1、NATIONALADVISORYCOMMITTEEFORAERONAUTICSTECHNICALINOTE3221STUDYOF TEE SUBSONICFORCESANDMOMENTSONANINCLINEDPLATE OF INFINITESPANBy Bradford H. WickAmes Aeronautical LaboratoryMoffett Field, Calif.WashingtonJune1954AFMBCProvided by IHSNot for ResaleNo reproduction or networking permitted without licens
2、e from IHS-,-,-TECHLIBRARYKAFB,NMIilllllllllllulllllllllNATIONALADVISORYCOMMITTEEFORAERONAUTIC.STUDYOFTHESUBSONICFORCESANDMOMENTSONANINCLINEDPLATEOFINFINITESPANByBradfordH.WickSUMMARYA studyhasbeenmadeofexistingexperimentalandtheoreticalresultsforaninclinedflatplateofinfinitespan,andoftheextenttowhi
3、chtheresultsareindicativeofthoseforthinairfoilsections.Thestudyincludedanexaminationoftheflowaboutaninclinedplate,theforcesontheplate,andtheadequacyoftheoryinpredictingtheforces.Theoriesconsideredwerethewell-knownthin-airfoiltheory,andthetheoryofdiscontinuouspotentialflowandmodificationsthereof.Thee
4、ffectsofcompressibilitywereexsmined.s Theresultsofthestudyindicatethattherearetwoimportantrangesofangleofattackdifferingbytheextentofflowseparationontheuppersurface.Atanglesofattackbelowabout8, flowseparation. andreattachmentoccur,andthewell-knownthin-airfoiltheoryisade-quateforpredictingtheliftandn
5、ormalforceontheplate.Similarresultswerenotedforthinairfoilsections.Atthehigheranglesofattacktheflowiscompletelyseparatedfromtheuppersurfaceasisassumedinthediscontinuouspatential-flowtheoryforaninclinedflatplate.Thetheory,however,isentirelyinadequate.A simpleempiricalmodificationofthetheoryissuggeste
6、d;themtiifiedtheoryprovidesagoodfirstapproximationoftheforcesandmomentsonthinairfoilsec-tionswiththeflowcompletelyseparatedfromtheuppersurface.Effectsofcompressibilitywereevidentfromtheavailableexperimentaldata;however,theeffectswerenotdefinedsufficientlyforevaluatingmethodsofprediction.INTRODUCTION
7、Theresultsofstudies,byearlyresearchersinhydrodynamics,oftheflowaboutandtheresultantforcesonaninclinedflatplateofinfinitespan,heretofore,havehadlittlepracticalapplication.Theh typeofflowconsidered,consistingofdetachedflowovertheuppersurface(i.e.,rearwardsurface)andattachedflowoverthelowersur-face,was
8、notencounteredonconventionalairfoilsintheangle-of-attackv rangeofpracticalinterest.Withtheintroductionofthinairfoilsand,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATN3221inparticular,thosewithsharpleadingedges,theforegoingcircumstance .nolo
9、ngerexists.Theseparatedtypeofflowhasbeenfoundtooccuronthinunsweptwingsatandabovetheangleofattackformaximumlift,onb“thinsweptbackwingsconsiderablypriortowingmaximumlift,andonthinpropellerswhenoperatingattake-offconditions.Itappearedworthwhile,therefore,tomakea study ofexistingtheoreticalandexperiment
10、alresultsfortheflatplateandtodeterminetheirapplicabilitytothinairfoilsections.Theresultsofthestudyarereportedherein.cdclcmc/4CnPPavZacMPPo%vV.XcpaNOTATIONsectiondragcoefficient,qocsectionliftcoefficient,theotheristhewell-knownthin-airfoiltheory(ref.1,pp.24-53)whichtreatsthecaseofunsepa-ratedflow.Sin
11、cetheformertheoryhasbeenoflittlepracticalinterestand,consequently,isnotsowellknown,thefollowingbriefdiscussionisbelievedinorder.Thefirstcompletetreatmentoftheseparatedtypeofflow,usingmethodsofclassichydrodynamicsappearstobethatpresentedbyRayleighin1876.Hetreatedboththecaseoftheplateobliquetothestres
12、mandnormaltothestream.Kirchhoffsomeyearsearlier(in1869)hadcon-sideredbothcasesbutpresentedcalculatedresultsonlyinthecaseoftheplatenormaltothestream.Althoughworkingindependently,theirapproachwasa commonone,makinguseofHelmholtzshypothesisofasurfaceofdiscontinuity(i.e.,a surfacewhichseparatestwostreams
13、ofdifferentvelocities).Asa consequenceoftheuseofthishypothesis,m theirapproachisknownintheliteratureasthemethodofdiscontinuouspotentialflow.v A completedescriptionofthemethodisgiveninreference1. Thesalientfeaturesofthemethodareasfollows:Itisassumedthatlinesofdiscontinuitystartattheleadingandtrailing
14、edgesoftheplateandextendtoinfinity.(Seefig.1.) Withinthetwolinesthefluidisassumedtobeatrestwithrespecttotheplate.Outsidetheselinestheflowisassumedtobesmoothandsteady.Asa resultoftheflowconditionsassumed,thepressureinthewake(i.e.,theregionboundedbythelinesofdiscontinuity)isconstantandequaltothefree-s
15、treamstaticpressure,andthevelocityoutsidethewakeisequaltothefree-streamvelocity.Thesolutionfortheforceontheplateduetoabouttheplateis,incoefficientforms2X sinsCn=4+ fisinaTheingThethecenter-of-pressurelocationinfractionsoftheedgeis= 0.50 - cos a0.75 4 + x sinaXcpderivationoftheequationfor Cn isgiveni
16、nequationforthecenter-of-pressurelocationisthedescribedflow(1)chordfromthelead-(2)references1and2;fromthederivationProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4giveninreference2,whereinratherthantheleadingedge.mcA TN3221.thelocationisreferredtoth
17、emidchordSincethereisonlya normalforceactingontionsforthecoefficientsofliftanddragareCz=2Y(sinu 0sa4+fisthedatawerenotcorrectedfortunnel-walleffects.Alsoshownarethevaluesofliftindicatedbythin-airfoiltheoryandtheRayleigh-Kirchhofftheory.Theextentoftheseparated-flowregionisindicatedinfigure4,whichis a
18、 reproductionofafigureinreference4. Theboundaryoftheseparated-flowregionwasdefinedbythezero-velocitypointinvelocitydistributionsabovethesurfacewhichweredetemninedbyrakesofconventionalstatic-andtotal-pressuretubes.Itisnotedfromfigure3 that,asfortheflatplate,theliftvariationwithangleofattackwasessenti
19、allythatspecifiedbythin-airfoiltheoryuptoabout7.5, andthendeviatedrapidly.Thedataontheextentofflowseparation(fig.4)showthattheflowsepar-atedfromtheleadingedgeata verysmallangleofattackandthenreattachedfartherbackalongthesurface.Thepointofreattachmentmovedfartherbackwfthincreasingangleofattackuntilat
20、7.5, theangleofthelift-curvedivergence,theflowwascompletelyseparatedProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-5NACATN3221fromtheupperdependentuponsurface.Thattheamountofliftdevelopedisprimarilytheflowatthetrailingetieis.ofcourse.tobeexpected,si
21、nceinthin-airfoiltheory-the-amo:405060708090P,Experimental-0.58-.80-.90-.98-1.04-1.04-1.05-1.05vTheoreticalo0000000P2avExperimentalITheoretical0.25 0.34.41 .56a7153 .67.62a7175.69 .81.75 .85.78 .87979 .88Itcanbeseenfromthetablethatthedifferencesbetweenexperi-. mentandtheoryarelargeinthecaseoftheuppe
22、rsurfaceandreltiveProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 mcA TN3221smallInthecaseofthelowersurface.Thedifferencebetweentheexperimentalandthetheoreticalvaluesoftheupr-surfacepressurecoefficientvariesfromabout60to70percent-ofthecorresponding
23、experimentalnormal-forcecoefficient,whereasforthelowersurfacethedifferencevariesfromabout5 to12percent.EffortstoimprovetheRayleigh-Kirchhofftheoryobviouslyshouldbeandhavebeendirectedtowardobtainingamethodofpredictingthewakeconditionsandtheireffectontheupper-surfacepressure.Theonlyexistingmodificatio
24、nknownisthatproposedbyD.Riabouchinsky.Hisproposalisbrieflydescribedinreference1. Itisstatedthereinthathesuggestedanassumptionofa secondplatedown-streamandthecalculationoftheshapeofthewakebetweenthetwoplates,thesizeandlocationofthesecondplatebeingchoseninsuchawaythatthepressureinthewakewasequaltothev
25、alue foundexperi-mentally.ThusRiabouchinskylsmethodisessentiallyempirical.Asimplerempiricalapproachissuggestedinthefollowingsectionofthereport.EmpiricalModificationoftheRayleigh-KirchhoffTheorySincetheRayleigh-Kirchhofftheoryadequatelyaccountsfortheaveragepressureoverthelowersurfaceofaplate,a simple
26、empiricalmodificationofthetheorywouldbetosubstituteexperimentalvaluesoftheupper-surfaceprssurecoefficientdirectlyinplaceofthetheoreti-cal. Theonlyvaluesfoundtobeavailablefora flatplatewerethosemeasuredbyFageandJohansen(ref.3) andgivenintheprecedingtable.A comparisonofthesevalueswiththoseavailablefor
27、airfoilsectionsathighanglesofattackindicatedthedesirabilityofobtainingadditionalvaluesfora flatplate.Inordertoprovideadditionalvalues,measure-mentsweremadeoftheaveragepressureovertheuppersurfaceofa2-inch-chordplateinawindtunnelwitha2-by5-feettestsection;theplatespannedthe2-footdimensionofthetest.sec
28、tion.Theresultingvaluesof PUavJ correctedfortunnel-walleffectsbythemethodgivenintheappendix,arepresentedinfigure5 alom withtheflat-platevaluesfromreference3. Alsoshowninfigure5 arethevaluesforseveralairfoilsectionswithcompletelydetachedupper-surfaceflow.ThevaluesfortheNACA0015sectionwereobtainedfrom
29、testsofthesectionthroughanangle-of-attackrangeof0to1800(ref.6);cor-rectionsfortunnel-walleffectswerenotrequired(seeAppendtiIIofref.6).Thevaluesforthe64A-seriessectionwereobtainedfromtestsofthesectionsatanglesofattackupto28,ataMachnumberofapproxi-mately0.3,andincludetunnel-wallcorrectionsbythemethodg
30、ivenintheappendixofthepresentreport;theMachnuniberisabout0.2higherthantheMachnumbersofthetestsoftheplatesandtheNAC!A0015section.(Theeffectoftheclifferenceissmallandhasbeenapproxi-matelyaccountedforbyusingthetheoreticalcompressibilityfactorsdiscussedlaterinthereport.).r,.L.a71a15Provided by IHSNot fo
31、r ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN3221a71Theflat-plate values ofvarious airfoil sections were7thepresentreportandthevaluesfortheusedinestablishingthecurveshownind figure. Itisbelievedthatthiscurveprovide;a reasonablygooddefinitionofvaluesof Ialsoshown
32、inthefigurearethin-airfoil-theoryvalues.(Althoughthevaluesof P%v tobeusedinequations(5)through(8)wereestablishedfromdataforbothairfoilsectionsandplates,theequationsarestrictlyapplicableonlytoa plateorairfoilsectionwitha flatlowersurface,sincetheRayleigh-Kirchhofftheoryappliesonlytoa flatlowersurface
33、.)Theindicationofapplicabilityislimitedsomewhatbytheangle-of-attackrangeandscatteroftheexperimentalvalues.Fortheangle-of-attackrangecovered,however,itisconcludedthatthemodifiedRayleigh-Kirchhofftheoryprovidesa goodfirstapproxhnationoftheforcesandmomentsonthinairfoilsectionswithcompletelydetacheduppe
34、r-surfaceflow.m A briefexaminationhasbeenmadeoftheeffectsofcompressibilityontheseparated(i.e.,discontinuous)typeofflowconsideredherein.Thecompressible-flowcounterpartoftheRayleigh-KirchhofftheorywasgivenbyChaplyginin1902(ref.9). Hissolutioncanbeappliedapproximatelyasa compressibilityfactorinamannera
35、nalogoustothatProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACATN3221usedinapplyingthewell-knownPrandtl-Glauertrelation.ThefactorfromChaplyginssolutionisapproximately1/1- (0.)2.A consider.ablysmallercompressibilityeffectisindicatedbyC!haplyginls
36、solutionthanwouldbeindicatedbythePrandtl-Glauertrelation.ItmayseemquestionabletoconsidertheuseofthePrandtl-Glauertrelationinthiscase,sinceitisnormallyassociatedwiththecontinuoustypeofsteadypotentialflow.Thereappearstobenoreason,however,whyitshouldbeinvalidbecauseofthediscontinuityintheflow(fig.1)ass
37、umedintheRayleigh-Kirchhofftheory,sincethetheoreticalforceisduetothecon-tinuoussteadypotentialflowoccurringoutsideoftheareaboundedbytheplateandwake.Inthecaseoftheactualflowandforceonaplate,thereisnotheoreticalbasisforapplyingeithertheChaplyginsolutionorthePrandtl-Glauertrelationbecauseofthepreviousl
38、ydiscussedlackofa theoreticaltreatmentofthelargewakeeffect.Itappearsofinterest,nevertheless,toexeminetheirapplicabilityinthelightofavailableexperimentalevidence.ValuesofliftcoefficientpredictedbyapplyingeithertheChaplygincompressibilityfactor,orthePrandtl-GlauertrelationtothemodifiedRayleigh-Kirchho
39、fftheoryarecomparedinfigure7withmeasuredvaluesforthree6-percent-thickairfoilsections.(Theexperimentaldata,fromreferences7and8, wererecor-rectedfortunnel-walleffectsbythemethodgivenintheappendixofthepresentreport.)Duetounaccountabledifferencesandscatterintheavailabledata,nodefiniteconclusioncanbereac
40、hed.ApplicabilityofthePrandtl-GlauertrelationisgenerallyindicatedbythedatafortheNACA64-oo6section,andtheChaplyginsolutionbythedatafortheothertwosections.CONCLUDINGREMARKSTheBtudyofexisthgexperimentalandtheoreticalresultsforaninclinedflatplateofinfinitespanrevealedthefollowingfactsregard-ingthetypeso
41、fflowoccurringabouttheplate,andtheadequacyoftheoryinpredictingtheforcesontheplate.Atlowanglesofattack,belowabout8,flowseparationandreattachmentoccursontheuppersurface,andforthisanglerangethin-airfoiltheoryisadequateforpredictingtheliftandnormalforceOHtheplate.Athigheranglesofattacktheflowiscompletel
42、yseparatedfromtheuppersurface,a condi-tionwhichisassumedintheRayleigh-Kirchhofftheoryforaninclinedplate.TheRayleigh-Kirchhofftheory,however,isentirelyinadequateforpredictingthemagnitudeoftheliftandnormalforceontheplatewithcompletedetachmentoftheupper-surfaceflow.ThedeficiencyoftheRayleigh-Kirchhofft
43、heoryisduetodiffer-encesbetweenassumedandactualwakeconditions;asa consequence,theaverageupper-surfacepressuregivenbytheoryisconsiderablydifferentfromexperimentalvalues.A simpleempiricalmodificationoftheRayleigh-Kirchhofftheorythatappearspromisingistosubstituteu.Provided by IHSNot for ResaleNo reprod
44、uction or networking permitted without license from IHS-,-,-,MNACATN3221.elqerimentallydetenedvaluesoftheupper-surfacepressureinplace. ofthetheoretical.Comparisonofvaluesoflift,normal-force,drag,andpitching-momentcoefficientgivenbythemodifiedtheorywithvaluesmeasuredforthinround-noseairfoilsectionsin
45、dicatesthatthemodifiedtheoryprovidesa goodfirstapproximationoftheforcesandmomentsonsuchairfoilectionswhentheflowiscompletelyseparatedfromtheuppersurface.Experimentaldataindicateaneffectofcompressibilityontheliftofairfoilsectionswithcompletelydetachedupper-surfaceflow;theeffectofcompressibilitywasnot
46、sufficientlydefined,however,formethodsofpredictiontobeevaluated.AmesAeronauticalLaboratoryNationalAdviBoryCommitteeforAeronauticsMoffettField,Calif.,May4,1954.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACATN3221APPENDIXTUNNEL-WALLCORRK!TIONS
47、FORANINCLINEDFLATPLATEOFINFINITESPANThemethodofcorrectionisa simpleextensionofthemethodgiveninreference10forcorrectingthedragofaninfinite-spanplateinclined90tothestreamina closedtunnel.Itisshowninreference10thattheeffectofthewallsc betreatedassimpleempiricallyestablishedthattheareablockedisplate.The
48、equivalentfree-airvelocityisthuswhereV. equivalentfree-airvelocityV. tunnelvelocityc chordlengthofplatewakeblockage.Itwasequaltotheareaoftheh dimensionoftunnelcrosssectionnormaltoplatespanToextendthisapproachtoanglesofattackotherthan90,itisassumedthatthewalleffectscanstill.betreatedassimpleblockagea
49、ndthattheblockedareaisequaltothefrontalareaoftheplate.(Thefactthatthisreductioninareadoesnotoccuratonestreamwisepositionisneglected.)Itisalsoassumedthattheapproachisapplicabletocom-pressiblesubsonicflow.Theresultingequationsforthevelocity,Machnumber,anddamicpressureare!l= l+K% 1- (M)2,.whereK=l( c/h)s
