EFFECTOFSHEARLAYERUNSTEADINESSONTHEAERODYNAMICSOFAPITCHING AIRFOIL By AlirezaSafaripour-Tabalvandani ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2020 ABSTRACT EFFECTOFSHEARLAYERUNSTEADINESSONTHEAERODYNAMICSOFAPITCHING AIRFOIL By AlirezaSafaripour-Tabalvandani Itiscommonpracticetoutilizeauniformapproach˛owinmostoftheproblemsinaerodynamics. However,innumerouscircumstancesthecomplexapproach˛owsfoundinnaturecanbesignif- icantlynon-uniformandevenincludespatiallynon-uniformtemporal˛uctuations.Motivatedby thesenon-uniformunsteady˛ows,thisexperimentalstudyinvestigatesthee˙ectsofnon-uniform approach˛owunsteadinessontheaerodynamicperformanceofanairfoil. Toisolatethee˙ectsofunsteadinessfromthoseofthemeannon-uniformapproach˛ow,two non-uniform(shear)˛owswithamatchingmeanvelocitypro˝learegeneratedinawatertunnel facility.Oneofthesepro˝lesismadethroughacanonicaltwo-streamshearlayer,whichisknownto containvorticalstructuresandexhibitnon-uniformvelocity˛uctuations.Amatchingshearvelocity pro˝lewithuniformlowlevel˛uctuationsisgeneratedutilizingamodi˝edshapedhoneycombshear generationmethodbasedontheoriginalmodelproposedbyKotansky(1966).Theseshear˛ows demonstratethesamemeanbehaviorandonlydi˙erintheir˛uctuationpro˝les. Behavioranddevelopmentofbothshear˛owsareexaminedthroughmeasurementsoftheir streamwisevelocitypro˝leatmultipledownstreamlocationsutilizingsinglecomponentmolecular taggingvelocimetry.Thesteadyandunsteadyshear˛owsarefoundtoproducethesamemean velocitypro˝lewithdi˙erentclassesofvelocity˛uctuations.Thesteadyshear˛owdemonstrates auniformlowlevelof˛uctuatingvelocitypro˝lewhiletheunsteadyshearlayervelocity˛uctua- tionsmimicthesignature˛uctuatingvelocitypro˝leofplanemixinglayers,withahighlevelof ˛uctuationsinthecenterandagradualdecreasein˛uctuationsmovingawayfromtheshearlayer centerline. ANACA0012airfoilispositionedatthecenterofeachshear˛owandtheaverageaerodynamic forcesonthestationaryairfoilaredirectlymeasuredacrossawiderangeofanglesofattack.The resultingliftanddragcoe˚cientcurvesarecomparedforeachoftheseshear˛owsaswellasthe referenceuniform˛ow.Theunsteadyshearlayerisfoundtogenerateapositiveliftatzeroangle ofattack,incontrasttothenegativeliftobservedunderthesameconditioninthesteadyshear ˛ow.Furthermore,inthepresenceoftheunsteadyshear˛ow,thelinearregionoftheliftcurve aroundzeroangleofattackshowsalargerslopeandextendsoverawiderrangeofanglesofattack comparedtothoseofthesteadyshearanduniform˛owliftcurves.Theunsteadyshear˛owresults inasmallermagnitudeofdragatsmallanglesofattackcomparedtotheothertwo˛owconditions. Forcemeasurementsarealsoperformedwiththeairfoilsettosinusoidallypitcharoundits quarterchordoverarangeofoscillationfrequenciesatthecenterofbothshear˛owsandthe referenceuniform˛ow.Themeanliftresultsshowthatatsmalloscillationfrequenciesthesteady andunsteadyshear˛owsproduceoppositesignliftforces(negativeliftinsteadyshear˛ow),but bothresultinpositiveliftcoe˚cientsofsimilarmagnitudesathigherfrequencies.Thepresenceof shearanditsunsteadinessseemstoonlyweaklya˙ectthemeanand˛uctuationsofthestreamwise force,withalmostnoe˙ectobservedonlift˛uctuations. Acloserlookatthe˛owaroundthesurfaceofthestationaryairfoilrevealsinterestingdi˙erences betweenthebehavioroftheairfoilboundarylayerinthepresenceofsteadyversusunsteadyshear ˛ow.Singlecomponentmoleculartaggingvelocimetryisusedtomeasurethestreamwisevelocity ofthe˛ownearthesurfaceoftheairfoilatmultipleanglesofattackineachshear˛ow.While thesteadyshearlayerresultscon˝rmthepresenceoflaminarseparationonthesuctionsideof theairfoilattheanglesofattackinvestigatedhere,nosignoflaminarseparationorareverse˛ow regionisfoundwhentheairfoilisplacedintheunsteadyshearlayer. Thewake˛owbehindthepitchingairfoilisalsovisualizedthroughmoleculartagging˛ow visualizationtoqualitativelyexaminehowtheseshear˛owsa˙ectthewake˛owbehavior.Itis observedthatbothsteadyandunsteadyshear˛owsresultinthewake˛owde˛ectingtowardsthe highspeedsideofthe˛owathighoscillationfrequencies,withmorecycle-to-cyclevariationand perturbationsobservedinthepresenceofunsteadyshear˛ow. Copyrightby ALIREZASAFARIPOUR-TABALVANDANI 2020 Thisthesisisdedicatedtomyfamily. v ACKNOWLEDGEMENTS Itisalmostimpossibletoachieveanythingmeaningfulinlifejustbyyourself.Duringmylong journeyasastudent,Ihavereceivedsupportandassistancethroughmanysources.Iwouldliketo acknowledgeandexpressmygratitudetowardsthosewhohelpedmegetwhereIamtoday. Firstofall,Iwouldliketothankmywifewithoutwhomnoneofthiswouldhavebeenpossible. Shehassupportedmeeverystepoftheway,eventhroughmanymany,veryverylatenightsof experimentalwork.Eventhoughsheisnotamechanicalengineer,sheseemstoalwaysknowthe solutiontoevenmyacademicproblems.Iwouldalsoliketothankmyfamily,especiallymyparents foralltheirsupportandsacri˝cesfromnearandfar.MyveryspecialgratitudegoestoDr.David Olson,whohasbeenverypatientandhelpfuldespitemyendlessrelevantorirrelevantquestions. Ourconversationshaveallowedmetovent,decompress,brainstormandgetthroughthehardtimes alittleeasier.Iverymuchappreciatehisinvaluablecontributions. IamalsothankfultomycurrentandformerTMUALinmateswhoIhaveenjoyedworkingwith, includingDr.PatrickHammer,MitchellAlbrecht,BorhanHamedaniandKianKalanwithavery specialmentiontoDr.ShahramPouyawhohadbeenapillarofTMUALforsolongthatitwas harderthanIthoughttogetusedtoworkingatTMUALwithouthim. Mostimportantly,myveryspecialgratitudegoestomyadvisor,Prof.Koochesfahaniwhohas beennotjustagreatadvisor,butalsolikeafathertomeforthepastfewyears,guidingmethrough myacademicjourneyaswellaspointingoutwaystoimprovemyselfforlifebeyondacademia.Itis alwaysablessingtohaveacompassionateadviserwhothinksofhisstudentsashissons,especially whenyourparentsarefarawayandyouneedallthesupportyoucanget.Iwouldalsoliketoextend mygratitudetoProf.Naguibforallhishelpandguidanceovertheyears. Lastly,thisworkwassponsoredbyAFOSRawardnumbersFA9550-10-1-0342andFA9550- 15-1-0224. vi TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ....................................... x CHAPTER1INTRODUCTION ............................... 1 1.1Background......................................1 1.2CurrentStudy.....................................7 1.3Outline.........................................9 CHAPTER2EXPERIMENTALMETHODS ........................ 10 2.1FlowFacility.....................................10 2.2ShearGenerationMethods..............................13 2.2.1UnsteadyShearGeneration..........................13 2.2.2SteadyShearGeneration...........................14 2.3MolecularTaggingVelocimetry...........................17 2.3.1Background..................................17 2.3.2SingleComponentMTVforShearLayerCharacterization.........18 2.3.3SingleComponentMTVforAirfoilBoundaryLayerCharacterization...21 2.3.4MolecularTaggingFlowVisualizationforAirfoilWakeVisualization...23 2.4ForceMeasurementSetup...............................24 CHAPTER3SHEARGENERATIONRESULTS ...................... 28 3.1UnsteadyShearFlowCharacterization........................30 3.2SteadyShearFlowCharacterization.........................33 3.3SensitivityofShearFlowstoDownstreamPerturbations...............36 CHAPTER4AERODYNAMICFORCEMEASUREMENTS ............... 39 4.1AerodynamicForcesonaStationaryAirfoil.....................40 4.1.1E˙ectofUpstreamUnsteadyShearFlowPro˝leonAerodynamicForces onaStationaryAirfoil............................45 4.2AerodynamicForcesonanOscillatingAirfoil....................48 4.2.1E˙ectofUpstreamUnsteadyShearFlowPro˝leonAerodynamicForces onanOscillatingAirfoil...........................53 4.3PhaseOrderedAerodynamicForcesonOscillatingAirfoil..............54 CHAPTER5VELOCIMETRYINSIDEAIRFOILBOUNDARYLAYER ......... 61 CHAPTER6PITCHINGAIRFOILWAKEVISUALIZATION ............... 71 CHAPTER7CONCLUSIONS ................................ 84 APPENDICES ......................................... 87 vii APPENDIXASHAPEDHONEYCOMBSHEARGENERATIONMETHOD ... 88 APPENDIXBEFFECTOFAIRFOILPRESENCEONUPSTREAMFLOW BOUNDARYCONDITION ...................... 103 APPENDIXCAERODYNAMICFORCESONOSCILLATINGAIRFOILWITH 0 = 4 ................................ 111 APPENDIXDSAMPLESOFPHASEORDEREDAERODYNAMICFORCES ONOSCILLATINGAIRFOIL .................... 115 APPENDIXEBOUNDARYLAYERMEASUREMENTALTERNATECON- TOURS ................................ 125 BIBLIOGRAPHY ........................................ 132 viii LISTOFTABLES Table3.1:Characteristicparametersderivedforthetwo-streamshearlayer.Displayed valuesareaveragevaluesbasedonmultiplemeasurementsintheself-similar region........................................32 Table3.2:Acomparisonbetweenthecharacteristicsofthesteadyversusunsteadyshear layer.Thevaluesforunsteadyshearlayerareextractedfromthetwo-stream shearlayermeanvelocitypro˝lemeasuredat94cmdownstreamofthesplitter plate( x x 0 = 107 cm)...............................35 Table4.1:Listofunsteadyforcemeasurementcases......................48 ix LISTOFFIGURES Figure1.1:Schematicofmeanand˛uctuatingvelocitypro˝lesofdi˙erentapproach˛ow conditionsconsideredinthiswork: (a) Uniform˛ow, (b) Steadynon-uniform (shear)˛owand (c) Unsteadynon-uniform(shear)˛ow..............2 Figure1.2: (a) Schematicofathree-segmentlinearvelocitypro˝leusedbyHammer etal.(2018) (b) Averageliftcoe˚cient, C L ,vsshearrate, K ,forinviscidand viscidsolutionoftheairfoilsinvestigatedinHammeretal.(2018).(Figures fromHammeretal.,2018)............................4 Figure1.3:SchematicofareversevonKármánvortexstreetbehindanairfoilpitching aboutitsquarterchord.Thesignofthecirculationofthevortexisindicated bythearrowandbythe(+)or(-)sign.......................4 Figure1.4:Computational˛owvisualizationoftraditionalvonKármánstreet (a) versus reversevonKármánstreet (b) withaccompanyingmeanwakevelocitypro˝le. (FiguresfromHammer,2016)...........................5 Figure1.5:Computedmeanand˛uctuatingloadsonanoscillatingairfoilversusreduced frequency k fordi˙erentshearrates( Û ).(FiguresfromHammeretal.,2019)..6 Figure1.6:Colorcontourmapofphase-averagedstreamwisevelocity,showingthevari- ationofvelocitypro˝leswithphaseduringonecycleofoscillationinthe wakeofthepitchingairfoil( k = 10 : 6 Hz, 0 = 2 ). (a) Thebaselinecase ofuniformapproach˛owshowstheexpectedsignatureofalternatingsign vortexpair. (b) Inthecaseofshearlayerapproach˛ow,thereisevidencefor onlyonevortexcore.(FigurecourtesyofKoochesfahani&Naguib)......8 Figure2.1:Ascaleddrawingofthe˛owfacility. (a) Topview, (b) Sideview.Quartz windowsarevisibleinthesideview.(FigurescourtesyofOlson,2017).....11 Figure2.2:Ascaleddrawingofthetestsetupshowing (a) Thethree-degree-of-freedom (3DOF)servomotionsystem,and (b) Theairfoilandskimmerplateplacement inthetestsection.(FigurescourtesyofOlson,2017)...............12 Figure2.3:Aschematicofthetwo-streamsheargenerationdevice..............14 Figure2.4:Aschematicoftheshapedhoneycombmodelalongwiththemainassumptions involved.(afterKotansky,1966).........................16 x Figure2.5:Anexamplecomparisonofthestreamwise (a) meanand (b) ˛uctuatingveloc- itypro˝lesinthegeneratedsteadyshearlayerversusthereferenceunsteady shearlayer......................................17 Figure2.6:TypicalMTVimagepairsandtheresultantvelocity˝eld(Gendrichetal., 1997).The˛owshownisfromavortexringimpactingona˛atwallat normalincidence.Theaxisofsymmetryisindicatedbythedashedlines. (a) Thegridimaged1 safterthelaserpulse. (b) Thesamegridimaged8ms later. (c) Thevelocity˝eldderivedfrom(a)and(b)................18 Figure2.7:Aschematicof1c-MTVstreamwisevelocitymeasurementsetuputilizedin two-streamshearlayergenerationcon˝guration(topview)............19 Figure2.8:Aschematicofhowavelocitycomponentparalleltoataggedlinecangenerate anerrorinthevelocitynormaltothelinein1c-MTV...............21 Figure2.9:Aschematicof˝eldofviewsizeandtheopticalsetupforairfoilboundary layervelocimetry.Thesolidanddashedlinesillustratethetwodi˙erent˝elds ofviewsusedforthesemeasurements.......................22 Figure2.10: (a) Asamplepairofmoleculartagging˛owvisualizationtagginglinesshown atinitial(violet)anddelayed(green)statesinthewakeofastationaryairfoil inauniform˛ow.Theinitialimageistaken ˘ 2 s afterthelaserpulse,while thedelayedimageiscaptured30msafterthelaserpulse. (b) Phaseaveraged moleculartaggingvisualizationofthe˛owinthewakeofanoscillating NACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequency of k = 8 inuniform˛owat4di˙erentphasesofoneairfoiloscillationcycle. Theoriginprescribesthelocationoftheairfoiltrailingedge...........24 Figure2.11: (a) 3Dmodelofthewatertunneltestsection,depictingtheverticalairfoil mountedtothe3DOFmotionsystemontopofthetestsection. (b) Detailsof the3DOFsystemandairfoilmounting.(FigurescourtesyofHammeretal.,2019)25 Figure3.1:Aschematicofaplanemixinglayer........................28 Figure3.2:Asampleplanemixinglayermeanvelocitypro˝le................29 Figure3.3:Developmentofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝les inthetwo-streamshearlayer............................30 Figure3.4: (a) Developmentoftheunsteadyshearlayervorticitythickness, ! . (b) Inclinationofunsteadyshearlayercenterlinepositiontowardsthelowspeedside.31 Figure3.5:Self-similarbehaviorofthenormalizedstreamwise (a) meanand (b) ˛uctu- atingvelocitypro˝lesinthetwo-streamshearlayer................32 xi Figure3.6:Comparisonofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝les inthesteadyshearlayerversusthereferenceunsteadyshearlayer.........33 Figure3.7:Comparisonofthenormalizedstreamwise (a) meanand (b) ˛uctuatingve- locitypro˝lesinthesteadyshearlayerversusthereferenceunsteadyshear layer.........................................34 Figure3.8:Developmentofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝les inthesteadyshearlayer...............................35 Figure3.9:E˙ectofairfoiloscillationonmean (a) and˛uctuating (b) velocitypro˝les twochordupstreamoftheairfoilleadingedgewhenairfoilislocatedat x = 94 cminthetwo-streamshearlayer..........................36 Figure3.10:E˙ectofairfoiloscillationonmean (a) and˛uctuating (b) velocitypro˝les twochordupstreamoftheairfoilleadingedgewhenairfoilislocatedinthe steadyshearlayer..................................37 Figure4.1:Schematicofforcemeasurementcoordinatesystem................39 Figure4.2:Extentofairfoilangulardisplacement( 20 )comparedtotheshearlayer thickness......................................40 Figure4.3:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoil placedinuniform,steadyshearandunsteadyshear˛ows: (a) meanlift coe˚cient, C L , (b) Zoomedupviewofmeanliftcoe˚cient, C L , (c) Mean dragcoe˚cient, C D , (d) Ratioofmeanlifttomeandrag, C L C D ..........42 Figure4.4:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoil placedinthesteadyshearlayerandtheresultsofOlsonetal.(2016)and Hammeretal.(2019): (a) meanliftcoe˚cient, C L , (b) Zoomedupviewof meanliftcoe˚cient, C L , (c) Meandragcoe˚cient, C D , (d) Ratioofmean lifttomeandrag, C L C D ...............................44 Figure4.5:Comparisonoftheshear˛owvelocitypro˝lesstudiedinthecurrentstudy withthelinearsteadyshearusedinOlsonetal.(2016)..............45 Figure4.6:Measurementlocationsforforcemeasurementsonstationaryairfoilintwo- streamshearlayer.Thesymbolsshowthe x locationswhereforcemeasure- mentshavebeenperformed,withthepro˝lesindicatingthemeasuredmean velocitypro˝leatthatlocation...........................46 xii Figure4.7:Extentofairfoilangulardisplacement( 20 )comparedtothetwo-stream shearlayerthicknessateachmeasurementlocation................46 Figure4.8:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoil placedinunsteadyshearlayeratdi˙erentdownstreamlocations: (a) mean liftcoe˚cient, C L , (b) zoomedupviewofmeanliftcoe˚cient, C L , (c) mean dragcoe˚cient, C D , (d) ratioofmeanlifttomeandrag, C L C D ..........47 Figure4.9:Extentofpitchingairfoilangulardisplacement( 2 )comparedtotheshear layerthickness....................................49 Figure4.10:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoil placedinuniform,steadyshearandunsteadyshear˛ows: (a) meanliftco- e˚cient, C L , (b) liftcoe˚cient˛uctuations, C L 0 , (c) meanthrustcoe˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .Theairfoilispitchingwithzero meanangleofattackandanoscillationamplitudeof 2 arounditsquarter chordpoint.....................................50 Figure4.11:Comparisonofaerodynamicforce˛uctuationsmeasuredonalowfrequency pitchingairfoilplacedinuniform,steadyshearandunsteadyshear˛ows: (a) liftcoe˚cient˛uctuations, C L 0 , (b) thrustcoe˚cient˛uctuations, C T 0 .The airfoilispitchingwithzeromeanangleofattackandanoscillationamplitude of 2 arounditsquarterchordpoint........................51 Figure4.12:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoil placedinsteadyshearwithexperimentalandnumericalresultsofHammer etal.(2019): (a) meanliftcoe˚cient, C L , (b) meanthrustcoe˚cient, C T . Theairfoilispitchingwithzeromeanangleofattack( m = 0 )andan oscillationamplitudeof 0 = 2 arounditsquarterchordpoint..........52 Figure4.13:Extentofairfoilangulardisplacement( 2 )comparedtothetwo-stream shearlayerthicknessateachmeasurementlocation................53 Figure4.14:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoil placedintwo-streamshearlayeratdi˙erentdownstreamlocations: (a) mean liftcoe˚cient, C L , (b) meanthrustcoe˚cient, C T .Theairfoilispitching withzeromeanangleofattack( m = 0 )andanoscillationamplitudeof 0 = 2 arounditsquarterchordpoint.......................54 Figure4.15:Highlightedcomparisonofmeanliftcurvebetweenunsteadyversussteady shearlayer......................................55 xiii Figure4.16:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 2 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................56 Figure4.17:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 2 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................56 Figure4.18:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 2 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................57 Figure4.19:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 5 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................57 Figure4.20:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 5 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................58 Figure4.21:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 5 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................58 Figure4.22:Comparisonof (a) rawand (b) normalizedaveragephasebinstandarddevi- ationinliftfordi˙erentupstream˛owconditionsasafunctionofreduced frequency......................................59 Figure4.23:Comparisonof (a) rawand (b) normalizedaveragephasebinstandarddevi- ationinthrustfordi˙erentupstream˛owconditionsasafunctionofreduced frequency......................................59 xiv Figure5.1:Schematicofairfoilanglesofattackinvestigatedthroughboundarylayer measurementssuperposedontheliftcoe˚cientsmeasuredforsteadyand unsteadyshear˛ows.Thecyanlineshighlightthecasesthatboththesuction andpressuresidesoftheairfoilhavebeenconsidered,whereasforthecases shownwithblacklinesonlythesuctionsideoftheairfoilisevaluated......62 Figure5.2:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 0 atthecenterofthesteadyshear layer.Thearrowindicatesdirectionofthefree-stream˛ow............63 Figure5.3:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 0 atthecenteroftheunsteady shearlayer.Thearrowindicatesdirectionofthefree-stream˛ow.........64 Figure5.4:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 2 inthesteadyshearlayer.The ˝eldofviewisrotatedsuchthattheairfoilisportrayedatzeroanglebutarrow indicatesdirectionofthefree-stream˛ow.....................65 Figure5.5:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 2 intheunsteadyshearlayer. The˝eldofviewisrotatedsuchthattheairfoilisportrayedatzeroanglebut arrowindicatesdirectionofthefree-stream˛ow..................65 Figure5.6:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatpositive anglesofattackinthecenterofthesteadyshearlayer.The˝eldofview isrotatedsuchthattheairfoilisportrayedatzeroanglebutarrowindicates directionofthefree-stream˛ow..........................66 Figure5.7:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatpositive anglesofattackinthecenteroftheunsteadyshearlayer.The˝eldofview isrotatedsuchthattheairfoilisportrayedatzeroanglebutarrowindicates directionofthefree-stream˛ow..........................67 Figure5.8:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatnegative anglesofattackinthecenterofthesteadyshearlayer.The˝eldofview isrotatedsuchthattheairfoilisportrayedatzeroanglebutarrowindicates directionofthefree-stream˛ow..........................68 xv Figure5.9:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatnegative anglesofattackinthecenteroftheunsteadyshearlayer.The˝eldofview isrotatedsuchthattheairfoilisportrayedatzeroanglebutarrowindicates directionofthefree-stream˛ow..........................69 Figure6.1:Asamplepairofmoleculartagging˛owvisualizationtaggedlinesshownat initial(violet)anddelayed(green)states.Theinitialimageistaken ˘ 2 s afterthelaserpulse,whilethedelayedimageiscaptured30msafterthelaser pulse.Theaxesarenormalizedbythechordlengthoftheairfoilwiththe trailingedgeoftheairfoilatzeroangleofattacksetastheorigin.........71 Figure6.2:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 6 inuniform˛owat4di˙erentphasesofoneairfoil oscillationcycle.Thelocationofthetrailingedgeishighlightedbyawhite circleandthedashedlinesindicatethedirectionofthe˛owfromthetrailing edge.Theredarrowsindicatethedirectionoftrailingedgemotion........73 Figure6.3:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 6 insteadyshearlayerat4di˙erentphasesofone airfoiloscillationcycle.Thelocationofthetrailingedgeishighlightedbya whitecircleandthedashedlinesindicatethedirectionofthe˛owfromthe trailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion....74 Figure6.4:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 6 inunsteadyshearlayerat4di˙erentphasesof oneairfoiloscillationcycle.Thelocationofthetrailingedgeishighlighted byawhitecircleandthedashedlinesindicatethedirectionofthe˛owfrom thetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion..75 Figure6.5:Instantaneousspanwisevorticity˝eldsfor (a) uniform˛owand (b) linear steadyshear˛owsimulatedbyHammeretal.(2019).Theairfoilisat = 0 andpitchingup.(FigureCourtesyofHammeretal.,2019)...........76 Figure6.6: (a) Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewake ofanoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 6 inunsteadyshearlayeratstartingphaseofone airfoiloscillationcycle. (b) through (c) showsampleinstantaneousmolecular taggingvisualizationimagesfromthesamecaseandphaseincycleusedto generate (a) .Thelocationofthetrailingedgeishighlightedbyawhitecircle andthedashedlinesindicatethedirectionofthe˛owfromthetrailingedge. Theredarrowsindicatethedirectionoftrailingedgemotion...........77 xvi Figure6.7:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 8 inuniform˛owat4di˙erentphasesofoneairfoil oscillationcycle.Thelocationofthetrailingedgeishighlightedbyawhite circleandthedashedlinesindicatethedirectionofthe˛owfromthetrailing edge.Theredarrowsindicatethedirectionoftrailingedgemotion........78 Figure6.8:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 8 insteadyshearlayerat4di˙erentphasesofone airfoiloscillationcycle.Thelocationofthetrailingedgeishighlightedbya whitecircleandthedashedlinesindicatethedirectionofthe˛owfromthe trailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion....79 Figure6.9:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 8 inunsteadyshearlayerat4di˙erentphasesof oneairfoiloscillationcycle.Thelocationofthetrailingedgeishighlighted byawhitecircleandthedashedlinesindicatethedirectionofthe˛owfrom thetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion..80 Figure6.10:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 10 insteadyshearlayerat4di˙erentphasesofone airfoiloscillationcycle.Thelocationofthetrailingedgeishighlightedbya whitecircleandthedashedlinesindicatethedirectionofthe˛owfromthe trailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion....81 Figure6.11:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeof anoscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 ata reducedfrequencyof k = 10 inunsteadyshearlayerat4di˙erentphasesof oneairfoiloscillationcycle.Thelocationofthetrailingedgeishighlighted byawhitecircleandthedashedlinesindicatethedirectionofthe˛owfrom thetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion..82 Figure6.12:Colorcontourmapofphase-averagedstreamwisevelocityresultsofNaguib& Koochesfahani(2012),showingthevariationofvelocitypro˝leswithphase duringonecycleofoscillationinthewakeofthepitchingairfoil( k = 10 : 6 , 0 = 2 ). (a) Thebaselinecaseofuniformapproach˛owshowstheexpected signatureofalternatingsignvortexpair. (b) Inthecaseofshearlayerapproach ˛ow,thereisevidenceforonlyonevortexcore.(FigurecourtesyofNaguib &Koochesfahani,2012)..............................83 FigureA.1:Aschematicofthevariablelengthhoneycombmodelalongwiththemain assumptionsinvolved................................89 xvii FigureA.2:Non-uniformhyperbolictangentvelocitypro˝leselectedwithcenterlineve- locityof u c = 10 cm/s................................94 FigureA.3:Calculatedhoneycomblengthdistributionforthehyperbolictangentpro˝le basedonthepresentmodelcomparedtotheestimatedentrancelengthofthe ˛owateach y (basedon L e ˇ 0 : 06 Re ¹ y º ).Thehorizontallinesdepictthe approximatediameterofhoneycombtubes.....................94 FigureA.4:Imageofthesheargenerationdevicecutaccordingtodesignedlengthpro˝le basedonthepresentmodelandshowninFigureA.3...............95 FigureA.5:Aschematicofthe˛owmeasurementsetupinthepresenceoftheshear generationdevice..................................96 FigureA.6:Asampleimagepairshowingaportionofthetaggedregioncapturedwitha timedelayof10ms.(i)thetaggedregionrightafterthelaserpulse(ii)the sametaggedregion10mslater.Theindividualhoneycombcellsareshown inyellowontheimages...............................96 FigureA.7:Normalizedvelocitypro˝lesshowingdevelopmentofthe˛owdownstream ofthehoneycombcomparedwiththedesignvelocitypro˝le...........97 FigureA.8:Temporalvelocity˛uctuationsofthegeneratedhyperbolictangentpro˝le comparedwiththatofthefreestream˛owwithoutthesheargeneration deviceatadownstreamlocationof x š d ˇ 62 ....................98 FigureA.9:Ahyperbolictangentcurve˝ttedtotheexperimentalvelocitypro˝leata downstreamlocationof x š d ˇ 24 ..........................98 FigureA.10:Developmentofspatialdeviationofthemeasuredvelocitypro˝leformthe hyperbolictangent˝tasafunctionofdistancedownstream............99 FigureA.11:Calculatedhoneycomblengthdistributionforthestreamwisevelocitycom- ponentofaconvectingGaussianvortexpro˝lebasedonthepresentmodel comparedtotheestimatedentrancelengthofthe˛owateachy(basedon L e ˇ 0 : 06 Re ¹ y º ).Thehorizontallinesdepicttheapproximatediameterof honeycombtubes..................................99 FigureA.12:Imageofthesheargenerationdevicecutaccordingtodesignedlengthpro˝le basedonthepresentmodel(showninFigureA.12)................100 FigureA.13:Normalizedexperimentalvelocityofthestreamwisecomponentofaconvect- ingGaussianvortexmeasuredatadownstreamlocationof x š d ˇ 72 compared tothedesignvelocitypro˝le............................100 xviii FigureA.14:ComparisonoftheshapeofthehoneycombdevicesdesignedusingKotan- sky'sandthemodi˝edmodeltogeneratethehyperbolictangentvelocity pro˝leshowninFigure2A.2)...........................101 FigureA.15:Estimatedvelocitypro˝lefromKotansky'shoneycombdesigncomparedto thatofthemodi˝edmodelandtheexperimentalvelocitypro˝le.........101 FigureB.1:E˙ectofstationaryairfoilangleonrawstreamwise (a) meanand (b) ˛uctu- atingvelocitypro˝lesinthesteadyshearlayer...................104 FigureB.2:E˙ectofstationaryairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesinthesteadyshearlayer................104 FigureB.3:E˙ectofstationaryairfoilangleonstreamwise (a) high-speedand (b) low- speedvelocitiesinthesteadyshearlayer......................105 FigureB.4:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinevelocityand (b) velocitydi˙erenceinthesteadyshearlayer....................105 FigureB.5:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinepositionand (b) vorticitythicknessinthesteadyshearlayer.....................106 FigureB.6:E˙ectofstationaryairfoilangleonrawstreamwise (a) meanand (b) ˛uctu- atingvelocitypro˝lesintheunsteadyshearlayer.................107 FigureB.7:E˙ectofstationaryairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayer..............107 FigureB.8:E˙ectofstationaryairfoilangleonstreamwise (a) high-speedand (b) low- speedvelocitiesintheunsteadyshearlayer.....................108 FigureB.9:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinevelocityand (b) velocitydi˙erenceintheunsteadyshearlayer...................108 FigureB.10:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinepositionand (b) vorticitythicknessintheunsteadyshearlayer...................109 FigureB.11:E˙ectofpitchingairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayerwith ! c = 0 : 5 and K max = 1 : 4 .....................................109 xix FigureB.12:E˙ectofpitchingairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayerwith ! c = 0 : 9 and K max = 0 : 8 .....................................110 FigureC.1:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoil placedinuniform,steadyshearandunsteadyshear˛ows: (a) meanliftco- e˚cient, C L , (b) liftcoe˚cient˛uctuations, C L 0 , (c) meanthrustcoe˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .Theairfoilispitchingwithzero meanangleofattackandanoscillationamplitudeof 4 arounditsquarter chordpoint.....................................112 FigureC.2:Extentofairfoilangulardisplacement( 4 )comparedtotheshearlayer thicknessateachmeasurementlocation......................113 FigureC.3:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoil placedinunsteadyshearlayeratdi˙erentdownstreamlocations: (a) mean liftcoe˚cient, C L , (b) liftcoe˚cient˛uctuations, C L 0 , (c) meanthrustco- e˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .Theairfoilispitching withzeromeanangleofattackandanoscillationamplitudeof 4 aroundits quarterchordpoint.................................114 FigureD.1:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 1 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................116 FigureD.2:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 2 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................116 FigureD.3:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 3 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................117 FigureD.4:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 4 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................117 xx FigureD.5:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 5 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................118 FigureD.6:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequency of k = 6 .Bluelinesrepresentthephaseordereddata,redlinesportraythe phaseaveragedvalueswiththeblacklineshighlightingthestandarddeviation ofthedataineachphasebin............................118 FigureD.7:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 1 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................119 FigureD.8:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 2 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................119 FigureD.9:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 3 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................120 FigureD.10:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 4 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................120 FigureD.11:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 5 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................121 xxi FigureD.12:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛ow forapitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfre- quencyof k = 6 .Bluelinesrepresentthephaseordereddata,redlinesportray thephaseaveragedvalueswiththeblacklineshighlightingthestandardde- viationofthedataineachphasebin........................121 FigureD.13:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 1 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................122 FigureD.14:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 2 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................122 FigureD.15:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 3 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................123 FigureD.16:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 4 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................123 FigureD.17:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 5 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................124 FigureD.18:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear ˛owforapitchingairfoilwithoscillationamplitudeof 0 = 2 andreduced frequencyof k = 6 .Bluelinesrepresentthephaseordereddata,redlinespor- traythephaseaveragedvalueswiththeblacklineshighlightingthestandard deviationofthedataineachphasebin.......................124 FigureE.1:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 0 atthecenterofthesteadyshear layer.........................................126 xxii FigureE.2:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 0 atthecenteroftheunsteady shearlayer......................................126 FigureE.3:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 2 inthesteadyshearlayer......127 FigureE.4:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednear thesurfaceoftheairfoilpositionedat = 2 intheunsteadyshearlayer.....127 FigureE.5:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatpositive anglesofattackinthecenterofthesteadyshearlayer...............128 FigureE.6:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatpositive anglesofattackinthecenteroftheunsteadyshearlayer..............129 FigureE.7:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatnegative anglesofattackinthecenterofthesteadyshearlayer...............130 FigureE.8:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycon- toursmeasurednearthesuctionsurfaceoftheairfoilpositionedatnegative anglesofattackinthecenteroftheunsteadyshearlayer..............131 xxiii CHAPTER1 INTRODUCTION Thefocusofthisworkistostudyhowtheunsteadybehaviorofanon-uniformapproach˛ow in˛uencestheaerodynamicperformanceofanairfoilinstationaryandoscillatingcircumstances. Themajorityofaerodynamicproblemsconsiderauniformapproach˛ow,anassumptionthat isnotalwaysrepresentativeofthecomplex˛owsfoundinnature.Inpractice,therearemany circumstanceswheretheapproachvelocitypro˝leisnon-uniform,suchaswhenanairplane encounterswindshearsorlarge-scaledisturbancesduetothetopographyoftheterrainbeneath. Furthermore,inmanyofthepracticalsituationswherenon-uniform˛owsareencountered,they couldincludeinherentvorticalstructuresandspatiallynon-uniformtemporalvelocity˛uctuations. These˛uctuations,ontopoftheirmeanvelocitygradient,mayfurthera˙ecttheaerodynamic performanceofanairfoil.Althoughtherehavebeenstudiesfocusingonthee˙ectofnon-uniform (shear)˛owsonairfoilaerodynamics,manyofthemarelimitedtotheoreticalinviscid˛owsand thereareveryfewsystematicexperimentalresearchonairfoilsinrealviscousshear˛ows.There areevenlesspublishedworksonthein˛uenceofunsteadyshear˛owsonairfoils. 1.1Background Itisbene˝cialto˝rstintroduceandde˝nethedi˙erentapproach˛owsthatarereferredto inthisdocument.Figure1.1presentsthethreedi˙erentupstream˛owconditionsstudiedinthis research.Auniformapproach˛owreferstothemostcommonupstreamboundaryconditionwhere boththemeanandlow-levelbackground˛uctuatingvelocitypro˝lesarespatiallyuniform(Figure 1.1a).Thispro˝lerepresentsthetypicalapproach˛owinmostexperimental˛owfacilities.A steadynon-uniform(shear)˛owhasanon-uniformmeanvelocitypro˝lewithaspatiallyuniform (andlowmagnitude)backgroundvelocity˛uctuationpro˝le(Figure1.1b).Thisindicatesthat theinstantaneousvelocitypro˝lesareallveryclosetothemeanvelocitypro˝le.The˛owsin whichthemeanand˛uctuatingvelocitypro˝lesarebothspatiallynon-uniform,withsigni˝cant 1 Figure1.1:Schematicofmeanand˛uctuatingvelocitypro˝lesofdi˙erentapproach˛ow conditionsconsideredinthiswork: (a) Uniform˛ow, (b) Steadynon-uniform(shear)˛owand (c) Unsteadynon-uniform(shear)˛ow. levelsofpeak˛uctuations,arereferredtohereasunsteadynon-uniform(shear)˛ows(Figure1.1c) inthisdocument.Intheseunsteadyshear˛ows,theinstantaneousvelocitypro˝lescanbevery di˙erentfromthemeanvelocitypro˝le,dependingonthemagnitudeanddistributionofvelocity ˛uctuations. Asoneofthe˝rstworksonthee˙ectofasteadynon-uniformapproachvelocityonthebehavior ofanairfoil,Tsien(1943)assumeda2Dsteadyinviscid˛owwithalinearvelocitypro˝le(uniform shearrate)approachingasymmetricJoukowskyairfoilandshowedthepresenceoftheshear˛ow resultedinashiftinthezero-liftangleofattackfortheairfoil.Themainconclusionfromhiswork wasthatasymmetricairfoilatzeroangleofattack(AoA)inapositivelyshearedapproach˛ow generatesapositiveliftandthemagnitudeofthisgeneratedliftincreaseswiththeshearrateofthe approach˛ow.SeveralstudieshaveexpandedthescopeofTsien'sworktomoregeneralvelocity pro˝les,butallofthesestudiesarelimitedtoinviscidsteady˛ows(seeJames,1951;Honda& Lighthill,1960;Nishiyama&Hirano,1970) Asmentionedpreviously,thereareverylimitedsystematicstudiesonthee˙ectsofsteady shear˛owsonairfoilsinrealviscous˛ows.ArareexperimentalstudybyPayne&Nelson(1985) 2 consideredasteadyNACA0018airfoilinbothuniformandsteadyshear˛owsinawindtunnel withchordReynoldsnumbersof75,000to200,000.Theyusedaforcebalancetomeasurelift anddragforcesandfoundthatwhiletheliftanddragcurvesexhibitedanasymmetryduetothe shear˛ow,theaerodynamiccharacteristicsoftheairfoilweremainlyuna˙ectedbytheintroduced velocitygradient.Theyobserveda˛owinducedasymmetryindragcoe˚cientcurves,witha positiveshearresultinginahigherdragcoe˚cientatpositiveanglesofattack.Theyconcludedthat thee˙ectsofshear˛owwasofthesameorderofmagnitudeasofthoseduetootherphenomena suchasadditionalturbulencegeneratedbytheirsheargenerationmethod. AmorerecentnumericalstudybyLeeetal.(2014)observedanincreaseintheliftcoe˚cient ofanS809airfoil(dedicatedtohorizontal-axiswindturbines,thisisanasymmetricairfoilwithan expectedpositiveliftatzeroAoA)inthepresenceofapositivesteadyshear˛ow,andareduction inthecaseofnegativeshear.Theyfoundthisincreasetobeindependentoftheangleofattackfor Reynoldsnumbersintheorderof 10 6 . OngoingnumericalandexperimentalstudiesinourowngroupintheTurbulentMixingand UnsteadyAerodynamicsLaboratoryatMichiganStateUniversityhaveshownsomeinteresting e˙ectsofviscousshear˛owsonaerodynamicperformanceofairfoils.Hammeretal.(2018) investigatedthee˙ectofasteadyshear˛owonastationaryNACA0012airfoilwithachord Reynoldsnumberof Re c = 1 : 2 10 4 through2Dnumericalsimulations.Theirsteadyshear˛ow wasathree-segmentedpro˝lewithalinearvelocitypro˝leextendingovera˝nitethickness, ,in betweentwouniformvelocityzones(Figure1.2a).Thisvelocitypro˝lewascharacterizedbya non-dimensionalshearrate K = du d y c U 1 and wasselectedlargeenough( ˇ 1 : 5 c )sothatthe ˝nitethicknessoftheshearzonewouldnota˙ecttheresults.Theyfoundthatapositivelinearshear ˛owresultsina negative liftcoe˚cientatzeroAoA(Figure1.2b),whichiscontrarytopredictions oftheinviscidtheorybyTsien(1943).Theirhypothesisforthisobservationwasthatwhilethe airfoilisgeometricallysymmetric,duetotheasymmetricdevelopmentofviscousboundarylayers ontheupperandlowersurfacesoftheairfoil,itbehaveslikeanairfoilwithanegativecamber. Theyalsoobservedthatthelocationoftheleadingedgestagnationpointmovesfurtherdownstream 3 (a) C L (b) Figure1.2: (a) Schematicofathree-segmentlinearvelocitypro˝leusedbyHammeretal.(2018) (b) Averageliftcoe˚cient, C L ,vsshearrate, K ,forinviscidandviscidsolutionoftheairfoils investigatedinHammeretal.(2018).(FiguresfromHammeretal.,2018) Figure1.3:SchematicofareversevonKármánvortexstreetbehindanairfoilpitchingaboutits quarterchord.Thesignofthecirculationofthevortexisindicatedbythearrowandbythe(+)or (-)sign. withincreasedshearrate,whichisconsistentwiththebehaviorofanegativelycamberedairfoil. SubsequentexperimentalforcemeasurementsreportedinHammeretal.(2019)alsoshoweda negativeliftcoe˚cientatzeroAoAforaNACA0012airfoilinasimilarshear˛ow,consistentwith theirpreviousnumericalresultsandoppositetoTsien'sinviscidtheory. Inadditiontoaerodynamicperformanceofastationaryairfoil,theunsteadyaerodynamicsof anoscillatingairfoilisalsoofinterest.Thepurepitchingoscillationofanairfoilwithchordlength c istypicallydescribedbyitspitchaxislocation o c ,oscillationamplitude( 0 ),meanangleof attack( m )andreducedfrequency k = ˇ fc u c ,where f istheoscillationsfrequencyand u c isthe ˛owspeed(SeeFigure1.3). Classicalstudies(Theodorsen,1949;vonKarman&Burgers,1935;vonKarman&Sears,1938) 4 haveshownthatasimpleoscillatorypitchingmotionoftheairfoilcanresultinthegenerationof thrustinsteadofadragforce.VonKarman&Burgers(1935)andLighthill(1969)associated thisthrustgenerationwithanarrayofalternating-signvorticesshedfromtheairfoil.Experiments ofKoochesfahani(1989)showedthecrossoverfromavonKármánvortexstreettoareverse vonKármánstreetathighenoughoscillationfrequenciesofapurelypitchingairfoil.Bohl& Koochesfahani(2009)foundexperimentallythatforaNACA0012airfoilatachordReynolds numberof Re c = 1 : 25 10 4 ,theswitchfromdragtothrusthappenedatanoscillationfrequency slightlyhigherthanthatofthecrossoverinvortexpattern.ExperimentalresultsofBohl& Koochesfahani(2009)werecloselyreplicatedbycomputationsofHammer(2016)throughhigh ˝delity2Dnumericalsimulations,wherehealsoinvestigatedthee˙ectsofReynoldsnumberand airfoilmotionasymmetryonthewakevortexproperties.Figure1.4showscomputational˛ow visualizationsofHammer(2016)comparingatraditionalvonKármánvortexstreet(resultofa lowfrequencypurepitchingoscillation)withareversevonKármánvortexstreet(resultofahigh frequencypurepitchingoscillation)alongsideasketchofthewakevelocitypro˝les. Thee˙ectsofasteadyshearapproach˛owontheaerodynamicperformanceofanunsteady airfoilinpurepitchingoscillationisreportedbyHammeretal.(2019).Inthisstudy,complimentary numericalandexperimentalinvestigationswereperformedonaNACA0012airfoilpitchinghar- monicallyaboutitsquarter-chord,withanoscillationamplitudeof 0 = 2 andreducedfrequencies upto k = 12 ,inapositivesteadyshear˛ow.Theirapproach˛owwassimilartothepro˝leusedin Figure1.4:Computational˛owvisualizationoftraditionalvonKármánstreet (a) versusreverse vonKármánstreet (b) withaccompanyingmeanwakevelocitypro˝le.(FiguresfromHammer, 2016) 5 Figure1.5:Computedmeanand˛uctuatingloadsonanoscillatingairfoilversusreduced frequency k fordi˙erentshearrates( Û ).(FiguresfromHammeretal.,2019) Hammeretal.(2018)withnon-dimensionalshearratevaluesvaryingbetween0to1andachord Reynoldsnumberof Re c = 1 : 2 10 4 .Thenumericalpartofthestudyshowedabreakofthe symmetryofthewakestructureforallreducedfrequencies,withthewakede˛ectingtowardsthe highspeedsideathighenoughreducedfrequencies.Consistentwiththeirprevious˝ndings,the presenceofshear˛owresultedinanegativemeanliftcoe˚cientatthelimitofstationaryairfoil (seeFigure1.5a).Theyobservedthatthemagnitudeofthisnegativeliftdecreasedwithincreasing oscillationfrequencyuntilitswitchedsignaroundareducedfrequencyof k = 3 ,andfromthat pointonthemagnitudeincreasedgoingtowardshigherfrequencies.Theyfoundthatforaconstant reducedfrequency,themagnitudeofthemeanliftcoe˚cientincreasedwithnon-dimensionalshear rate(Figure1.5a),however,themagnitudeofliftcoe˚cient˛uctuationsdidnotexhibitanychange withshearrate(Figure1.5c).Inaddition,Meanand˛uctuationsofthrustcoe˚cientwereweakly a˙ectedbythepresenceofshearintheapproach˛ow(Figures1.5b&1.5d).Theseconclusions 6 werebasedonbothnumericalandexperimentalresults. Allthestudiesmentionedabovepertainedtosteadyshear˛ows.Oneoftheonlyexperimental measurementsavailableonthee˙ectofanunsteadyshear˛owonanunsteadyairfoilisfromNaguib &Koochesfahani(2012).Theyexaminedthee˙ectofacanonicalplanemixinglayerapproaching aharmonicallypitchingNACA0012airfoilatareducedfrequencyof k = 10 : 6 andoscillation amplitudeof 0 = 2 withachordReynoldsnumberof Re c = 1 : 2 10 4 .Thephase-resolved spatialmapofthestreamwisevelocityforoneoscillationcycleatonechordlengthdownstreamof thetrailingedgeisshowninFigure1.6forboththeuniform˛owandunsteadyshear˛ow.The caseofuniformapproach˛owshowedtheexpectedsignaturefromanalternatingvortexarrayin thewake(Figure1.6a)andthecorrespondingregionsofvelocityundershoot/overshoot(evidence oftwooppositesignvorticesinoneoscillationperiod).However,theresultsfromtheunsteady shear˛owcaseexhibitedonlyoneregionofvelocityundershoot/overshoot,withoutanysignof asimilarvelocitysignatureforanoppositesignvortex(Figure1.6b).Thisobservationseemsto implythatinthepresenceoftheunsteadyshear˛ow,theairfoilshedonlyonevortexstructureper oscillationcycle,contrarytothetwofortheuniform˛ow.Anotherinterestingobservationwasthe simultaneouspresenceofbothmeanmomentumexcessandde˝citinthewake˛ow. 1.2CurrentStudy ThemainobjectiveofthecurrentPh.D.thesisistoinvestigatehowtheunsteadybehaviorofa shear˛owalterstheaerodynamicperformanceofalowReynoldsnumber( Re c = 1 : 2 10 4 )airfoil, inbothstationaryandpitchingsituations.Inordertoisolatethisissue,itisrequiredtogeneratetwo separatematchingshear˛owswithidenticalmeanvelocitypro˝les,whereoneshowsanunsteady natureresultinginspatiallynon-uniformhigh-leveltemporalvelocity˛uctuationswhiletheother hasasteadyvelocitypro˝lewithverylowandspatiallyuniformtemporalvelocity˛uctuations. Thegoalistobeabletoseparatethee˙ectsthatareattributedtothemeanshearpro˝lefromthose causedbyunsteadinessoftheshear˛ow. Regardingthee˙ectsofinterest,theaerodynamicforcesonstationaryandpitchingairfoils 7 Figure1.6:Colorcontourmapofphase-averagedstreamwisevelocity,showingthevariationof velocitypro˝leswithphaseduringonecycleofoscillationinthewakeofthepitchingairfoil ( k = 10 : 6 Hz, 0 = 2 ). (a) Thebaselinecaseofuniformapproach˛owshowstheexpected signatureofalternatingsignvortexpair. (b) Inthecaseofshearlayerapproach˛ow,thereis evidenceforonlyonevortexcore.(FigurecourtesyofKoochesfahani&Naguib) 8 (oscillatingoverarangeofamplitudesandfrequencies)aretobecomparedforthesetwodi˙erent shearlayersaswellastheuniformapproach˛ow.Moreover,boundarylayerresolvedsingle componentvelocimetryontheairfoilsurfacewillbeusedtoinvestigatethee˙ectofunsteady andsteadyshearlayersonboundarylayerbehavioroftheairfoil.Anotherpointofinterest istoqualitativelylookattheimpactofthesetwoshearlayersonthewakecharacteristicsand vortexarrangementbehindanoscillatingairfoilbymeansofmoleculartagging˛owvisualization. Thesemeasurementswillprovidethe˝rstinsightintotheroleofshearlayerunsteadinessonthe aerodynamicperformanceofairfoilsinunsteadyshearlayersastheupstreamboundarycondition. 1.3Outline Theremainderofthisworkispresentedinsixchapters.Chaptertwo,brie˛yintroducesthe ˛owfacilityandexperimentaltechniquesusedtoperformvelocimetryandforcemeasurementsin thetestfacility.Italsodiscussesthedi˙erentmethodsusedtogeneratethesteadyandunsteady non-uniformvelocitypro˝les.Theseexperimentaltechniquesareusedtogenerateandcharacterize thebehaviorofapairofmatchingunsteadyandsteadyshearlayers.Theresultsoftheseshear ˛owcharacterizationsarepresentedinChapterthree.Chapterfourdiscussesthestationaryand pitchingaerodynamicperformanceofanairfoilpositionedinbothoftheseshearlayers,through directmeasurementsofliftanddragcoe˚cientsoverarangeofmeananglesofattack,reduced frequenciesandoscillationamplitudes.Theseresultsarecomparedwiththosecorrespondingtothe referenceuniform˛owaswellaspreviousworksintheliterature.E˙ectsofshear˛owunsteadiness onthebehaviorofairfoilboundarylayerishighlightedinChapter˝ve,whereboundarylayer resolvedstreamwisevelocitymeasurementsonastationaryairfoiloverarangeofanglesofattack provideaninsightintotheunderlyingmechanismsthroughwhichtheunsteadinessoftheshear ˛owcana˙ectairfoil'saerodynamicperformance.Chaptersixqualitativelycomparesthewake structuresofapitchingairfoilinsteadyversusunsteadyshear˛owsaswellasthereferenceuniform approach˛ow.Finally,Chaptersevenconcludesthedissertationwithabriefsummaryofthe importantresults. 9 CHAPTER2 EXPERIMENTALMETHODS 2.1FlowFacility Alltheexperimentsforthisprojectareperformedina10,000literclosed-returnfreesurface watertunnelfacility(EngineeringLaboratoryDesignInc.,LakeCity,MN)locatedintheTurbulent MixingandUnsteadyAerodynamicsLaboratory(TMUAL)atMichiganStateUniversity.Thetest sectionofthefacilitymeasures61cm 61cm 244cm(showninFigure2.1)andhasfulloptical accessinthevisiblespectrumonbothsides,bottomandafullheightend-viewwindow.Asapart ofapreviousmodi˝cation,twoquartzwindows(each41cm 84cm)wereinstalledtoprovide opticalaccessforultraviolet(UV)wavelengthsrequiredforMolecularTaggingVelocimetry,as describedlateron.This˛owfacilitycomeswitha˛owmanagementsystemconsistingoftwo perforatedplatesintheentrancedi˙user,a 1 / 4 "celldiameterhoneycomband˝nemeshscreenin thesettlingchamber,a6:1contractionchanneland˝nallya 1 / 8 "celldiameterhoneycomband˝ne meshscreenattheentranceofthetestsection. TheimpellerofthewatertunnelispoweredbyaToshiba20hpmotorandcontrolledwitha ToshibaVF-AS1drivewhichensuresday-to-dayrepeatabilityofthesamefreestreamvelocity;this repeatabilityisaveryimportantrequirementsinceallthevelocimetryandforcemeasurements wereperformedonmultipledi˙erentoccasionsspanningovera2yeartimeperiod.Whilethe˛ow facilitycanoperateuptoamaximumfreestreamvelocity( U 1 )of1m/s,itwassettoanominal valueof U 1 = 10 cm/sforallthemeasurementsperformedduringthiswork.Theactualmagnitude ofvelocitywasslightly˝netunedforeachcasetomaintainthedesiredReynoldsnumberamidst year-roundtemperaturechangesoflaboratoryenvironmentandconsequently,water.Freestream turbulenceintensity( FSTI = u rms š U 1 )ofthetunnelwasmeasuredtobe ˘ 1 : 9% atthisoperating conditions.Thisvalueincludestheknowncontributionsfromtheverylowfrequency( f < 0 : 2 Hz)"slushing"ofthefacilityandisconsistentwiththepreviousmeasurementsbyKatz(2010)and 10 (a) (b) Figure2.1:Ascaleddrawingofthe˛owfacility. (a) Topview, (b) Sideview.Quartzwindowsare visibleinthesideview.(FigurescourtesyofOlson,2017) Olson(2011)forthesamefacility.Olson(2011)reportedthatthee˙ectsofthisslushingmotion canberemovedbysubtractingthespatialmeanvelocityateachinstanceintimetoisolatethe contributionsfrombroadbandturbulenceandfoundthebroadband FSTI tobearound0.5%for thisfacility. Athree-degree-of-freedom(3DOF)servomotionsystemismountedontopofthetestfacility toprovidethecapabilitytoproduceacombinationofpitching,plungingandsurgingmotionsofthe airfoil(Figure2.2a).ThismotionsystemwasinitiallydesignedandbuiltbyDr.BrunoMonnier andlaterre˝nedandutilizedbyDr.DavidOlsonforexperimentsreportedinOlson(2017)and Hammeretal.(2019).Thecurrentstudyonlyutilizedthepitchingcapacityofthemotionsystemto oscillateaNACA0012airfoilwithachordlengthof c = 12 cmandaspectratioof AR = 5 : 14 about itsquarterchordpoint.Inordertominimizethee˙ectsoffreesurfacedisturbancesduringairfoil oscillation,askimmerplatespanningthewidthofthetunnelisused.Thisskimmerplatealso 11 (a) (b) Figure2.2:Ascaleddrawingofthetestsetupshowing (a) Thethree-degree-of-freedom(3DOF) servomotionsystem,and (b) Theairfoilandskimmerplateplacementinthetestsection.(Figures courtesyofOlson,2017) providesawell-de˝nedboundaryconditiononthetopsideoftheairfoil.Thesystemisdesigned sothatthereisalessthan0.5mmgapinbetweentheairfoiltopsurfaceandtheskimmerplate, aswellasbetweenairfoilbottomsurfaceandbottomplateofthetestsection(Figure2.2b).The actualairfoilpitchingmotiontrajectoryismonitoredthroughahighresolutionencoderwithhigh resolutionof0.003degrees. 12 2.2ShearGenerationMethods 2.2.1UnsteadyShearGeneration Duetotheirwell-studiedbehavior,aplanemixinglayerisselectedasarepresentativeofanunsteady shearlayerforthiswork(Liepmann&Laufer,1947;Wygnanski&Fiedler,1970;Brown&Roshko, 1974).Suchshearlayersareknowntoproduceahighlevelofspatiallynon-uniform˛uctuations andhavebeenshowntocontainvorticalstructures. Preliminaryexperimentsina6"watertunnel(a 1 / 4 scaleofthewatertunneldescribedabove) showedthatadevicewithacon˝gurationoftwoblocksofhoneycombwithdi˙erenttubelengths separatedbyasplitterplatecanbeusedtogenerateatwo-streamshearlayerwithabehavior veryclosetotheclassicaltwostreamshearlayers.Followingthoseresults,asimilardevicewas fabricatedtobeusedinthefullscalewatertunnelfacilitythatwouldgenerateashearlayerwith atwo-to-onevelocityratio.Thissheargenerationdeviceconsistedoftwouniformhoneycomb blocksoflengths1.5"and6"withasplitterplateseparatingthem,extending3"frombothends (seeFigure2.3). Thesplitterplateisgluedtobothhoneycombblocksusingurethaneadhesiveandthehoneycomb orientationisalignedsothatthestraightrowsoftubesareparalleltothesplitterplate.The honeycombblocks(PC2PolycarbonateHoneycomb,Plascore,Zeeland,MI)havecircular 1 / 8 " diametertubesandcomeinmaximumtubelengthof12". Inordertomakesuretherangeofairfoiloscillationfrequencieswillnotexcitethesplitterplate itself,anaturalfrequencyanalysisisperformedforanassumedcantileverplatelengthof3".Itis foundthatusinga0.03"thick( ˇ 0.76mm)highlycorrosion-resistant316stainlesssteelsheetwould resultina˝rstnaturalfrequencyof85Hzwhichiswellbeyondtherangeofairfoiloscillation frequencies( f < 3 Hz)andwatertunnel'snaturalfrequency( ˘ 0.2Hz).Thecharacteristicsofthe unsteadyshearlayergeneratedbythisdeviceisdescribedinsection3.1. 13 Figure2.3:Aschematicofthetwo-streamsheargenerationdevice 2.2.2SteadyShearGeneration Variousmethodshavebeenintroducedtoaltertheuniform˛owoffacilitiesbyinsertingan additionaldeviceinthepathofthe˛ow,downstreamofitsoriginal˛owmanagementsystem. Owen&Zienkiewicz(1957)werethe˝rsttodevelopamodeltogeneratealinearvelocitypro˝le usingagridofparallelrodswithvaryingcross-streamspacing,whichwasfurthermodi˝edby Cockrell&Lee(1966)toproducenonlinearvelocitypro˝les.Anothermodelutilizingshaped screenstogeneratearbitraryvelocitypro˝leswasproposedbyElder(1959)andwaslaterre˝ned byLau&Baines(1968),Turner(1969)andLivesey&Laws(1973).Takingadi˙erentapproach, Kotansky(1966)developedasimpli˝edmodeltoemployhoneycombswithvaryingtubelengths tocreateaspeci˝edvelocitypro˝le.Asetofequal-widthandequal-lengthparallelchannelswith variableinternalresistanceshavealsobeenusedbyChampagneetal.(1970)toproduceanear linearmeanvelocitypro˝le. Besidesthefactthatallofthesemethodscanproducenon-uniformvelocitypro˝les,some practicallimitationsareinvolvedwitheachmethodthatareworthconsidering.Maintainingthe 14 designedpro˝leoftheshapedscreensacrossthetestsectionisverychallenging,especiallyin largertestfacilities.Furthermore,screenscanbeeasilydamagedorobstructedwhichcouldcause distortionsintheresultingvelocitypro˝le.Thegridofparallelrodsareslightlyeasiertomaintain inlargerfacilities,butthismethodintroducesadditionaltemporalvelocity˛uctuationsintothe˛ow duetothevortexsheddingaroundeachindividualcylinder.Theshapedhoneycombtechniqueis relativelyhardertofabricate,butitmaintainsitsstructureovertimeandislesssusceptibletominor mishaps.Also,theresultingvelocitypro˝leshavealowleveloftemporalvelocity˛uctuations (Rose,1970;Kiyaetal.,1980). Duetothelowleveloftemporalvelocity˛uctuationsrequiredforthecurrentstudy,andwith repeatabilityoftheexperimentsinmind,theshapedhoneycombtechnique(Kotansky,1966)was chosenasthebasisforthesheargenerationmethodemployedinthiswork.Insteadoftheoriginal model'sassumptionofaconstantfrictionfactorinallhoneycombtubes,thismodelismodi˝ed toutilizeavariablefrictionfactormodeltoestimatethepressuredropinsidehoneycombtubes. Thisfrictionfactormodel(afunctionoftubelengthandReynoldsnumber)isapplicableinsidethe tubeentrancelengthaswellasinthefullydevelopedregion,inthelimitoflaminar˛owinsidethe tubes( Re d 2300 ).Additionally,themodelisfurtherre˝nedtoeliminatetheneedtocalibrate thepressuredropexperimentallyforeachdesign. Figure2.4presentsaschematicofthesheargenerationmodelalongwithitsunderlyingassump- tions.Kotansky(1966)assumedthe˛owfarupstreamofthehoneycombhastheknownuniform inletvelocity, u ,withauniformpressuredistribution.Fardownstream,the˛owhasdeveloped intothedesirednon-uniformvelocitypro˝le, u ¹ y º ,andhasrecoveredauniformpressure.Based ontheseassumptions,theoverallpressurechange( P )betweenfarupstreamandfardownstream is: P = P P + 1 = h P P ¹ 0 ; y º i + h P ¹ 0 ; y º P ¹ L ; y º i + h P ¹ L ; y º P + 1 i : (2.1) The˝rsttermontherighthandsideofequation2.1representsthepressurechangeupstream ofthehoneycomb.Kotansky(1966)treatsthisdomainasapotential˛owregionandhencethis pressurechangecanbefoundusingBernoulli'sequation.Thesecondtermisthepressuredrop 15 Figure2.4:Aschematicoftheshapedhoneycombmodelalongwiththemainassumptions involved.(afterKotansky,1966) insidethehoneycombtubeswhichcanbeestimatedusingafrictionfactorandDarcy-Weisbach equation.Thelastterm,pressurechangedownstreamofthehoneycomb,ispresumedtobe negligible,basedontheassumptionofparallelstreamlinesafterthehoneycomb.Thisleadstothe followingequationforthehoneycomblengthdistribution: L ¹ y º = d 4 f 2 6 6 6 6 4 2 ˆ P + u 2 v ¹ y º 2 u ¹ y º 2 1 3 7 7 7 7 5 ; (2.2) where L ¹ y º isthehoneycombtubelengthateach y , d thehoneycombtubediameter, f theFanning frictionfactorand ˆ the˛uiddensity. Distributionofcross-streamcomponentofvelocitypro˝leattheentranceofthehoneycomb, v ¹ y º ,isdeterminedthroughthesolutionofpotential˛owregion,whilethefrictionfactor, f is estimatedbasedonanasymptoticfrictionfactormodel.Finally,aonepointboundarycondition withareferencehoneycomblengthisusedto˝ndtheoverallpressuredropvalue, P .Adetailed descriptionofthismodi˝edshapedhoneycombmodel,itsperformanceandthecharacteristicsof thegeneratedpro˝lesispresentedinAppendixA. Thisshapedhoneycombsheargenerationmethodisusedtoreproducethemeanstreamwise velocitypro˝leofthetwo-streamshearlayerwithauniformdistributionof˛uctuationsatthesame 16 (a) (b) Figure2.5:Anexamplecomparisonofthestreamwise (a) meanand (b) ˛uctuatingvelocity pro˝lesinthegeneratedsteadyshearlayerversusthereferenceunsteadyshearlayer. levelasthefreestream˛ow.Anexampleofmeasuredmeanand˛uctuatingstreamwisevelocity pro˝lesforthesesteadyandunsteadyshear˛owsisshowninFigure2.5. 2.3MolecularTaggingVelocimetry 2.3.1Background MolecularTaggingVelocimetry(MTV)techniqueisusedthroughoutthisworktomeasurethe velocityofthe˛ow.MTVisawhole˝eldnon-intrusiveopticaltechniquethatturnspremixed (ornaturallypresent)moleculesinthe˛uidintolonglifetimetracersthroughphotonexcitation. Typically,apulsedlasersourceisusedtoexcite(tag)speci˝cpatternsintotheareaofthe˛ow thatisofinterest.Thesetaggedregionsareinterrogatedtwicewithaprescribedtimedelayto formanimagepair(Figures2.6a&2.6b).Theextracteddisplacement˝eldfromtheimagepair, alongwiththeknowntimedelay,yieldstheestimateofthevelocity˝eldacrossthetaggedregion (Figure2.6c).ForfurtherdetailsaboutMTVtechnique,interestedreaderscanrefertoGendrich& Koochesfahani(1996)forMTVimageprocessingproceduresandKoochesfahani&Nocera(2007) foracomprehensivereviewofthedevelopment,photochemistry,andseveralapplicationsofMTV. Inthepresentwork,awater-solublephosphorescentsupramoleculetracer(Gendrichetal.,1997) 17 (a)(b)(c) Figure2.6:TypicalMTVimagepairsandtheresultantvelocity˝eld(Gendrichetal.,1997).The ˛owshownisfromavortexringimpactingona˛atwallatnormalincidence.Theaxisof symmetryisindicatedbythedashedlines. (a) Thegridimaged1 safterthelaserpulse. (b) The samegridimaged8mslater. (c) Thevelocity˝eldderivedfrom(a)and(b). ispremixedthewatertunnelfacility.Thechemicalcompositionofthissupramoleculeconsists ofthefollowingchemicals: 1 10 4 Mofmaltosyl- -cyclodextrin,0.055Mofcyclohexanol, andasaturatedsolutionof1-bromonaphthalene( ˘ 1 10 5 M).Thelifetimeofthissolutionwas measuredtobenominally ˘ ˝ = 3 : 5 ms.ACoherentCOMPexPro205XeCl,anexcimerlaserwith awavelengthof308nmandpulsedurationof20ns,isusedtotagthischemicalcomplex. 2.3.2SingleComponentMTVforShearLayerCharacterization Basedontherequirementsofthisstudy,asinglecomponentversionoftheMTVtechniquewith paralleltaggedlines(normaltothefreestream˛ow)isimplementedtomeasurethestreamwise velocityofthe˛owacrossthewidthofthetestsection.SinglecomponentMTV(1c-MTV)delivers thecapabilitytomeasureonecomponentof˛owvelocity(normaltothetaggedlines)ateverypixel alongthetaggedlinestoprovideahighspatialresolution. Togeneratethetaggingpattern,theinitiallyrectangularbeamoftheexcimerlaserispassed throughacylindricallenscombinationtoreducethebeamthicknessalongitsshorteraxis.This resultsinathinandwidelasersheetthatcanbefocusedwhereneeded.Inthisimplementation, theopticsare˝ne-tunedtofocusthelasersheet( ˘ 0.5mmthick)aroundthecenterlineofthetest 18 Figure2.7:Aschematicof1c-MTVstreamwisevelocitymeasurementsetuputilizedin two-streamshearlayergenerationcon˝guration(topview) section.Thetaggingpatternisgeneratedbypassingthislasersheetthroughabrassbeamblocker withverticalslotsofdesiredthicknessandspacing.Theresultingtaggingpatternbecomesaseries oflaserlines(typicallyabout5)eachwithafullwidthhalfmaximum(FWHM)widthof ˘ 0.7mm andspaced ˘ 6mmfromeachother. Accordingtothedesireddomainofmeasurementsrequiredtocharacterizetheshearlayer behavior,theimagingsystemisdesignedtoconsistoftwoCCDcameras(pco.pixel˛y)with35mm lenses(f/1.2NikonNikkor)mountedsidebysideunderneaththetestsection.Thesecamerasare positionedsothatthereisa ˘ 5%overlapbetweentheir˝eldsofviews.Utilizingthisarrangement, eachcameraimagesanareaabout18cm 13cm(witharesolutionof ˘ 114 µ m /px),whichwould cover ˘ 60%ofthetestsectionwidthwhencombined(showninFigure2.7).Thisregionislarge enoughtoincludetheentireshearlayersandextendsintotheir˛atvelocityregions. Theopticalsetupisdesignedinafashiontoallowforboththelaseropticsandcameras totranslateconcurrentlyinthestreamwisedirections.Additionallythecamerasetupcanmove independentlyinthecrossstreamdirectiontobeabletocoverdi˙erentregionsinthewidthof thetestsection,whenneeded.Asmallertraversesystemprovidesthecapabilitytomovethelaser 19 opticsrelativetothecameraopticsinthestreamwisedirections,tomakeitpossibletofurther ˝ne-tunetheplacementofthetagginglinesinthecamera˝eldsofview. Foreach˛owcondition,thestreamwisecomponentofthe˛owvelocityismeasuredatmultiple downstreamlocationstocharacterizethestreamwisebehavioranddownstreamdevelopmentofthe shearlayers.Ateachdownstreamposition,undelayedimagesaretakenashorttime( ˘ 2 µ s )after thelaser˝restoprovidetheinitialpatternlocation.Delayedimagesaretaken8msafterthe˝rst imagestoproducethedisplacedpatterns.Thistimedelayisselectedbasedonthespatialresolution oftheopticalsetupandtheaveragespeedofthe˛owtoresultinanaveragedisplacementof ˘ 7 pixelsinthetaggedregions.Basedoncameralimitations,theimageacquisitionratewassetat 6.27Hzduringallofvelocimetrymeasurementsinthiswork. Inthecurrentimplementationof1c-MTV,correspondingundelayedanddelayedimagesarenot acquiredfromtheexactsamelaserpulse.Instead,separatesetsofundelayedanddelayedimages arerecordedseparatelyandtheneachoftheindividualdelayedimagesiscorrelatedtotheaverageof thesetofundelayedimages.Thisapproachisacceptablebasedontheassumptionthatthetiming jitterandpulsetopulsepointinginstabilityofthelaserarerandomandtheircontributionsare negligiblecomparedtothedisplacementbetweentheundelayedanddelayedimages.Additionally, itisassumedthattheexposuretimeoftheundelayedimagesareshortenoughthattheinstantaneous intensitypro˝lesofthetaggedlineshardlychangeduoto˛ow˛uctuationsandthesepro˝lesarevery closetotheiraveragerepresentation.Anydeviationfromthesepremiseswillmanifestasanincrease inthemeasuredvelocity˛uctuationsofthe˛ow.Thisincreasecanbeapproximatedbyperforming asimilarcorrelationbetweentheinstantaneousundelayedimagesandtheirmean.The˛uctuations levelsextractedfromtheseundelayedcorrelationscanserveasanestimateoftheuncertaintiesin instantaneousstreamwisevelocitiesduetothecombinede˙ectofprocessingnoiseandusingthe averageundelayedimage.Asetof512undelayedimagesusedinthesemeasurementsarefound toresultinanassociateduncertaintyoflessthan0.057cm/sforinstantaneousstreamwisevelocity measurements.Byacquiringasetof2048delayedimages,thistranslatesintoanuncertaintyof 0.0025cm/sinmeanstreamwisevelocityvalues. 20 Figure2.8:Aschematicofhowavelocitycomponentparalleltoataggedlinecangeneratean errorinthevelocitynormaltothelinein1c-MTV. TheimageprocessingprocedureusedhereissimilartotheoneusedbyKatz(2010)andfurther improveduponbyOlson(2011),whichperformsaline-by-line,row-by-rowcrosscorrelation betweentheintensity˝eldsofundelayedanddelayedtaggedlinesandthenutilizesa7thorder polynomial˝tto˝ndthepeakofthecorrelationmapwithsub-pixelaccuracy.Formoreaccurate valuesofvelocity˛uctuationsin˛ow,Thecontributionsofwhitenoise(frombothcameraand processing)tothe˛uctuatingvelocitymeasurementsareremovedbyusingautocorrelationofthe instantaneousvelocitytime-series,asproposedbyOlson(2011). Duetotheinherentassumptionofunidirectional˛owin1c-MTVtechnique,itispronetoerrors whenthereisacomponentofvelocityparalleltothetaggedlines,(asillustratedinFigure2.8). FollowingtheanalysisofHill&Klewicki(1996),thiserrorinmeasuredinstantaneousstreamwise velocityovertimedelay t canbeexpressedas: u u = v u du d y t ; (2.3) where u and v representtheinstantaneousstreamwiseandcross-streamvelocitycomponents,and du š d y istheinstantaneousstreamwisevelocitygradient. 2.3.3SingleComponentMTVforAirfoilBoundaryLayerCharacterization SimilartotheworksofKatz(2010),Olson(2011)andOlsonetal.(2013,2015)1c-MTVisutilized tomeasurethestreamwisecomponentof˛owvelocityataveryhighspatialresolutioninthe 21 Figure2.9:Aschematicof˝eldofviewsizeandtheopticalsetupforairfoilboundarylayer velocimetry.Thesolidanddashedlinesillustratethetwodi˙erent˝eldsofviewsusedforthese measurements. immediateproximityofairfoilsurface.ThistechniquetakesadvantageofthefactthattheMTV imagesarecapturedwithashortdelayaftereachlaserpulseandhencethereisnosigni˝cantwall glareobstructingtheviewclosetothesurface. Thelaseropticsandcamerasetuparesimilartotheonedescribedintheprevioussubsection. Theonlydi˙erenceinthelaseropticssetupisthatthelasersheetispassedthroughanotherset ofcylindricallensestoexpandthewidthofthesheetbeforegoingthroughabeamblocker.This generatesaseriesof31laserlineseach0.7mmthick(FWHM)andspaced2.8mmfromeach other.Forthesemeasurements,onlyasinglecameraisutilizedwhichispositionedclosertothe planeofinterestandusesa58mmlens(f/1.2Nikkor)toproduceamorezoomedup˝eldofview approximately9.5cm 7.1cm(witharesolutionof ˘ 68 µ m /px). Thesenearwallmeasurementsareperformedfortheairfoilpositionedatdi˙erentanglesof attackineachshear˛ow.Sincethesizeofthissetup's˝eldofview(andlasersheetwidth)is smallerthanthechordlengthoftheairfoil,foreachcasetwo˝eldofviewsarerequiredtocover theentiresurfaceoftheairfoil.Figure2.9showsasketchofthisopticalcon˝guration.Foreach ˝eldofview,asetof512undelayedand4096delayedimagesarerecordedwitha5msdelay(to generate ˘ 7pixelsnominaldisplacement)betweenthematarateof6.27Hz. 22 Theseimagesareprocessedinasimilarfashiontothesinglecomponentmeasurementsfor shearlayercharacterizationdescribedintheprevioussubsection.Afterprocessing,thelocationof thesurfaceoftheairfoilateachtaggedlineisfoundmanuallybasedontherawimagesaswell asrecoveredvelocitypro˝les.Furthermore,themeasurementpointsatwhichtheinterrogation windowispartiallyblockedbytheairfoilsurfaceora˙ectedbytheglarefromtheairfoilsurface havebeenremoved. Sincethequartzwindowsareinstalledonlyononesideofthewatertunnelfacility,UVoptical accessisonlypossiblefromonedirectionintothetestsection.Thislimitstheopticalaccessfor measurementstoonlyonesideoftheairfoil.Tocompensateforthisandacquire˛owvelocities fromtheoppositesideaswell,theorientationoftheshear˛owitselfcanbeadjustedthrough modifyingtheplacementsheargenerationdevices.Thisstrategycanprovide˛owinformationin theproximityofboththesuctionandpressuresidesoftheairfoilwhenitispositionedateither negativeofpositiveanglesofattackinapositiveshear˛ow. 2.3.4MolecularTaggingFlowVisualizationforAirfoilWakeVisualization Thesamelaseropticsandcamerasetupdescribedintheprevioussubsectionisusedtovisualizethe wake˛owbehindanoscillatingairfoil,similartotheworkofKoochesfahani&Bohl(2002).The opticalsetupismovedtotagandrecordtheregionjustdownstreamofthetheairfoiltrailingedge anddelayedimagesareacquiredforarangeofairfoiloscillationparameters.Inordertoenhance the˛owfeaturesthroughlargedistortionsofthetaggedlines,adelaytimeof30msisusedinthese measurements. Sincethisdelaytimeof30msiswellbeyondthenominallifetimeoftheMTVsolution,the camerapixelsarebinned 2 2 toimproveimagequalityandacquisitionrate.Thisresultsina resolutionof ˘ 136 µ m /pxandasamplingrateof12.27Hz.Foreachairfoilmotioncaseineach ˛ow,theairfoilisallowedtooscillatethelargerof10airfoilconvectivetimesor60airfoiloscillation periodsandthen1024delayedimagesareacquired.Theinstantaneouspositionoftheairfoilateach frameisalsorecorded,whichislaterusedtophaseordertheimagesbasedonairfoiloscillation 23 (a) (b) Figure2.10: (a) Asamplepairofmoleculartagging˛owvisualizationtagginglinesshownat initial(violet)anddelayed(green)statesinthewakeofastationaryairfoilinauniform˛ow.The initialimageistaken ˘ 2 s afterthelaserpulse,whilethedelayedimageiscaptured30msafter thelaserpulse. (b) Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 8 inuniform˛owat4di˙erentphasesofoneairfoiloscillationcycle.Theoriginprescribes thelocationoftheairfoiltrailingedge. frequency.A2Dmedian˝lterwitha 3 3 ˝ltersizeisusedtoreduceimagenoisebeforephase averaging. Figure2.10ashowsablownupdemonstrationoftheamountofdisplacementobservedbetween theundelayedtaggedlinesandthe30msdelayedimagesbehindastaticairfoilsetatzeroangleof attack.Asampleimageofphaseaveragedwake˛owbehindanoscillatingairfoilisalsopresented inFigure2.10b. 2.4ForceMeasurementSetup TheforcemeasurementsetupconsistsofanATImini40six-componentforce/torquesensor whichisassembledsothatitconnectstheNACA0012airfoiltothepitchingmotor.Thissetup (showninFigure2.11)wasdesignedandassembledbyDr.DavidOlsonandwasusedforforce measurementsperformedinOlsonetal.(2016)andHammeretal.(2019).Presentstudyonly utilizestheliftanddragforcereadingsforboththestationaryandpitchingairfoilmeasurements. Inthecaseofforcemeasurementsonastationaryairfoil,dragandliftforcesaremeasuredover 24 Figure2.11: (a) 3Dmodelofthewatertunneltestsection,depictingtheverticalairfoilmounted tothe3DOFmotionsystemontopofthetestsection. (b) Detailsofthe3DOFsystemandairfoil mounting.(FigurescourtesyofHammeretal.,2019) arangeofanglesofattackwiththefollowingprocedurerepeatedforeachangleofattack: 1. Withthewatertunnelturnedo˙,theairfoilissettotheprescribedangleofattack. 2. 20secondswaittomakesureanyvibrationsfromtheairfoilmotionhavediminished. 3. Withthetunnelstillo˙,forcesaremeasuredfor120stoestablishaninitialbiasmeasurements. 4. Tunnelisturnedonwitha3minuteswaitforeverythingtoreachasteadystate. 5. Theaerodynamicforcesaremeasuredfor180s 6. Tunnelisturnedo˙againwitha3minuteswaitforeverythingtosettledown. 7. Withthetunnelo˙,forcesaremeasuredfor120stoestablishapost-measurementbias. Theinitialand˝nalbiasmeasurementsareusedtoremovethebiaserrorandquantifythedrift ofthesensorduringeachmeasurement.Additionally,themeasurementdurationisdesignedtobe 25 lessthan15minutesforeachcasetominimizetheimpactofsensordrift(towithinthemeasure- mentresolution).Thereportedinstantaneousresolutionaccordingtothesensorspeci˝cationsis 0.005Nforliftanddragmeasurements,correspondingtoaresolutionof0.014inliftanddrag coe˚cients(basedonthemeasurementoperatingparameters).Hammeretal.(2019)havefound theuncertaintyofthemeanforcecoe˚cientmeasurementstobedominatedbysensordriftduring eachmeasurementwithanestimatedvalueof0.005innormalizedcoe˚cients.Duringthecourse ofthesemeasurementstheoriginalsensorwasdamagedandreplaced.Whilethereplacement sensorwastheexactsamemodel,itwasobservedthatthenewsensoroccasionallyexperienced slightlylargermagnitudesofdrift. Asimilarprocedureisperformedforpitchingairfoilmeasurementswiththeairfoiloscillating overarangeofoscillationamplitudesandfrequencies: 1. Withthewatertunnelturnedo˙,theairfoilissettotheprescribedmeanangleofattack. 2. 20secondswaittomakesureanyvibrationsfromtheairfoilmotionhavediminished. 3. Withthetunnelstillo˙,forcesaremeasuredfor60stoestablishaninitialbiasmeasurements. 4. Tunnelisturnedonwitha3minuteswaitforeverythingtoreachasteadystate. 5. Withthetunnelon,aerodynamicforcesonthestationaryairfoilatmeanangleofattackare measuredfor120s. 6. Theairfoilstartsoscillationwiththeprescribedmotionparameters. 7. Afterawaitof10convectivetimesplus20airfoiloscillationperiods,theunsteadyaerody- namicforcesaremeasuredfor180s 8. Tunnelisturnedo˙againwitha3minuteswaitforeverythingtosettledown. 9. Withthetunnelo˙,forcesaremeasuredfor60stoestablishapost-measurementbias. Forcemeasurementdataforthestaticairfoilinthereferenceuniformapproach˛owisusedto establishthereferencezeroangleofattackinthesetup.Thisistheangleatwhichthenetforce 26 (intheplanenormaltoairfoilspan)ontheairfoilisatitsminimummagnitude.Thesymmetry oftheliftanddragforcesofthesamedatasetareutilizedto˝ndandcorrectforanypossible smallmisalignmentsbetweentheaxisofthesensorand˛owcoordinates.Allofthereportedforce measurementareperformedwithasamplingrateof2kHz. Hammeretal.(2019)measuredtheinertiaforcesduetoanyslightmisalignmentbetweenthe centerofmassoftheairfoil(includingthesupportshaftandothercomponents)andthepitchingaxis bypitchingtheairfoilinair.Theyfoundthatthecontributionsofinertiaforcestothe mean forces werenegligibleduetoharmonicnatureofthemotion.However,theseinertiaforcesresultedinan increaseof ˘ 0.15inliftcoe˚cient˛uctuationsand0.016fordrag/thrustcoe˚cient˛uctuationsin theirmeasurements. Foroscillatingairfoilmeasurements,frequenciesbeyond5timestheoscillationfrequencyand thesetupnaturalfrequencyareremovedfromtherawdatabeforecalculatingthereportedstatistical data.These˝ltereddataarealsophaseorderedandphaseaveragedbasedontherecordedmotion oftheairfoil. 27 CHAPTER3 SHEARGENERATIONRESULTS Figure3.1depictsaschematicofthefamilyofplanemixinglayersthatarestudiedinthiswork.In theclassicaltwo-streamshearlayers,twoplane˛ows(streams)withuniformvelocities U 1 and U 2 areinitiallyseparatedbyasplitterplate.Thesetwoparallelstreamscometogetherasthesplitter plateendsat x = 0 andthestreamsbegintomix.Theresultingshearlayergrowsfromthisvelocity di˙erenceandwidensasit˛owsdownstream.This˛owinitiallyentailsawakepro˝leduetothe presenceofthesplitterplate,butasitdevelopsdownstream,thewakecomponentdissipatesand the˛owapproachesself-similarityinvariable = c y š x farenoughdownstream. Atypicalaveragestreamwisevelocityoftwo-streamshearlayersintheself-similarregionis presentedinFigure3.2.Thismeanvelocitypro˝lecanusuallybeapproximatedbyahyperbolic tangentpro˝le,wherethecenterlineofthepro˝leisde˝nedasthe y locationwherethevelocityis equaltothecenterline(oraverage)velocity, U c = U 1 + U 2 2 .Therearetwowidelyusedparameters todescribethewidth,orthickness,oftheshearlayerateachdownstreamlocation:momentum Figure3.1:Aschematicofaplanemixinglayer. 28 Figure3.2:Asampleplanemixinglayermeanvelocitypro˝le. thickness, ,andvorticitythickness, ! ,eachde˝nedasfollows: = ¹ + 1 U 1 u ¹ y º U u ¹ y º U 2 U d y ; (3.1) ! = U @ u @ y max : (3.2) Usingeitherofthesede˝nitionsforshearlayerthickness,ithasbeenshownthattheshearlayer spreadslinearlywithdownstreamdistance.Here,vorticitythicknesswillbeusedasthemeasureof shearlayerthickness,whichforahyperbolictangentpro˝leis4timeslargerthanthemomentum thickness.Brown&Roshko(1974)reportedthatthegrowthrateofthevorticitythicknesscanbe approximatedby: d ! dx = ! x x 0 = 0 : 181 ; (3.3) where, x 0 istheapparentoriginand representsthevelocitydi˙erenceparameter,asproposedby Abramovich(1963)andSabin(1965): = U 1 U 2 U 1 + U 2 : (3.4) Theshiftoforiginto x 0 ismostlytocorrectforthee˙ectsof˝niteboundarylayerthicknesson thesplitterplateat x = 0 .Themagnitudeandsignofthiso˙setinoriginishighlydependenton theinitialboundaryconditionsattheedgeofthesplitterplate. 29 3.1UnsteadyShearFlowCharacterization Astheunsteadyshearapproach˛owinthiswork,behavioranddevelopmentofthetwo-stream shearlayerisstudiedbymeasuringitsstreamwisevelocitypro˝leasafunctionofdistancefrom splitterplate,utilizing1c-MTV.Figure3.3depictsthedevelopmentofthemeanand˛uctuating velocitypro˝lesoftheshearlayeratselectdownstreamlocations.Itisobservedthatrightatthe edgeofthesplitterplate(blacklineinFigure3.3a),the˛owconsistsoftwouniformvelocity pro˝lesofdi˙erentmagnitudes,separatedbyawakepro˝leresultingfromthepresenceofthe splitterplate.Theslightlyrippledpro˝leatthelowervelocitysideisduetothefactthatthisstream hasnothadenoughspacetocompletelyrecoverfromtheinitialjetscomingoutofthehoneycomb andtheremnantsofthesejetsarestillpresentinthe˛ow.Thesenon-uniformitiesarealsovisible inthe˛uctuatingvelocitypro˝leatthislocation(seeFigure3.3b). Travelingdownstreamfromthesplitterplate,boththewakecomponentandnon-uniformities onthelowspeedsideofthemeanvelocitypro˝ledisappearratherquickly,afteradistanceof ˘ 20 cm(redlineinFigure3.3a).Fromthispointon,themeanvelocitypro˝lesinFigure3.3aexhibitthe typicalshapeoftheplanemixinglayers,withtheshearlayerwideningasitcontinuesdownstream. The˛uctuatingvelocitypro˝lesinFigure3.3balsoshowtheexpectedbehaviorofthetemporal (a) (b) Figure3.3:Developmentofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesinthe two-streamshearlayer. 30 ˛uctuationsintwo-streamshearlayers,withthemaximummagnitudeofthe˛uctuationaround thecenteroftheshearlayerandgradualdecreasein˛uctuationsmovingawayfromthecenterof theshearlayer.Outsidetheshearlayer,theamplitudeof˛uctuationsdropsdowntothelevelof background˛uctuationpresentinthetestfacility.Thegrowthoftheshearlayerisalsonoticeable inthevelocity˛uctuationpro˝les. Figure3.4illustratesthelineargrowthofthegeneratedtwo-streamshearlayerthroughdevel- opmentofvorticitythickness(Figure3.4a)andshearlayercenterlinelocation(Figure3.4b).A linear˝ttothecalculatedvaluesofvorticitythicknessisusedtoestimatethegrowthrate, d ! š dx , andapparentorigin, x 0 ,oftheshearlayer.Thegrowthrateofthisshearlayerisfoundtobe d ! š dx = 0 : 071 = 0 : 208 ,where = 0 : 34 .Thisvalueisconsistentwiththegrowthratereported byBrown&Roshko(1974), d ! š dx = 0 : 181 .Theslightinclinationoftheshearlayercenterline locationtowardsthelowspeedsideisdemonstratedinFigure3.4b,whichisaknownbehaviorof thecanonicalshearlayers. Thenormalizedbehaviorofthetwo-streamshearlayermeanand˛uctuatingvelocitypro˝les arepresentedinFigure3.5.Theself-similarnatureofthisshearlayerisevidentfromthefact thattheproperlynormalizedcurvesfromdi˙erentdownstreamlocationsfallontopofeachother, (a) (b) Figure3.4: (a) Developmentoftheunsteadyshearlayervorticitythickness, ! . (b) Inclinationof unsteadyshearlayercenterlinepositiontowardsthelowspeedside. 31 (a) (b) Figure3.5:Self-similarbehaviorofthenormalizedstreamwise (a) meanand (b) ˛uctuating velocitypro˝lesinthetwo-streamshearlayer. forboththemeanand˛uctuatingvelocitypro˝les.Forcomparison,ahyperbolictangentpro˝le issuperposedonthemeanvelocitypro˝lesinFigure3.5a.TheshearlayerReynoldsnumber basedonvorticitythicknessandvelocitydi˙erence Re ! = U ! valuesrangefrom2700to 6800forthelocationsshowninFigure3.5.Itisworthmentioningthatthemaximumvalueof measurednormalized˛uctuatingvelocity, ¹ u 0 š U º max = 0 : 23 ,ishighercomparedtothereported valueof0.17intheliteraturetypicalforthese˛ows.Thisdiscrepancycouldbeduetotheerrors introducedbytheinstantaneousvaluesoflateralcomponentofvelocity, v ,intotheinstantaneous measurementsofthestreamwisevelocity. Parameter Value Parameter Value U 1 13.35cm/s U 1 U 2 2.01 U 2 6.65cm/s u 0 U max 0.23 U c 10.00cm/s 0.34 U 6.71cm/s d ! dx 0 : 208 Table3.1:Characteristicparametersderivedforthetwo-streamshearlayer.Displayedvaluesare averagevaluesbasedonmultiplemeasurementsintheself-similarregion. 32 Morerelevantparametersderivedfromthegeneratedtwo-streamshearlayerarepresentedin Table3.1.Itshouldbenotedthatthedisplayedvaluesareaveragevaluesbasedonalltheself-similar measurementsatvariousdownstreamlocations. 3.2SteadyShearFlowCharacterization Inordertoisolatethee˙ectsofshear˛owunsteadinessfromthoseofthemeanshear,asteady shearlayerisgeneratedtomatchthemeanstreamwisevelocitypro˝leofthetwo-streamshearlayer ataspeci˝clocation.Forthispurpose,themeanvelocitypro˝leofthetwo-streamshearlayerat 94cmdownstreamofthesplitterplate( x x 0 = 107 cm)isselectedasthereferencemeanvelocity pro˝le.Themodi˝edshapedhoneycombmodel(describedinAppendixA)isusedtodesigna variablelengthhoneycombdevicetoproducethisdesiredvelocitypro˝le.Figure3.6presentsthe meanand˛uctuatingvelocitypro˝lesofthisgeneratedsteadyshearlayer,comparedtothoseof thereferencetwo-streamshearlayer.Itisobservedthatwhilethemeanvelocitypro˝leofthe steadyshearlayerisverysimilartotheunsteadyshearlayer,their˛uctuatingvelocitypro˝lesare distinctivelydi˙erent.The˛uctuationsofthesteadyshearlayerarespatiallyuniformandatthe levelofthebackground˛uctuationsinthetestfacility,incontrasttothebellshapedpro˝leofthe ˛uctuationsintheunsteadyshearlayerwhichreachesamplitudesashighas7timesthebackground (a) (b) Figure3.6:Comparisonofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesinthe steadyshearlayerversusthereferenceunsteadyshearlayer. 33 (a) (b) Figure3.7:Comparisonofthenormalizedstreamwise (a) meanand (b) ˛uctuatingvelocity pro˝lesinthesteadyshearlayerversusthereferenceunsteadyshearlayer. ˛uctuationslevel. Figure3.7furtherhighlightsthesimilarityofthemeanvelocityandthecontrastin˛uctuations pro˝lethroughshowingthenormalizedmeanand˛uctuatingvelocitypro˝lesforboththesteady andunsteadyshearlayers.Amoredetailedcomparisonofthesetwoshearlayersintermsoftheir characteristicparametersisprovidedinTable3.2.Basedontheseobservations,thetwoshear layersdiscussedherewillbeusedasthepairofmatchingshearlayersforalloftheensuingforce measurementsandvelocimetryexperimentscarriedoutinthisproject.Fromhereon,theterms steadyshearlayer and unsteadyshearlayer willsimplyrefertotheseshearlayers,unlessstated otherwise. Althoughthesteadyshearlayergeneratedthroughtheshapedlengthhoneycombmethodis notexpectedtogrowasit˛owsdownstream,itsdevelopmentisinvestigatedtobettercharacterize itsbehavior.Figure3.8depictsthemeanand˛uctuatingvelocitypro˝lesmeasuredatmultiple downstreamlocations( x valuesmeasuredfromthehoneycombexitplane).Figure3.8asuggests thatthemeanvelocitypro˝leofthesteadyshearlayerremainsalmostidenticalasitprogresses downstream.Theslightincreaseinthe˛uctuationslevelatthecenteroftheshearlayer( ˘ 10% asshowninFigure3.8b)indicatesthatgivenenoughtravelingdistance,thissteadyshearlayer couldbecomeunstable.Toavoidanycomplicationsdueto˛owinstability,allthesubsequent 34 Parameter UnsteadyShearLayer SteadyShearLayer RelativeDi˙erence U 1 13.34cm/s 13.14cm/s 1.08% U 2 6.64cm/s 6.79cm/s 2.33% U c 9.99cm/s 9.96cm/s 0.26% U 6.71cm/s 6.35cm/s 5.40% ! 8.62cm 7.82cm 9.26% U 1 U 2 2.01 1.94 3.8% 0.34 0.32 5.15% u 0 U max 0.23 0.04 82.61% Table3.2:Acomparisonbetweenthecharacteristicsofthesteadyversusunsteadyshearlayer. Thevaluesforunsteadyshearlayerareextractedfromthetwo-streamshearlayermeanvelocity pro˝lemeasuredat94cmdownstreamofthesplitterplate( x x 0 = 107 cm). measurementsforthesteadyshearlayerareperformedwithindownstreamdistancesshownin Figure3.8. (a) (b) Figure3.8:Developmentofthestreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesinthe steadyshearlayer. 35 3.3SensitivityofShearFlowstoDownstreamPerturbations Itiswellknownthattheevolutionoftwo-streamshearlayerscanbestronglya˙ectedby perturbationsintroducedtothe˛ow,andathighforcingfrequencies,downstreamdisturbancescan evenin˛uencethebehaviorofvorticalstructuresofthetwo-streamshearlayeruptothetrailing edgeofsplitterplate(Ho&Huerre,1984;Koochesfahani&Dimotakis,1989).Therefor,before lookingatthee˙ectsoftheseshearlayersonaerodynamicperformanceofstaticandoscillating airfoils,itisbene˝cialto˝rstinvestigatehowtheperturbationsduetoairfoiloscillationsmight a˙ecttheshearlayeritself(i.e.doespitchingtheairfoilchangetheupstreamboundarycondition?). Thisissueisinvestigatedbymeasuringthestreamwisevelocitypro˝leoftheshearlayersupstream oftheairfoil(twoairfoilchordlengthsupstreamoftheairfoilleadingedge),whentheNACA0012 airfoilisplacedatthecenteroftheshearlayerandoscillatesarounditsquarterchordpointwith arangeofoscillationamplitudesandfrequencies.Thesesensitivitymeasurementsarecarriedout forbothoftheshearlayersandatmultipleairfoildownstreamplacementsfortheunsteadyshear layer. Figure3.9presentsnormalizedmeanand˛uctuatingvelocitypro˝lesmeasuredtwochord (a) (b) Figure3.9:E˙ectofairfoiloscillationonmean (a) and˛uctuating (b) velocitypro˝lestwochord upstreamoftheairfoilleadingedgewhenairfoilislocatedat x = 94 cminthetwo-streamshear layer. 36 (a) (b) Figure3.10:E˙ectofairfoiloscillationonmean (a) and˛uctuating (b) velocitypro˝lestwo chordupstreamoftheairfoilleadingedgewhenairfoilislocatedinthesteadyshearlayer. lengthsupstreamoftheairfoilwhenairfoilwasplacedinthecenteroftheunsteadyshearlayer atdownstreamlocation x = 94 cm(thedesignatedunsteadyshearlayerlocation),forselect oscillationamplitudeandfrequencycombinations.Itisobservedthattheperturbationsintroduced bythepitchingairfoilhavenonoticeablee˙ectonthebehaviorofbothmeanand˛uctuating velocitypro˝lesoftheunsteadyshearlayertwochordlengthsupstreamoftheairfoil.Whilethese plotsonlyshowtheresultsfortheairfoilpositionedat x = 94 cm,similarmeasurementsatlocations furtherdownstreamwiththickerandupstreamwiththinnerunsteadyshearlayersdidnotshowany e˙ectonthenormalizedmeanand˛uctuatingvelocitybehaviorupstreamoftheairfoileither.The resultsofthesemeasurementsalongwiththee˙ectsofstationaryairfoilpositionedatdi˙erent anglesofattackontheupstreamshear˛owareprovidedinAppendixB. Repeatingthesemeasurementswiththeairfoilplacedatthecenterofthesteadyshearlayer resultedinasimilarconclusion(SeeFigure3.10forvelocitypro˝lesmeasuredattwochordlengths upstream).Again,airfoiloscillationsdonotchangethenormalizedmeanand˛uctuatingvelocity behaviorupstreamoftheairfoil. Accordingtotheseresults,placingandpitchingtheairfoilatthecenteroftheseshearlay- ersdoesnota˙ecttheupstream˛owboundarycondition.Thismeansthatsincetheupstream boundaryconditionsareindependentofairfoiloscillationfrequencyandamplitude,unsteadyforce 37 measurementsforpitchingairfoilsshouldonlyre˛ectthee˙ectsoftheairfoiloscillationwithno rami˝cationsfromchangesintheapproach˛ow. 38 CHAPTER4 AERODYNAMICFORCEMEASUREMENTS Afterestablishingthebehavioranddevelopmentofbothsteadyandunsteadyshear˛ows,theload cellsetupisusedtomeasuretheaerodynamicforcesonstationaryandoscillatingairfoilspositioned atthecenterofeachshearlayer.Figure4.1showsaschematicoftheforcemeasurementsetup coordinatesystemusedforreportingtheaerodynamicforces. Afterintroductionoftheairfoiltothesetup,thechordlengthoftheairfoil, c ,willbeusedasthe lengthscalefordescribingtheapproach˛owandairfoilmotionproperties.Hencetheshear˛ows arecharacterizedbytheirnon-dimensionalmaximum(centerline)shearrateof K = du d y c U c = 1 : 0 andnormalizedthicknessof ! c = 0 : 7 .Themeasuredliftanddragforcesarenormalizedbythe centerlinevelocityoftheshearlayertoyieldliftanddragcoe˚cients, C L = L š¹ 1 š 2 ˆ U 2 c cs º and C D = D š¹ 1 š 2 ˆ U 2 c cs º ,where s istheairfoilspan.ThenominalchordReynoldsnumberofthe˛ow iskeptas Re c = 12 ; 500 . Ineachofthesectionsonstationaryandoscillatingairfoilsthatfollows,˝rsttheresultsofthe Figure4.1:Schematicofforcemeasurementcoordinatesystem. 39 matchingsteadyandunsteadyshearlayerpair(with K max = 1 : 0 and ! c = 0 : 7 )arecomparedto eachotherandthereferenceuniform˛ow.Then,theresultsfromonlytheunsteadyshearlayerat twomoredownstreamlocations(correspondingto K max = 1 : 4 and ! c = 0 : 5 ,and K max = 0 : 8 and ! c = 0 : 9 respectively)areshowntohighlightthee˙ectsofupstreamshearlayerwidthandshear rate. 4.1AerodynamicForcesonaStationaryAirfoil Toprovidethe˝rstinsightintothee˙ectsofunsteadinessofashear˛owontheaerodynamic performanceofastationaryairfoil,theresultsofforcemeasurementsinthesteadyandunsteady shear˛owsarecompared.Additionally,aerodynamicforcesmeasuredforauniform˛owareused asareferencetohighlightthemaindi˙erencesduotothepresenceofthemeanshearitself. AschematicoftheairfoiltrailingedgedisplacementisportrayedinFigure4.2,whereitis overlaidontopofthemeanstreamwisevelocitypro˝leoftheshear˛ows.Itisobservedthatthe Figure4.2:Extentofairfoilangulardisplacement( 20 )comparedtotheshearlayerthickness. 40 trailingedgeoftheairfoilspansapproximately65%oftheshearlayerthickness. Theaerodynamicperformanceofthestationaryairfoilintheshearlayersisevaluatedthrough averageliftanddragcoe˚cientsaswellaslifttodragratio.Here,thesymbolsshowtheaverage valuesmeasuredover180sofmeasurementsateachangleofattack,whiletheerrorbarsindicate theuncertaintyofeachmeasurements,whichasmentionedinexperimentalmethodschapter,is primarilydominatedbydriftofthesensorduringmeasurementperiod.Theerrorbarsaremostly withinthesymbolsize,however,thereareafewcasesthatexperiencedmoredrift.Thee˙ectof theseuncertaintiesismorepronouncedinlifttodragratioplots,accordingtoerrorpropagation. Figure4.3ashowsthatthegeneraltrendofthemeanliftcoe˚cientcurveinsteadyshearexhibits thetypicalbehavioroftheNACA0012airfoilinlowReynoldsnumberwheretheregionaround zeroangleofattackshowsalowerslopefollowedbyaregionofhigherslopeatmoderateangles, beforereachingstalloraplateauatlargerangles.Thisbehaviorissigni˝cantlydi˙erentinthe unsteadyshear˛ow,whereitshowsarelativelylinearbehavioracrossamoderaterangeofangles ofattackandwithalargerslope.Bothsteadyandunsteadyshear˛owcurvesshowanasymmetry inmaximumliftcoe˚cient,withalargeramplitudeofmaximumlifthappeningatnegativeangles ofattack. Thiscontrastinbehavioroftheliftcurveinunsteadyshearlayercanbeattributedtothehigh levelof˛uctuationspresentinthis˛ow.Thecentralregionoflowerslopeintheliftcurve(as observedinFigure4.3afortheuniform˛owandsteadyshearlayercases)isusuallyassociated withlaminarseparationonthesurfaceoftheairfoilwhenitissetatsmallanglesofattack(Cleaver etal.,2010;Wangetal.,2014).Thislaminarseparationisknowntobesensitiveto˛owconditions andOlson(2011)hasshownthatanincreaseinfreestreamturbulencelevelinauniform˛owcan greatlya˙ectthelaminarseparationbubblebyshorteningitinbothlengthandheightonaSD7003 airfoil.Henceitispossiblethatthehighlevelof˛uctuationsinthecentralregionoftheunsteady shearlayerhavea˙ectedthedevelopmentoflaminarseparationontheairfoil,andconsequently,its liftcharacteristics. Acloserlookattheliftcoe˚cientcurvearoundzeroangleofattack(Figure4.3b)showsa 41 (a) (b) (c) (d) Figure4.3:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoilplaced inuniform,steadyshearandunsteadyshear˛ows: (a) meanliftcoe˚cient, C L , (b) Zoomedup viewofmeanliftcoe˚cient, C L , (c) Meandragcoe˚cient, C D , (d) Ratioofmeanlifttomean drag, C L C D . negativeliftcoe˚cientatzeroangleofattackforsteadyshear˛ow,similartothe˝ndingsof Hammeretal.(2016,2018,2019)andOlsonetal.(2016).Incontrast,theunsteadyshear˛owis foundtogenerateapositiveliftatzeroangleofattack.Thischangeofliftforcesigncanalsobe relatedtothe˛uctuationsoftheunsteadyshearlayer.Hammeretal.(2018,2019)havehypothesized thatinthecaseofthesteadyshearlayer,thenegativeliftisaresultofasymmetricviscousboundary layerdevelopmentontheairfoilupperandlowersurfaces,whichcausesageometricallysymmetric 42 airfoiltoe˙ectivelybehavelikeonewithnegativecamber.Inthecaseofunsteadyshearlayer, however,thehighlevelof˛uctuationsseemstomakethe˛owbehaveclosertotheinviscidtheory ofTsien(1943),whichpredictsapositiveliftforapositiveshear˛ow. Theoverallbehaviorofthedragcoe˚cient(Figure4.3c)forbothsteadyandunsteadyshear layersisasymmetricwithlargermaximumdragcoe˚cientsmeasuredatnegativeanglesofattack. Theonlynoticeabledi˙erenceisthattheunsteadyshear˛owcurveexhibitssmallerdragcoe˚cients overalargerregionofanglesofattack,comparedtobothsteadyshearlayeranduniform˛ow.This dragreductionisconsistentwiththehypothesisofthelaminarseparationbeinga˙ectedbythe ˛uctuationsoftheunsteadyshearlayer,sincelaminarseparationisknowntoincreasethedragof theairfoilsatlowReynoldsnumber(Lissaman,1983). LifttodragratiocurvesareshowninFigure4.3d,wherealargeasymmetryinthepeakvalues oflifttodragratioisnoticeableforthecaseofsteadyshearlayer.Suchasymmetryisnotperceived intheunsteadyshear˛owcase,whichproducesanalmostsymmetricalpro˝le. Figure4.4providesacomparisonbetweentheresultsofthecurrentstudyinsteadyshearlayer withexperimentalresultsofOlsonetal.(2016)andnumericalresultsofHammeretal.(2019).It shouldbenotedthattheirsteadyshear˛owresultsarenotfromameanvelocitypro˝lematching theoneinthiswork,butfromathreesegmentedlinearvelocitypro˝le(asdescribedinHammer etal.,2019)with K = 0 : 51 , ! š c = 1 : 7 andavelocityratioof U 1 U 2 = 3 operatingatthesame centerlinevelocity, U c = 10 cm/sresultinginachordReynoldsnumberof Re c = 1 : 2 10 4 .This velocitypro˝leisshowninFigure4.5andcomparedtoshear˛owsgeneratedinthiswork.The comparisonwiththesesamplesteadyshearresultsareonlytohighlightthesimilaritiesingeneral trendsofliftcoe˚cientaroundzeroangleofattack. Theagreementbetweentheforcemeasurementsinthesteadyshear˛owwithhyperbolictangent velocitypro˝legeneratedhereandthelinearvelocitypro˝leofOlsonetal.(2016)andHammer etal.(2019)isveryinteresting;especiallysincetheirshearrate,shearlayerthicknessandeven theshearlayerpro˝lesarecompletelydi˙erent.Fromtheperspectiveofshearrate,itisexpected thattheshear˛owwiththehighershearratewouldproducealargermagnitudeofnegativeliftat 43 (a) (b) (c) (d) Figure4.4:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoilplaced inthesteadyshearlayerandtheresultsofOlsonetal.(2016)andHammeretal.(2019): (a) mean liftcoe˚cient, C L , (b) Zoomedupviewofmeanliftcoe˚cient, C L , (c) Meandragcoe˚cient, C D , (d) Ratioofmeanlifttomeandrag, C L C D . zeroangleofattack,duetoanincreased˛owinducedasymmetryintheboundarylayerthicknesses onthetwosidesoftheairfoil(SeeHammeretal.,2018).However,unpublishedsimulationsof Hammerhaveshownthatforalinearvelocitypro˝le(similartotheonesusedinOlsonetal.2016 andHammeretal.2016,2018,2019)themagnitudeofgeneratednegativeliftatzeroangleof attackisalsodependentonshearlayerthicknessandincreaseswith ! c untilthisratioislarge enoughthatthe˝nitesizeoftheshearzonedoesnotin˛uencetheoutcomeanymore.Hefound 44 Figure4.5:Comparisonoftheshear˛owvelocitypro˝lesstudiedinthecurrentstudywiththe linearsteadyshearusedinOlsonetal.(2016). thatthishappenswhen ! c > 2 ,whichmeansthatthe˝nitethicknessoftherelativelythinshear layerusedhere ! c = 0 : 7 wouldgenerateasmallermagnitudeofnegativeliftatzeroangle ofattack.Furthermore,hiscomparisonsofahyperbolictangentvelocitypro˝leversusalinear velocitypro˝leofthesamethickness bothwith ! c = 1 : 5 andequalcenterlineshearrates( K max ) showedaslightlysmallermagnitudeofliftgeneratedbyhyperbolictangentpro˝leatzeroangle ofattack.Thisisconsistentwiththefactthatinahyperbolictangentvelocitypro˝le,theaverage shearrateoverthethicknessofairfoilisalwayssmallerthanitscenterline(maximum)shearrate. Consequently,thecombinationoftheseopposinge˙ectsmayhavecontributedtothesimilarityof theobservedliftforcebehaviordespitetheobviousdi˙erencesbetweenthesetwoshearlayers. 4.1.1E˙ectofUpstreamUnsteadyShearFlowPro˝leonAerodynamicForcesonaStation- aryAirfoil Thissubsectionfocusesonlyontheforcemeasurementresultsintheunsteadyshear˛owtolook atthee˙ectofthedownstreampositioningoftheairfoilinthegrowingunsteadyshearlayer.For thispurpose,theairfoilispositionedatthecenterofthetwo-streamshearlayeratthreedi˙erent downstreamlocations,whereboththeshearrateandthicknessoftheshearlayerchangewiththis 45 Figure4.6:Measurementlocationsforforcemeasurementsonstationaryairfoilintwo-stream shearlayer.Thesymbolsshowthe x locationswhereforcemeasurementshavebeenperformed, withthepro˝lesindicatingthemeasuredmeanvelocitypro˝leatthatlocation. streamwiseposition.Figure4.6showsthethreedownstreamlocationsatwhichforcemeasurements havebeenperformedonthestationaryairfoil.Figure4.7comparestheextentofairfoildisplacement toshearlayerthicknessateachoftheselocationsforthemaximumangleofattackof20 ° .Itcan (a) (b) (c) Figure4.7:Extentofairfoilangulardisplacement( 20 )comparedtothetwo-streamshearlayer thicknessateachmeasurementlocation. 46 (a) (b) (c) (d) Figure4.8:Comparisonofaerodynamicforcecoe˚cientsmeasuredonastationaryairfoilplaced inunsteadyshearlayeratdi˙erentdownstreamlocations: (a) meanliftcoe˚cient, C L , (b) zoomedupviewofmeanliftcoe˚cient, C L , (c) meandragcoe˚cient, C D , (d) ratioofmeanlift tomeandrag, C L C D . beseenthatthedisplacementoftheairfoiltrailingedgebecomescomparabletotheshearlayer thicknessatthelargestangleofattack,especiallyforthethinnestshearlayerwith ! c = 0 : 5 . Theaerodynamicforcemeasurementresultsfordi˙erentdownstreamlocationsinunsteady shearlayerarepresentedinFigure4.8.Themainobservationformtheliftcoe˚cientcurvesisthat theoverallbehavioroftheliftcoe˚cientisverysimilarforallthedownstreamlocations(Figure 4.8a).Theonlydi˙erenceisthatthemagnitudeofthepositiveliftatzeroangleofattackdecreases withdistancedownstream(Figure4.8b).AccordingtoFigure4.8c,thechangesinshearrate(and 47 shearlayerthickness)donotseemtonoticeablya˙ectthebehaviorofthedragcoe˚cient.Figure 4.8dindicatesthatwhilealltheunsteadyshearlayersdisplayamostlysymmetricpatterninliftto dragratio(especiallycomparedtothatofthesteadyshearlayer),goingfurtherdownstreamslightly increasestheweakasymmetryofthecurve. 4.2AerodynamicForcesonanOscillatingAirfoil Correspondingtothemeasurementsdescribedearlier,aerodynamicforcesaremeasuredona purelypitchingNACA0012airfoilplacedatthecenterofbothsteadyandunsteadyshearlayers. Similarly,themeasurementsarecarriedoutontheairfoilpitchinginauniformapproach˛owas welltoprovideareferenceforcomparisonwithsteadyandunsteadyshearlayerresults.Ineach˛ow theairfoilissettobepurelypitchingarounditsquarterchordpointwithoscillationamplitudesof 2 and 4 andreducedfrequencies, k ,rangingfrom1to6.Table4.1liststheoscillationfrequency detailsforallofthepitchingairfoilforcemeasurementsperformedinthiswork.Forbrevity,only theresultsfrom 0 = 2 oscillationsareshownhereandtheresultsfor 0 = 4 areprovidedin AppendixC. Theextentofthetrailingedgedisplacementduringpitchingoftheairfoiliscomparedtothe thicknessoftheshearlayerinFigure4.9,whereitisevidentthatthedisplacementoftheairfoilis wellwithinthecentralregionoftheshearlayer. Sinceinmoststudiesonoscillatingairfoilstheinterestisinthepossiblethrustforcegenerated bytheairfoilmotionandtobeconsistentwithpreviousworks,thestreamwisecomponentof oscillatingaerodynamicforcesmeasuredinthisworkispresentedintheformofthrustcoe˚cient, C T = C D . k = ˇ fc u c k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 0 = 2 f = 0 : 26 Hz f = 0 : 53 Hz f = 0 : 79 Hz f = 1 : 06 Hz f = 1 : 32 Hz f = 1 : 59 Hz 0 = 4 f = 0 : 26 Hz f = 0 : 53 Hz f = 0 : 79 Hz Table4.1:Listofunsteadyforcemeasurementcases. 48 Figure4.9:Extentofpitchingairfoilangulardisplacement( 2 )comparedtotheshearlayer thickness. Aerodynamicforcemeasurementresultsforthepitchingairfoilsareexploredthroughthe behaviorofmeanliftandthrustcoe˚cientsaswellastheir˛uctuationsasafunctionofreduced frequencyforasingleoscillationamplitude.SuchresultsaredepictedinFigure4.10forthe caseswithoscillationamplitudeof 2 ,wheresymbolsdepicttheaveragevalueof4independent measurementrepetitionsforeachcombinationof˛owandoscillationfrequency.Theerrorbars formeanvaluesarebasedonthepropagationofsensordriftduringeachmeasurement,whilethe uncertaintyin˛uctuationsisshownasthestandarddeviationofthe˛uctuationmagnitudesbetween the4repetitions. The˝rstnoticeableobservationfromtheplotofmeanliftcoe˚cientversusairfoiloscillation reducedfrequency(Figure4.10a)isthatfortheuniform˛owcases,themeanliftstaysveryclose tozero(withinexperimentalerrors)forallreducedfrequencies.Thisistheexpectedresultfor asymmetricairfoiloscillatingsymmetricallyaroundzeroangleofattackinauniform˛ow,and rea˚rmstheaccuracyofmeanliftmeasurements.Inthecaseofsteadyshearlayer,themean 49 (a) (b) (c) (d) Figure4.10:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoilplaced inuniform,steadyshearandunsteadyshear˛ows: (a) meanliftcoe˚cient, C L , (b) liftcoe˚cient ˛uctuations, C L 0 , (c) meanthrustcoe˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .The airfoilispitchingwithzeromeanangleofattackandanoscillationamplitudeof 2 aroundits quarterchordpoint. liftgraduallydecreasesinmagnitudefromitsinitialstateofnegativeliftuntilittransitionsto positiveliftaround k = 2 : 5 .Afterthispointthemagnitudeofnowpositivemeanliftincreaseswith frequency.ThispatternisconsistentwiththenumericalandexperimentalobservationsofHammer etal.(2019).Forunsteadyshearcase,themeanliftremainsrelativelyconstant(aroundthevalue ofstaticliftatzeroangleofattack)forsmallreducedfrequencies( k 2 )andgraduallyincreases afterwards.Thebehaviorofmeanliftinunsteadyshearlayerbecomessimilartothatofsteady 50 (a) (b) Figure4.11:Comparisonofaerodynamicforce˛uctuationsmeasuredonalowfrequencypitching airfoilplacedinuniform,steadyshearandunsteadyshear˛ows: (a) liftcoe˚cient˛uctuations, C L 0 , (b) thrustcoe˚cient˛uctuations, C T 0 .Theairfoilispitchingwithzeromeanangleofattack andanoscillationamplitudeof 2 arounditsquarterchordpoint. shearlayerbeyond( k 4 ). Thecomparisonofliftcoe˚cient˛uctuationsindi˙erent˛ows(Figure4.10b)suggeststhat neitherthepresenceofshearnoritsunsteadinessseemtoa˙ectthe˛uctuationsintheliftcoe˚cient, similarto˝ndingsofHammeretal.(2019).Meanand˛uctuatingthrustcoe˚cientsarefound tobeweaklya˙ectedbythechangeinshear˛owparameters(Figures4.10cand4.10d),whichis againinagreementwiththeconclusionsofHammeretal.(2019). Acloserlookatbothliftandthrust˛uctuationsatlowoscillationfrequenciesemphasizesthe slightlyhighermagnitudeof˛uctuationsinunsteadyshearlayercomparedtosteadyshearlayer anduniform˛ow(Figure4.11).Thisisconsistentwiththehigherlevelofinherent˛uctuationsin theunsteadyshear˛ow.Astheoscillationfrequencyofthepitchingairfoilincreases,these˛ow induced˛uctuationsbecomenegligiblecomparedtothe˛uctuationsgeneratedbyairfoiloscillation inmeasuredforces. Theseobservationssuggestthatatlowfrequencies,themeanliftgeneratedbyunsteadyshear layerismostlygovernedbytheunsteadinessofthe˛ow,butasthefrequencyofoscillation increasestheimpactofairfoiloscillationsbecomesmoredominantandathighenoughfrequencies, 51 (a) (b) Figure4.12:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoilplaced insteadyshearwithexperimentalandnumericalresultsofHammeretal.(2019): (a) meanlift coe˚cient, C L , (b) meanthrustcoe˚cient, C T .Theairfoilispitchingwithzeromeanangleof attack( m = 0 )andanoscillationamplitudeof 0 = 2 arounditsquarterchordpoint. theairfoiloscillationalonewilldictatetheaerodynamicperformance.Thishypothesiswillbe furtherinvestigatedlaterthroughphaseorderingtheforcemeasurementresults. Figure4.12comparesthemeanaerodynamicforcecoe˚cientsmeasuredinthesteadyshear layerofthecurrentworkwithnumericalsimulationsandexperimentalresultsofHammeretal. (2019)forasimilarNACA0012airfoilpitchingwiththesameparameters.Asmentionedpreviously, theirsteadyshear˛owresultsarenotfromameanvelocitypro˝lematchingtheoneinthiswork, butfromathreesegmentedlinearvelocitypro˝le.Withthisinmind,boththemeanliftandthrust coe˚cientsshowaqualitativeagreementwiththeirresults. Theresultsfromforcemeasurementsonanairfoilpitchingwithanoscillationamplitudeof 0 = 4 (presentedinAppendixC)showasimilartrend.Whilethemeanliftofthesteadyand unsteadyshearlayersinitiallyhaveoppositesigns,theyseemtoproducesimilarvaluesofliftat reducedfrequenciesof k > 2 : 5 . 52 4.2.1E˙ectofUpstreamUnsteadyShearFlowPro˝leonAerodynamicForcesonanOscil- latingAirfoil Theunsteadyaerodynamicforcesarealsomeasuredwiththepitchingairfoilpositionedatthetwo otherdownstreamlocationsinthetwo-streamshearlayer(showninFigure4.6)mentionedinthe stationaryairfoilsection.Figure4.13illustratesthesweepoftheairfoiltrailingedgeduringthe oscillatingmotionincomparisonwiththewidthoftheshearlayerateachoftheselocations.A maximumof ˘ 20%oftheshearlayerthicknessiscoveredfortheworstcaseof 4 oscillationsin themostupstreamlocationwith ! c = 0 : 5 (seeAppendixC). Themeanliftresultsofthesemeasurementsfortheoscillationamplitudeof 0 = 2 are comparedwiththereferenceunsteadyshearlayercaseinFigure4.14.Themeanliftcoe˚cient curvesexhibitasimilarbehaviorofrelativelyconstantmagnitudeofliftatlowfrequencyoscillations andthengraduallyincreasingafterareducedfrequencyof k = 2 : 5 .Themaindi˙erencebetween thedi˙erentdownstreampositionsseemstobeafairlyconstantshiftinthecurvebasedontheirlift magnitudeatzeroangleofattack.Thiswouldsuggestthattothe˝rstorder,theliftperformance (a) (b) (c) Figure4.13:Extentofairfoilangulardisplacement( 2 )comparedtothetwo-streamshearlayer thicknessateachmeasurementlocation. 53 Figure4.14:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoilplaced intwo-streamshearlayeratdi˙erentdownstreamlocations: (a) meanliftcoe˚cient, C L , (b) meanthrustcoe˚cient, C T .Theairfoilispitchingwithzeromeanangleofattack( m = 0 )and anoscillationamplitudeof 0 = 2 arounditsquarterchordpoint. oftheairfoilintheunsteadyshearseemscloselyrelatedtotheshearrateandtheinitialliftofthe stationaryairfoil. 4.3PhaseOrderedAerodynamicForcesonOscillatingAirfoil Becauseoftheoscillatorynatureofforcesinthisproblemandhighrepeatabilityofthemeasure- ments,phaseordering/phaseaveragingthemeasuredforcetimeseriescanprovidemoreinsightinto thedynamicsoftheforces.Themeanforceresultspresentedintheprevioussectionhaveshown tworegionswherethebehavioroftheunsteadyshearlayerwasdi˙erentfromthatofsteadyshear layer;atlowoscillationfrequenciestheunsteadyshearlayerproducedapositiveliftasopposed tothenegativeliftobservedinthesteadyshearlayer,whereasathigherfrequenciesthetwoshear layersbehavedclosertoeachotherwithbothgeneratingpositiveliftwithsimilarmagnitudes.These tworegionsarehighlightedinFigure4.15.Tofurtherinvestigatethedi˙erenceinaerodynamic performancebetweenthesetworegions,twosamplefrequencies( k = 2 and5,eachinoneregion) areselectedtocomparethephaseordered/averagedbehaviorofthemeasuredforces. Figures4.16,4.17and4.18showthephaseorderedandphaseaveragedliftandthrustcoe˚cients foruniform˛ow,steadyshearlayerandunsteadyshearlayerrespectively,withtheairfoilpitching 54 withareducedfrequencyof k = 2 andoscillationamplitudeof 0 = 2 .Eachoftheseplotsare fromasingleforcemeasurementexperimentdatasortedinto128phasebinsbasedontheairfoil angularposition(withaminimumof ˘ 1300 pointsperbinforcaseswiththelargestoscillation period).Intheseplots,thebluelinesrepresentthephaseordereddata,whiletheredlinesportray thephaseaveragedvaluesandtheblacklineshighlightthestandarddeviationofthedatagrouped ineachphasebin.Figures4.19,4.20and4.21depictthesameresultsforthereducedfrequencyof k = 5 . Throughthe˝rstglanceatthephaseorderedliftcoe˚cientresultsfor k = 2 cases,itbecomes obviousthatthecycletocyclevariationismuchlargerintheunsteadyshearlayercomparedto thesteadyshearanduniform˛ows.Whilethesameobservationistrueforthelargerreduced frequencyof k = 5 ,theratioofcycletocyclevariationtotheactualforceoscillationsissmaller inthiscase.Thisdi˙erencecanbedemonstratedbetterbycomparingtheaveragevalueofall thephasebinvariationsforeach˛owateachreducedfrequency h C L i 0 .Figure4.22displays thesevaluesbothstraightawayandnormalizedbytheoverallliftcoe˚cient˛uctuations h C L i 0 C L 0 ! inthatcase.Itisobservedthatalthoughtheaveragephasebinvariationsaresigni˝cantlyhigher forunsteadyshearlayeracrossallreducedfrequencies,itsnormalizedmagnitudebecomesmuch smallermovingtowardsthehigherfrequencies.Forexample,at k = 1 thephasebinvariation Figure4.15:Highlightedcomparisonofmeanliftcurvebetweenunsteadyversussteadyshear layer. 55 Figure4.16:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. Figure4.17:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 56 Figure4.18:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. Figure4.19:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 57 Figure4.20:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. Figure4.21:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 58 (a) (b) Figure4.22:Comparisonof (a) rawand (b) normalizedaveragephasebinstandarddeviationin liftfordi˙erentupstream˛owconditionsasafunctionofreducedfrequency. inunsteadyshearlayeris ˘ 56%oftheoverall˛uctuations,however,at k = 5 thisratiodrops to ˘ 7%.Thisindicatesthatthecontributionoftheupstream˛owunsteadinesstoliftdiminishes signi˝cantlyathigheroscillationfrequenciesandthe˛owbecomesdominatedbytheperturbations duetotheairfoiloscillations.Itisinterestingtoseethatthesameconclusiondosnotapplyto thrustcoe˚cient˛uctuations(seeFigure4.23),whereitwaspreviouslyobserveditisonlyweakly a˙ectedbythethechangesinapproach˛owconditions.Figure4.23bsuggeststhatalargeportion (a) (b) Figure4.23:Comparisonof (a) rawand (b) normalizedaveragephasebinstandarddeviationin thrustfordi˙erentupstream˛owconditionsasafunctionofreducedfrequency. 59 ofthrust˛uctuationsareduetothecontributionofcycle-to-cyclevariation,evenfortheuniform ˛owandsteadyshearlayer. 60 CHAPTER5 VELOCIMETRYINSIDEAIRFOILBOUNDARYLAYER Comparisonoftheforcesmeasuredonthestationaryairfoilpositionedatthecenterofunsteadyand steadyshearlayersshowedsigni˝cantdi˙erences,especiallyinthebehavioroftheliftcurve.These observationsleadtothehypothesisthattheaerodynamicperformanceoftheairfoilisa˙ectedby theunsteadyshearlayerthroughthein˛uenceofitshigherlevelof˛uctuationsonthelaminar separationonthesurfaceoftheairfoil. Toexaminethishypothesis,streamwisecomponentofthevelocityismeasuredusing1c-MTV nearthesurfaceofastationaryairfoilplacedinbothsteadyandunsteadyshearlayersoverarange ofanglesofattack.Theresultsofthesemeasurementsareusedtoprovideanoverviewofthe behaviorofthe˛owaroundtheairfoil. Sinceitwasobservedfromtheforcemeasurementsthattheaerodynamicperformanceofthe airfoilwasasymmetricinthepresenceofapproachshear˛ows,theobjectivehereistoperform boundarylayerresolvedmeasurementsonthesuctionsideoftheairfoilwhenitisplacedatboth positiveandnegativeanglesofattack.Furthermore,itisdesiredtocarryoutsimilarmeasurements onthepressuresideoftheairfoilatacoupleofselectanglesofattack.However,asdescribed earlierintheexperimentalmethodschapter,opticalaccessislimitedtoonlyonesideoftheairfoil duetophysicalconstraintsoftheexperimentalsetup.Innormalcon˝gurationofthesetupthis willrestrictthemeasurementstothesuctionsideoftheairfoilwhenitisplacedatapositiveangle intheapproach˛owwithpositiveshear.Toovercomethislimitation,aportionofmeasurements areperformedwiththesheargenerationdevicesusedinareversearrangement(i.e.shearlayers generatedwiththeplacementofthehighandlowspeedsidesreversedinthewatertunnelframe ofreference).Thisstrategymadeitpossibletogainopticalaccesstothesuctionsideoftheairfoil whenplacedinnegativeanglesofattackinthepositiveshear,aswellastothepressuresideof theairfoilatpositiveanglesofattack.Theconsistencyofgeneratedshearlayersinreverseversus normalarrangementiscon˝rmedbydoublecheckingthemeanand˛uctuatingvelocitypro˝lesof 61 Figure5.1:Schematicofairfoilanglesofattackinvestigatedthroughboundarylayer measurementssuperposedontheliftcoe˚cientsmeasuredforsteadyandunsteadyshear˛ows. Thecyanlineshighlightthecasesthatboththesuctionandpressuresidesoftheairfoilhavebeen considered,whereasforthecasesshownwithblacklinesonlythesuctionsideoftheairfoilis evaluated. eachreverseshearagainstitsoriginalarrangement.Additionally,aerodynamicperformanceofthe airfoilineachreverseshear˛owwasmeasuredtobeinagreementwiththatoftheoriginalshear layers.Theseresultsarenotshowninthisdocumentforbrevity. TheairfoilanglesofattackthathavebeeninvestigatedinthischapterareillustratedinFigure5.1 withverticallinesoverlaidontheplotofcorrespondingliftcoe˚cientsmeasuredattheseanglesof attackinsteadyandunsteadyshearlayers.Thecyanlineshighlightthecasesthatboththesuction andpressuresidesoftheairfoilhavebeenconsidered,whereasfortherestofthecases(shownwith blacklines)boundarylayervelocimetryisonlyperformedonthesuctionsideoftheairfoil. Figures5.2presentsthemeanand˛uctuatingvelocitycontoursmeasuredaroundtheairfoil atzeroangleofattackinthesteadyshearlayer.Thecolormapusedforgeneratingthecontour plotsaredesignedtoshowasharpchangefromdarkbluetopurpleatavelocityofzerotofurther highlighttheregionswithreverse˛ow.Themeanstreamwisevelocityvaluesarenormalizedby thecenterlinevelocityoftheshear˛ow( U c )whiletheshearlayervelocitydi˙erence, U ,isused 62 (a) (b) Figure5.2:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 0 atthecenterofthesteadyshearlayer.Thearrow indicatesdirectionofthefree-stream˛ow. tonormalizethevelocity˛uctuations.Thewhitepatchesinthecontoursareregionsthatareeither partiallyblockedbytheairfoilorhavebeena˙ectedbytheglowfromtheairfoilsurface. ThemeanvelocitycontourinFigure5.2aindicatesthatinthepresenceofsteadyshear,the boundarylayersseparateasymmetricallyontheupperandlowersurfacesoftheairfoil,withthe separationoftheboundarylayerontheuppersideoftheairfoil(thesurfacetowardsthehighspeed sideoftheshear˛ow)˝rstdetectedatamoreupstreamlocation.This˝ndingisinagreementwith numericalsimulationsofHammeretal.(2018)forasimilarNACA0012airfoilinauniformshear ˛ow,wheretheyfoundthatthisasymmetryincreaseswiththeshearrate.Theasymmetryofthe ˛owisalsonoticeableintheslightlylargerboundarylayerthicknessontheuppersidetowardsthe trailingedgeoftheairfoil. Figure5.2bdisplaysalopsideddistributionofvelocity˛uctuationsinthepresenceofthesteady shear˛ow,withawiderbandofhigher˛uctuationmagnitudesneartheuppersideoftheairfoil.The verticalstreakobservedinthe˛uctuatingvelocitycontourisanartifactoftheexperimentalerrors duetothelowersignaltonoiseratioofthetaggedlinecombinedwithlowlevelsof˛uctuationsin this˛ow. Correspondingresultsfortheairfoilsetatzeroangleofattackintheunsteadyshearlayerare showninFigure5.3.Whilethemean˛ow˝eldfurtherawayfromtheairfoilmaylooksimilar 63 (a) (b) Figure5.3:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 0 atthecenteroftheunsteadyshearlayer.Thearrow indicatesdirectionofthefree-stream˛ow. tothatofthesteadyshearlayerinthe˝rstglance,thereisacrucialdi˙erencethatnoboundary layerseparationorreverse˛owisdetectedintheunsteadyshearlayer(Figure5.3a).Thismeans thateitherthereisnoseparationhappeninginthis˛oworthisregionisthinnerthanthenearwall spatialresolutionoftheseexperiments.The˛uctuatingvelocitycontourfortheunsteadyshear layershowninFigure5.3bexhibitsanasymmetricdistributionofvelocity˛uctuationssimilarto thesteadyshearlayercase,however,itshouldbenotedthatthemaximumlevelof˛uctuationis roughly6timeshigherintheunsteadyshear˛ow. Fornon-zeroanglesofattackcontoursandtoprovideabettervisualcomparisonofthe˛ow ˝eldoverdi˙erentanglesofattack,the˝eldsofviewarerotatedsuchthattheairfoilisportrayedat zeroangle.However,thecontourisstillshowingtheactualstreamwisecomponentofthevelocity, whosedirectionisindicatedbythearrowshownoneachplot.Di˙erentversionsoftheseplotswith theairfoilshownatitsactualangleofattackarepresentedinAppendixE. Acomparisonbetweenthe˛ow˝eldaroundtheairfoilsetat 2 angleofattackinthesteady shearlayer(showninFigure5.4)andasimilargeometryintheunsteadyshearlayer(shownin Figure5.5)leadstoaparallelobservation.Thereisaclearreverse˛owregiononthesuctionside oftheairfoilinthesteadyshearlayer,butnosignofsuchregionorseparationisspottedinthe unsteadyshear˛ow.Thegeneralbehaviorofthevelocity˛uctuationsinthetwo˛owsaremostly 64 (a) (b) Figure5.4:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 2 inthesteadyshearlayer.The˝eldofviewisrotated suchthattheairfoilisportrayedatzeroanglebutarrowindicatesdirectionofthefree-stream˛ow. di˙erentintheamplitudeofoscillations(withhigherlevelsof˛uctuationsintheunsteadyshear ˛ow).Sincenonoticeabledistinctionisobservedonthepressuresideoftheairfoilbetweenthe steadyandunsteadyshear˛owsandthe˛ow˝eldsaremostlydi˙erentonthesuctionside,therest ofthemeasurementsareonlyperformedonthesuctionsideoftheairfoil. Figure5.6depictsthe˛ow˝eldforastationaryairfoilplacedatmultiplepositiveanglesof attackinthesteadyshear˛ow.Comparisonofthemeanvelocitycontoursrevealsthattheseparation (a) (b) Figure5.5:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 2 intheunsteadyshearlayer.The˝eldofviewisrotated suchthattheairfoilisportrayedatzeroanglebutarrowindicatesdirectionofthefree-stream˛ow. 65 (a) (b) (c) (d) (e) (f) Figure5.6:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatpositiveanglesofattackinthecenter ofthesteadyshearlayer.The˝eldofviewisrotatedsuchthattheairfoilisportrayedatzeroangle butarrowindicatesdirectionofthefree-stream˛ow. pointonthesuctionsideoftheairfoilmovesupstreamandtheseparatedregiongrowsinbothwidth andheightastheairfoilangleofattackincreases.Themainfeaturesofthevelocity˛uctuations remainthesame,butthemagnitudeof˛uctuationsincreaseswithairfoilangleofattack. Thee˙ectofairfoilangleofattackonthe˛ow˝eldintheunsteadyshearlayerisfoundtobe muchsubtler,asshowninFigure5.7.Thereisstillnoexplicitsignofseparationinanyofthe geometriesandthemeanvelocitycontoursarefairlyclosefortheportrayedanglesofattack,with 66 (a) (b) (c) (d) (e) (f) Figure5.7:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatpositiveanglesofattackinthecenter oftheunsteadyshearlayer.The˝eldofviewisrotatedsuchthattheairfoilisportrayedatzero anglebutarrowindicatesdirectionofthefree-stream˛ow. anincreaseinthesizeoftheshearedregionabovethetrailingedge.Theincreaseinangleofattack seemstoshifttheregionofmaximum˛uctuationsupstreamtowardstheleadingedgeoftheairfoil, inadditiontoraisingthemaximumlevelof˛uctuations. The˛ow˝eldaroundtheairfoilatnegativeanglesofattackinthesteadyshear˛owispresented inFigure5.8.itisobservedthatat = 2 and 4 the˛owbehavessimilartothepositiveangles ofattackwithaopenseparationgrowinginheightandtheseparationpointmovingupstreamwith 67 (a) (b) (c) (d) (e) (f) Figure5.8:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatnegativeanglesofattackinthe centerofthesteadyshearlayer.The˝eldofviewisrotatedsuchthattheairfoilisportrayedat zeroanglebutarrowindicatesdirectionofthefree-stream˛ow. anincreaseinangleofattack.At = 6 ,however,themeanvelocitycontourindicatesaclosed laminarseparationbubble.Thesignatureofvelocity˛uctuationsisalsoverydi˙erentatthisangle, wheretheinitiallythinregionofhigh˛uctuationsshowsanabruptincreaseinheightandquickly extendsuptotheairfoilsurface. Oncemore,noseparationorreverse˛owregionisdetectedintheunsteadyshear˛owwiththe airfoilsetatnegativeanglesofattack,asshowninFigure5.9.Theprimaryfeatureofthese˛ow 68 (a) (b) (c) (d) (e) (f) Figure5.9:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatnegativeanglesofattackinthe centeroftheunsteadyshearlayer.The˝eldofviewisrotatedsuchthattheairfoilisportrayedat zeroanglebutarrowindicatesdirectionofthefree-stream˛ow. ˝eldsisfoundtobeanincreaseinbothmeanand˛uctuatingvelocitiesneartheleadingedgeofthe airfoil,withthemagnitudesrisingwithnegativeangleofattack. Basedontheseobservations,itwascon˝rmedthatunsteadinessoftheshear˛owa˙ectsthe laminarseparationonthesurfaceoftheairfoiltotheextentthatnoseparationregionwasobserved withinthenearwallresolutionofcurrentmeasurements.Thisabsenceoflaminarseparationbubble cancontributetothedi˙erencesobservedintheliftcharacteristicsoftheNACA0012airfoilatlow 69 Reynoldsnumberintheunsteadyversusthesteadyshearlayer. 70 CHAPTER6 PITCHINGAIRFOILWAKEVISUALIZATION Moleculartaggingvisualizationisusedforaqualitativeexplorationofthee˙ectoftheshear˛ow unsteadinessonthewake˛owbehindasymmetricallypitchingairfoil.Theresultingimagesfrom bothshear˛ows(steadyandunsteadyshearlayerswith K max = 1 : 0 and ! c = 0 : 7 )arealso comparedtothoseofauniformapproach˛owtohighlightthesimilaritiesanddi˙erencesbetween allthesecases.Figure6.1illustratesasampleoftheinitialtaggingpattern(shownasviolet)anda delayedstateoftheselines(shownasgreen)onastaticairfoilinuniform˛ow.Afewofthelines aresettostrikethesurfaceoftheairfoiltoprovideareferencepointforthepositionoftheairfoil trailingedge.Delayedimagesaretaken30msaftereachlaserpulsetogivethetaggedregionsof ˛uidenoughtimetodistortandunderlinethemain˛owfeatures.Toimprovethevisibilityof˛ow structuresthroughthedeformeddelayedlines,theinitialtaggedlinesarenotshownintherestof the˝guresinthischapter. Sincethese˛owvisualizationsimagesaretakenatsuchlongdelaytimesafterthetagginglaser Figure6.1:Asamplepairofmoleculartagging˛owvisualizationtaggedlinesshownatinitial (violet)anddelayed(green)states.Theinitialimageistaken ˘ 2 s afterthelaserpulse,whilethe delayedimageiscaptured30msafterthelaserpulse.Theaxesarenormalizedbythechordlength oftheairfoilwiththetrailingedgeoftheairfoilatzeroangleofattacksetastheorigin. 71 pulse(30mscomparedto3.5msnominalphosphorescencelifetimeoftheMTVsolution),the resultsarephaseorderedinto64phasebinsbasedonthefrequencyofairfoiloscillationandthen phaseaveragedovereachphasebintoreducethenoiseandincreasethequalityoftheimages.A sampleofthesephaseaveragedimagesarepresentedinFigure6.2fortheairfoiloscillatingwith oscillationamplitudeof 0 = 2 andreducedfrequencyof k = 6 inuniform˛ow.Theaxesofthese ˝guresarenormalizedbythechordlengthoftheairfoilwiththetrailingedgeoftheairfoilatzero angleofattacksetastheorigin.Thelocationofthetrailingedgeishighlightedbyawhitecircle andthedashedlinesindicatethedirectionofthe˛owfromthetrailingedge.Thephasedaveraged imagesarecoloredgreentomimicthephosphorescencecoloroftheactualtaggedregionsinthe ˛ow. These4imagesportraytheaveragestateofthewake˛owatdi˙erentphasesofoneairfoil oscillationperiod,denotedby ˚ varyingfrom0to1.Figure6.2acorrespondstothestartofeach oscillationcycle,describedwith ˚ = 0 andde˝nedaswhentheairfoilisat = 0 andispitching upwardwith d dt > 0 .Thismeansthatthetrailingedgeoftheairfoilismovingdownwardinthis picture.Thelocationandmotiondirectionofthetrailingedgearemarkedoneachimageaswell. Figures6.2band6.2dshowthe˛ow˝eldwhenairfoilisat =+ 2 and = 2 ,respectively ( ˚ = 0 : 25 and0.75),whileFigure6.2ccoincideswith = 0 and d dt < 0 ( ˚ = 0 : 5 ),wherethe airfoilispitchingdownwardandthetrailingedgeismovingup. Tofacilitatefollowingthedevelopmentofthe˛owinthese˝gures,theapproximatelocations ofthetwomostrecentlyformedvorticescanbedetectedat ˚ = 0 andtrackedinthesequenceof images.Forexample,inFigure6.2a,thesetwovorticesareapproximatelylocatedat x š c ˘ 0 : 02 and0.27nearthecenterline,whichthenmovetoaround x š c ˘ 0 : 14 and0.42inthesecondimage (Figure6.2b). Correspondingimagesforthesameoscillationparametersinsteadyshearandunsteadyshear layersarepresentedinFigures6.3and6.4respectively.Itshouldbenotedthatinthecaseswith non-uniform˛ow,thehighspeedsideispassingoverthetopsideoftheairfoilasshowninthese ˝gures,meaningat ˚ = 0 theairfoilleadingedgemovestowardsthehighspeedsideanditstrailing 72 (a) (b) (c) (d) Figure6.2:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 6 inuniform˛owat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationofthe trailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe˛ow fromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. edgegoesinthedirectionofthelowspeedside. Inthephaseaveragedimagesequencesforboththeuniformandsteadyshear˛ows(Figures6.2 and6.3),thevorticestravelparalleltothe˛owandarenearlyalignedwiththewakecenterline.This observationisinagreementwithexperimentsofBohl(2002)andBohl&Koochesfahani(2009) forthisairfoilmotioninuniform˛owandnumericalresultsofHammeretal.(2019)forbotha similaruniform˛owandalinearsteadyshearlayer(seeFigure6.5). 73 (a) (b) (c) (d) Figure6.3:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 6 insteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationofthe trailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe˛ow fromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. Whilethewake˛ow˝eldintheunsteadyshearlayerindicatessimilar˛owfeaturesasthe previoustwo˛ows,itishardertofollowthestructuresclearlyduetotheblurryaveragedlines.This isadirectresultofthehighlevelof˛uctuationsintheapproach˛owwhichintroduceslargecycle tocyclevariationstothemostlyoscillatory˛ow.Figure6.6showsthephaseaveragedimageofthe ˛ow˝eldinthetwostreamshearlayerat ˚ = 0 (thesameasFigure6.4a)alongside3samplesof theinstantaneousimagesatthisphasethatareusedtogeneratethephaseaveragedimage.Itcanbe 74 (a) (b) (c) (d) Figure6.4:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 6 inunsteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationof thetrailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe ˛owfromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. seenthatthetwovorticeslocatedaround x š c = 0 : 22 and0.5donotappearatthesamelocationin thesethreeimagesandtheirfeaturesarealsonotidentical.Di˙erencessimilartotheonesshown inFigure6.6betweentheinstantaneoussamplesleadtothephaseaveragedimagesnotbeingas sharpasfortheothertwo˛owsshownhere. Figures6.7,6.8and6.9comparethe˛ow˝eldinthewakeoftheairfoiloscillatingwithareduced frequencyof k = 8 inuniform˛ow,steadyshearandunsteadyshearlayerrespectively.Thewake 75 Figure6.5:Instantaneousspanwisevorticity˝eldsfor (a) uniform˛owand (b) linearsteadyshear ˛owsimulatedbyHammeretal.(2019).Theairfoilisat = 0 andpitchingup.(Figure CourtesyofHammeretal.,2019) intheuniform˛owshowsthesignatureofareversevonKármánvortexstreetpattern,whichtravels paralleltothefree-stream˛ow.Thesteadyshear˛owdisplaysasimilarvortexarrangementpattern asthereversevonKármánvortexstreet,butthevorticesde˛ectupwardtowardsthehighspeedside oftheshearlayerastheyprogressdownstream.ThisisconsistentwithsimulationsofHammer etal.(2019)whichdemonstratedafter k = 8 ,theangleofwakede˛ectionincreaseswithreduced frequency(seeFigure6.5).Thisincreaseinde˛ectionangleisobservedincurrentvisualizations aswell,asshowninFigure6.10forareducedfrequencyof k = 10 inthesteadyshearlayer. Anotherinterestingfeatureofthe˛owinthesteadyshearlayeratthisreducedfrequency ( k = 10 )isthatthetwovorticesshedinthesameairfoiloscillationgetclosetoeachotherwhile theytraveldownstream,andupwardtowardsthehighspeedside.Thispatternalsooccursinthe unsteadyshear˛ow(Figure6.11),whichmainlydisplayssimilargeneralfeaturestothatofthe steadyshear˛ow.Althoughperformedondi˙erenttwo-streamshearlayers,theairfoiloscillation andother˛owparametersofthiscaseareclosetotheexperimentsofNaguib&Koochesfahani 76 (a) (b) (c) (d) Figure6.6: (a) Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 6 inunsteadyshearlayeratstartingphaseofoneairfoiloscillationcycle. (b) through (c) showsampleinstantaneousmoleculartaggingvisualizationimagesfromthesamecaseandphase incycleusedtogenerate (a) .Thelocationofthetrailingedgeishighlightedbyawhitecircleand thedashedlinesindicatethedirectionofthe˛owfromthetrailingedge.Theredarrowsindicate thedirectionoftrailingedgemotion. (2012),wheretheirphase-resolvedspatialmapofthestreamwisevelocityforoneoscillationcycle atonechordlengthdownstreamofthetrailingedgeshowedevidenceofonlyoneregionofvelocity undershoot/overshoot,suggestingonlyonevortexwasshedduringeachairfoiloscillationcyclein thepresenceoftheunsteadyshear˛ow(seeFigure6.12).BasedontheindicationsfromFigure 6.11,itispossiblethatthecloseproximityofthetwovorticesinthewakeoftheoscillatingairfoil 77 (a) (b) (c) (d) Figure6.7:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 8 inuniform˛owat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationofthe trailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe˛ow fromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. intheunsteadyshearlayercombinedwiththehighlevelof˛uctuationsinthe˛owmayhave contributedtoonlyonevortexbeingdetectedintheworkofNaguib&Koochesfahani(2012). 78 (a) (b) (c) (d) Figure6.8:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 8 insteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationofthe trailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe˛ow fromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. 79 (a) (b) (c) (d) Figure6.9:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 8 inunsteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationof thetrailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe ˛owfromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. 80 (a) (b) (c) (d) Figure6.10:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 10 insteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationof thetrailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe ˛owfromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. 81 (a) (b) (c) (d) Figure6.11:Phaseaveragedmoleculartaggingvisualizationofthe˛owinthewakeofan oscillatingNACA0012airfoilwithoscillationamplitudeof 0 = 2 atareducedfrequencyof k = 10 inunsteadyshearlayerat4di˙erentphasesofoneairfoiloscillationcycle.Thelocationof thetrailingedgeishighlightedbyawhitecircleandthedashedlinesindicatethedirectionofthe ˛owfromthetrailingedge.Theredarrowsindicatethedirectionoftrailingedgemotion. 82 Figure6.12:Colorcontourmapofphase-averagedstreamwisevelocityresultsofNaguib& Koochesfahani(2012),showingthevariationofvelocitypro˝leswithphaseduringonecycleof oscillationinthewakeofthepitchingairfoil( k = 10 : 6 , 0 = 2 ). (a) Thebaselinecaseof uniformapproach˛owshowstheexpectedsignatureofalternatingsignvortexpair. (b) Inthecase ofshearlayerapproach˛ow,thereisevidenceforonlyonevortexcore.(Figurecourtesyof Naguib&Koochesfahani,2012) 83 CHAPTER7 CONCLUSIONS Motivatedbymanysituationsofcomplex,non-uniformandmostlyunsteady˛owsencounteredin nature,thisstudyexperimentallyexploreshowtheunsteadybehaviorofashear˛owa˙ectsthe aerodynamicperformanceofaNACA0012airfoilinlowReynoldsnumbers( Re = 1 : 2 10 4 ).This objectiveisachievedbyexaminingtheperformanceoftheairfoilinstaticconditionoverarange ofanglesofattack,aswellaswhenitispitchingsinusoidallyarounditsquarterchordwithasmall oscillationamplitudeoverarangeofoscillationfrequencies. Thee˙ectsofshear˛owunsteadinessareisolatedfromthoseofthemeanshear˛owitselfby performingthemeasurementin2di˙erentshear˛owswithmatchingmeanvelocitypro˝leswhere onehasauniformdistributionoflowlevel˛uctuationswhiletheotherexhibitslargescalespatially non-uniform˛uctuations.Thiswayitwouldbepossibletodiscernbetweenthee˙ectsattributed tothemeanshearpro˝lefromthosecausedbyunsteadinessoftheshear˛ow. Aplanemixinglayerisselectedasarepresentativeofanunsteadyshearlayer,sincethey areverywellstudiedandareknowntocontainvorticalstructuresandhighlevelof˛uctuations. Suchtwo-streamshearlayerwithacenterlinevelocityof10cm/sandatwo-to-onevelocityratio isgeneratedinthewatertunnelfacility.Thebehavioroftheresultingtwostreamshearlayer ischaracterizedthroughmeasurementsofitsstreamwisevelocitypro˝leatmultipledownstream locationsutilizingsinglecomponentmoleculartaggingvelocimetry.Afteradevelopmentlength, thetwo-streamshearlayerisfoundtoproducethetypicalself-similarhyperbolictangentmean velocitypro˝leandaspatiallynon-uniform˛uctuationspro˝lewithahighlevelof˛uctuationsin thecenterandagradualdecreasein˛uctuationsmovingawayfromtheshearlayercenterline.The twostreamshearlayerwidthincreaseslinearlyasafunctionofdownstreamdistancewithagrowth ratesimilartotheonesreportedinliteraturefortwo-streamshearlayers. Aftervalidatingthebehavioroftheunsteadyshearlayer,themeanvelocitypro˝leofthistwo- streamshearlayerataspeci˝cdownstreamlocation( x x 0 = 107 cm)isusedasareferencefor 84 generationofaduplicatesteadyshear˛ow.Thesteadycounterpartofthisshearlayerisproducedby ashapedhoneycombsheargenerationmethodbasedontheoriginalmodelproposedbyKotansky (1966).Thismodi˝edmodelutilizesavariablefrictionfactormodeltoestimatethepressure dropofthelaminar˛owinsidehoneycombtubes(applicableinsideandafterthetubeentrance length).Additionally,themodelisfurtherre˝nedtoeliminatetheneedtocalibratethepressure dropexperimentallyforeachdesign.Thee˙ectivenessofthemodi˝edmethodisdemonstrated throughitsabilitytogeneratethedesiredvelocitypro˝lewithhigh˝delitywhilemaintaining velocity˛uctuationlevelsatorbelowthatofthefreestreampriortoinstallationofthedevice.Mean streamwisevelocitypro˝lesmeasuredinthissteadyshear˛owshowagoodagreementwiththe meanvelocitypro˝leofthetargettwo-streamshearlayer,whileitsstreamwisevelocity˛uctuations arespatiallyuniformandatthelevelofthebackground˛uctuationsinthetestfacility Withapairofmatchingsteadyandunsteadyshear˛ows,aerodynamicforcemeasurements areperformedonastationaryairfoiloverawiderangeofanglesofattack,placedatthecenter ofeachshearlayeraswellasinthereferenceuniformfreestream˛ow.Inthepresenceofsteady shearapproach˛ow,theliftcoe˚cientcurveexhibitsthesamegeneraltrendasthatoftheuniform ˛owanditproducesanegativeliftatzeroangleofattack,whichisconsistentwiththeresultsof Hammeretal.(2016,2018,2019)andOlsonetal.(2016).Howevertheunsteadyshearapproach ˛owresultedinasigni˝cantlydi˙erentbehavioroftheliftcurve,wheretheregionaroundzero angleofattackshowsalinearbehaviorwithalargerslopeandextendingoveralargerrangeof anglesofattackcomparedtothatofsteadyshearanduniform˛ow.Furthermore,theunsteady shear˛owinducesapositiveliftatzeroangleofattack.Thisbehavioroftheliftcurveinthe unsteadyshearlayerishypothesizedtobeduetoe˙ectsofhighlevelsof˛uctuationspresentinthe approach˛owonthelaminarseparationonthesurfaceoftheairfoil.Theunsteadyshear˛owalso generatesasmalleramplitudeofdragoversmallanglesofattackcomparedtobothuniformand steadyshear˛ows. Forcemeasurementsonthepitchingairfoilsshowthatatsmalloscillationfrequencies( k < 2 : 5 ), thesteadyandunsteadyshear˛owsproduceoppositesignliftforcesconsistentwiththeirstatic 85 performance,butbothresultinpositiveliftcoe˚cientsofsimilarmagnitudesathigherreduced frequencies( k > 4 ).Thepresenceofshearanditsunsteadinessseemtoonlyweaklya˙ectthe meanand˛uctuationsofthestreamwiseforce,withalmostnoe˙ectobservedonlift˛uctuations. Theseobservationsareinagreementwith˝ndingsofHammeretal.(2019). Tolookfurtherintothehypothesisofthelaminarseparationbeinga˙ectedbytheunsteady shearlayer,streamwisevelocityofthe˛owaroundtheairfoilismeasuredinbothshear˛owswith thestationaryairfoilsetatmultipleanglesofattack.Inthesteadyshearapproach˛ow,allofthe anglesofattackinvestigatedhere( 6 << + 6 ),demonstratearegionofreverse˛owonthe suctionsideoftheairfoil,indicatingthepresenceofalaminarseparation.However,noneofthe anglesofattackintheunsteadyshear˛owshowanysignofareverse˛owregion.Thissuggeststhat thehighlevelof˛uctuationsintheunsteadyshear˛owhavesigni˝cantlya˙ectedtheformationof laminarseparation. Flowvisualizationsinthewakeofthepitchingairfoilthroughmoleculartagging˛owvisual- izationshowsthatatthehighreducedfrequenciesvisualizedhere( k 6 ),theoverallbehaviorof thewake˛owandvorticalstructuresissimilarforbothsteadyandunsteadyshear˛ows,withthe wake˛owde˛ectingtowardsthehighspeedsideofthe˛owathigherfrequencies( k > 7 ).The maindi˙erencebetweentheobservationsfromthesteadyvsunsteadyshearlayeristhefactthatthe unsteadyshearlayerresultsexhibitlargerperturbationstothevortexarrangementandconsiderable cycletocyclevariations. 86 APPENDICES 87 APPENDIXA SHAPEDHONEYCOMBSHEARGENERATIONMETHOD Withthepurposeofimprovingtheperformanceoftheoriginalvariablelengthhoneycombmodel, ourproposedmodi˝edmodelutilizesavariablefrictionfactorcoe˚cientmodeltoestimatethe pressuredropinsidehoneycombtubes.Thisfrictionfactorcoe˚cientmodelisapplicableinside thetubeentrancelengthaswellasinthefullydevelopedregion,inthelimitoflaminar˛owinside tubes( Re d 2300 ),andisafunctionoftubelengthandReynoldsnumber.Additionally,the modelisfurtherre˝nedtoeliminatetheneedtocalibratethepressuredropexperimentallyfor eachdesign.Withthesemodi˝cations,thenewmethodcanbeusedintwofashions:todesigna honeycombshaperequiredtogenerateadesirednon-uniformvelocitypro˝le;andtopredictthe velocitypro˝leresultingfromagivenhoneycombshapepro˝leunderdi˙erentoperatingconditions. Theperformanceoftheproposedtechniqueisinvestigatedthroughexperimentalmeasurementsof streamwisevelocitydownstreamoftwosamplehoneycombdevices.Furthermore,thedownstream evolutionofthe˛owafterthehoneycombdeviceisstudiedtodeterminethedistancerequiredfor the˛owtodevelopintothesmoothdesiredvelocitypro˝le. A.1TheoreticalModel ThepresentmodelfollowsthesamemainassumptionsasintheoriginalmodelbyKotansky (1966).AschematicofthemodelanditsmainassumptionsisshowninFigureA.1.The˛ow farupstreamofthehoneycomb( x )isassumedtobeauniform˛owwithknowninletvelocity ( u )anduniformpressuredistribution( P ).Fardownstreamofthehoneycomb( x + 1 ),the velocitypro˝leisexpectedtohavedevelopedintotheknownanddesiredvelocitypro˝le, u ¹ y º , withtheuniformpressuredistribution( P + 1 ).Furthermore,Kotansky(1966)presumedthatthe ˛owentering/exitingeachhoneycombtubehasthestreamwisevelocityequaltothedesiredvelocity pro˝le, u ¹ y º andthe˛owexitinghoneycombtubesisonlyinthestreamwisedirection. Followingtheseassumptions,theoverallpressurechange( P )betweenfarupstreamandfar 88 FigureA.1:Aschematicofthevariablelengthhoneycombmodelalongwiththemain assumptionsinvolved. downstreamcanbedividedinto3segmentsas: P = P P + 1 = h P P ¹ 0 ; y º i + h P ¹ 0 ; y º P ¹ L ; y º i + h P ¹ L ; y º P + 1 i : (A.1) The˝rsttermontherighthandsideofEquationA.1isthepressurechangeupstreamofthe honeycomb,thesecondtermrepresentsthepressuredropwithinthehoneycombtubes,andthelast termstandsforthepressurechangedownstreamofthehoneycomb.Similartotheoriginalmodel, upstreamofthehoneycombistreatedasapotentialregionandthepressurechangeinthispotential region(the˝rstterm)isfoundusingBernoulli'sequation.Thepressuredropinsidethehoneycomb tubes(thesecondterm)isestimatedusingafrictionfactorcoe˚cientandtheDarcy-Weisbach equation.Thepressurechangedownstreamofthehoneycomb(thethirdterm)isassumedtobe negligiblecomparedtotheothertwoterms,consistentwithKotansky(1966)assumptionofparallel streamlinesdownstreamofthehoneycomb.Theentranceandexitlossesareignoredinthisanalysis. Theterms P P ¹ 0 ; y º and h P ¹ 0 ; y º P ¹ L ; y º i inEquationA.1canbesubstitutedfrom Bernoulli'sequationonastreamline(betweenfarupstreamandtheentranceofthehoneycomb) andDarcy-Weisbachequationrespectively,whichthenafterrearrangementyields: 89 L ¹ y º = d 4 f 2 6 6 6 6 4 2 ˆ P + u 2 v ¹ y º 2 u ¹ y º 2 1 3 7 7 7 7 5 ; (A.2) where L ¹ y º isthehoneycombtubelengthateach y , d thehoneycombtubediameter, f theFanning frictionfactorand ˆ the˛uiddensity.InordertouseEquationA.2tocalculatethehoneycomb lengthdistribution,onestillneedsto˝ndthedistributionoflateralcomponentofthevelocity enteringthehoneycombtubes, v ¹ y º ,Fanningfrictionfactor, f ,andtheoverallpressuredropvalue, P . A.1.1DistributionofCross-StreamVelocityComponent, v ¹ y º Asmentionedpreviously,Kotansky(1966)consideredthe˛owupstreamofthehoneycomb,ex- cludingwallboundarylayers,asapotential˛owwith r 2 = 0 ,where isthetwo-dimensional streamfunction.Theboundaryconditionsofthispotential˛owregionincludeno˛ownormal tothesidewalls,undisturbedparallel˛owatfarupstream,and u ¹ y º astheknownstreamwise componentofvelocityenteringthehoneycombat x = 0 .Applyingtheseboundaryconditionsto thegeneralpotential˛owsolution,thecross-streamcomponentofvelocityattheentranceofthe honeycomb, v ¹ y º canbeestimatedas: v ¹ y º = 1 Õ n = 1 a n n ˇ H sin n ˇ y H ; (A.3) where a n arecoe˚cientsoftheFourierseriesrepresentingthedi˙erencebetweenstreamwise componentofvelocityenteringthehoneycombandtheuniformvelocityoffarupstream: a n = 2 n ˇ ¹ H 0 cos ˇ y H » u ¹ y º u ¼ d y : (A.4) A.1.2FrictionFactor, f TheoriginalmodelofKotansky(1966)assumedthatthe˛owwasfullydevelopedinallofthe honeycombtubesandusedanexperimentallymeasuredconstantfrictionfactorcoe˚cientforallof thehoneycombtubes.However,asmentionedpreviously,thisassumptionisnotalwaysvalid.The 90 twomostprobableinstancesthatthisassumptionfailsarewhenthehoneycombtubesarenotlong enough(duetomaterial,fabricationorfacilitysizeconstraints),orwhenlowerReynoldsnumber rangesareofinterest.Utilizingshortertubesmeansthereisnoguaranteethatthe˛owisfully developedinallofthetubes,whileoperatinginlowerReynoldsnumberrangesentailsthatthe variationofevenfullydevelopedfrictionfactorcoe˚cientwithReynoldsnumberisnotnegligible, asReynoldsnumbercanvarybyafactorof2-3acrossthehoneycombdevice.Moreover,based ontherelationshipsuggestedbyShah(1978)foralaminardeveloping˛owinacirculartube,a tubelengthofapproximately5timesthecommonlyde˝nedentrancelength(wherethecenterline velocityreaches99%offullydevelopedvelocitylimit)isrequiredfortheapparentfrictionfactorto approachwithin5%ofthefullydevelopedfrictionfactorvalue.Amorerecentscalinganalysisfor internallaminar˛owsbyMuzychka&Yovanovich(2009)suggeststhatitcantakeabout10times theentrancelengthforallboundarylayere˙ectstobelostinthepressuredrop.Thesehighlight theshortcomingsofaconstantfrictionfactorcoe˚cientassumptionwhenlaminar˛owcouldbe presentinsidehoneycombtubes. Thus,foramoregeneralandaccuratevariablelengthhoneycombmodel,itisrequiredtoutilize avariablefrictionfactorcoe˚cientmodelthatisvalidfortheentirerangeofdeveloping˛owto fullydeveloped˛ow.Thisdemandsthefrictionfactorcoe˚cientmodeltobeafunctionofbothtube lengthandReynoldsnumber.Fortheproposedvariablelengthhoneycombmethod,suchavariable frictionfactormodelproposedbyDuPlessis&Collins(1992)isemployed.Thismodelestimates theapparentfrictionfactorcoe˚cientofalaminar˛owwithintubesofdi˙erentcross-sectionsin therangeofdevelopingtofullydeveloped˛ows.Forasimplecirculartube,theapparentfriction factorcanbeapproximatedby: f ¹ y º = 16 Re d ¹ y º 2 6 6 6 6 6 4 1 + 0 : 0462 x + y ! 2 : 17 2 3 7 7 7 7 7 5 1 2 : 17 ; (A.5) where Re d ¹ y º istheReynoldsnumberbasedontubediameterand x + y thedimensionlessaxial 91 distancefromtubeentranceateachy: x + y = » L ¹ y º š d ¼ Re d ¹ y º : (A.6) A.1.3OverallPressureDrop, P InordertouseKotansky'soriginalmethod,oneneedsto˝ndthevalueofoverallpressuredrop, P , throughcalibration,byexperimentallymeasuringthepressuredi˙erencebetweenthefarupstream andfardownstreamlocationsinthetestsectionofthe˛owfacility.Thisisnotthebestapproach foradesignmethod,though,sinceintroducingthehoneycombdeviceintothefacilitya˙ectsthis overallpressuredropvalue.Thiscalibrationrequirementleadstoaniterativeprocessforamore accuratedesignprocess.Foramorerobustapproachinourmodi˝edmodel,thisoverallpressure droptermiscalculatedusingareferencehoneycomblengthandotherdesignspeci˝cationsto eliminatetheneedforapriorirequirementof P duringthedesignofthehoneycombtubelength distribution. Considering P = P P + 1 isuniformacrossthechannelwidth,onceitsvalueisfound atonepoint,onecanusethesamevaluefortherestofthechannelwidth.Usingthisapproach, aprescribedreferencehoneycomblengthcanbeassignedtoaspeci˝cpointiny,whereboth thevelocitycomponentsareknown.Afterestimatingthefrictionfactorcoe˚cientbasedonthis information,areorganizedversionofEquationA.2canbeusedto˝nd P forthispoint.Inpractice itisusuallyconvenienttousethemaximumhoneycomblength(apracticalconstraintfromraw material,manufacturing,orfacilitydimensionlimitations)asthereferencelength.Thislength typicallycoincideswithwheretheloweststreamwisevelocitywouldoccur.Thisyields: P = P 0 = 4 f 0 L 0 d + 1 1 2 ˆ u 2 0 + 1 2 ˆ v 2 0 1 2 ˆ u 2 ; (A.7) where L 0 isthereferencehoneycomblength,and u 0 , v 0 and f 0 arevelocitycomponentsandfriction factorcoe˚cientthatcoincidewiththeylocationofthisreferencehoneycomblength. 92 A.1.4FinalModel Theincorporationofavariablefrictionfactorcoe˚cientmodelcomesattheexpenseofmakingthe ˝nalexpressionforthehoneycomblengthdistributionimplicit.Tosolveforthe˝nalhoneycomb tubelengthdistribution,onecanrearrangeEquationA.2toisolatetheunknownvariablesinthe lefthandsideoftheexpression,as: f ¹ y º L ¹ y º = d 4 2 6 6 6 6 4 2 ˆ P + u 2 u ¹ y º 2 v ¹ y º 2 u ¹ y º 2 1 3 7 7 7 7 5 : (A.8) Foranyspeci˝edvelocitypro˝le, u ¹ y º ,therighthandsideofEquationA.8, C ¹ y º ,isknownforevery ylocation;throughusingEquationA.3forcross-streamvelocitycomponent, v ¹ y º ,andEquation A.7alongwithareferencehoneycomblengthforoverallpressuredrop, P .Finally,inputtingthe expressionforfrictionfactorcoe˚cientfromEquationsA.5intothelefthandsideofEquationA.8 yieldsthe˝nalexpressionforhoneycombtubelengthdistribution, 16 Re d ¹ y º 2 6 6 6 6 6 6 6 4 1 + © « 0 : 0462 L ¹ y ºš d Re d ¹ y º ª ® ® ¬ 2 : 17 2 3 7 7 7 7 7 7 7 5 1 2 : 17 = C ¹ y º ; (A.9) where L ¹ y º istheonlyunknown.Thisexpressioncanbenumericallysolvedtoobtainthehoneycomb lengthdistribution, L ¹ y º ,thatwouldgeneratethespeci˝eddesiredvelocitypro˝le, u ¹ y º . A.2ExperimentalMethod Theperformanceoftheproposedmodelisdemonstratedbyconsideringahyperbolictangent velocitypro˝le(showninFigureA.2)asthetargetvelocitypro˝letobegeneratedinawater tunnelfacilitywithtestsectiondimensionsof15 Ö 15 Ö 46cm.Thesheargenerationdevice wasdesignedforamean˛owvelocityof10cm/sandamaximumnormalizedshearrateof K max = du d y max H u c = 2 : 1 .Apolycarbonatehoneycomb(PC2PolycarbonateHoneycomb, Plascore,Zeeland,MI)withcircular3.175mmdiametertubesandamaximumtubelengthof15 cmwasusedtofabricatethesheargenerationdevice.Thismaximumtubelengthwasselectedso 93 FigureA.2:Non-uniformhyperbolic tangentvelocitypro˝leselectedwith centerlinevelocityof u c = 10 cm/s. FigureA.3:Calculatedhoneycomblength distributionforthehyperbolictangent pro˝lebasedonthepresentmodel comparedtotheestimatedentrancelength ofthe˛owateach y (basedon L e ˇ 0 : 06 Re ¹ y º ).Thehorizontallines depicttheapproximatediameterof honeycombtubes. thatthesheargenerationdevicewouldnotoccupymorethan 1 š 3 ofthetestsectioninthefacility, leavingenoughspacedownstreamofthedevicetoobservethe˛owdevelopment. FigureA.3presentsthedesignedhoneycomblengthdistributionusingourproposedmodel. AnestimateoftheentrancelengthdistributionisalsoshowninFigureA.3tohighlightthatmore than 1 š 3 ofthedesignedhoneycombtubesareshorterthantheapproximateentrancelength,and hence,donothavefullydeveloped˛ows.Aspecializedbandsawblade(SimmonsKnife&Saw, GlendaleHeights,IL)wasusedtocutthedesignedhoneycombshapeformarectangularblock withauniformlengthof15cm.The˝nalcutsheargenerationdeviceisshowninFigureA.4. Singlecomponentmoleculartaggingvelocimetry(1c-MTV)isusedtocharacterizetheresulting meanand˛uctuatingvelocitypro˝lesandtheirstreamwisedevelopment.Inthesingle-component implementationofMTVusedhere,straightlaserlinesareusedasthetaggingpattern,withwhich, 94 FigureA.4:Imageofthesheargenerationdevicecutaccordingtodesignedlengthpro˝lebased onthepresentmodelandshowninFigureA.3 onlythestreamwisecomponentofthevelocity˝eldisobtained.Aschematicofthemeasurement setupisprovidedinFigureA.5.Basedonthe˛owvelocityrangeandspatialcorrelationtechnique considerations,atimedelayof10mswasusedtocapture1024undelayedanddelayedimages atarateof12.27Hz.Anaveragesampleofthe1c-MTVimagepairispresentedinFigureA.6, wherethegreenlinesarethetaggedregionsinthe˛uidcapturedrightafterthelaserpulse(Figure A.6i)andaftertheprescribeddelayof10ms(FigureA.6ii).Theyellowlinesdemonstratethe approximatelocationofthehoneycombtubes.Visuallycomparingthetwoaverageimagesof FigureA.6quicklyyieldsthe˛owpatternofsmallindividualjetsfromeachhoneycombtubeand howtheymixandbecomesmootherasthe˛owdevelopsdownstreamofthehoneycomb. A.3Results Thestreamwisedevelopmentoftheresultingnon-uniform˛owispresentedinFigureA.7,where themeanvelocitypro˝lesmeasuredfromthehoneycombexitupto20cm x d ˇ 62 downstream arecomparedtothedesignvelocitypro˝le.Consistentwithexpectations,the˛owattheexitof thedeviceisdominatedbytheindividualjetscomingoutfromeachhoneycombcell(seeFigure A.7a),andquicklysmoothsout(FigureA.7b-c).Thepresenceofindividualjetsisnon-existent andthemeanvelocitypro˝leshowsaverygoodagreementwiththedesignpro˝leafteradistance 95 FigureA.5:Aschematicofthe˛ow measurementsetupinthepresenceofthe sheargenerationdevice. FigureA.6:Asampleimagepairshowinga portionofthetaggedregioncapturedwitha timedelayof10ms.(i)thetaggedregion rightafterthelaserpulse(ii)thesame taggedregion10mslater.Theindividual honeycombcellsareshowninyellowonthe images. ofapproximately60tubediameters(FigureA.7d). FigureA.8comparestemporalvelocity˛uctuationsofthehyperbolictangentvelocitypro˝le atthedownstreamlocationwherethemeanvelocityhasbecomesmooth, x d ˇ 62 ,withfreestream temporalvelocity˛uctuationsinthesamelocationwithoutthesheargenerationdevicepresent. Notonlytheadditionofthehoneycombdevicedoesnotinduceanyadditionaltemporalvelocity ˛uctuation,itevenslightlyreducesthese˛uctuations,anotsounexpectedresultconsideringthe typicalusageofhoneycombsas˛owmanagementdevices. Thesmoothnessofthevelocitypro˝leateachdownstreamlocationcanbequanti˝edby˝tting ahyperbolictangentcurvetothemeasuredvelocitypro˝leandcalculatingtherootmeansquare ofitsspatialdeviations, ˙ ,fromthiscurve.Thereasonbehindusingahyperbolictangent˝t insteadoftheactualdesignvelocitypro˝leistoavoidintroducingabiaserror,especiallyatthe initiallocations,wherethevelocitypro˝lehasyettodevelopintothedesignvelocitypro˝le.This 96 (a) (b) (c) (d) FigureA.7:Normalizedvelocitypro˝lesshowingdevelopmentofthe˛owdownstreamofthe honeycombcomparedwiththedesignvelocitypro˝le. 97 FigureA.8:Temporalvelocity˛uctuations ofthegeneratedhyperbolictangentpro˝le comparedwiththatofthefreestream˛ow withoutthesheargenerationdeviceata downstreamlocationof x š d ˇ 62 . FigureA.9:Ahyperbolictangentcurve ˝ttedtotheexperimentalvelocitypro˝leat adownstreamlocationof x š d ˇ 24 . isevidentbyconsideringthevelocitypro˝leinFigureA.9for x d ˇ 24 ,wherethe˛owhasnot developedintoits˝nalpro˝leyet,butisstillrepresentedwellbyahyperbolictangentcurve.The developmentofthesespatialdeviationsbetweenthemeanvelocitypro˝leandahyperbolictangent ˝ttothedata,isshownasafunctionofthedistancedownstreamofthehoneycombexitinFigure A.10.Afteradistanceofroughly60honeycombtubediameters,spatialdeviationsfromasmooth hyperbolictangentpro˝lefallwithin1%ofthecenterlinevelocity. The˛exibilityofthemodelisdemonstratedwiththefabricationofaseconddevicetogenerate amorecomplicatedvelocitypro˝le,modeledtomimicthestreamwisevelocitycomponentof aGaussianvortexconvectinginauniform˛ow.FiguresA.11andA.12presentthedesigned honeycomblengthdistributionpro˝leandanimageofthe˝nalhoneycombdevice,respectively. Acomparisonbetweenthedesignvelocitypro˝leandtheexperimentalmeasurementsata downstreamlocationof x š d ˇ 72 isshowninFigureA.13.Overall,consistentwiththepre- 98 FigureA.10:Developmentofspatial deviationofthemeasuredvelocitypro˝le formthehyperbolictangent˝tasafunction ofdistancedownstream. FigureA.11:Calculatedhoneycomblength distributionforthestreamwisevelocity componentofaconvectingGaussianvortex pro˝lebasedonthepresentmodel comparedtotheestimatedentrancelength ofthe˛owateachy(basedon L e ˇ 0 : 06 Re ¹ y º ).Thehorizontallines depicttheapproximatediameterof honeycombtubes. vioushyperbolictangentdevice,verygoodagreementbetweenthedesignvelocitypro˝leand experimentallygeneratedvelocitypro˝leisobserved.Anotherexampleofanon-uniform˛ow generatedusingthismodi˝edmodelispresentedinHammeretal.(2019),wheretheystudieda harmonically-pitchingairfoilinauniform-shearapproach˛ow. Inadditiontothedescribedhoneycombdesignmethod,theprocedurepresentedinthisarticle canalsobeutilizedinareversefashiontoestimatetheexpectedvelocitypro˝legeneratedbya knownhoneycomblengthpro˝leandoperatingconditions.Thisperquisiteprovidestheopportunity topredicttheperformanceofasheargenerationhoneycombdeviceino˙-designsituationslike usingadi˙erentapproachvelocityoranentirelydi˙erent˛uid. 99 FigureA.12:Imageofthesheargeneration devicecutaccordingtodesignedlength pro˝lebasedonthepresentmodel(shown inFigureA.12). FigureA.13:Normalizedexperimental velocityofthestreamwisecomponentofa convectingGaussianvortexmeasuredata downstreamlocationof x š d ˇ 72 compared tothedesignvelocitypro˝le. A.4ComparisonwithOriginalModel Abriefcomparisonofourmodi˝edmodelversustheoriginalmodelisperformedthrough usingaconstantfrictionfactorcoe˚cient(equaltotheaveragevalueestimatedbyourmodel)to designanotherhoneycombtogeneratethepreviouslydescribedhyperbolictangentvelocitypro˝le. FigureA.14showsthisconstantfrictionfactorcoe˚cienthoneycombalongsidetheearlierdesign basedonourmodi˝edmodel.Theassumptionofconstantfrictionfactorresultsina13%larger variationintubelengthacrossthehoneycombcomparedtoourmodi˝edmodelwithavariable frictionfactor.Varyingthevalueoftheconstantfrictionfactorcoe˚cientby ± 20%changedthe tubelengthdi˙erenceacrossthedevicelessthan ± 2%. Finally,theresultingvelocitypro˝lesfromthetwodi˙erenthoneycombdesignsareevaluated againsttheexperimentalresultsbyusingourmethodinreverse(i.e.givenacertainhoneycomb lengthdistributiontocomputetheresultingvelocitypro˝le).Thedevicewasevaluatedatthe operatingconditionssimilartotheexperiment.AsexhibitedinFigureA.15,theconstantfriction 100 FigureA.14:Comparisonoftheshapeof thehoneycombdevicesdesignedusing Kotansky'sandthemodi˝edmodelto generatethehyperbolictangentvelocity pro˝leshowninFigure2A.2). FigureA.15:Estimatedvelocitypro˝le fromKotansky'shoneycombdesign comparedtothatofthemodi˝edmodeland theexperimentalvelocitypro˝le. factorhoneycombwouldproduceavelocitydi˙erencethatis20%largerthandesired. A.5Discussion TheimprovementspresentedinthisstudytoKotansky'soriginalvariablelengthhoneycomb modelenhancetheaccuracyandtractabilityofthetechnique.Thereare,however,afewkey limitationsobservedthatareprudenttodiscuss.Themostimportantpointtokeepinmindisthe factthatthevariablefrictionfactorcoe˚cientmodelusedinthisworkisbasedontheassumption oflaminar˛owinhoneycombtubes.Onewaytoside-stepthisissueforhighspeedapplications canbethroughemployinghoneycombswithsmallenoughtubediametertosatisfythelaminar ˛owconditioninsidethetubes.Regardingthefabricationofthedevice,careshouldbetakento useamethodthatproducesaclean-cutsurfacewithoutdamagingthehoneycombtubes,while providingsu˚cientmaneuverability.Specializedbandsawbladesusedinthisworksatis˝edthese conditionsforthedesiredshapepro˝les,butformorecomplicatedgeometriesinvolvinghigher 101 curvatures,wesuggestusingawater-jetcuttingtechniqueinstead.Whenusedinwatertunnel facilities,itiscriticaltothoroughlyremoveallthebubblestrappedinsidethehoneycombtubes,as thesebubbleswouldsigni˝cantlya˙ectthepressuredropdistributionacrossthehoneycomband altertheresultingvelocitypro˝le. 102 APPENDIXB EFFECTOFAIRFOILPRESENCEONUPSTREAMFLOWBOUNDARYCONDITION Figurespresentedinthisappendixdemonstratethee˙ectsofpresenceofastationaryorpitching airfoilontheupstreamshear˛owconditions.Theseplotsarebasedonsinglecomponentvelocime- trymeasurementsperformed2chordlengthsupstreamoftheairfoil.Rawandnormalizedmean and˛uctuatingvelocitypro˝lesarepresentedtohighlightthee˙ectsofairfoilontheupstream ˛ow.Inthecaseofstationaryairfoils,themagnitudeofhighspeedandlowspeedvelocities,as wellascenterlinelocationshearlayerthicknessarealsoreportedasafunctionofairfoilangleof attack, . B.1E˙ectofStationaryAirfoilonUpstreamSteadyShearLayer Itisobservedthatwhilethenormalizedbehaviorofthesteadyshearlayerdoesnotchangewith airfoilangleofattack,theshearlayercenterpositionmovesslightlytowardsthehightspeedside withpositiveanglesofattackandtowardsthelowspeedsidewithnegativeanglesofattack.Thisis consistentwithtrailingedgeoftheairfoilblockingpathoftheshearlayeratlargeanglesofattack. Forexample,whentheairfoilisplacesatapositiveangleofattack,thetrailingedgeextendsinthe pathofthelowspeedside,resultinginadecreesinthelowspeedsidevelocity.Thisinturn,shifts theshearlayertowardsthehighspeedside.Thethicknessoftheshearlayerdoesnotseemtobe a˙ectedbytheangleofthestationaryairfoil. 103 (a) (b) FigureB.1:E˙ectofstationaryairfoilangleonrawstreamwise (a) meanand (b) ˛uctuating velocitypro˝lesinthesteadyshearlayer. (a) (b) FigureB.2:E˙ectofstationaryairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesinthesteadyshearlayer. 104 (a) (b) FigureB.3:E˙ectofstationaryairfoilangleonstreamwise (a) high-speedand (b) low-speed velocitiesinthesteadyshearlayer. (a) (b) FigureB.4:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinevelocityand (b) velocity di˙erenceinthesteadyshearlayer. 105 (a) (b) FigureB.5:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinepositionand (b) vorticitythicknessinthesteadyshearlayer. B.2E˙ectofStationaryAirfoilonUpstreamUnsteadyShearLayer Similartotheobservationsinthesteadyshearlayer,whilethenormalizedbehaviorofthe steadyshearlayerdoesnotchangewithairfoilangleofattack,theshearlayercenterpositionmoves slightlytowardsthehightspeedsidewithpositiveanglesofattackandtowardsthelowspeedside withnegativeanglesofattack.Thisisconsistentwithtrailingedgeoftheairfoilblockingpathof theshearlayeratlargeanglesofattack.Forexample,whentheairfoilisplacesatapositiveangle ofattack,thetrailingedgeextendsinthepathofthelowspeedside,resultinginadecreesinthelow speedsidevelocity.Thisinturn,shiftstheshearlayertowardsthehighspeedside.Thethickness oftheshearlayerdoesnotseemtobea˙ectedbytheangleofthestationaryairfoil. 106 (a) (b) FigureB.6:E˙ectofstationaryairfoilangleonrawstreamwise (a) meanand (b) ˛uctuating velocitypro˝lesintheunsteadyshearlayer. (a) (b) FigureB.7:E˙ectofstationaryairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayer. 107 (a) (b) FigureB.8:E˙ectofstationaryairfoilangleonstreamwise (a) high-speedand (b) low-speed velocitiesintheunsteadyshearlayer. (a) (b) FigureB.9:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinevelocityand (b) velocity di˙erenceintheunsteadyshearlayer. 108 (a) (b) FigureB.10:E˙ectofstationaryairfoilangleonstreamwise (a) centerlinepositionand (b) vorticitythicknessintheunsteadyshearlayer. B.3E˙ectofPitchingAirfoilonUpstreamUnsteadyShearLayerwith ! c = 0 : 5 (a) (b) FigureB.11:E˙ectofpitchingairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayerwith ! c = 0 : 5 and K max = 1 : 4 . 109 B.4E˙ectofPitchingAirfoilonUpstreamUnsteadyShearLayerwith ! c = 0 : 9 (a) (b) FigureB.12:E˙ectofpitchingairfoilangleonnormalizedstreamwise (a) meanand (b) ˛uctuatingvelocitypro˝lesintheunsteadyshearlayerwith ! c = 0 : 9 and K max = 0 : 8 . 110 APPENDIXC AERODYNAMICFORCESONOSCILLATINGAIRFOILWITH 0 = 4 FigurespresentedinthisappendixdemonstratetheaerodynamicperformanceofaNACA0012 airfoilsinusoidallypitchingarounditsquarterchordwithanoscillationamplitudeof 0 = 4 invarious˛owconditionsconsideredinthiswork.Inallofthese˝guresthesymbolsdepict theaveragevalueof4independentmeasurementrepetitionsforeachcombinationof˛owand oscillationfrequency.Theerrorbarsformeanvaluesarebasedonthepropagationofsensordrift duringeachmeasurement,whiletheuncertaintyin˛uctuationsisshownasthestandarddeviation ofthe˛uctuationmagnitudesbetweenthe4repetitions. 111 C.1E˙ectofShearFlowUnsteadiness (a) (b) (c) (d) FigureC.1:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoilplaced inuniform,steadyshearandunsteadyshear˛ows: (a) meanliftcoe˚cient, C L , (b) liftcoe˚cient ˛uctuations, C L 0 , (c) meanthrustcoe˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .The airfoilispitchingwithzeromeanangleofattackandanoscillationamplitudeof 4 aroundits quarterchordpoint. 112 C.2E˙ectofUnsteadyShearFlowThicknessandShearRate (a) (b) (c) FigureC.2:Extentofairfoilangulardisplacement( 4 )comparedtotheshearlayerthicknessat eachmeasurementlocation. 113 (a) (b) (c) (d) FigureC.3:Comparisonofaerodynamicforcecoe˚cientsmeasuredonapitchingairfoilplaced inunsteadyshearlayeratdi˙erentdownstreamlocations: (a) meanliftcoe˚cient, C L , (b) lift coe˚cient˛uctuations, C L 0 , (c) meanthrustcoe˚cient, C T , (d) thrustcoe˚cient˛uctuations, C T 0 .Theairfoilispitchingwithzeromeanangleofattackandanoscillationamplitudeof 4 arounditsquarterchordpoint. 114 APPENDIXD SAMPLESOFPHASEORDEREDAERODYNAMICFORCESONOSCILLATING AIRFOIL Thefollowing˝guresprovideonesampleofthephaseorderedandphaseaveragedliftandthrust coe˚cientsforeachoftheoscillationcasesperformedinuniform˛ow,steadyshearlayerand unsteadyshearlayer.Intheseexperimentstheairfoilispitchingwithareducedfrequencyof varyingfrom k = 1 to6andoscillationamplitudesof 0 = 2 .Eachoftheseplotsarefromasingle instanceofforcemeasurementsortedinto128phasebinsbasedontheairfoilangularposition.In theseplots,thebluelinesrepresentthephaseordereddata,whiletheblacklinesportraythephase averagedbehavior( h C L i and h C T i )withtheredlineshighlightingthestandarddeviationofthe dataineachphasebin( h C L i 0 and h C T i 0 ). 115 FigureD.1:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 1 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.2:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 116 FigureD.3:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 3 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.4:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 4 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 117 FigureD.5:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.6:Sampleofphaseorderedforcemeasurementsperformedinuniform˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 6 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 118 FigureD.7:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 1 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.8:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 119 FigureD.9:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 3 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.10:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 4 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 120 FigureD.11:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.12:Sampleofphaseorderedforcemeasurementsperformedinsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 6 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 121 FigureD.13:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 1 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.14:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 2 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 122 FigureD.15:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 3 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.16:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 4 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 123 FigureD.17:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 5 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. FigureD.18:Sampleofphaseorderedforcemeasurementsperformedinunsteadyshear˛owfora pitchingairfoilwithoscillationamplitudeof 0 = 2 andreducedfrequencyof k = 6 .Bluelines representthephaseordereddata,redlinesportraythephaseaveragedvalueswiththeblacklines highlightingthestandarddeviationofthedataineachphasebin. 124 APPENDIXE BOUNDARYLAYERMEASUREMENTALTERNATECONTOURS Thefollowing˝guresprovidealternativeversionsoftheboundarylayerresolvedvelocimetryresults inafashionthattheairfoilshownatitsactualangleofattackandthe˛owdirectionisfromleft toright,asshownwiththearrowinplots.Thecolormapusedforgeneratingthecontourplots aredesignedtoshowasharpchangefromdarkbluetopurpleatavelocityofzerotofurther highlighttheregionswithreverse˛ow.Themeanstreamwisevelocityvaluesarenormalizedby thecenterlinevelocityoftheshear˛ow( U c )whiletheshearlayervelocitydi˙erence, u ,isused tonormalizethevelocity˛uctuations.Thewhitepatchesinthecontoursareregionsthatareeither partiallyblockedbytheairfoilorhavebeena˙ectedbytheglowfromtheairfoilsurface. 125 (a) (b) FigureE.1:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 0 atthecenterofthesteadyshearlayer. (a) (b) FigureE.2:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 0 atthecenteroftheunsteadyshearlayer. 126 (a) (b) FigureE.3:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 2 inthesteadyshearlayer. (a) (b) FigureE.4:Contourof (a) meanand (b) ˛uctuatingstreamwisevelocitiesmeasurednearthe surfaceoftheairfoilpositionedat = 2 intheunsteadyshearlayer. 127 (a) (b) (c) (d) (e) (f) FigureE.5:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatpositiveanglesofattackinthecenter ofthesteadyshearlayer. 128 (a) (b) (c) (d) (e) (f) FigureE.6:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatpositiveanglesofattackinthecenter oftheunsteadyshearlayer. 129 (a) (b) (c) (d) (e) (f) FigureE.7:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatnegativeanglesofattackinthe centerofthesteadyshearlayer. 130 (a) (b) (c) (d) (e) (f) FigureE.8:Comparisonofmean(left),and˛uctuating(right)streamwisevelocitycontours measurednearthesuctionsurfaceoftheairfoilpositionedatnegativeanglesofattackinthe centeroftheunsteadyshearlayer. 131 BIBLIOGRAPHY 132 BIBLIOGRAPHY Abramovich,G.N.1963. 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