ARC-R ~~No. 1566 WIND TUNNEL INTERFERENCE ONWINGS,BODIESAND AIRS CREWS GLAUERT 1933 ~IIISTRY ForOfficialUse AERONAUTICALRESEARCHCOMMITTEE REPORTSANDMEMORANDANo.1566 (T3434) WindTunnelInterferenceon Wings,BodiesandAirscrews ByH.GLAUERT F.R.S. Communicatedby THEDIRECTOROFSCIENTIFICRESEARCH,AIRMINISTRY SEPTEMBER1933 Crow .. CopyrightR""'VIa LONDON _ PRINTEDANDPUBLISHEDBYHISMAJESTY'SSTATIONERYOFFICE TobepurchaseddirectlyfromH.M.STATIONERYOFFICEatthefollowingaddresses AdastralHouse,Kingsway,London,W.C.2 ;120,GeorgeStreet,Edinburgh2 YorkStreet,ManchesterI;I ,St.Andrew'sCrescent,Cardiff 80,Chiches terStreet,Belfast orthroughanyBookseller 1933 Price41.6d.Net ii WIND TUNNEL INTERFERENCE ON WINGS, BODIES ANDA IRSCREWS ByH.GLAUERT,F.R.S. Communicatedby the Director ofScientificResearch, AirMinistry ReportsandMemorandaNo.1566 13thSeptember,1933* Summary.-Thisreportprovidesacomprehensivesurveyof thesubjectofwindtunnelinterferenceonwings,bodiesandair-screws.Thebasisofthetheoreticaltreatmentofthesubj ectis examined critically and the method ofanalysing particular problems isexplainedindetail,butthereaderisreferredtotheoriginal papersforthemorecomplexpartsofthemathematicalanalysis. Experimentalresultsarequotedtojustifythetheoreticalformulae ortoderiveempiricalvaluestocompletethetheoreticalanalysis. Theresultsrequiredforthepracticalapplicationofthecorrection formulaearegiveninsuitabletablesandfigures,andafulllist ofreferencesisappendedtothereport,groupedaccordingtothe fourmainpartsofthereportandarrangedchronologicallyineach group. * R.A.E.Report,June,1933. iii CONTENTS tio 1.Generalintroduction PARTI Wings,ThreeDimensions 2.Generaldiscussion Ithodof images Interference flow Closedtunnels and freejets Circular tunnels 7.Effect of lift distribution Plane walls.. 9.Rectangular tunnels 10.Elliptic tunnel 11.Down,,-ahandIbettinu 12.la.ximumlif. coefficient PART2 Wings ,TwoDimensions 13.Induced curvature of the flow 14.Free jets 15.Closedwindtunnels PART3 SymmetricalBodies 16.General discussion 17.Twodimensions 18.Three dimensions 19.Pressuregradient PART4 Airscrews 20.Generaldiscussion.. 21.Closedwindtunnels 22.Free jets 23.Airscrewandbody (:8983) Page 1 3 5 8 11 13 17 19 21 29 35 38 41 46 48 50 52 57 61 66 67 70 71 A2 1 WINDTUNNELINTERFERENCEONWINGS,BODIES ANDAIRSCREWS ByH.GLAUERT,F.R.S. CommunicatedbytheDirectorofScientificResearch, AirMinistry ReportsandMemorandaNo.1566 13thSeptember,1933 1.GeneralI ntroduction.-It iswellknownthat theaerodynamic forceexperiencedbyabodymaybeseriouslymodifiedbythe proximityofanotherbody,evenwhenthereisnodirectcontact. Thestudyofsuchinterferenceisanimportantbranchofaero-dynamics,but in the firstplace it is necessary to know the behaviour ofthebodyapartfromanyinterference.Themostconvenient methodofexperimentistoinvestigatethebehaviourofamodel intheartificialstreamofawindtunnel,andthelimitedextentof thisstream,boundedeitherbytherigidwallsofaclosedtypeof windtunnelorby thefreesurfaceofanopenjet,inevitablyleads tosomeconstraintoftheflowandtosomeinterferenceonthe behaviourofthemodel.Thisinterferencecouldbeminimised byusingverysmallmodels,butitisdesirableformanyreasons thatthemodelshouldbeaslargeaspossible.Thestudyofwind tunnelinterferenceisthereforeofgreatimportance,sincesome interferenceisinevitable,andanaccurateknowledgeofthisinter-ferencewilljustifytheuseoflargermodelsthanwouldotherwise bepermissible. Thegeneralnatureoftheinterferencecanbeappreciatedmost readilybyconsideringtheconditionsinaclosedtunnel.If alarge bodyisplacedinthestream,thefirstandmostobviousconstraint imposedby therigidwallsof thetunnel isthat thestream is unable toexpandlaterallyasfreelyasitwouldinanunlimitedfluid,and inconsequencethat thevelocityofflowpast thebodyisincreased, leadingtoanintensificationofthe forcesexperienced bythebody. Anotherchokingconstraintofadifferentcharacterarisesifthere isawakeofreducedorincreasedvelocitybehindthebody,as occursrespectivelywithabluffbodyor anairscrew.Thenecessity ofmaintainingcontinuityofflowinthetunnelthenimpliesthat t hevelocityandpressureofthestreamsurroundingthewake 2 willdifferfromtheundisturbedvaluesfarinfrontofthebody, andthischangeofpressurereactsbacktocauseachangeinthe forceexperiencedbythebody.Theinterferenceexperiencedby aliftingbody,suchasawing,isofadifferentcharacter.The liftofawingisassociatedwithageneraldownwardmovement oftheairbehindthewing,andtheconstraintofthetunnelwalls on this downwash modifies the behaviour and aerodynamic character-isticsofthewing.Finallyafourthtypeofinterferenceoccurs ifthereisagradientofstaticpressuredownthestreamofthe windtunnel.This pressure gradient arisesowing to the development ofthefrictionalboundary layerofreducedvelocityalongthewalls ofthetunnel,whichleads toanincreaseofvelocityandadecrease ofpressurealongtheaxisofthetunnel.Anybodyistherefore tested inaslightlyconvergentstream,and experiencesanincreased dragowingtothedropofstaticpressurefromnosetotail. Thesevariousinterferenceeffectsinaclosedtunnel,together withthecorrespondingeffectsinafreejet,willbediscussedin detail fordifferent types of body, whichcan be grouped conveniently undertheheadingsofwings,symmetricalbodiesandairscrews, butbeforeproceedingtothisanaiysisitisdesirabletoconsider theprecisenatureoftheboundaryconditions,andthelimitations of the theoretical treatment ofthesubject. Thepressuregradientcorrectionwillbereservedforspecial considerationinSection19ofPart3,sinceitisimportantonly forbodiesoflowdragandmaybeneglectedintheconsideration ofwingsandairscrews.Moreover,thepressuregradient,which is due in a closed tunnel to the development of the frictional boundary . layeralongthewallsandtoleakagethroughthewalls,canbe eliminatedbydesigningthewindtunnelwithaslightexpansion inthedirectionofthestream,andissensiblyzeroinafreejet. Thediscussionoftheothertypesoftunnelinterferenceisbased ontheconceptionof an ideal streamwithoutanypressuregradient alongitsaxis,andneglectsboththeboundarylayeralongthe wallsofaclosedtunnelandtheanalogousdisturbedregionatthe boundaryofafreejetwherethestreammingleswithanddraws alongsomeofthesurroundingair. The boundary condition at any wall of a closed tunnel is expressed preciselybytheconditionthatthenormalcomponentofthefluid velocitymustbezero.Thecorrespondingconditionforafree jet is that the pressure at the boundary must haveaconstant value, equaltothepressureofthesurroundingair,butitisinpractice impossibletousethisexactconditionintheanalysisowingtothe distortionoftheshapeofthejetcausedbythepresenceofabody inthes t r e a m ~Theapproximationisthereforeadoptedofapplying thisconditionofconstantpressureattheundisturbedposition oftheboundary of thejet.Asanadditionalapproximation,which isofthesameorderofaccuracyasthepreviousone,itisassumed thatthedisturbancetothetunnelvelocityVattheboundary 3 of the jet issmall.If (u,v,w)are the components of the disturbance, the pressure P at the boundary of thejet is obtainedfromBernoulli's equationas P =Po+ !e V2- !e {(V+ u) 2 + v2 + w2} =Po- e Vu- !e(u2 + v2 + W2)(1.1) andtothefirstorderofthedisturbancetheconditionofconstant pressureissimplythatuisconstant.Butuisevidentlyzerofar infrontofthebody in theundisturbedstream,andhenceumust be zero at all points of the boundary.This implies that the velocity potential1>,whichrepresentsthechangeofflowfromtheuniform undisturbedstream,musthaveaconstantvalueoverthewhole boundaryofthefreejet,andtheboundaryconditionsassumed intheanalysisarenowsimply :-a1>_0} a-;=constant Closedtunnel (1.2) Freejet Theboundaryconditionforaclosedtunnel isexactandprecise, exceptforanyeffectsduetothefrictionalboundarylayeralong thewalls.Theboundaryconditionforafreejetisapproximate only,beingappliedattheundisturbedpositionoftheboundary andbasedontheassumptionofsmalldisturbingvelocities.There isalsooneotherpointinwhichthetreatmentofaclosedtunnel ismoreprecisethanthatofafreejet.Aclosedtunnel'usually extends foraconsiderablelengthwithconstantcross-sectionbefore andbehindthemodel,whereasafreejetusuallyissuesfroma closedcylindricalmoutbimmediatelyinfrontofthemodeland is receivedinto acollector at amoderatedistancebehindthemodel. Thustheconditionsdifferappreciablyfromthoseofthelongfree jet,envisagedintheanalyticaltreatmentofthesubj ect,andthe validityotthetheoreticalinterferencecorrectionsmustrestulti-matelyonexperimentalconfirmationoftheiraccuracy. PART1 Wings,ThreeDimensions 2.Generaldiscussion.-Tbe methodofanalysingtheinterference experiencedbyawinginaclosedtunnelorinafreejetisdueto Prandtl1.Thenatureoftheboundaryconditionsandtheapproxi-mateassumptionsmaderegardingtheseconditi.onsforafreejet have beendiscussedpreviouslyinSection1,butin the development oftheanalysisitisnecessarytomakesomefurtherassumptions regardingtheflowpastthewingitself.Theliftofthewingis intimatelyrelatedtothecirculationoftheflowrounothewing, andineffectthewingcan beregardedasagroupofboundvortices runningalongitsspan.Ingeneraltheliftandcirculationhave 4 their maximum values at the centre of the wingandfall off gradually tozeroatthewingtips.Thislateraldecreaseofcirculationfrom thecentreofthewingoutwardsisaccompaniedby thecreationof freetrailingvorticeswhichspringfromthetrailingedgeofthe wingandpassdownstream.Thesetrailingvorticesaredeflected downwardswiththegeneraldownwashbehindthewing,and, sinceavortexsheetisunstable,theyultimatelyrollupintotwo vorticessomewhatinboardofthewingtips.Inthedevelopment ofaerofoiltheory,however,theseeffectsare ignoredandthe trailing vorticesareassumedtoliealongstraightlinespassingdownstream fromthewing.Thissameassumptionismadeintheanalysisof windtunnelinterference,andtheanalysisisthereforestrictly applicableonlytolightlyloadedwings. Theproblemofawinginasmalltunnelinvolvesmotionin threedimensions,butPrandtllhasshownthatitcanconveniently bereducedtoaproblemintwodimensionsonly,whenthewing isregardedasaliftinglineextendingfromwingtiptowingtip. Consideringfirsttheflowpastthewinginanunlimitedfluidand takingthexco-ordinatedownstream,thevelocitypotential,due tothewingandadditionaltothatoftheundisturbedstream,is oftheform r/>f(y,z)+ F(x,y,z) (2.1 ) wherethefirstfunctionrepresentsthevelocitypotentialinthe transverseplanecontainingthewingandthesecondfunction changessignwithx.Sincer/>mustbezerofarinfrontofthe wing, itfollowsthatfarbehindthewingthevelocitypotentialwillhave thevalue 2f(y,z) (2.2) Turningnowtotheproblemofawinginafreejet,theinterference duetothelimitedextentotthestreamwillberepresentedbythe additionofaterm1/tothevelocitypotential,whichmustsatisfy theconditionsthatr/>'isfiniteatallpointsinthelimitedstream andthat(r/>+ r/>')mustbezeroatallpointsottheboundary. It followsthat1/canbedividedintotwotermsofthesameform asr/>,andthatitsvalueintheultimatewakewillbedoubleits valueatthecorrespondingpointinthetransverseplanecontaining thewing.Theinterferenceexperiencedby thewingdependssolely ontheflowinthistransverseplane,andhenceasaconvenient methodofanalysisitsufficestoanalysetheflowinatransverse planeofthedistantwake,whichissimplyatwo-dimensionalpro-blem,andtodeducethat theinterferenceexperiencedby thewing ishalfthatwhichoccursinthedistantwake.Asimilarargument canbeappliedtotheproblemofaclosedwindtunnelwherethe boundaryconditionisnow 'Or/>'Or/>' an+ an o (2.3) 5 anditisagainpossibletoreducetheproblemtothatoftwo-dimensionalflowinatransverseplaneofthedistantwake. Theinterferenceduetothelimitedextentofthestreamwill ingeneralmodifythedistributionofliftacrossthespanofthe wingandinanexactanalysisoftheproblemitwouldbenecessary totakedueaccountofthiseffect.Thisaspectoftheproblemis discussedinSection7below;butunlessthespanofthewingisa verylargefractionofthewidthofthetunneltheresultingchange ofliftdistributionisverysmallandmaybeneglected.Generally itsufficestoassumethattheliftisdistributedellipticallyacross thespan,asonawingofellipticplanfor Ill,andoftenitispossible ' to proceedto the even simpler approximation of uniform distribution ofliftacross ' thespanofthewing.Whenthespanofthewingis averysmallfractionofthewidthofthetunnel,theformassumed fortheliftdistributionisquiteimmaterialanditthensuffices to assumethat thetotal liftforceofthe wing isconcentrated at the centreofitsspanandtocalculatetheinterferenceatthissingle point.Thistypeofsolutionwillsubsequentlybereferredtoas thatapplicabletosmallwings. 3.Methodojimages.Theconceptionofimages,asusedin aerodynamicalproblems,canbeappreciatedbyconsideringafew simpleexamples.Iftwoaeroplanesareflyinghorizontallyside bysidetherewillevidentlybenoflowacrosstheverticalplane ofsymmetrymidwaybetweentheaeroplanes,andthisplanecould bereplacedbyarigidwallwithoutalteringtheflowin any way. Thustheproblemofanaeroplaneflyingparalleltoaverticalwall canbesolvedby introducing theimageaeroplaneontheother side of the wall and by considering the new problem of the two aeroplanes flyingsidebyside.Similarlytheinterferenceexperiencedbyan aeroplaneflyingclosetothegroundcanbesolvedbyintroducing theinvertedimageaeroplanebelowtheground.Thismethodof introducingtheappropriateimageorsetofimagestorepresent theconstraintoftheboundaryofthestreamiscapableofvery wide application, and isthe methodused for analysing mostproblems ofwindtunnelinterference. ThediscussionofSection2hasshownthattheproblemof theinterferenceexperiencedbyawinginaclosedtunnelorin afreejetcanbesolvedbyconsideringthetransverseflowina sectionofthedistantwake.Inthisplanethewingisrepresented solelybythesystemofitstrailingvortices,whichnowappear aspointvorticesandextendalongalineoflengthequaltothe span of the wing,and theproblem to be solved isthedetermination oftheflowwhichmustbesuperimposed on thatdue to thevortices inordertosatisfytheappropriateconditionattheboundaryof the stream.Inthesimpleassumptionofawingofuniform loading acrossthespanthevortexsystemscomprisemerelytwopoint vorticesofequalandoppositestrengthsatadistanceapartequal 6 to the span of the wing.Moregenerally any wing can be represented byadistributionofsuchpairsofvorticesextendingacrossthe wholespan,andhencetheproblemofawingwithanytypeot lift distributionacrossits span can be derived fromthe simpler problem ofauniformly loadedwing by aprocessofintegration. The method of images can be used directly when the cross-section of the boundary of the stream is rectangular or circular, and solutions forsomeotherformscanbederivedfromtheseprimarysolutions bymeansofsuitableconformaltransformations.Theformof imagerequiredinasinglerectilinearboundaryisillustratedin Fig.1.The imageofasinglevortex inarigidrectilinearboundary isanequalvortexofoppositesign(Fig.1a),sincethispairof vortices will by symmetry give zero normal velocity at the boundary. Eachvortexwillgivethesamecqmponentofvelocityparallel totheboundary,andhencetheconditionofzeroflowalongafree boundary can be obtained by reversing the sign of the image (Fig.1b), andvortexandimagehavenowthesamesign.Thustheimage systemforafreeboundarycanbederivedfromthatforarigid boundarybyreversingthesignoftheimages.Theimagesystem ofauniformlyloadedwinginthepresenceofarigidverticalwall isshowninFig.1c,andthecorrespondingsystemforahorizontal boundaryinFig.1d.Here,andgenerallythroughoutthereport, itisassumedthatthespanofthewingishorizontal.Theimage intheverticalwallisareplicaofthewingitself,buttheimage inthe horizontalwallisaninvertedwing.Byapplying thefunda-mentalconceptionsillustratedinFig.1itispossibletobuildup theimagesystemrequiredforapair ofwallsor foranyrectangular boundary.Theseproblemsusuallyinvolveinfiniteseriesofimages andexamplesofsuchsystemswillbediscussedlaterinthereport. Acircularboundary(Fig.2)canalsoberepresentedquite simply by the method ofimages.Considering firstarigid boundary, theimageofapointvortexofstrengthKatanypointAisan equalandoppositepointvortexattheinversepointA'.The streamfunction'lfJatanypointPduetothesetwopointvortices issinlply KA'P =2nlogAP sincethecomplexpotentialfunctionofapointvortexofstrength Kandanti-clockwiserotationatthepointZois iK - 2n log(z--zo) (3.1) NowifPisapointofthecircularboundary,thetrianglesOAP and OPA' are similar, and hence A'POP APOA=const. 7 whichprovesthat thecircularboundaryisastreamlineoftheflow andmay thereforeberegardedasarigidboundary. Byanalogywiththeproblemofarectilinearboundarywe mightanticipatethatthesameimagecouldbeusedforafree boundaryifthesignoftheimagewerereversed.Thevelocity potentialofthevortexatAandofanequalimageatA'is,apart from the addition of any arbitrary constant, andsince A RA'p=AOP+ OP A' =AOP+ OAp weobtain K =2n (n+a) where()denotestheangularpositionoftheradiusOP.Thusthe velocitypotentialofthevortex andits imageisnotconstantalong thecircumferenceofthecircleandthenecessaryconditionfora freeboundaryisnotsatisfied.If, however,weintroduceasecond vortexofequalandoppositestrengthatanyotherpointinside thecircle,thevelocitypotential of thissecondvortex andits image will be of the form K =- 2n (n+ ()fJ) andonadditiontothepreviousexpressionthevariableterm() disappears.Thusthenecessaryconditionforafreeboundaryis satisfiedbyapairofvorticesanywhereinsidethecircleandby thecorrespondingimages.Sincethevorticesrepresentingawing alwaysoccurinpairsofthiskind,themethodofimagescanbe usedforafreecircularboundary.Theimagesystemisidentical, exceptforthechangeofsignofthe vorticity,with the systemfora rigidboundary. Awingwithuniformdistributionofliftacrossitsspanisrepre-sented by a pair of vortices at its tips, and any form of lift distribution canberepresentedbyasystemofpairsofvorticesdistributed alongthespanofthewing.Thustheconditionsforanyarbitrary distributionofliftcanbederivedbyintegrationfromthesimpler conditionofawingwithuniform distributionofliftacrossits span. Ingeneraltheliftdistributionissymmetricalaboutthecentre ofthewing.Taketheorigin0atthecentreofthewingandthe axisOx tostarboardalongthespan.If risthecirculationat any 8 pointxofthewing,thestrengthofthetrailingvortexspringing froman element