NATIONAL ADVISORY COMMITTEE FORAERONAUTICS TECHNICALI NOTE3221 STUDYOFTEE SUBSONICFORCESANDMOMENTSONAN INCLINEDPLATE OFINFINITESPAN By Bradford H. Wick Ames Aeronautical Laboratory Moffett Field, Calif. Washington June1954 AFMBC
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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.
* , ,
.-— — ——-
EI 0 MICA 64- Wt5, rz?ference 8
❑ NXGX 64AO06, mfemrxw 7no
“QEI o A!ACA 84A406, refemce 7
––––– Thrn-OirfmY_.— Abdified K@k+igh- Kkchhoff theory
\\
A,
.
\\ \
w
. , 1 *
L6
U*M= o
l..? /
& $ ‘
y Om %,.8 u
CJ
,4
00 2 ,4 .6M
.6 10
.4 0 AMC4 64-- m- 8
0 hWA 64AW, refetwn.w 7
0 IWIC.4 MAW. mfenm2? 7I
00I
2 .4M
“’ w
h)
R1-
F@ure 7.- Comparison of the measured effects of cmupreaaibilityon the lift characteristicsofseveral thin airfoil sectionsat high angles of attack with the effects predicted for a flatplate by applying either the Prandtl-Glauertremtion or we cbkw~n solution to me @-
fled Rayleigh-’K3.rchhofftheory. %
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