyindicatethattherearetwoimportantrangesofangleofattackdifferingby theextentof flowseparationontheuppersurface.At anglesofattackbelowabout8°, flowseparation

. andreattachmentoccur,andthewell-knownthin-airfoiltheoryisade-quateforpredictingtheliftandnormalforceon theplate.Similarresultswerenotedforthinairfoilsections.At thehigheranglesofattacktheflowiscompletelyseparatedfromtheuppersurfaceas isassumedinthediscontinuouspatential-flowtheoryforan inclinedflatplate.Thetheory,however,is entirelyinadequate.A simpleempiricalmodificationofthetheoryis suggested;themtiifiedtheoryprovidesagoodfirstapproximationof theforcesandmomentson thinairfoilsec-tionswiththeflowcompletelyseparatedfromtheuppersurface.Effectsof compressibilitywereevidentfromtheavailableexperimentaldata;however,theeffectswere notdefinedsufficientlyforevaluatingmethodsofprediction.

INTRODUCTION

Theresultsof studies,by earlyresearchersinhydrodynamics,oftheflowaboutandtheresultantforcesonan inclinedflatplateofinfinitespan,heretofore,havehadlittlepracticalapplication.The

h typeof flowconsidered,consistingof detachedflowovertheuppersurface(i.e.,rearwardsurface)andattachedflowoverthelowersur-face,wasnotencounteredon conventionalairfoilsin theangle-of-attack

v rangeofpracticalinterest.Withtheintroductionof thinairfoilsand,

2 NACATN 3221

inparticular,thosewithsharpleadingedges,theforegoingcircumstance.

no longerexists.Theseparatedtypeof flowhasbeenfoundtooccuronthinunsweptwingsatandabovetheangleofattackformaximumlift,on b“

thinsweptbackwingsconsiderablypriortowingmaximumlift,andonthinpropellerswhenoperatingat take-offconditions.Itappearedworthwhile,therefore,tomakea study ofexistingtheoreticalandexperimentalresultsfortheflatplateandtodeterminetheirapplicabilitytothinairfoilsections.Theresultsofthestudyarereportedherein.

cd

cl

cmc/4

Cn

P

P‘av

‘Za~

c

M

P

Po

%

v

V.

Xcp

a

NOTATION

sectiondragcoefficient,~qoc

sectionliftcoefficient,&

sectionpitching-momentcoefficient,momentcenterat c/4,pitchingmoment

%C2

sectionnormal-forcecoefficient,‘0-1 ‘orc~.-

pressurecoefficient,

averageupper-surface

averagelower-surface

chord

P-POT

pressurecoefficient

pressurecoefficient

Machnumberoffreestream

localstaticpressure

free-streamstaticpressure

free-streamdynamicpressure

localvelocity

free-streamvelocity

center-of-pressurelocation,distancealong chordfromleadingedge,fractionsofchordlength

angleofattackof chordplane,deg

k

.

NAcATN3221

RESULTSANDDISCUSSIONOFSTUDY

Therearetwoof infinitespan.

theorieswhicharepertinenttoanOneis theso-calleddiscontinuous

inclinedflatplatepotential-flow

theory(ref.1,pp.330-336)whichtreatsthecasewheretheflowis com-pletelydetachedfromtheuppersurface;theotheris thewell-knownthin-airfoiltheory(ref.1,pp.24-53)whichtreatsthecaseofunsepa-ratedflow. Sincetheformertheoryhasbeenof littlepracticalinterestand,consequently,isnotsowellknown,thefollowingbriefdiscussionisbelievedin order.

Thefirstcompletetreatmentoftheseparatedtypeof flow,usingmethodsof classichydrodynamicsappearstobe thatpresentedby Rayleighin 1876. Hetreatedboththecaseof theplateobliqueto thestresmandnormalto thestream.Kirchhoffsomeyearsearlier(in1869)hadcon-sideredbothcasesbutpresentedcalculatedresultsonlyinthecaseoftheplatenormalto thestream.Althoughworkingindependently,theirapproachwasa commonone,makinguseofHelmholtz’shypothesisofasurfaceofdiscontinuity(i.e.,a surfacewhichseparatestwostreamsofdifferentvelocities).As a consequenceof theuseofthishypothesis,

m theirapproachisknownintheliteratureas themethodofdiscontinuouspotentialflow.

v A completedescriptionof themethodisgiveninreference1. Thesalientfeaturesofthemethodareas follows:It isassumedthatlinesofdiscontinuitystartat theleadingandtrailingedgesof theplateandextendto infinity.(Seefig.1.) Withinthetwolinesthefluidisassumedtobe at restwithrespectto theplate.Outsidetheselinestheflowisassumedtobe smoothandsteady.As a resultof theflowconditionsassumed,thepressurein thewake(i.e.,theregionboundedby thelinesofdiscontinuity)isconstantandequaltothefree-streamstaticpressure,andthevelocityoutsidethewakeisequalto thefree-streamvelocity.

Thesolutionfortheforceontheplateduetoabouttheplateis,incoefficientforms

2X sinsCn= 4+ fisina

Theing

Thethe

center-of-pressurelocationinfractionsof theedgeis

= 0.50 - cos a0.75 4 + x sinaXcp

derivationoftheequationfor Cn isgiveninequationforthecenter-of-pressurelocationis

thedescribedflow

(1)

chordfromthelead-

(2)

references1 and2;fromthederivation

4

giveninreference2,whereinratherthantheleadingedge.

mcA TN3221

.

thelocationisreferredtothemidchord

Sincethereis onlya normalforceactingontionsforthecoefficientsof liftanddragare

Cz= 2Y(sinu 0sa4+fis&Z%rsin2cd = CLk+ fisina

h

theplate,theequa-

(3)

(4)

Theforegoingequationsarepresentedingraphicalforminfigure2,togetherwiththethin-airfoil-theoryresultsandtheexperimentalresultsfora flatplateasmeasuredby FageandJohansen(ref.3). (Theexperimentalresultsareuncorrectedfortheconstraintofthetunnelwalls.Itis statedinthereferencereportthatthemeasuredvaluesofthenormal-forcecoefficientshouldbe reducedby amountsvaryingfrom8 percentata = 30°to 13.5percentata = 900.) Theadequacyofthin-airfoiltheoryinaccountingforthemagnitude,ofthenormalforceandtheliftontheplateinthelowangle-of-attackrange,andtheinadequacyoftheRayleigh-Kirchhofftheorythroughouttheentireangle-of-attackrangearereadilyapp=entfromthefigure.Inthecaseofdragcoef-ficientandcenter-of-pressurelocation,boththeoriesareinadequatethroughouttheangle-of-attackrange.

Thatthin-airfoiltheorywouldbe applicableinpredictingtheliftofa flatplateat lowanglesofattackmayseemsurprisinginviewoftheprobableseparationof flowfromtheleadingedgeoftheplate.Itappears,however,fromtheoreticalconsiderationsandan examinationoftheliftandflowmeasurementsona thinsharp-edgeairfoilsection(ref.4),thattheapplicabilityofthin-airfoiltheoryisdeterminedprimari.lybytheflowconditionat thetrailingedge. Theliftmeasure-mentsasgiveninreference4 forthethinsharp-edgeairfoilsectionarereproducedinfigure3; thedatawerenotcorrectedfortunnel-walleffects.Alsoshownarethevaluesofliftindicatedby thin-airfoiltheoryandtheRayleigh-Kirchhofftheory.Theextentoftheseparated-flowregionisindicatedin figure4,whichis a reproductionofafigureinreference4. Theboundaryof theseparated-flowregionwasdefinedby thezero-velocitypointinvelocitydistributionsabovethesurfacewhichweredetemninedby rakesofconventionalstatic-andtotal-pressuretubes.Itisnotedfromfigure3 that,as fortheflatplate,theliftvariationwithangleofattackwasessentiallythatspecifiedby thin-airfoiltheoryup toabout7.5°, andthendeviatedrapidly.Thedataontheextentofflowseparation(fig.4) showthattheflowsepar-atedfromtheleadingedgeata verysmallangleofattackandthenreattachedfartherbackalongthesurface.Thepointofreattachmentmovedfartherbackwfthincreasingangleofattackuntilat 7.5°, theangleofthelift-curvedivergence,theflowwascompletelyseparated

5NACATN 3221

fromtheupperdependentupon

surface.Thattheamountof liftdevelopedisprimarilytheflowatthetrailingetieis.of course.tobe

expected,sincein thin-airfoiltheory-the-amo&tof liftisestablishedby satisfyingtheKuttaconditionat thetrailingedge. Leading-edgeflowseparationcould havean effecton theamountofliftdeveloped,however,througha changeintheboundary-layerthicknessat thetrailingedge.Anotherwaythattheleading-edgeflowseparationcouldpossiblyinfluencetheamountof liftisthatitproduces,ineffect,a camberedairfoilformedby theplateandtheseparated-flowregion.If suchwerethecase,thethin-airfoil-theorysolutionforliftduetoangleofattackmightnotbe expectedtobe applicable.However,intiewoftheliftresultsobtained,it isapparentthatleading-edgeseparationhadlittleeffectonthecirculationata givenangleofattackas longastheflowreattachedtothesurfacewellaheadof thetrailingedge.

Withcompletedetachmentof theupper-surfaceflow,a flowconditionassumedintheRayleigh-~rchhofftheoryis satisfied,but,aswasnoted,thetheoryfailstodefinetheforcesandmomentsontheplate.Ithasbeenfairlywellestablishedthatthefailureisduetodifferencesbetweentheassumedandactualwakeconditions.As notedinreference1,flowobservationshaveshownthefluidbehindtheplatetohavea definiteverticalmotionratherthanbeingat restas assumedinthetheory.

●

Further,thewakeboundariesareactuallyvortexsheetsratherthansur-facesofdiscontinuitiesas assumedinthetheoreticaltreatment.(See

● reference5 fortheresultsofa detailedstudyofthestructureofthesheets.)Dueto thepresenceof thevorticesinthewake,a pressurelowerthanthatof thefreestreamisdevelopedat theuppersurfaceoftheplate.Howmuchthepressurediffersfromthatof thefreestreamisindicatedinthefollowingtable.Alsoshownaretheexperimentalvaluesoftheaveragelower-surfacepressurecoefficientandthetheo-reticalvaluesforbothsurfaces.Theexperimentalvaluesarefromreference3 andhavebeencorrectedforwind-tunnel-walleffects.(Seetheappendixforthemethodof correction.)

9

a,deg

;:405060708090

P,

Experimental

-0.58-.80-.90-.98

-1.04-1.04-1.05-1.05

vTheoretical

o0000000

P2av

ExperimentalITheoretical

0.25 0.34.41 .56●53 .67.62 ●75.69 .81.75 .85.78 .87979 .88

Itcanbe seenfromthetablethatthedifferencesbetweenexperi-.mentandtheoryarelargeinthecaseof theuppersurfaceandrel~tive~

6 mcA TN3221

smallInthecaseofthelowersurface.Thedifferencebetweentheexperimentalandthetheoreticalvaluesoftheup~r-surfacepressurecoefficientvariesfromabout60to70percent-ofthecorrespondingexperimentalnormal-forcecoefficient,whereasforthelowersurfacethedifferencevariesfromabout5 to 12percent.Effortsto improvetheRayleigh-Kirchhofftheoryobviouslyshouldbe andhavebeendirectedtowardobtaininga methodofpredictingthewakeconditionsandtheireffectontheupper-surfacepressure.

Theonlyexistingmodificationknownisthatproposedby D.Riabouchinsky.Hisproposalisbrieflydescribedinreference1. It isstatedthereinthathe suggestedan assumptionofa secondplatedown-streamandthecalculationoftheshapeofthewakebetweenthetwoplates,thesizeandlocationofthesecondplatebeingchosenin suchawaythatthepressureinthewakewasequaltothevalue foundexperi-mentally.ThusRiabouchinskylsmethodisessentiallyempirical.Asimplerempiricalapproachis suggestedinthefollowingsectionofthereport.

EmpiricalModificationoftheRayleigh-KirchhoffTheory

SincetheRayleigh-Kirchhofftheoryadequatelyaccountsfortheaveragepressureoverthelowersurfaceofa plate,a simpleempiricalmodificationofthetheorywouldbe to substituteexperimentalvaluesoftheupper-surfacepr~ssurecoefficientdirectlyinplaceofthetheoreti-cal. Theonlyvaluesfoundtobe availablefora flatplatewerethosemeasuredby FageandJohansen(ref.3) andgivenintheprecedingtable.A comparisonof thesevalueswiththoseavailableforairfoilsectionsathighanglesofattackindicatedthedesirabilityofobtainingadditionalvaluesfora flatplate.Inordertoprovideadditionalvalues,measure-mentsweremadeoftheaveragepressureovertheuppersurfaceofa 2-inch-chordplateina windtunnelwitha 2-by 5-feettestsection;theplatespannedthe2-footdimensionofthetest.section.Theresultingvaluesof PUavJ correctedfortunnel-walleffectsby themethodgivenintheappendix,arepresentedinfigure5 alom withtheflat-platevaluesfromreference3. Alsoshowninfigure5 arethevaluesforseveralairfoilsectionswithcompletelydetachedupper-surfaceflow.ThevaluesfortheNACA0015sectionwereobtainedfromtestsofthesectionthroughanangle-of-attackrangeof 0°to 1800(ref.6);cor-rectionsfortunnel-walleffectswerenotrequired(seeAppendtiIIofref.6). Thevaluesforthe64A-seriessectionwereobtainedfromtestsofthesectionsatanglesofattackup to28°,ata Machnumberofapproxi-mately0.3,andincludetunnel-wallcorrectionsby themethodgivenintheappendixofthepresentreport;theMachnuniberisabout0.2higherthantheMachnumbersofthetestsoftheplatesandtheNAC!A0015section.(Theeffectoftheclifferenceis smallandhasbeenapproxi-matelyaccountedforby usingthetheoreticalcompressibilityfactorsdiscussedlaterinthereport.)

.

r,

.—L.

—

●✎

NACATN 3221

●

Theflat-plate values ofvarious airfoil sections were

7

thepresentreportandthevaluesfortheusedinestablishingthecurveshownin

d figure~. It isbelievedthatthiscurveprovide;a reasonablygooddefinitionofvaluesof I&v touseinthemodificationof theRayleigh-Kirchhofftheory.AlthoughthecurveisbasedondatacoveringonlyaReynoldsnumberrangeof 0.15to 1.23million,thecurveshouldbeapplicabletohigherReynoldsnumber.

Usingthevaluesof Puav fromthefairedmodifytheRaylelgh-Kirchhofftheory,theforcearegivenby thefollowingequations:

%sina -pcn=~ +nsina %v

Cz=(2s(sina

4+ fisina )- Puav Cosa

(2Ycsinacd=

4+ fisina )- ‘% ‘ina

curveof figure5, toandmomentcoefficients

2fisinac~/A= 4 +fisina (0.25 -X=p) +%

. where ~p is givenbyequation(2). Theresultsgivenequationsarein goodagreementwiththeflat-platedatarectedfortunnel-walleffects.

Inorderto indicatethedegreeofapplicabilityof

(5)

(6)

(7)

(8)

by these(ref.3) cor-

themodifiedflat-platetheoryto thinairfoilsections,thecoefficientvaluesgivenby theforegoingequationsarecomparedin figure6 withcorrespondingmeasuredvaluesforseveralthinairfoilsections(refs.7 and8);alsoshowninthefigurearethin-airfoil-theoryvalues.(Althoughthevaluesof P%v tobe usedinequations(5)through(8)wereestablishedfromdataforbothairfoilsectionsandplates,theequationsarestrictlyapplicableonlytoa plateorairfoilsectionwitha flatlowersurface,sincetheRayleigh-Kirchhofftheoryappliesonlytoa flatlowersurface.)Theindicationofapplicabilityis limitedsomewhatby theangle-of-attackrangeandscatterof theexperimentalvalues.Fortheangle-of-attackrangecovered,however,itis concludedthatthemodifiedRayleigh-Kirchhofftheoryprovidesa goodfirstapproxhnationof theforcesandmomentsonthinairfoilsectionswithcompletelydetachedupper-surfaceflow.

m A briefexaminationhasbeenmadeof theeffectsofcompressibilityontheseparated(i.e.,discontinuous)typeofflowconsideredherein.Thecompressible-flowcounterpartoftheRayleigh-Kirchhofftheorywasgivenby Chaplyginin 1902(ref.9). Hissolutioncanbe appliedapproximatelyasa compressibilityfactorina manneranalogoustothat

8 NACATN 3221

usedinapplyingthewell-knownPrandtl-Glauertrelation.ThefactorfromChaplygin’ssolutionisapproximately1/[1- (0.~)2].A consider.ablysmallercompressibilityeffectisindicatedby C!haplyginlssolutionthanwouldbe indicatedby thePrandtl-Glauertrelation.ItmayseemquestionabletoconsidertheuseofthePrandtl-Glauertrelationin thiscase,sinceitisnormallyassociatedwiththecontinuoustypeof steadypotentialflow.Thereappearstobe no reason,however,whyit shouldbeinvalidbecauseofthediscontinuityintheflow(fig.1)assumedintheRayleigh-Kirchhofftheory,sincethetheoreticalforceisdueto thecon-tinuoussteadypotentialflowoccurringoutsideoftheareaboundedbytheplateandwake. Inthecaseoftheactualflowandforceona plate,thereisno theoreticalbasisforapplyingeithertheChaplyginsolutionor thePrandtl-Glauertrelationbecauseofthepreviouslydiscussedlackofa theoreticaltreatmentof thelargewakeeffect.Itappearsofinterest,nevertheless,toexeminetheirapplicabilityinthelightofavailableexperimentalevidence.Valuesof liftcoefficientpredictedby applyingeithertheChaplygincompressibilityfactor,orthePrandtl-GlauertrelationtothemodifiedRayleigh-Kirchhofftheoryarecomparedinfigure7 withmeasuredvaluesforthree6-percent-thickairfoilsections.(Theexperimentaldata,fromreferences7and8, wererecor-rectedfortunnel-walleffectsby themethodgivenintheappendixofthepresentreport.)Duetounaccountabledifferencesandscatterintheavailabledata,nodefiniteconclusioncanbe reached.Applicabilityof thePrandtl-Glauertrelationisgenerallyindicatedby thedatafortheNACA64-oo6section,andtheChaplyginsolutionby thedatafortheothertwosections.

CONCLUDINGREMARKS

TheBtudyofexisthgexperimentalandtheoreticalresultsforaninclinedflatplateofinfinitespanrevealedthefollowingfactsregard-ingthetypesof flowoccurringabouttheplate,andtheadequacyoftheoryinpredictingtheforcesontheplate.At lowanglesofattack,belowabout8°,flowseparationandreattachmentoccursontheuppersurface,andforthisanglerangethin-airfoiltheoryisadequateforpredictingtheliftandnormalforceOHtheplate.At higheranglesofattacktheflowiscompletelyseparatedfromtheuppersurface,a condi-tionwhichisassumedintheRayleigh-Kirchhofftheoryforan inclinedplate.‘TheRayleigh-Kirchhofftheory,however,isentirelyinadequateforpredictingthemagnitudeof theliftandnormalforceontheplatewithcompletedetachmentof theupper-surfaceflow.

ThedeficiencyoftheRayleigh-Kirchhofftheoryisduetodiffer-encesbetweenassumedandactualwakeconditions;asa consequence,theaverageupper-surfacepressuregivenby theoryis considerablydifferentfromexperimentalvalues.A simpleempiricalmodificationoftheRayleigh-Kirchhofftheorythatappearspromisingisto substitute

u

.

,MNACATN 3221

.

elqerimentallydete~ned valuesoftheupper-surfacepressureinplace. of thetheoretical.Comparisonofvaluesof lift,normal-force,drag,

andpitching-momentcoefficientgivenby themodifiedtheorywithvaluesmeasuredforthinround-noseairfoilsectionsindicatesthatthemodifiedtheoryprovidesa goodfirstapproximationof theforcesandmomentsonsuchairfoil~ectionswhentheflowiscompletelyseparatedfromtheuppersurface.Experimentaldataindicatean effectof compressibilityon theliftofairfoilsectionswithcompletelydetachedupper-surfaceflow;theeffectof compressibilitywasnotsufficientlydefined,however,formethodsofpredictiontobe evaluated.

AmesAeronauticalLaboratoryNationalAdviBoryCommitteeforAeronautics

MoffettField,Calif.,May4, 1954

.

10 NACATN 3221

APPENDIX

TUNNEL-WALLCORRK!TIONSFORAN INCLINEDFLAT

PLATEOFINFINITESPAN

Themethodofcorrectionisa simpleextensionof themethodgiveninreference10forcorrectingthedragofan infinite-spanplateinclined90°tothestreamina closedtunnel.Itis showninreference10thattheeffectof thewallsc~ be treatedas simpleempiricallyestablishedthattheareablockedisplate.Theequivalentfree-airvelocityisthus

where

V. equivalentfree-airvelocity

V.‘ tunnelvelocity

c chordlengthofplate

wakeblockage.Itwasequalto theareaofthe

h dimensionoftunnelcrosssectionnormaltoplatespan

To extendthisapproachtoanglesofattackotherthan90°,itisassumedthatthewalleffectscanstill.be treatedas simpleblockageandthattheblockedareaisequaltothefrontalareaoftheplate.(Thefactthatthisreductioninareadoesnotoccurat onestreamwisepositionisneglected.)It isalsoassumedthattheapproachisapplicableto com-pressiblesubsonicflow.Theresultingequationsforthevelocity,Machnumber,andd~amicpressureare

!l~—= l+~K%’ 1- (M’)2

,

.

.

where

K=l( c/h)sina- (c/h)sins

NACATN 3221 I-1

c andh areaspreviouslydefined,andtheprimedsymbolsaretheuncor-rectedvalues.Theratiosof correctedtouncorrectedvaluesof thelift,normal-force,drag,andpitching-momentcoefficientareequalto thereciprocalofthecorrespondingvaluesof ~/~’; forexample

c1 g—=Cz’ q.

Thecorrectedvalueof thepressurecoefficient

2K + P’P=m

%/% ‘

is

Itistobe notedthattheforegoingmethodof correctionneglectsanypossibleeffectsofthetunnelwallson theangleofattackor thecenterofpressure.Itisbelieved,however,thatsucheffectsaresmall.

12 NACATN 3221

R3ZFERENCES.

.

1.

2.

39

4.

6.

8.

9.

10.

vonK&-ma’n,Th., andBurgers,J.M.: GeneralAerodynamicTheory-PerfectFluids.Vol.11ofAerodynamicTheory,div.E.,W. F.Durand,cd.,JuliusSpringer(Berlin),1935..

Lamb,Horace:Hydrodynamics,CambridgeUniv.Press,1932,pp.99-103.

Fage,A.,andJohansen,F.C.: OntheFlowofAirBehindan InclinedFlatPlateof InfiniteSpan.R&MNo.1104,BritishA.R.C.,1927.

Rose,LeonardM.,andAltmanJohnM.: Iow-SpeedInvestigationoftheStallingofa Thin,Faired,Double-WedgeAirfoilwithNoseFlap,IfM2ATN2172,lg50.

Fage,A.,andJohansen,F.C.: TheStructureofVortexSheets.PhilosophicalMagazine,S. 7,vol.5,no.28,Feb.1%8, pp.417-441.

Pope,Alan: TheForcesandPressuresOveranNACA0015AirfoilThrough1800AngleofAttack.DanielGuggenheimSchoolofAero-nautics,GeorgiaSchoolofTechnology,Tech.Rep.E-102,1947.Seealso,Aero.Digest,vol.58,no.4,Apr.1949,p. 76.

Stivers,LouisS.,Jr.: EffectsofSubsonicMachnumberontheForcesandPressureDistributionsonFourNACA6bA-SeriesAirfoilSectionsatAnglesofAttackasHighas28°. ~CA TN 3162,1954.

Wilson,HomerB.,Jr.,andHorton,ElmerA.: AerodynamicCharacter-isticsatHighandLowSubsonicMachNwbersofFourNACA6-SeriesAirfoilSectionsatAnglesofAttackfrom-2° to 310. NACARML53c20,1953.

Chaplygin,S.: GasJets. NACATM 1063,1944,PP.72-m8.

Glauert,H.: Wind-TunnelInterferenceonWings,BodiesandAir~screws.R&MNo.1566,BritishA.R.C.,1933,pp.55-57.

, I

P=h Mes ofV=o discontinuity

Q

J!——— ———— —

Figure1.-Flow about an Incltned flat plate of tnflniteKirchhoff theory.

P’POv= v~ v

ep.m, as assumedIn the Rayleigh-

2.4

2.0

1.6

C“

u

.6

.4

0

0 Mea$m?dh O fbt j)btt?_ — #@@igh- Klk?hhofftheory––––– Thrn- oidoil ttiy

tI

~--~

/ ‘/

//

//

.- -- —- .- --u Iw Zo .30 40 50 60 iv 80 90(2

(a) cn vs. a

Figure 2.- Ccuprison of the measured characteristicsof a flat plate (ref. 3) with thecharacteristicsgiven by the Rayleigh-Kirchhofftheory aud by thin-airfoiltheory.

} 4 . ,

i ,, 1,,

, .

-1

2.G

L6

c’

1.2

.8

.4

0

. I

=>

—— —_

-------

/

0 Meosiwd fbr a flat pkn%— — Rayle@h- Kircbhoff theory——. — Thin- oirfoil fkary

— \

o 10 20 30 40 50 m m(2’ ‘-w&$m

(b) Cz VS. u

Figure 2.- Continued.

..—

2.4

2.0

i.6

&

1.2

.8

.4

00

I I I

/ /-?

—. ~tiQ@h - K-f tky––––– TM- okfoil hwny

/— ——

//

/

110 20 m 40 50 60 m 80 5W

Q(c) cd vs. a

Figure 2.- Continued.

I-Jcm

●

.6

.5

.4

Xcp

.3

.2

.1

—

< ~- //-----

r- ~ ~/-

/----.----’

/d

.—-—-.—-

0 IWYSun9dbr o flot plate— — RJyk@b- Kimhboff themy––––– Thlil- akfoil theory

I

.— -. .

I , E“

(2

(d) qp Vs. a

Figure 2.- Concluded.

P03

L2

.8

cl

.4

007////

Figure 3.. Comparison ol’section (ref. 1) withthin-airfoiltheory.

T.,/

“//

//

/

——kyleigh-Kimhhff -y

——––-Ttin -airfoil theory

n

4 8 /2a

the measured lift characteristicslift characteristicsgiven by the

16 m

of a faired double-wedgeairfoilRaylei@-Kirchhoff theory ad by

(

,,#

I

#

0 .2 .4 .8 .8 LoDistance along chard, fraction of chord

=!S=

,

Ftgure 4.- The extent or upper-surfaceflow separation on a faired double-wedge airfoil sectton(ref. 4).

-2.4

-2.6

- /.6

ho,

- I.z

-. 8

-. 4

0

0 Flat pbte, reference 3n Fbt pbte, present re~rtA NAL210(25, & e@ tirmmd, refmwuw 6B IU4GX W5, sharp s@@tbrwr~ m%renm 6

MA(M 64AO06, reference 7z AMC4 64A406, m%rence 7h IWC4 64AOI0, reference 7

I

o Jo 20 30 40 50 W m 80 m(2

Figure ~.- Average upper-surfacepressure coefficientson plates or airfoil sectionswith tom.pletely separatedupper-surfaceflow.

, b

2.0

1.6

L?

C“

.8

.4

*

~ —“ —

/ -

/

/ ‘//;

/ o AWA 64- W6, refenwce 8

/ A!A(M64AO06, reference 7

I A!ACA64A406, reference 7/’ – – --- Thin- airfoil theory

—. Modified ~yleigh - Kirchhoff theory1’

0(o 10 m 30 40 50 60 iv 80 99

Q

(a) cn vs. a

Figure 6.- Meammed force and moment characteristicsfor Beveralangles of attack and theoreticalcharacterlaticBgiven by thetheory and thin-airfoiltheory.

thin airfoil sections at highmcilifiedRayleigh-K3rchhoff

2.C

1.6

1.2

Cj

.&

.4

00

A!XCA64- @6, reference 8AUICA64AO06, mftweme 7hlACA~A406, mfer~m 7Thin- airfoil theoryModifdd RoyWgh- Kimhhoff thwry

‘=----

\ \

\

\

\

\

\1 I I

/0 20\

30 40 50 60 m &v SWa

(b) c1 VS. a

Fi~ 6.- Continued.

2.0

i.6

1.2

cd

.8

.4

0

.-

8 I F ,

+..uzii?.?:&y-- “

—— Modified Rvleig; - Kirchhoff theoryq

.—— —_ ——0 10 20 30 40 50 60 m 80

(2 VW

(c) cd v13.a

Figure 6.- Continued.

—.

o

-. I

-. 2

Cmti

-. 3

-. 4

-.50 Iv 20 .30 40 50 W 70 w 90

L? ‘- -(d) ~ vs. u

Figure 6.- Concluded.

* , ,

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