Flow field of flexible flapping wings

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Title:
Flow field of flexible flapping wings
Physical Description:
1 online resource (194 p.)
Language:
english
Creator:
Sallstrom,Erik M
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Ukeiley, Lawrence S.
Committee Members:
Ifju, Peter
Lind, Richard C
Bloomquist, David G

Subjects

Subjects / Keywords:
aerodynamics -- dynamics -- flapping -- flexible -- flight -- fluid -- image -- interaction -- particle -- piv -- structure -- unsteady -- velocimetry -- wings
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
The agility and maneuverability of natural fliers would be desirable to incorporate into engineered micro air vehicles (MAVs). However, there is still much for engineers to learn about flapping flight in order to understand how such vehicles can be built for efficient flying. The goal of this study is to develop a methodology for capturing high quality flow field data around flexible flapping wings in a hover environment and to interpret it to gain a better understanding of how aerodynamic forces are generated. The flow field data was captured using particle image velocimetry (PIV) and required that measurements be taken around a repeatable flapping motion to obtain phase-averaged data that could be studied throughout the flapping cycle. Therefore, the study includes the development of flapping devices with a simple repeatable single degree of freedom flapping motion. The acquired flow field data has been examined qualitatively and quantitatively to investigate the mechanisms behind force production in hovering flight and to relate it to observations in previous research. Specifically, the flow fields have been investigated around a rigid wing and several carbon fiber reinforced flexible membrane wings. Throughout the whole study the wings were actuated with either a sinusoidal or a semi-linear flapping motion. The semi-linear flapping motion holds the commanded angular velocity nearly constant through half of each half-stroke while the sinusoidal motion is always either accelerating or decelerating. The flow fields were investigated by examining vorticity and vortex structures, using the Q criterion as the definition for the latter, in two and three dimensions. The measurements were combined with wing deflection measurements to demonstrate some of the key links in how the fluid-structure interactions generated aerodynamic forces. The flow fields were also used to calculate the forces generated by the flapping wings using momentum balance methods which yielded details of where along the wing the forces were generated. As expected, these results indicated that the spanwise location of where the forces were generated depended upon the wings membrane material and reinforcement pattern, but in general it was in the outer third of the wing.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Erik M Sallstrom.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Ukeiley, Lawrence S.

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UFRGP
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Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0042772:00001


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FLOWFIELDOFFLEXIBLEFLAPPINGWINGS By ERIKS ¨ ALLSTR ¨ OM ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011ErikS ¨ allstr ¨ om 2

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ACKNOWLEDGMENTS IwouldliketothankmyadvisorProf.LawrenceUkeileyforgiv ingmetheopportunity todomydoctoralstudiesandresearchatUniversityofFlori da.Iwouldalsoliketothank themembersofmysupervisorycommittee;Prof.DavidBloomqui st,Prof.PeterIfju,and Prof.RickLind. Ihaveenjoyedandlearnedmuchfromworkingwiththeotherme mbersofthe groupworkingwithProf.UkeileyattheUniversityofFlorida ResearchandEngineering EducationFacility(REEF)andwouldthereforeliketothankDie goCampos,JonDudley, ParvezKhambatta,TaylorLusk,AdamHart,GeorgeShumway,Amory Timpe,andProf. CharlesTinney.IwouldalsoliketothankthestaffattheREEFf ormakingresearchand studiestherepossible. IwouldliketoshowmyappreciationtoDicWu,SatishChimakur thi,andHikaro AonothatIgottheopportunitytoworkwiththroughtheMURIpr ojectleadbyProf.Wei Shyy.IwouldalsoliketothankJustinMcIntireformanufactu ringtheskeletonsforthe wingsusedinmylastsetofexperiments. IwouldliketoacknowledgethesupportoftheAirForceOfceo fScientic ResearchundertheMURIprograminajointeffortwiththeUni versityofMichiganand theUniversityofMaryland,andfromtheFloridaCenterforAd vancedAeroPropulsion. 3

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TABLEOFCONTENTS Page ACKNOWLEDGMENTS ..................................3 LISTOFTABLES ......................................8 LISTOFFIGURES .....................................9 ABSTRACT .........................................13 CHAPTER 1INTRODUCTION ...................................15 1.1FlightinNature .................................16 1.1.1HummingbirdFlight ...........................16 1.1.2InsectFlight ...............................19 1.2EngineeredSystems ..............................20 1.3LowReynoldsNumberFlightAerodynamics .................22 1.3.1StrokePlaneAngle ...........................22 1.3.2Quasi-SteadyAnalysis .........................22 1.3.3TheWagnerEffect ...........................25 1.3.4LeadingEdgeVortex ..........................25 1.3.5Delayedstall ..............................25 1.3.6VirtualMass ...............................28 1.3.7WingTwistandAngleofAttack ....................28 1.4Objectives ....................................29 1.5ExpectedContributions ............................30 2PARTICLEIMAGEVELOCIMETRY(PIV) .....................32 2.1Background ...................................32 2.2DigitalParticleImageVelocimetry ......................33 2.3DigitalCross-Correlation ............................33 2.4VelocityMeasurementAccuracy .......................34 2.4.1Sub-PixelAccuracy ...........................35 2.4.2Peak-Locking ..............................36 2.4.3TracerParticleRelaxationTime ....................36 2.4.4TerminalVelocityofTracerParticles .................37 2.5Super-ResolutionPIV .............................37 2.6StereoscopicPIV ................................38 2.7ScheimpugCondition .............................38 2.8Calibration ...................................39 2.8.1PinholeModel ..............................39 2.8.2RadialDistortion ............................41 2.9PIVAlgorithm ..................................41 4

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2.9.1Pre-Processing .............................41 2.9.2Grid ...................................42 2.9.3Correlations ...............................42 2.9.4CorrelationPeaks ............................44 2.9.5ComputingThree-ComponentDisplacementVectors ........46 2.9.6RejectingOutliers ............................47 2.9.7FillingGaps ...............................48 2.9.8ReningGrid ..............................48 2.9.9Iterations ................................49 2.10Post-Processing ................................49 2.10.1MethodForOutlierRejection,Smoothing,andInterpo lation ....49 2.10.2FittedPhase-Average .........................50 3DATAANALYSISMETHODS ............................53 3.1EstimationsofDifferentialQuantities .....................53 3.1.1Vorticity .................................53 3.1.2Strain ..................................55 3.2VortexIdentication ..............................56 3.2.1QCriterion ...............................56 3.3LineIntegralConvolution ...........................57 3.3.1FastImplementation ..........................58 3.3.2ModicationstoFastImplementation .................60 3.3.3Example .................................61 3.3.4Renements ...............................61 3.4AerodynamicForces ..............................62 3.4.1PressureEstimation ..........................63 3.4.2MomentumBalance ..........................64 4EXPERIMENTALEQUIPMENT ...........................68 4.1PIVEquipment .................................68 4.1.1DantecSystem .............................68 4.1.1.1Software ...........................69 4.1.1.2IDTMotionProX3cameras .................69 4.1.1.3Leelaser ...........................69 4.1.2LaVisionSystem ............................70 4.1.2.1LaVisionDaVissoftware ..................70 4.1.2.2ImagerproX4Mcameras .................71 4.1.2.3LitronNanoLlaser .....................71 4.1.3Seeding .................................71 4.1.3.1LaVisionDSaerosolgenerator ...............72 4.1.3.2Dantechighvolumeliquiddropletseedinggenerato r ...72 4.1.3.3Expancel TM microspheres ..................72 4.2FlappingDevices ................................73 4.2.1Type1 ..................................73 5

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4.2.2Type2 ..................................74 4.2.3MaxonEC1615Wbrushlessmotor .................74 4.2.4Type3 ..................................75 4.2.5Type4 ..................................75 4.2.6MaxonEC1640Wbrushlessmotor .................76 4.3Wings ......................................76 4.3.1ZimmermanPlanform .........................77 4.3.1.1Aluminumwings .......................77 4.3.1.2Anisotropicwings ......................77 4.4ForceMeasurements .............................78 4.4.1ATINano17ForceTransducer .....................78 4.4.2ATIGammaForceTransducer .....................78 4.4.3NI-DAQ6220DataAcquisitionCard .................78 4.5FlappingPhaseofPIVDataSnapshots ...................79 4.6HoveringTestingEnvironment .........................80 5EXPERIMENTALRESULTS .............................85 5.1NormalizedQuantities .............................85 5.2RigidAluminumWings .............................86 5.2.1ExperimentalSetup ..........................87 5.2.1.1Wing .............................87 5.2.1.2Flappingdevice .......................88 5.2.1.3Particleimagevelocimetrysetup ..............88 5.2.2DataAnalysisMethods .........................88 5.2.2.1Particleimagevelocimetry .................88 5.2.2.2Synchronization .......................89 5.2.3Results .................................89 5.2.4Summary ................................92 5.3L m B1Wings ..................................94 5.3.1ExperimentalSetup ..........................94 5.3.1.1Wing .............................94 5.3.1.2Flappingdevice .......................95 5.3.1.3Particleimagevelocimetrysetup ..............95 5.3.2DataAnalysisMethods .........................96 5.3.2.1Particleimagevelocimetry .................96 5.3.2.2Synchronization .......................96 5.3.2.3Masking ............................97 5.3.3Results .................................98 5.3.4Summary ................................102 5.4AxxxxWings ..................................103 5.4.1ExperimentalSetup ..........................103 5.4.1.1Wing .............................103 5.4.1.2Flappingdevice .......................104 5.4.1.3Particleimagevelocimetrysetup ..............104 6

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5.4.1.4Synchronization .......................104 5.4.1.5Forcemeasurements ....................105 5.4.2DataAnalysisMethods .........................105 5.4.2.1Particleimagevelocimetry .................105 5.4.2.2Wingtracking .........................108 5.4.3Results .................................109 5.4.3.1Flowelds ..........................109 5.4.3.2Forceestimations ......................111 5.4.4Summary ................................117 5.5VaryingMembraneWings ...........................118 5.5.1ExperimentalSetup ..........................118 5.5.1.1Wing .............................119 5.5.1.2Wingmembranematerials .................119 5.5.1.3Particleimagevelocimetrysetup ..............119 5.5.1.4Flappingdevices .......................120 5.5.1.5Forcemeasurements ....................120 5.5.2DataAnalysisMethods .........................120 5.5.2.1Particleimagevelocimetry .................120 5.5.2.2Wingtracking .........................121 5.5.3Results .................................121 5.5.3.1Forcemeasurements ....................122 5.5.3.2Floweldmeasurements ..................126 5.5.4Summary ................................132 6SUMMARY ......................................167 6.1WingPlanform .................................171 6.2ParticleImageVelocimetry ..........................171 6.3FutureWork ...................................173 APPENDIX:PIVALGORITHMACCURACY .......................177 A.1Assumptions ..................................177 A.2GeneratingArticialPIVImages .......................179 A.3Results .....................................181 REFERENCELIST .....................................187 BIOGRAPHICALSKETCH ................................194 7

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LISTOFTABLES Table Page 4-1ATINano17ratedsensingrange. ..........................80 4-2ATINano17typicalresolutionfora16-bitDAQ. ..................81 4-3ATIGammaratedsensingrange. ..........................81 4-4ATIGammatypicalresolutionfora16-bitDAQ. ..................81 5-1PIVcapturesettingsforrigidwingoweldmeasurements ...........133 5-2PIVsnapshotstakenattheoriginalverticallocationoft herigidwing. ......133 5-3PIVsnapshotstakenoftherigidwingwithdomainshiftedb y50mmvertically. 134 5-4PIVcapturesettingsfortheL m B1wings. .....................134 5-5Membranematerialproperties. ...........................134 5-6ScopeofPIVdataforwingswithdifferentmembranemateria ls. .........134 5-7Forcesfromwingsonthesemi-linearapper. ...................134 5-8Forcesfromwingsonthesinusoidalapper. ....................135 5-9Membranematerialpropertiesinthenumericalstudy. ..............135 8

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LISTOFFIGURES Figure Page 1-1Horizontalandverticalstrokeplanes. .......................31 2-1StereoPIVschematic. ...............................51 2-2SchematicoftheScheimpugcondition. .....................51 2-3Coordinatetransformationstepsforthepinholemodel. .............52 3-1Stagnationoweldvectors. ...........................65 3-2LineIntegralConvolution(LIC)ofstagnationoweld. .............66 3-3LICofstagnationoweldwithlowerstreamlinedensity ............66 3-4Two-dimensionalcolorscaleforLIC. .......................66 3-5LICofstagnationoweldwithcolorindicatingvelocit ymagnitude. ......66 3-6LICofstagnationoweldwithmorepronouncedstreamli nesathighvelocities. 66 3-7 y momentumleavingacontrolvolume. ......................67 4-1LaVisioncalibrationtarget. .............................81 4-2Flappingdevicetype1. ..............................82 4-3Flappingdevicetype2. ..............................82 4-4Encoderoutput. ...................................82 4-5Flappingdevicetype3. ..............................83 4-6Flappingdevicetype4. ..............................83 4-7Predictedangle,angularvelocity,andangularaccelera tionversusphase. ..84 4-8Zimmermanprolewing. ..............................84 5-1PIVsetupschematicforrigidwingoweldmeasurements. ..........136 5-2PIVdatagridforrigidwingoweldmeasurements. ..............136 5-3Coordinatesystemforrigidwingowelddata. .................136 5-4Chordwiselasersheetlocationsforrigidwingmeasurem ents. .........137 5-5Iso-surfacesofphase-averagedchordwisevorticityof rigidwing. .......137 5-6Iso-surfacesofphase-averagedspanwisevorticityofr igidwing. ........138 9

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5-7CombinedPIVdomainforrigidwingowelddata. ...............138 5-8Phase-averagedvelocityandvorticityofrigidwingat5H z. ..........139 5-9Phase-averagedvelocityandvorticityofrigidwingat10 Hz. ..........140 5-10Timeversuswingangleofrigidwing. .......................141 5-11L2B1winganglesinairandvacuum. .......................141 5-12AnL m B1wing. ...................................141 5-13ThrustproducedbytheL m B1wingsatdifferentappingfrequencies. ....142 5-14L m B1wingappinginlasersheet. ........................142 5-15PIVsetupschematicforoweldmeasurementsontheL m B1wings. ....142 5-16SpanwiselasersheetlocationsforL m B1wingmeasurements. ........143 5-17PIVdatagridforL m B1wingoweldmeasurements. .............143 5-18CoordinatesystemforL m B1wingowelddata. ................143 5-19VorticityandvelocityeldsaroundtheL1B1wing. ...............144 5-20VorticityandvelocityeldsaroundtheL2B1wing. ...............145 5-21VorticityandvelocityeldsaroundtheL3B1wing. ...............146 5-22WingdeectionsatthequarterchordoftheL m B1wings. ...........147 5-23VorticityaroundtheL2B1wingat z =50mm. ..................147 5-24Motionofthequarterchordpoint65mmfromtherootofth eL m B1wings. ..148 5-25Time-averagedstreamwisemomentumintegratedover y and z .......148 5-26Axxxxwings. ....................................148 5-27PIVsetupschematicforoweldmeasurementsontheAxxxx wings. ....149 5-28ValidPIVdatagridsaftercalibrationformeasurements ontheAxxxxwings. .149 5-29CoordinatesystemforAxxxxwingowelddata. ................150 5-30SpanwiselasersheetlocationsforAxxxxwingoweldmea surements. ...150 5-31 Q aroundtheA0001wing. .............................151 5-32VorticityeldsaroundtheA0001wing. ......................152 5-33 Q aroundtheA1101wing. .............................153 10

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5-34VorticityeldsaroundtheA1101wing. ......................154 5-35Time-averagedcontributionstoforcefortheAxxxxwing s. ...........155 5-36Phase-averagedcontributionstoforcefortheAxxxxwing s. ..........155 5-37SpanwisecontributionstoforcefortheAxxxxwings. ..............156 5-38 z and Q fortheA0001wingattwophaseinstances. ..............156 5-39 z and Q fortheA1101wingattwophaseinstances. ..............156 5-40Wingswitha0.010inchthicklatexmembranes. .................157 5-41Thrustcoefcient C T asafunctionofeffectivewingstiffness 1 ........157 5-42Quadrilateralelementwinggrid. .........................158 5-43Exampleofdeformedwingshape. ........................158 5-44Relationshipbetweenpressureandwingdeection. ..............158 5-45Highlydeformedwingwithmembranematerial2. ................159 5-46Highlydeformedwingwithmembranematerial3. ................159 5-47Thrustcoefcient C T asafunctionofreferencewingdeformation. .......160 5-48Spanwisecontributionstoforceforwingswithvaryingm embranes. ......161 5-49Time-averagedcontributionstoforceforwingswithva ryingmembranes. ...162 5-50Vorticityeldsofthe0.010”latexmembranewingonthe semi-linearapper. .163 5-51Vorticityeldsofthe0.010”latexmembranewingonthe sinusoidalapper. .164 5-52Vorticityeldsofthe0.001”PETmembranewingonthesemi -linearapper. .164 5-53Vorticityeldsofthe0.001”PETmembranewingonthesinu soidalapper. .165 5-54Vorticityeldsofthe0.002”PFAmembranewingonthesem i-linearapper. .165 5-55Vorticityeldsofthe0.002”PFAmembranewingonthesin usoidalapper. .166 5-56Vorticityeldsofthe0.001”PFAmembranewingonthesem i-linearapper. .166 5-57Sketchoftrustproducingvortexstructures. ...................166 A-1ComparisonbetweenarealandanarticiallygeneratedPIV image. .....183 A-2Particleintensitydistributionsforrealandarticial PIVimages. ........183 A-3FloweldsfortestingPIValgorithm. .......................184 11

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A-4Swirlingowbiaserror. ..............................184 A-5Swirlingowmeanerrorandfractionvalidvectors. ...............185 A-6Vortexowbiaserror. ...............................185 A-7Vortexowmeanerrorandfractionvalidvectors. ................186 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy FLOWFIELDOFFLEXIBLEFLAPPINGWINGS By ErikS ¨ allstr ¨ om August2011 Chair:LawrenceUkeileyMajor:AerospaceEngineering Theagilityandmaneuverabilityofnaturalierswouldbede sirabletoincorporate intoengineeredmicroairvehicles(MAVs).However,thereis stillmuchforengineers tolearnaboutappingightinordertounderstandhowsuchv ehiclescanbebuiltfor efcientying.Thegoalofthisstudyistodevelopamethodo logyforcapturinghigh qualityowelddataaroundexibleappingwingsinahover environmentandto interpretittogainabetterunderstandingofhowaerodynam icforcesaregenerated.The owelddatawascapturedusingparticleimagevelocimetry (PIV)andrequiredthat measurementsbetakenaroundarepeatableappingmotionto obtainphase-averaged datathatcouldbestudiedthroughouttheappingcycle.The refore,thestudyincludes thedevelopmentofappingdeviceswithasimplerepeatable singledegreeoffreedom appingmotion.Theacquiredowelddatahasbeenexamined qualitativelyand quantitativelytoinvestigatethemechanismsbehindforce productioninhoveringight andtorelateittoobservationsinpreviousresearch.Speci cally,theoweldshave beeninvestigatedaroundarigidwingandseveralcarbonbe rreinforcedexible membranewings.Throughoutthewholestudythewingswereac tuatedwitheithera sinusoidalorasemi-linearappingmotion.Thesemi-linea rappingmotionholdsthe commandedangularvelocitynearlyconstantthroughhalfof eachhalf-strokewhilethe sinusoidalmotionisalwayseitheracceleratingordeceler ating.Theoweldswere investigatedbyexaminingvorticityandvortexstructures ,usingtheQcriterionasthe 13

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denitionforthelatter,intwoandthreedimensions.Theme asurementswerecombined withwingdeectionmeasurementstodemonstratesomeofthe keylinksinhowthe uid-structureinteractionsgeneratedaerodynamicforce s.Theoweldswerealso usedtocalculatetheforcesgeneratedbytheappingwingsu singmomentumbalance methodswhichyieldeddetailsofwherealongthewingthefor cesweregenerated.As expected,theseresultsindicatedthatthespanwiselocati onofwheretheforceswere generateddependeduponthewingsmembranematerialandrei nforcementpattern,but ingeneralitwasintheouterthirdofthewing. 14

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CHAPTER1 INTRODUCTION Naturaliersexhibitimpressivefeaturesofagilityandma neuverabilitywhich wouldbedesirabletoincorporateintoengineeredyingsys tems.Therefore,nature isanobvioussourceofinspirationfordesignersofmicroai rvehicles(MAVs)looking toimprovethescopeofthemissionsthattheirvehiclescana ccomplish.Although bio-mimicryaloneshouldnotbeexpectedtorevealallthefu ndamentalphysics behindight,detailedstudiesofcertaincharacteristics ofbiologicalightmayhelp toincreaseourknowledge.Tothatend,thisstudywasconduc tedaspartofalarger multidisciplinaryeffortindevelopingabetterunderstan dingoftheeffectsofexibilityin thewingsofappingsystems.Specically,thisstudywillco ncentrateonexperimental investigationsoftheoweldaroundwingswithanisotropi cbendingcharacteristics underhoveringconditions.Thiswillallowforabetterunde rstandingoftheowbehavior anduid-structureinteractionsonwingsdeemed“good”bas edontheamountofthrust theycangenerate. Thelargereffortalludedtoaboveincludesacomputational effortthatisgoing oninparallel[ 1 – 3 ].Thateffortisaimedatcreatingaframeworkofcomputatio nal modelsoptimizedforstructuralandaerodynamiccondition sandusesresultsfromthe experimentallygeneratedstudieshereandbyWu[ 4 ]forvalidation.However,thework beingconductedaspartofthisdissertationservesasastan daloneexperimentaleffort tostudytheeffectsofwingexibilityontheaerodynamicso fappingwings. Theremainderofthischaptercontainsareviewofresearchp rimarilyrelatedto appingwingightatascalerelevanttothatbeingstudied, andestablishesthegoalsof thenaloutcomesofthiswork.ItisfollowedbyChapter2whi chdescribestheparticle imagevelocimetry(PIV)methodthatwasusedtoobtainuidow elddata.Chapter3 describestheanalyticalmethodsthatwereusedtoinvestig ateexperimentaldata. Chapter4providesdetailsoftheexperimentalequipmentus edforthesestudies,which 15

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includesthePIVsystemsaswellasthemechanicalappingdev icesandthewings. Chapter5describestheexperimentsthathavebeenconducte dandpresentsresults fromthoseexperiments.Finally,Chapter 6 summarizestheworkthathasbeendone. 1.1FlightinNature Naturecontainsierswithawiderangeofsizesandabilitie swhichdemonstrate someofwhatispossibleintermsofappingight.Thisstudy willstartbyreviewing previousresearchonnaturaliersandrelatedtoight.Espe ciallyinterestingfor thescopeofthisstudyarenaturalierswithamassontheord erofafewgrams,a maximumdimensionontheorderofafewinches,andthathavet heabilitytohover. Thiswilllaythegroundworkforthefeaturesoftheuidowa roundwingsthatwillbe investigatedthroughoutthisstudy. Thereisalargebodyofpublishedresearchavailableinvolv ingavianandinsect ightforanimalsofthisscale.Mostoftheearlystudieswer econductedbybiologistsbut morerecentlyuiddynamicistshavestartedgettinginvolv ed.Manyaspectshavebeen investigatedbothnumericallyandexperimentallytounder standvariousmechanismsof theowandtheaerodynamicforcesgenerated[ 5 ].Inparticular,hummingbirdandlarge insectightareofinterest,asthesizesofthesegroupsofs peciessuitsthescopeofthis study.TheyalsosharesomeimportantlowReynoldsnumberfe aturessuchasdynamic stallandleadingedgevortices[ 6 ],whichwillbediscussedbelow. 1.1.1HummingbirdFlight Weis-Fogh[ 7 ]studiedhoveringightofhummingbirds( Amazihambriatauviatilis )assumingquasi-steadyaerodynamics(seeSection 1.3.2 )tounderstandthe forcesgeneratedbythewings.Hummingbirdshoverwiththei rwingsfullyextendedin contrasttootherspeciesofsmallbirds.Thismakesthemuni que.Thestrokeplaneofa hummingbirdinhoveristiltedforwardontheorderof10 fromahorizontalstrokeplane. Themotionofthewingtipdescribesagure-eightmotionwit hthewingtipmoving downwardstowardstheendofeachwingstroke.Byassumingsim pliedsteady-state 16

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aerodynamicsandusingknowndataforaspecies, Weis-Fogh couldprescribea minimumsteady-stateliftcoefcient.Heassumedthatifth eminimumsteady-statelift coefcientdidnotexceedthemaximumliftcoefcientobtai nableattheinvestigated Reynoldsnumber,steady-stateaerodynamicsissufcientf orexplainingthelift.Using thismethod,heconcludedthatunsteadyaerodynamicsdoesn otseemtosignicantly affecttheliftofthehoveringhummingbird.However,itsho uldbenotedthataminimum steady-stateliftcoefcientsmallerthanthemaximumobta inableliftcoefcientisonlya necessarybutnotsufcientconditionforprovingthatunst eadyeffectsarenotimportant. Weis-Foghalsofoundnoevidencethatkineticenergyiselas ticallystoredattheendof thewingstrokeashasbeenpostulatedbyothers. IncontrasttoWeis-Fogh[ 7 ],severalstudieshaveshownevidencethathummingbirds doindeedstoreelasticenergywhiledeceleratingthewing. Wells[ 8 ]investigated hummingbirdightbysimultaneouslycollectingmetabolic andkinematicdata.The metabolicdatawascollectedbymeasuringoxygenconsumpti on.Anaerodynamic analysiswasalsodoneasdescribedbyEllington[ 9 ].Thenecessarypoweroutput forhoveringbroad-tailed( Selasphorusplatycercus )andrufous( Selasphorusrufus )hummingbirdswasestimated.Bycomparingthemusclepowero utputderived frommetabolicdatawiththepowerrequirements, Wells coulddrawtheconclusion hummingbirdmostlikelydostoreelasticenergyattheendof thehalf-stroke. Wells[ 10 ]continuedbystudyingtheeffectsofhoveringoverawideve rsusanarrow owertoseehowblockageofthedownwashaffectsthewaythat hummingbirdsap.A widerowerwasshowntosignicantlyincreasethemetaboli cpowerinputrequiredfor hoverbymoreeffectivelyblockingtheowunderthewings.T hestrokeangleandwing beatfrequencyincreasedandtheamplitudedecreasedwithi ncreasingowersize.He alsoinvestigatedtheeffectofaddingmasstohummingbirds .Theincreasedloadwas handledbyanincreasedwingbeatamplitudeandadecreaseds trokeplaneangle,but thewingbeatfrequencyremainedconstant.Thisdemonstrat edthatthehummingbird 17

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preferschangingappingamplituderatherthanappingfre quencytoincreasethrustin freeight. Tobalskeetal.[ 11 ]usedhighspeedcamerastomeasurethree-dimensional kinematicsofrufoushummingbirdsyingatvelocitiesrang ingfrom0to12m/sinawind tunnel.Theyshowedthatwhentheightvelocityincreased, thebodyangledecreased andthestrokeplaneanglefromthehorizontalincreased.Fu rthermore,thewingchord angleasafunctionofappingphasechangedsignicantlywh enalteringightvelocity. Wingbeatamplitudealsochangedwithightvelocityanddemo nstrateditslowest valuesatmid-rangeightvelocities.Anothereffectonthew ingkinematicsthatwas determinedfromthisstudywasthatthetimespentonthedown strokeportionofthe appingcyclewasincreasedwithvelocity.Flappingfreque ncy,however,remained roughlyconstantforallvelocities. Altshuleretal.[ 12 ]measuredforcesonmodelandrealfemaleruby-throated hummingbird( Archilochuscolubris )wingswhilerotatedbyaspinningdevice.They foundthatliftcoefcientsnormallyincreasedaswingmode lsbecamemorerealisticwith sharpleadingedgesandcamber.Realwingsproducedsignic antlyhigherlift-to-drag ratiosthanmodelwings,indicatingthatfeathersmaybeess entialforimprovingwing performance. Throughthesestudiesofactualhummingbirds,onecandevel opanideaofthe featuresofthewingkinematicsnecessaryforhoveringigh tofnaturaliers.They demonstratethatightcanbeperformedefcientenoughtob efeasibleinnatureusing mainlypassivedeformationsofthewingsurfaceandthatthe actuationkinematicsare farmoreelaboratethanwillbeinvestigatedinthisstudy.T heyalsoshowthatnatural ierstypicallyvarystrokeamplitudeandkinematicsbetwe endifferentightconditions, whileleavingappingfrequencyconstant.However,therei sstillaneedtobetter understandwhatthesemotionsdototheowsothatfeatureso fengineeredsystems withsimilarcapabilitiescanbedeveloped. 18

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1.1.2InsectFlight Weis-Fogh[ 13 ]continuedhisstudiesofnaturalierswithquasi-steadya nalysis ofinsectighttodetermineifitissufcientforexplainin gtheliftofinsects.Usingthe samemethodasforhummingbirds,heconcludedthattheliftf orcesdependmostlyon quasi-steadyaerodynamiceffects,withtheexceptionofth esmallestinsectsinthestudy. Furthermore,heestablishedthatthelift-to-dragratiode creaseswithreducedReynolds numberandthatthemajorityoftheinvestigatedinsectsdep endonelasticstorageof energy. Ellington[ 9 14 15 16 17 18 ]studiedhoveringinsectightinaseriesofpapers. Theseriesincluded,butwasnotlimitedto,avortextheoryr elatingthelifttothe generationofcirculationduringhoveringightandaninve stigationof25individuals fromseveralspeciesintermsofmorphologyandwingkinemat ics.Theinvestigated individualshadwinglengthsrangingfrom7.7mm(ahovery, Episyrphusbalteatus ) to51.8mm(ahawkmoth, Manducasexta ).Ellington[ 14 ]usedbladeelementtheory (seeSection 1.3.2 )foraquasi-steadyanalysisandcomparedthemeanliftcoef cient C L tothemaximumsteady-stateliftcoefcient C L ,max forseveralspecies.Forthecases where C L wasgreaterthan C L ,max ,whichincludedseveralinsectspeciesalongwith humingbirdsandbats,thequasi-steadyanalysiswasnotsuf cienttoexplainthelift coefcient.Unsteadyeffectsmustthereforebetakenintoa ccount.Thisisincontrast withtheconclusionbyWeis-Fogh[ 13 ]. TheinsectsstudiedbyEllington[ 16 ]alltiltedtheirstrokeplanebeforeaccelerating ordecelerating,bothinlongitudinalandlateraldirectio ns,similartohowahelicopter maneuvers.Furthermore,thereisastrongrelationshipbet weenstrokeplaneangleand ightvelocity.Thestrokeplanerotatessignicantlymore thantheresultantforcewhich isduetotheasymmetrybetweentheupstrokeandthedownstro kethatiscausedbya forwardightvelocity. 19

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HighspeedPIVat2100fpswasusedbyMountcastleandDaniel[ 19 ]toinvestigate theeffectofwingcomplianceofthehawkmoth( Manducasexta ).Sincethewing complianceofthehawkmothwingdecreaseswithtimeafterit isextractedfromthe insect,theeffectofcompliancecanbestudiedonasinglewi ng.Theirresultsshowed thatfreshlyextractedwings,i.e.morecompliantwings,cr eatemoreadvantageousow magnitudesandorientationsthanthesamewingswhentheyar estiffer.Thisreinforces theideathatwingexibilityisakeyaspectofinsectight. Moreadvantageousow orientationsmeanthatahigherratioofthemomentumaddedt otheowisorientiedin thedirectionofthrust.Asaresult,alowerratioofthemomen tumisaddedindirections normaltothedirectionofthrust,sinceaddingthatmomentu mcostsenergybutthe momentumcontributionsnormaltothedirectionoftheavera geforcecancelsoutovera appingcycle. Thesestudiesshowthatquasi-steadyanalysismaysometime sbeusefulfornding roughestimatesofliftanddragforsmallscaleappingigh t,butmorerenedmethods areneededforaccuratepredictions.Furthermore,insects areofinterestastheyrelyto alargeextentonpassivedeformations.Hence,moredetaile dmeasurementsoftheow eldaroundappingwingsanditsdependenceonwingdeforma tionsarenecessary. 1.2EngineeredSystems Section 1.1 describedstudiesofnaturaliersandspecicallytriedto highlight featuresofwingkinematicsandforcegeneration.Thissect ionwilltrytohighlight selectedresultsfromrecentstudiesofengineeredapping devices.Specically,it willhighlightresultsofadvancednumericalsimulationsa ndexperimentalstudiesof congurationswhichcloselyresemblefeaturesdescribeda boveorthatareotherwise deemedrelevantforthecasesincludedinthisstudy. Aerodynamicsandvorticalowstructuresinsimulationsof appinghoveringwings havebeenreportedinmanyrecentnumericalstudies.Tanget al.[ 20 ]usednumerical simulationstostudyatwo-dimensionalhoveringappingmo tionatReynoldsnumbers 20

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upto1700.Theyidentieddelayedstall(seeSection 1.3.5 )asthemostimportant unsteadyeffectforgeneratinglift. Ansarietal.[ 21 22 ]investigatedtheinuenceofwingkinematicsandgeometry by developinganon-linearunsteadyaerodynamicmodelforhov eringinsect-likeapping wings.Oneoftheirndingswasthataslightlyadvancedwing rotationincreasedliftand drag.Advancedrotationiswhenthetemporalcenterofthewin grotationoccursbefore thecenterofthewingreversalattheendofthewingstroke.T hewinggeometrystudy showedadecreasedlift-to-torqueratiobothwithincrease daspectratioandincreased winglength.Ofthe12geometriescompared,theyfoundthata semi-ellipticalwing withanaspectratioontheorderof8wasoneofthebestperfor ming,bothintermsof lift-to-dragandlift-to-torqueratio.Itisinterestingt onotethatofalloftheirwings,that wing'splanformisgeometricallythemostsimilartothewin gplanformbeingusedinthe currentstudy. Vanellaetal.[ 23 ]usednumericalsimulationstostudythesimpliedcaseofa two-dimensionaltwo-linkmodeltoinvestigatetheinuenc eofwingexibilityonliftfor ahoveringwing.Theyachievedthebestperformancewhenthe wingwasexcitedbya non-linearresonanceat1/3ofthenaturalfrequency.AtReyn oldsnumbers75,250,and 1,000basedonchordlengthandthemaximumtranslationalve locityatthelinkjunction, theirresultsshowedanincreaseinlift-to-dragratioof21 -28%comparedtoarigidwing actuatedwiththesamekinematics.Theseresultsaresigni cantastheydemonstrate howpassivewingexibilitycanbeusedtoincreaselift. Wuetal.[ 24 25 ]measuredkinematicsandstructuraldeformationofsevera l miniatureappingwingswithvariousdesignsusingadigita limagecorrelation(DIC) system.Someofthesewingsareinvestigatedinthisstudytoo .Measurements weretakeninbothstillairandvacuumtocomparedeformatio nswithandwithout aerodynamiceffects.Theresultsshowedthatthewingdefor mationfortheinvestigated 21

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wingsincreasedandwassignicantlyalteredduetothepres enceofair.Theresults highlighttheneedforcouplinguidandstructuraldynamic s. 1.3LowReynoldsNumberFlightAerodynamics Thissectionwillintroduceaerodynamicterms,theories,a ndphenomenarelated tothescopeofthisstudy.Thisdiscussionisnotexhaustive butrepresentsanattempt tocoversomeoftherelevanttopicswhichwillhelptolaythe groundworkforthework beingconductedhere.1.3.1StrokePlaneAngle Hummingbirds,aswellasmostinsects,hoverwithastrokepl anethatiscloseto horizontal[ 11 13 ]asillustratedinFigure 1-1A .Bats,aswellassomebirdsandinsects, hoverwithaninclinedstrokeplane.Somespeciesevenhoverw ithaverticalstroke plane[ 14 ]asshowninFigure 1-1B .Onlythe(approximately)horizontalstrokeplanewill beconsideredinthisstudy.1.3.2Quasi-SteadyAnalysis Quasi-steadyanalysisofuidowisbasedontheassumption thatliftanddragina dynamicowcanbeapproximatedusingliftanddragfromstat icowconditionsforeach instantaneousconguration. Bladeelementtheory Bladeelementtheoryasdiscussedin[ 14 ]wasattributedtothepioneeringworkof Drzewiecki[ 26 ].Itwasoriginallydevelopedforaerodynamictreatmentof propellersby assumingaquasi-steadyanalysis.However,generalequati onsthatmakethistheory applicabletoappingighthavebeenderivedbyOsborne[ 27 ].Thetheorydividesthe wingintomultiplespanwiseelements,whereeachelementsp ansbetween r and r +d r r isthespanwisecoordinate,andd r isthewidthoftheelement.Thelift L 0 (normalto thespanwisedirectionofthewing)andproledrag D 0 pro ,bothperunitspanofawing 22

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section,arethen[ 14 ]: L 0 = 1 2 air c j U r j 2 C L ,(1–1) D 0 pro = 1 2 air c j U r j 2 C D ,pro .(1–2) air isthedensityofthesurroundingairand c isthechordlength. U r isthesumofthe appingvelocityandtheinducedvelocityduetotheboundan dwakevorticities.The spanwisecomponentoftherelativevelocityisassumedtoha venoeffectonliftand drag. C L and C D ,pro areliftandproledragcoefcients,respectively.Quasisteady theoryassumesthatthesecoefcientsareonlydependenton Reynoldsnumberand angleofattack.Thistypeofmodelrequiresaprioriknowled geof C L and C D ,pro ,which canbemeasuredthroughindependentexperiments.Thistype ofmodelcannotpredict inducedvelocities.Themodelcanbeexpectedtoperformbet teratfastforwardight, wheretheinducedvorticityismorerapidlyconvecteddowns tream,thusaffectingthe ownearthewinglessthanatlowightvelocityorhovering ight.Theinducedvelocity w 0 canbeestimatedusingtheRankine-Froudeaxialmomentumth eoryofpropellers. WhitneyandWood[ 28 ]coupledrotationaldynamicswithabladeelementmodel topredictliftgenerationandpassiverotationsforasmall roboticinsect-likewing.The resultsfromthreeexperimentswerecomparedtopredictedw ingrotationandlift.Their resultsshowexcellentagreementbetweenmodelandexperim ent.However,notall experimentsshowsuchgoodagreement.Quasi-steadyanalys isfailstoaccountfor delayedstallandthesheddingofvorticityatwingreversal .HubelandTropea[ 29 ] performedsimultaneousforceandPIVmeasurementsinthewak eofmodeledgoose ightatReynoldsnumbersintherange28,000-141,000andre ducedfrequencies between0.04and0.20.Undercertainassumptions,theycoul dsubtractadded massandinertialcomponentsfromthetotalforce,whichall owedfortheextractionof aerodynamicforce.Theirresultsshowedsignicantlyhigh erliftcoefcientsforapping ightcomparedtothemaximumxedwingliftcoefcients,in dicatingtheoccurrence 23

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ofdelayedstall.Thismeansthat,fortheirtestcases,theq uasi-steadyassumptionis inappropriate.Theassumptionsarelikelytobeevenworsea tlowerReynoldsnumbers. Anothershortfallofquasi-steadyanalysisisthatitmaysig nicantlyunderestimate dragandthereforepowerrequirements.Tosomeextent,thep redictionscanbe improvedbyextendingthemodel.SaneandDickinson[ 30 ]measuredstroke-averaged lift,drag,andprolepowerforadynamicallyscaledappin gfruitywingasafunction ofstrokeamplitudeandangleofattackintheranges60-180 and0-90 ,respectively. Theparametermappingpointswereseparatedby10 alongbothparameters.The valueswerecomparedtoquasi-steadyestimatestoevaluate theinuenceofunsteady effects.Thehighestlift-to-dragratiowasmeasuredat180 strokeamplitudeand30 angleofattack,whereasthequasi-steadyestimatewasinde pendentofstrokeamplitude andpeakedat20 angleofattack.Bothmeasuredandestimatedvaluesindicate d thatpeaklift-to-dragratiooccurswhenthewingipdurati onisshort,withthehighest valuesatthelowestmeasuredipdurationof0.1appingcyc ledurations.Aslightly advancedrotation,ontheorderof0.05appingcycledurati ons,alsoappearedtobe advantageous.Theirestimatesalsoshowthatthelift-to-p owerratioishighestatthe lowestinvestigatedstrokeamplitudes,intherange60-80 ,atanglesofattackinthe range40-60 .Itwasalsofoundthatalthoughquasi-steadyanalysisprov idesreasonable estimatesforlift,itseverelyunderestimatesthemeandra g.Furthermore,Saneand Dickinson[ 31 ]measuredaerodynamicforcesonatranslatingandrotating scaledinsect modelandcomparedthevalueswithquasi-steadybladeeleme nttheoryestimates. Themeasureddatawasusedtoextendthequasi-steadymodelt oincluderotational, translational,andaddedmassinertia.Theresultingmodel accuratelypredictsliftand drag,exceptduringsometimeafterwingrotationandatthee ndofeachhalf-stroke.The resultingdifferencesbetweentheimprovedmodelandmeasu redvaluesareattributedto wakecapture. 24

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1.3.3TheWagnerEffect Wagner[ 32 ]analyzedthegrowthofcirculationaroundawingsectionwh enthe quasi-steadyconditionssuddenlychange[ 17 ].Wagner'stheoryshowshowthesection hastotravelmanychordlengthsbeforethemagnitudesoflif tandcirculationcomeclose toquasi-steadyestimates. Forappingight,theWagnereffectmaymeanthattheowaro undaappingwing neverreachesquasi-steadyconditions,whichwillhaveane gativeimpactonlift.The Wagnereffectcanperhapsbereducedifthewingmotionatthe endofthewingstroke cancauseenoughcirculationtobuildupatthebeginningofe achwingstrokethrough rotation,wingdeformation,oraninteractionbetweenthet wowings. 1.3.4LeadingEdgeVortex Awingmeetingaowatanangleofincidencethatistoolargef ortheowtostay attachedwillcausetheowtoseparateattheleadingedge.Wh enawinginitiallymeets suchconditions,vorticitywillbeshedonthesuctionsideo fthewing,creatingalow pressureleadingedgevortexthatcontributestolift.Ifth evortexbecomestoolarge,it becomesunstableandvorticityisshedtothewakeofthewing .Thedynamicsofthis leadingedgevortexarekeytotheunderstandingoflowReyno ldsnumberunsteady aerodynamiceffects,hencetheyhavebeenthesourceofagre atwealthofstudies. 1.3.5Delayedstall Anairfoilwhoseconditionsaresuddenlychanged,suchasint heWagner experiment,maytravelseveralchordlengthswithoutstall ingatanglesofattack higherthanthecriticalangleatwhichthewingstallsatste adymotion.Thiseffectis calleddelayedstall[ 17 ].Furthermore,beforestallactuallyoccurs,signicantl ylarger circulationscanbedevelopedthanatsteadystall.Thisisr elatedtothegrowthofthe leadingedgevortex. Delayedstallcanincreasetheliftsignicantlyandimprov eaerodynamicperformance ofappingiersbutcontributestomakingpredictionsmore difcultassimplequasi-steady 25

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modelscannotbetrustedtogiveaccuratepredictions.Nume ricaldatabyWuandSun [ 33 ]onamodelfruitywinginaReynoldsnumberrangefrom13to1 ,500showed thatwhentheReynoldsnumberwasaboveapproximately100,l iftwasmainlycreated byaleadingedgevortexanddelayedstall.Theyusedasinuso idalappingmotion, andwhencomparingwithresultsfromtrapezoidalappingmo tiontheyfoundthatthe averageliftcoefcientwaslargelyunchanged.Thisdoesno tnecessarilyapplytothe experimentsthatareconductedinthisstudy.Thedifferenc eisthatthewingangleof incidenceintheircaseiskeptconstantexceptattheendofe achhalf-stroke,whereas inthisstudyitisafunctionof,amongotherthings,theangu larwingkinematicsthrough passivewingdeformation. Wingrotationtypicallyhelpsstabilizetheleadingedgevor texbycreatingan outwardspanwiseowwhichallowssomeofthevorticityabov ethewingtoleavethe leadingedgevortexatthewingtip.Thiscontributestofurt herdelayingtheonsetof stall.Inaseriesofstudies,HongandAltman[ 34 35 ]usedahingedappingmotion tosimplifythekinematicsoftheappingcycleandstudyvor ticalfeatures.They calculatedstreamwisevorticitycreatedfromasimplied appingmotiontoexamine liftgenerationusingPIVdatafromseveralplanesalongthes panofaatplate.Their resultsdemonstratethevorticityeld'sdependenceonthe wing'saspectratio,andthat streamwise(chordwise)vorticityincloseproximitytothe wingtipcontributestothelift. Inthemorerecentofthetwostudies,theyusedbothatandsp anwisecamberedwings toinvestigatetheeffectonliftduetospanwiseowoverrec tangularwings.Spanwise owwasshowntohaveapositivecontributiontolift.Theyal soobservedthatthe spanwisecambercanchangesignicantlyclosetothepointi ntheappingcyclewhen peakliftoccurs. Experimentaloweldresultsonappingightthatisthreedimensionaland time-dependentarestillrare,butdoexistinarchivallite rature.Therstpublished three-dimensionaltime-resolvedPIVdataaroundaappingw ingmaybethatbyPoelma 26

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etal.[ 36 ].Theycapturedphase-lockedPIVdataforanimpulsivelysta rtedapping modelfruitywinginmineraloil.Theyfoundtheowarounda appingwingtobe veryreproducibleandcouldthereforereconstructthefull time-resolvedoweldfrom separatemeasurements.Bystudyinganimpulsivelystartedw ing,theycoulddividethe owintotworegimes,wheretherstoneshowsanthattheinit ialgrowthofcirculation aroundthewingisnearlylinearandthereisnospanwiseow. Atalatertime,anearly quasi-steadyowisreachedwherethegrowthofcirculation attheleadingedgeis balancedbyacirculationleavingthewingatthetipduetosp anwiseow,causingthe leadingedgevortextobestable. LuandShen[ 37 ]presentedthree-dimensionaldataforthreedifferentpha sesofa appingmodeldragonywinginwatertoinvestigatetheevol utionoftheleadingedge vortexstructure.Theirresultsshowaleadingedgevortexs ystemconsistingofone majorandthreeminorvortices.Theresultshighlightthele adingedgevortexsystemas animportantfactorforliftandthatitisanecessitytobeab letoseethreedimensionsto revealallimportantfeaturesaowstructure. Undersomeconditionsthelowpressureontheliftingsideof thewingcaninstead causeaninwardspanwiseow,whichreducesthestabilityof theleadingedgevortex. GopalakrishnanandTafti[ 38 ]showedcomputationally,forowaroundaforwardying rigidappingrectangularwingataReynoldsnumberof10,00 0andanadvanceratio of0.5,thatcreationofanegativespanwiseowclosetothew ingtippreventedvorticity frombeingremovedfromtheleadingedgevortexatthewingti p.Asaresult,theleading edgevortexbecameunstableandseparated.Thishighlights theimportanceofwing geometryfortheabilitytosustainaleadingedgevortex.In addition,severalrotation timingswereinvestigated.Undertheinvestigatedconditi ons,advancedrotationandlong durationrotationresultedinhighthrustandpropulsiveef ciency. Introductionofturbulence,oratleastsomesteadyfeature s,toowoverawing withanangleofincidencehelpstheowtostayattached.Thi scanbedoneeitherby 27

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modifyingtheincomingoworbyintroducingnon-smoothfea turestothewingsurface. MurphyandHu[ 39 ]measuredtheairowaroundthreewingprolesusingPIV.The y comparedliftanddragcoefcientsbetweenaatplate,adra gonyinspiredcorrugated airfoil,andaproledairfoilinaReynoldsnumberrange,ba sedonchord,from58,000 to125,000atvariousanglesofattack.Theirresultsshowth atwhiletheliftcoefcient oftheproleairfoilishighlydependentonReynoldsnumber ,theliftcoefcientforthe corrugatedairfoilisnearlyindependentofReynoldsnumbe ruptothehighestmeasured angleofattackof20 .ThePIVdatashowsthatthecorrugatedairfoilhelpstheow stayattachedandisthereforealsoabletoperformwellathi ghanglesofattack.These benetscomewithapenaltyofhigherdragatlowanglesofatt ack. 1.3.6VirtualMass Thevirtualmass(sometimescalledaddedmass)ofawingisth emassofthe surroundingairthathastobeaddedtothewing'smasstoacco untforitseffecton theinertiaofthewing.Itisbelievedtoplayanimportantro leinlowReynoldsnumber unsteadyaerodynamics.Thevirtualmasscanbeestimatedas themassoftheair containedinanimaginarycylinder,withthewingchordasth ediameterandthewing spanasitslength[ 15 ].Fortheplanforminthisstudy,thevirtualmassisapproxi mately 0.09gestimatedthewayjustmentioned.1.3.7WingTwistandAngleofAttack Thewingtwistaffectsthewing'slocalangleofattack.Inth emiddleofahalf-stroke, largeinsectsandhummingbirdshavebeenfoundtohaveawing twistanglethatis highatthewingrootandapproximatelylinearlydecreasing towardsthewingtipwhere thevelocityofthewingrelativetotheoncomingowishighe r.Thisallowsthelocal angleofattacktobeclosetooptimalalongthespanofthewin g,similartoapropeller. Furthermore,itiscommoninnatureforthewingtwisttobese tinthebeginningofa half-strokeandremainnearlyconstantthroughmostoftheh alf-stroke[ 40 ]. 28

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SinghandChopra[ 41 ]measuredforcesonseveralscaledupappinginsect-like wingsunderhoveringconditions.Theyfoundthatawingpitc hingaboutthe20%chord locationfromtheleadingedgecouldproduceconsiderablym orethrustthanawing pitchingaboutthe50%chordlocation.Furthermore,theyfo undthatamembrane wingwithafreetrailingedgeandreinforcedwingrootandle adingedgecouldproduce signicantthrustwithonlypassivepitching. 1.4Objectives Todatetherehavebeenonlyafeweffortswiththree-dimensi onaloweld measurementsaroundappingwingsinair.Thisstudypresen tsexperimentallyobtained three-dimensionalthree-componentphase-averagedaerod ynamicdata.Someofthe aerodynamicdataispresentedincombinationwithstructur aldeformationswhichwere extensivelystudiedinWuetal.[ 42 ].Thisdataisanalyzedandrelatedtoprevious knowledgeaboutappingight.Theaimistounderstandthep henomenabehind thrustproduction,whichwillinturnenabledesignerstoma nufactureefcientapping iers.Thisrequiresdevelopingabetterunderstandingoft heuid-structureinteraction betweenthewinganditssurroundingoweldandrelatingvo rticityaroundthewing andinthewaketoforceproduction. AlthoughtomographicPIVissteadilyimproving,thepointwhe retheentireow eldaroundaappingwingcanbecapturedinasinglesnapsho tcannotbeexpectedto comeanytimesoon.Inarecentstudy,Buchmannetal.[ 43 ]demonstratedwhatcurrent tomographicPIViscapableof.Theirstudyisveryimpressive ,buttheirresultsdoreveal someconstraintsofthetechnique,suchasthattheinvestig atedboxsizeisanorderof magnitudesmallerinonespatialdimensionthanintheother two.Inmanycasesitis undesirabletohavesuchanarrowdomain.Hence,inthefores eeablefutureonemust relyoncombiningmeasurementsfromseparateplanestoobta inthree-dimensionalow elddata,ifsuchdataistobeobtainedatall.Theexperimen talaspectofthisstudy dealswithparticularproblemsthatarisewhentryingtoobt ainsuchthree-dimensional 29

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data.Inparticular,thephaseofeachsnapshotmustberelat edtoallothersnapshotsin thesamemeasurementaswellasmeasurementsatotherlocati onsonthesamewing. Inaddition,someaerodynamicmeasurementshavebeencombi nedwithdeformation measurementsincooperationwithWuetal.[ 24 25 42 44 ].Apartfrombeingableto synchronizethedata,carefulattentionmustbepaidtothed urabilityandrepeatability ofallequipmentforthecollecteddatatobecoherent.Smallc hangesinproperties betweenmeasurementsdoesnotnecessarilyaltertheowel dmuch.However, derivativescomputedbetweendifferentphysicallocation stakenatdifferenttimesmay besignicantlyalterediftheoweldchangesjustslightl y. 1.5ExpectedContributions Thisstudydevelopsamethodologyforacquiringthree-comp onentandthree-dimensional owelddatathatcanbeusedforanyperiodicow.Thecombin edaerodynamicand structuralexperimentaldataacquiredtogetherwithWueta l.[ 44 ]representaunique datasetthatallowsfordetailedinterpretationoftheuid -structureinteractions. Thisstudyalsousestheacquireddatatolookathoveringex ibleappingightand deciphersomeoftheowphysicsbehindit.Specically,byre latingwingdeformations toowelddata,thequestionswhereonthewing,wheninthe appingcycle,andhow thrustiscreatedhavebeenpartiallyanswered.Thepresent eddatawillalsoprovidean additiontothesmallbodyofexperimentalthree-dimension alowelddataacquiredfor appingight. 30

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z y x AHorizontalstrokeplane. z y x BVerticalstrokeplane. Figure1-1.Horizontalandverticalstrokeplanes.Thehori zontalplaneisnormaltothe z axis. 31

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CHAPTER2 PARTICLEIMAGEVELOCIMETRY(PIV) Thebasisfortheworkconductedinthisstudyinvolvestheme asurementofvelocity eldsaroundbiologicallyinspiredappingwingsandthefo rcesthatthewingsgenerate. Theprimarymeansforacquiringthevelocityelddataisano n-intrusiveparticle imagevelocimetrytechnique.Inwhatfollows,thegenerali tiesofthetheorybehind thistechniquewillbediscussedandthespecicsoftheactu alsystemsusedwillbe presentedinChapter4. 2.1Background Particleimagevelocimetry(PIV)isanopticaltechniqueform easuringuid velocitiesinaeld.Thevelocitiesaremeasuredbyseeding theuidwithsmall tracerparticlesanddeterminingtheirmovement.Underthe rightconditions,the tracerparticlescanbeassumedtofollowtheuid(seeSectio ns 2.4.3 and 2.4.4 ).The investigatedeldisilluminatedbytwoshortbutintensela serpulsesandphotographed. Thepulsesmustbeshortenoughfortheparticlesnottomoves ignicantlyduringa pulse,yetenergeticenoughtomakethelightscatteredbyth etracerparticlesvisibletoa camera.Thelocalvelocityiscalculatedfromthedisplacem entoftheparticlesbetween thetwolaserpulsesandthetimeinbetweenthepulses.Adrian [ 45 ]providesagood reviewofthedevelopmentofthetechniqueandRaffeletal.[ 46 ]discussmuchofwhat shouldbeknownwhenusingPIV. ThetechniquewasrstmentionedbythisnamebyAdrian[ 47 ].Previousefforts comefromtheeldofsolidmechanicsandweretargetedattra ckingspeckledpatterns tomeasuresurfacemotion.BarkerandFourney[ 48 ]suggestedthatthetechnique couldbeusedformeasuringuidvelocities.However,Adrian showedthatforfeasible seedingdensities,particleswouldbetrackedratherthanp atterns,andinsteadused localauto-correlationtoextractuidvelocities[ 47 49 ].Theauto-correlationcouldbe foundopticallythroughmethodsdescribedbyKov asznayandArman[ 50 ]. 32

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Whenthetechniquewasrstdeveloped,photographiclmwasu sedtocapture imagesoftracerparticles.Thesamelmwouldcapturethesc atteredlightfromboth laserpulsesofthesamemeasurement,whichleadtotheuseof theauto-correlation techniqueforprocessingofimagestondparticledisplace ments.Animportant limitationincapturingbothpulsesonthesameimageisthat evenifbothexposures ofthesameparticlearefoundintheimage,itisnotpossible totellwhichcomesfrom therstandsecondlaserpulse.Thisintroducesanambiguit y,sinceevenifboththe angleandthemagnitudeofparticledisplacementsareknown ,informationaboutthe directionofthemotioncannotbeextractedfromtheimagesw ithoutadaptingspecial techniques,suchasintentionallyshiftingthesecondimag eaknowndistance[ 51 ]. 2.2DigitalParticleImageVelocimetry Withtheintroductionofdigitalcameras,thetwoscatteredl ightimagesofparticles couldbeseparated.Therefore,insteadoftheauto-correla tiontechniquealready mentioned,onecoulduseacross-correlationtechniquebet weentwoindependent images.Thiseliminatedthedirectionalambiguityproblem .Someearlydemonstrations ofdigitalparticleimagevelocimetry(DPIV)arediscussedin Utamietal.[ 52 ],Willertand Gharib[ 53 ]andWesterweel[ 54 ]. IntheearlydaysofDPIV,whentwosubsequentsnapshotsweres eparatedon differentimages,thetimedifferencebetweenthetwoimage swasdependentonthe framerateofthecamera,whichlimitedtheapplicationtolo wspeedows.ModernDPIV camerasarebuilttohavetheabilitytotaketwoseparatesna pshotsinrapidsuccession, independentoftheframerateofthecamera.ThismakesDPIVsu itablemeasuringboth lowandhighspeedows.Intheremainderofthisstudy,PIVwil lrefertodigitalparticle imagevelocimetry. 2.3DigitalCross-Correlation WillertandGharib[ 53 ]wereamongthersttouseFastFourierTransforms(FFT) forthenumericalimplementationwhencalculatingcross-c orrelationsforPIV.Usinga 33

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spatialFFTinsteadofnumericallyintegratingthecross-c orrelationinastraightforward mannerreducesthecomputationaltimeofPIValgorithmssign icantly.Adownside isthat,becauseoftheFFT'speriodicity,themaximumspati aldisplacementishalfan interrogationwindowunlesstheinterrogationwindowispa dded.Duetonoise,this limitisinpracticeusuallyevenlower.Thisproblemcanbee asedbyusingamulti-pass algorithmandshifttheinterrogationwindowbetweenthesn apshotsaccordingtothe displacementobtainedinthepreviousiteration.Theproce ssingadvantageofusing FFT'shasledtothembeingthestandardprocessingtechniqu eforcommercially availablePIVsoftware. 2.4VelocityMeasurementAccuracy Ifthetracerparticleimagediameter,i.e.theapparentsiz eonthecameraimage (seeSection 2.4.2 ),islargerthanapixel,sub-pixelaccuracycanbeachieved as describedinSection 2.4.1 .WillertandGharib[ 53 ]presentedresultsbyBrowand andPlocher[ 55 ],whichshowedthatparticledisplacementaccuracydownto aafew 1/100thsofapixelcanbeobtained,usinga32 32interrogationwindowandacamera withan8-bitdynamicrange.Thedisplacementuncertaintyi ncreaseswithdisplacement, butlessthanlinearly.Additionallyincreasingtheseeding densitytypicallyreducesthe uncertainty. Thefunctionttingaroundthepeakinvolvessomeerrorasdi scussedinWillert andGharib[ 53 ].Sinceeachparticleonlyshowsitsownlocalvelocity,ther esults maybebiasedifthereisavelocitygradient.Anothersourceo ferroristhatthePIV cross-correlationdoesnottakeparticlepathcurvaturein toaccount. Westerweel[ 54 ]investigatedthePIVtechniqueanalyticallyforlowpixelr esolutions. Theconditionsheinvestigatedareinmanywaystypicalform odernPIVmeasurements withthetracerparticleimagediameterlargerthanbutonth esameorderofmagnitude asthepixelsizeandinterrogationareasinmostofthecompu tedcasesat32 32 pixels.Heshowedthatthereisacertainbiasinvolvedincom putingvelocityvectors 34

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fromcross-correlationthroughFFTandsub-pixelestimati onofthecorrelation peak.Hecould,however,presentacorrectionforremovingt hebias,whichwas experimentallyconrmedtobeefcient.Thecenter-of-mas sestimatorwascompared totheGaussian-testimatorforsub-pixelpeakestimation .TheGaussian-testimator wasshowntoyieldbetterresults,asthecenter-of-massest imatorshowedastrong biastowardsproducingintegervaluesofdisplacement.Thi sphenomenaiscalled peak-lockingandwillbediscussedinsection 2.4.2 .TheGaussian-testimatorwas alsoshowntoperformoptimallywhentheparticleimagediam eterisapproximately thesizeofapixel.Prasadetal.[ 56 ]recommendsaparticleimagediameterthat isapproximatelytwicethepixelsize,butthatsizewasfoun dtobeoptimalusinga center-of-massestimator.TheGaussian-testimatorisus edbydefaultbytheDaVis FlowMastersoftware(seeSection 4.1.2.1 )whichisthesoftwareusedformostofthe PIVprocessinginthisstudy. ResultsbyFinchamandSpedding[ 57 ]conrmthatabiaserrorisintroducedwhen avelocitygradientispresent.Theyalsoshowthattheoptim umparticleimagediameter canbesignicantlylargerundersuchconditions.2.4.1Sub-PixelAccuracy Thescatteringoflightfromasphericaltracerparticlefro mthepointofviewof anobserverlookingattheparticlesasapartofatwo-dimens ionalimagecanbe approximatedbyaGaussiandistribution[ 49 ]: I ( x )= I 0 exp ( x x 0 ) 2 2 2 ,(2–1) where I ( x )istheintensityoftheimageatthetwo-dimensionallocati on x I 0 isthe intensityofthelightscatteredinthedirectionoftheobse rver, x 0 isthecenterofthe particle,and isparticleimageradius.Aftercross-correlatinginterrog ationwindows fromtwoPIVimages,thecorrelationpeakwillhaveaGaussian shapetoo.Hence,by 35

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ttingaGaussiancurvearoundthecorrelationpeak,sub-pi xelaccuracycanbeobtained byinterpolatingthefunction.2.4.2Peak-Locking Thephenomenonofabiastowardsthenearestintegerdatapoi ntiscalled peak-lockingandmustbeavoidedtomaintaingoodaccuracya nddynamicrangeof PIVdata.Tracerparticlesizes,seedingdensity,thechoice ofcorrelationpeakestimator, andvelocitygradientsallaffectpeak-lockingasdescribe dbyFinchamandSpedding [ 57 ]. Theparticleimagediameter,i.e.,theapparentsizeofasph ericalparticleinthe imageplaneis d e = q M 2 d 2 p + d 2 s .(2–2) Here, d s =2.44(1+ M )( f = #) (2–3) isthediffraction-limitedspotdiameter, f = #isthef-numberofthelens, M isthe magnication,and isthewavelengthofthelaserlight[ 47 ].AsdiscussedinSection 2.4 d e shouldbeapproximatelythesizeofapixeltoachieveoptima laccuracyusinga Gaussian-testimator.If d e issmallerthanapixel,almostallthelightscatteredfroma particlemayhitonlyasinglepixelinsteadofbeingspreada roundasmallneighborhood ofpixels.Thatmakesitimpossibletodeterminethelocatio nofaparticlewithsub-pixel accuracyandwillcauseseverepeak-locking.2.4.3TracerParticleRelaxationTime Itisusuallyassumedthatacloseapproximationoftheuidm otionisobtained whenmeasuringthemotionoftracerparticlesduringPIVexpe riments.Thatassumption maynotholdtrue,especiallyiftheuidandtracerparticle shavedifferentdensities.If thetracerparticledensityismuchgreaterthantheuidden sityandStokes'owcan beassumed,thevelocityofaparticle U p atatime t aftertheuidwasinstantaneously 36

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acceleratedfromzeroto U ,canbeestimatedas[ 46 ] U p ( t )= U 1 exp t s ,(2–4) wheretherelaxationtimeis s = d 2 p p 18 .(2–5) Therelaxationtimeisameasureofhowfasttheparticlesres pondtochangesinow velocityordirectionandshouldbemuchsmallerthanthetim escalesoftheow. 2.4.4TerminalVelocityofTracerParticles Anotherissuewhenworkingwithparticleswhosedensityisdi fferentfromtheuid isthevelocityinducedduetogravity.IfStokes'owcanbeas sumed,thisvelocitycan beestimatedbycalculatingtheterminalvelocityincreepi ngowconditionsforatypical particle.Theterminalvelocityisdenedas[ 58 ] U t = gd 2 p 18 p where g 9.81m/s 2 willbeusedforgravity, d istheparticlediameter, 1.82 10 5 m 2 /s willbeusedasthedynamicviscosity, p isthetracerparticledensity,and 1.205kg/m 3 willbeusedfordensityofair.Allareapproximatevaluesatn ormaltemperatureand pressure(NTP). 2.5Super-ResolutionPIV Severalsuper-resolutiontechniqueshavebeendevelopedto increasethespatial resolutionofprocessedPIV.Theseareallprocessingtechni questhatresultinahigher spatialresolutionthanwhatisobtainedbysimplycross-co rrelatinganinterrogationarea betweentwosnapshots.Thesemethodsresultinspatiallyde nservelocityeldsand allowsforamoredetailedstudyoftheow. Insteadofjustcorrelatinganinterrogationwindowforeac hvelocityvector,itis possibletousearecursivealgorithmthatstartswithalarg einterrogationwindowand 37

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thenusesthedisplacementfromtheeachpreviousinterroga tionwindowasaninitial shiftbetweenthetwosnapshotsbeforecross-correlatingt henextsmallerinterrogation window[ 59 ].Themainbenetofthismethodisreductionofin-planedro pout,i.e. thatlessoftheparticlesthatareintheinboththerstandt hesecondimageofaPIV snapshotareoutsidethenalinterrogationareas.Thiswil lbecallediterativecorrelation inthisstudy. Severalothersuper-resolutionmethodsexistwhichcanbefo undinthePIVreview byAdrian[ 45 ].Therearemethodsforadaptingtheinterrogationwindows izelocally dependingonlocalseedingdensityandfortrackingindivid ualparticles.However,only themethoddescribedabovewillbeusedinthePIVprocessingf orthiswork. 2.6StereoscopicPIV ByusingaPIVsetupwithtwocameras,whicharelookingatthesa meplanefrom differentangles,three-componentvelocitydatacanbecal culated.Figure 2-1 showsa schematicofthesetupwithtwocamerasandalasersheet.Thr ee-componentvelocity vectorsareobtainedbyrstcomputingthetwo-dimensional displacementsfromthe imagesfromthetwocamerasseparately.Then,sincethelase rsheethassomenite thickness,thetwo-componentdisplacementsfromthetwoca merascanbemappedto three-componentvelocitiesinthethinbutnitevolumecre atedbytheintersectionofthe lasersheetandthecameraviews.Thisalsocorrectsforthee rrorthatresultsfromthe combinationofperspectiveandtheout-of-planevelocityc omponent[ 60 ]. 2.7ScheimpugCondition Whenmountingthecamerasatananglenotperpendiculartothe lasersheet,one endofthelasersheetinthecameraviewwillbefurtherawayf romthecameraimaging arraythantheother.Thedifcultyisthentohavetheentire lasersheetinfocus.One waytoovercomethisistoincreasethef-numbertoincreaset hedepthofeld.With agivenmagnication,thismeansreducingtheaperture,whi chinturnincreasesthe signal-to-noiseratioanderrorsofthemeasurements. 38

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Abettersolutionistolettheobject,lens,andimageplanes allintersectata singlepointasshowninFigure 2-2 .Thisway,theentireobjectplanecanbeinfocus simultaneously,withouthavingtoincreasethedepthofel d[ 61 ].Thisiscalledthe Scheimpugcondition.PIVcamerasintendedforstereoscopic measurementstherefore usuallyhaveScheimpugmounts,whichallowsforthelenstob erotatedcomparedto thecameramount. 2.8Calibration Onlyasinglemeasureistypicallyneededforcalibratingwh entakingtwo-component PIVmeasurementsinairwithasinglecamerathatismountedpe rpendiculartothe objectplane.Thiscanbedonebysimplyphotographingarule rintheobjectplanewith thePIVcameraandfromthatimagecalculatingpixelsperunit length. Withtwocameras,whentryingtoresolvethreecomponents,ca libratingismore complicated.AcommonwaytocalibratetheviewforstereoPIV isatwolayertargetand apinholemodelasdescribedbelow.2.8.1PinholeModel Acoordinatetransformationfromthetwocameraimages,loo kingattheobject planeatanangle,canbeperformedusingthepinholemodel[ 62 ].Themodelassumes thatthecoordinatetransformationcanbeperformed asif apinholecamerawasused. Worldcoordinates,asshowninFigure 2-3A ,canbetransformedtocoordinateson thecamera'simageplaneintwosteps.Firstthecoordinates ystemisrotatedsothat the z -axiscoincideswiththeprincipalaxis,i.e.thelineperpe ndiculartotheimageplane goingthroughthecameracenter: x W 7! x R =R x W .(2–6) x W isapointintheworldcoordinatesystemand x R isthecorrespondingpointinthe rotatedcoordinatesystem.Figure 2-3 showsasimpliedtwo-dimensionalcase,but therotationinthisstepandthetranslationinthenextstep canbothbedoneinthree 39

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dimensions.Therotationmatrixcanbedenedusingthreean gles,sothat R x ( x )= 266664 1000cos x sin x 0sin x cos x 377775 ,R y ( y )= 266664 cos y 0sin y 010 sin y 0cos y 377775 ,R z ( z )= 266664 cos z sin z 0 sin z cos z 0 001 377775 .(2–7) Then,arotationmatrixcanbecombinedto R=R( x y z )=R z ( z )R y ( y )R x ( x ).(2–8) Thenextcoordinatetransformationmovestheoriginfromth eobjectplane,asshownin Figure 2-3B ,totheimaginedpinholeasshowninFigure 2-3C : x W 7! x C =R x W + T ,(2–9) where x C isincameracoordinates.Now,apointincameracoordinates canbemapped ontotheimageplanebyusingcentralprojection,sothat x C 7! x I = 0B@ fx C = z C fy C = z C 1CA ,(2–10) where x I isthetwo-dimensionalcoordinateontheimageplane.Itisi gnoredherethat theimagewouldbeippedifitwasreallygoingthroughapinh ole.Finally,sincedigital camerasareused,amappingtothecorrespondingpixelissou ght: x I 7! x pixel = 0B@ x I = d x y I = d y 1CA + x P .(2–11) where x pixel isthecoordinateinpixelsonthecameraimage. d x and d y arephysical spacingofthepixelsonthecameraCCDin x and y directionsrespectively.The principalpointiswheretheprincipalaxisgoesthroughthe imageplane. x P isthe principalpointinpixelcoordinates. 40

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2.8.2RadialDistortion Inreality,neitherapinholecameranoraninnitelythinle nsisused.Thereforethe thicknessofthelenswillintroduceadistortiontotheimag e.Thedistortionisafunction oftheradialdistancefromtheprincipalpointintheimagep lane.Tocorrectforthiserror, acorrectionfunction L ( r )canbeintroduced[ 62 ]: ˆx = x I + L ( r ) x pixel x R ,(2–12) where ˆx isthecorrectedpixelcoordinate, x R isthecenterofradialdistortion,and r = j x pixel x R j .(2–13) ThecorrectionfunctioncanbeapproximatedbyaTaylorexpa nsion L ( r )=1+ 1 r + 2 r 2 +....(2–14) Whenthiscorrectionfunctionisappliedwithappropriateco efcientstotheparticle images,theeffectofradialdistortionissignicantlyred uced. 2.9PIVAlgorithm Thenalsetofexperimentalaerodynamicdatadiscussedint hisdissertationwas processedwiththeauthor'sownPIVcode,forgreaterspeed,c ontrol,andexibility.It allowsthedatatobeprocessedinaconsiderablyshortertim eandformoreinformation tobestoredabouteachprocessedsnapshot.Additionally,th euseofanin-housecode allowsforthealgorithmtobeadjustedifneeded.Thealgori thmwillbedescribedinwhat followsnext.Astudyoftheaccuracyofthealgorithmundert heconditionsthatitwillbe usedispresentedintheAppendix.2.9.1Pre-Processing Duringthevalidationofthealgorithmthatispresentedint heAppendix,itwas discoveredthatasignicantlylargernumberofsuccessful correlationscouldtypicallybe obtainedbyrsttakingtheconvolutionofanimageandaGaus siankernel.Therefore, 41

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beforecorrelatingtheimages,theyareconvolvedwiththe3 3pixeltwo-dimensional Gaussiandistributionwithastandarddeviationofhalfapi xelthatisgivenby: K 3 3Gaussian = 0.15770.68450.1577 T 0.15770.68450.1577 (2–15) 5 5pixelGaussiankernelshasalsobeentestedwithstandardd eviationsof p 1 = 2 and1pixels.Thesekernelsdonotresultinasignicantamou ntofadditionalsuccessful correlations,butincreasesthemeanabsoluteerror,where asakernelwithastandard deviationof0.25pixelshasalmostnoeffectcomparedtonop re-processingatall.A Gaussiankernelisusedforsmoothingsincetheconvolution oftwoGaussianfunctions resultsinanotherGaussianfunction,sothattheshapeofth elightscatteredfrom particlesaftersmoothingcanstillbeassumedtobeGaussia n. 2.9.2Grid ThecalibrationdatageneratedbyLaVisionDaVissoftwarecon tainsthehorizontal andverticaldimensionsinphysicalspaceofthephysicaldo mainofthedataaswellas apixeldensityinphysicaldimensions.Thisisroughlythea veragenumberofcamera pixelspermmifprojectingthepixellocationsontheobject plane.Usingapixelsize correspondingthepixeldensity,aCartesiangridforveloc ityvectorsiscreated,with gridpointsseparatedby N = 2pixelsalongeachaxis,with N N beingthesizeofthe interrogationareathatwillbeused.Thedistancebetweent hedomainedgeandtherst gridpointisequalonbothsidesofthedomainalongthesamea xistowithin 1pixel andisatleast3 N = 2pixels. 2.9.3Correlations Whenagridpointisprocessed,halfofaninitialdisplacemen tguessissubtracted andaddedtothegridpointlocation G kl =( G ( x ) k G ( y ) l )toobtain“before”and“after” convectionlocationsofthegridpoint.Duringtherstiter ationtheinitialdisplacement guessiszeroandduringsubsequentiterationsitistheresu ltfromthepreviousiteration. Theseconvectedlocationsofthegridpointareprojectedon tobothcameraplanes, 42

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scaledtopixelcoordinates,androundedto( g ( cf ) x g ( cf ) y ).Thecoordinatesareroundedto theclosestvaluessothat g ( cf ) x +0.5, g ( cf ) y +0.5 2Z .Here,cameraindex c iseither1or 2and f isrstorsecondframe,indicatedbyAandBrespectively.Ne xt,the N N pixel neighborhoodcenteredaround( g ( cf ) x g ( cf ) y )iscopiedfromeachimage I ( cf ) tothecenterof a3 N = 2 3 N = 2pixelmatrix M ( cf ) .Pixelsthatareoutsidethedomainoftheimageorthat aremarkedundenedarereplacedbyzeros.Theprojectedini tialdisplacementis: ( ( c ) x ( c ) y )=( g ( c 2) x g ( c 2) y ) ( g ( c 1) x g ( c 1) y )(2–16) Theclosertheprojectedinitialdisplacementistothedisp lacementthatwillbefound, themorepixelvaluesareusedtocomputethecorrelationand hencethemoreaccurate theresultsareexpectedtobe.Amajorspeedupofthealgorit hmwhenusingmultiple iterationsisachievedbynotcomputinganewvectorfromcor relationsagainif( ( c ) x ( c ) y ) isidenticaltothepreviousiteration. Amapofwhichpixelsaredenedin M ( cf ) iscreated,suchthat V ( cf ) i j = 8>><>>: 1,if M ( cf ) i j isdened, 0,otherwise. (2–17) Anewmatrix M ( cf ) isthencreatedbysubtractingthemeanofthealldenedpixe ls in M ( cf ) fromeachdenedpixelin M ( cf ) .Thestepistakentoremovethebackground intensity,whichisnecessarytoimprovetheresultswhenes timatingdisplacementswith sub-pixelaccuracy,aswillbediscussedinSection 2.9.4 Twocorrelations,oneforeachcamera,canthenbecomputed: C ( c ) =ifft2(conj(fft2( M ( c A) )) fft2( M ( c B) )),(2–18) wherefft2andifft2aretheforwardandinversetwo-dimensi onaldiscreteFastFourier Transformsnormalizedsothat F =ifft2(fft2( F ))foratwo-dimensionalmatrix F .conjis thecomplexconjugate,and denotestheelement-wise(Hadamard)matrixproduct. 43

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TheFouriertransformsarecomputedusingtheFFTWlibraryc ompiledfordouble precision. C ( c ) isthecorrelationbetweentheinterrogationareasofther standsecond frame.Similarly,weightscanbecomputeddependingonhowma nydenedvaluesare usedforeachelementin C ( c ) : W ( c ) =ifft2(conj(fft2( V ( c A) )) fft2( V ( c B) )).(2–19) W ( c ) isonlycomputedusingFFT'swhenthereareundenedpixels, sinceitisalways thesameforagiven N whentherearenoundenedpixels. Usingtheseweights,normalizedcorrelations C ( c ) canbedenedsuchthat C ( c ) i j = 8>><>>: C ( c ) i j = W ( c ) i j ,if W ( c ) i j > 0, 0,otherwise. (2–20) C ( c ) i j isdenedherefor i j 2 [ 3 N = 4,3 N = 4),sothatifthemaximumpeakislocatedat ( i j ),thatcorrespondstoadisplacementof( i j )pixelsbetweentherstandthesecond image.2.9.4CorrelationPeaks Apeakin C ( c ) i j isdenedasapointwhosecorrelationstrengthishighertha n thatofeachofitseightclosestneighborpoints.Itisalsor equiredtocorrespondtoa displacementthatislessthanorequaltoanabsolutepixeld istanceof N = 4fromthe initialdisplacement( ( c ) x ( c ) y )alongeachaxisandthat W ( c ) i j N 2 = 2.Allpeaksfromthe samecameraareputinalisttogetherwiththeircorrelation strengths. Fromthelistofcorrelationpeaksforeachcamera,thetwope akswiththestrongest correlationstrengthsfromeachcameraarekept.Thefourpo ssiblecombinationsare putinyetanotherlistandsortedindescendingorderbythep roductofthecorrelation strengthfrombothcameras.Iftheratiobetweenthecorrela tionstrengthproductsofthe rstandsecondpairinthelistisbelow1.2,theresultingve ctorissettoanundened 44

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statetopreventcreatingvectorsfromambiguouscorrelati ons.Otherwisethealgorithm proceedstoreningthepeakpairwiththehighestcorrelati onstrengthproduct. AGaussian-testimatorisusedtodeterminedisplacements withsub-pixel accuracy.Ifthepeakvalueof C ( c ) isfor( i j ),thenthesub-pixelestimateofthe incrementindisplacementcomparedtotheinitialguessisg ivenby[ 54 ]: ( c ) x = i + ln j +1 X j 0 = j 1 C ( c ) i 1, j 0 ln j +1 X j 0 = j 1 C ( c ) i +1, j 0 2 0@ ln j +1 X j 0 = j 1 C ( c ) i 1, j 0 +ln j +1 X j 0 = j 1 C ( c ) i +1, j 0 2ln j +1 X j 0 = j 1 C ( c ) i j 0 1A ,(2–21) ( c ) y = j + ln i +1 X i 0 = i 1 C ( c ) i 0 j 1 ln i +1 X i 0 = i 1 C ( c ) i 0 j +1 2 ln i +1 X i 0 = i 1 C ( c ) i 0 j 1 +ln i +1 X i 0 = i 1 C ( c ) i 0 j +1 2ln C ( c ) i 0 j .(2–22) Theestimatorisexactifthevaluesaroundthepeakcorrespo ndtopointvalues ofatwo-dimensionalGaussiansurfaceandanybackgroundin tensityhasbeen subtracted.Inreality,valuesaroundpeakscanmoreaccura telybedescribedas integralsoversquaresectionsofatwo-dimensionalGaussi ansurface.Thatdiscrepancy mayintroduceaslightbiaserror.Approximatebackgroundin tensityisremovedby subtractingthemeanintensityofeveryinterrogationarea .Thedifferencebetweenthe exactandapproximatedbackgroundintensitymayalsointro duceabiaserror. Oncetheincrementaldisplacementisknownwithsub-pixelp recision,itisaddedto theprojectedinitialguess,sothat: ( g ( c 1) x g ( c 1) y )=( g ( c 1) x g ( c 1) y ) 1 2 ( ( c ) x ( c ) y ),(2–23) ( g ( c 2) x g ( c 2) y )=( g ( c 2) x g ( c 2) y )+ 1 2 ( ( c ) x ( c ) y ).(2–24) Thepoints g ( cf ) arethenconvertedtophysicalunitsandprojectedbacktoth eobject planeas x ( cf ) .Fromthesetwo-dimensionalpoints,thethree-componentd isplacement 45

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iscomputedaswillbedescribedinSection 2.9.5 .Iftheerror,givenbyEquation 2–28 correspondstolessthanapixel,thepeakcombinationisdet erminedtovalidbeandis stored,otherwisevectorwillbesettotheundenedstate.2.9.5ComputingThree-ComponentDisplacementVectors Thedisplacementsinpixelsretrievedbyndingthecorrela tionpeaksatagrid pointcanbeconvertedtoin-planedisplacementsinphysica lcoordinates, ,byusing calibrationdata.Thecalibrationisalsousedtoobtaintra nslationvectors T 1 and T 2 .The translationvectorsaredenedasthevectorsfromthein-pl anecoordinatesystemorigin tothecameracentersfortherespectivecamerasaccordingt othepinholemodel. CalluaudandDavid[ 63 ]describehowtwo-componentdisplacementsfromthe camerascanbeusedtocomputeathree-componentdisplaceme nt.Thetwo-dimensional displacements ( c ) arethedisplacementsprojectedontothetwo-dimensionalp lane where z =0: ( c ) = x ( c B) x ( c A) .(2–25) Normalizedvectorsfromthecameracenterstothesepointsa re: v ( cf ) = x ( cf ) T ( c ) = T ( c ) z .(2–26) Byassumingthatthedisplacementiscenteredaround z =0,sothatthebeforeandafter pointsinthreedimensionsareatanequaldistancefromando noppositesidesofthe objectplane,thesevectorscanbeusedtoformulateanexpre ssionforthree-component displacementsasafunctionof w : ( c ) ( w )= ( c ) + w 2 v ( c B) v ( c A) .(2–27) 46

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Thedisplacementsdetectedbythetwocamerasshouldbethes ame.Therefore,an errorisdenedas: ( w )= j (1) ( w ) (2) ( w ) j = (1) (2) + w 2 v (1B) v (1A) v (2B) + v (2A) .(2–28) w canbesolvedforbyminimizingthesquarederror 2 ,whichyields: w =2 (1) (2) v (2B) v (2A) v (1B) + v (1A) v (2B) v (2A) v (1B) + v (1A) v (2B) v (2A) v (1B) + v (1A) .(2–29) Thethree-componentdisplacementisthenestimatedas: 1 2 (1) + (2) + w 2 v (1B) v (1A) + v (2B) v (2A) .(2–30) 2.9.6RejectingOutliers The N vectorsina3 3regionaroundacentervectorthatarevalidandcomputed fromcross-correlations(asopposedtovectorsthatarecre atedbyinterpolatingover gaps)arepickedout.Themedianvalueofeachvelocitycompo nentiscomputedas u median .Thenaresidual r i iscomputedforeachvalidvectoras: r i = j u i u median j .(2–31) Theunbiasedestimatorofthestandarddeviationoftheresi dualiscomputed r = P Ni =1 ( r i r ) 2 N 1 .(2–32) Anormalizedresidual r center isthencomputedforthecentervector: r center = r center r r .(2–33) After r iscomputedforeverynode,thenodewiththelargest r isfound.Ifitislarger thanalimitsetto1.5,thevectorismarkedasinvalid. r isthenupdatedfortheaffected neighborsandtheprocessisrepeateduntiltherearenooutl iersoverthespeciedlimit. 47

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Thenextstepistocompute r forallvectorsthatwereinvalidatedintheprevious step.Notethatthistimearound,thecentervectorisnotuse dwhencomputing U median r and r forthenode,sincethecentervectorhasbeeninvalidated.I fthelowest r forany invalidatedvectorislowerthan2.5,itisre-insertedand r ofanyneighboringpreviously invalidatedvectorsisupdated.Thisprocessisrepeatedun tiltherearenomorevectors whoseresidualisunderthespeciedlimit.Themethodforve ctorrejectionissimilarto theschemeproposedbyNogueiraetal.[ 64 ]andtheoneusedbyDaVis[ 65 ],butdoes notremovesmallgroupsofvectorsasthatcanbedoneasapost -processingstep. 2.9.7FillingGaps Afterthelastcross-correlationiterationandifatleaston epairofthenearestor nearestdiagonalpairofneighboringvectorslocatedsymme tricallyaroundamissing vectoraredened,thevectorisreplacedbyaninterpolatio nofitsneighbors.Ifnosuch pairexists,thevectorisleftundened. Whengapsarelledanditisnotthelastiteration,thealgori thmislessstrictand replacesthevectorwiththeaverageofallitsneighboringv ectorsthataredened, evenifonlyoneexists,whichcanyieldsomeextrapolatedve ctors.Extrapolatedvalues willonlybeusedfortheinitialdisplacementofthenextite rationandareneverkept inthenalresult.Ifanundenedvectorhasnoneighbors,it isreplacedwithazero displacementvector.2.9.8ReningGrid ItcanbeadvantageoustostartthePIVprocessingwithalarge rinterrogation areathanisnallysoughttoobtainbetterinitialdisplace ments( x y ),asdescribedin Section 2.9.3 .Thereningalgorithmusedheresimplyaddsgridpointshal fwaybetween thealreadyexistingonesandinterpolatesthedisplacemen tsthatwillbeusedforthe nextiteration.Additionalgridnodesarealsoaddedtotheed gesaslongastheyarenot closertotheborderthanwhatisspeciedin 2.9.2 .Displacementvaluesareinterpolated ontothenewgrid. 48

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2.9.9Iterations Thealgorithmstartsoffwiththreeiterationsusing64 64pixelinterrogation areas.Thegridisthenrened,afterwhich10iterationsare performedusing32 32 pixelinterrogationareastoproducethenalresults.Thes tatusofeachvector(i.e.ifit comesfromacorrelation,ifitisinterpolated,orifitisin valid)isstored,aswellasthe strengthofthecorrelationandtheratiobetweenthecorrel ationstrengthproductsoftwo strongestpeakcombinations.Additionally,thesameinform ationisstoredforthevector eldcomputedusing64 64pixelinterrogationareasbeforeitwasrened. 2.10Post-Processing Thevectoreldsnapshotsgeneratedusingthealgorithmdes cribedinSection 2.9 willbepost-processedbyrstreducingtheinuenceofbadv ectorsineachsnapshot asdescribedinSection 2.10.1 andthencomputingphase-averagesaccordingtothe methodinSection 2.10.2 2.10.1ASingleMethodForOutlierRejection,Smoothing,andI nterpolation Garcia[ 66 ]presentedasinglemethodforminimizingtheinuenceofou tliers, smoothing,andllingingapsinPIVvectorsnapshots.Itisba sedonasmoothing methodthatusesDiscreteCosineTransformsandpenalizedl eastsquarestosmooth dataandcorrectforoutliers[ 67 ].Themethodiscapableofautomaticallydetermining asmoothingparameter.Thatfunctionalityisnotusedinthi swork,sinceitvisuallywas determinedthatitwouldsmooththevelocitysomuchthatanu nnecessaryamount ofowstructuredetailwaslost.Thesmoothingparameterwa sinsteadsettoasmall value,andthemethodisusedmainlyforoutlierrejectionan dgaplling.Additionally,the algorithmhadtobeslightlymodiedtoallowforitsapplica tiontothethree-component dataacquiredinthisstudy.Themodicationconsistedofco mputingresidualsasthe squarerootofthesumofthesquareerrorfromeachcomponent ,insteadofasingle component.Whenusingtwocomponents,themethodisequivale nttoputtingthetwo 49

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componentsastherealandimaginarypartsofasinglevaluea ndcomputingresiduals astheabsolutevalueoftheerrorofacomplexnumberaswasdo nebyGarcia[ 66 ]. Afterthepost-processingalgorithmhasrunonavectoreld, vectorsarepresent atallgridpoints,nomatterhowfarfromtheclosestvalidgr idpointintheoriginalvector eld.Therefore,amapsettooneateachvalidgridpointandz eroeverywhereelseis createdbeforepost-processing.Then,everypointsettoze rothatcantintoatriangle createdbythreevalidpointsandwhoselongestsideislesst henasetthresholdisset toone.Hence,theresultingmapisusedtopickoutonlytheve ctorsthatareeitherat alocationthathasdatadened,orwheredataisinterpolate dfromneighborsthatare closeenough.2.10.2FittedPhase-Average Tocreatettedphase-averages,allsnapshotsthatcomefro mthesame1 = 50 th of aappingcyclearecollected.If i 2 [0,1)isthephaseofeachsnapshotand c isthe phaseforwhichtheaveragedvelocityvectoreldissought, thenaphase i centered around c canbedenedas: i =[( i c +0.5)(mod1)] 0.5.(2–34) Then,ateachgridpoint,arstorderpolynomialisttedusi ngleastsquarestothevalid velocityvectors,sothat: u ( )= u 0 + u .(2–35) Residualsfromthetarecomputed: r i = u ( i ) u i .(2–36) 50

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Thestandarddeviationofeachcomponentof r iscomputedas ~ r .Normalizedresiduals canthenbecomputedas: r i =max r ij r j .(2–37) Ifthelargestnormalizedresidualatagridpointislargert hanaspeciedlimit,the correspondingvectorisdiscardedandtheprocessisrepeat ed.Iftherearelessthan20 validvectors,lessthan5validvectorswith i < 0,orlessthan5validvectorswith i > 0 afterrejectingdeviantvectors,thettedaveragevalueis settoundened.Ifthereare enoughvalidvectorsthettedaverageissetto u 0 Figure2-1.StereoPIVschematic. imageplanelensplaneobjectplane Figure2-2.SchematicoftheScheimpugcondition. 51

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pinhole objectplane z x AWorldcoordinatesystem. T z x BRotatedsystem. f imageplane z x CCameracoordinatesystem. Figure2-3.Coordinatetransformationstepsforthepinhol emodel. 52

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CHAPTER3 DATAANALYSISMETHODS Thischaptercontainsthemathematicalbasisthatwillbeus edforanalyzing data.ThederivationsanddescriptionswilluseaCartesian coordinatesystem,with x =( x y z ) T astheaxiscoordinates. u =( u v w ) T willbeusedforowvelocitiesinthe correspondingthreedimensions. 3.1EstimationsofDifferentialQuantities AfterprocessingPIVimagesintovelocityvectors,theresult ingdataconsistsof thevelocityvectorsonatwo-dimensionalgrid.Toanalyzet hese,itisusefultoobtain differentialquantities.Thissectiondescribeshowvorti cityandstraintensorswillbe computed.3.1.1Vorticity Theaveragevorticityinaregion S onaplanenormaltothe z axisis x S z d x d y = S = I @ S u d l ,(3–1) where z isthe z componentofvorticity, S isthecirculationaround S ,and u isthelocal velocityvector[ 46 ]. @ S istheboundaryof S and d l isthedifferentialalong @ S inthe counterclockwisedirectionaround S Withdataonanorthogonaltwo-dimensionalgrid,Equation 3–1 canbeusedto relatethecirculationaroundacell: i +0.5, j +0.5 1 2 ( u i j + u i +1, j )( x i +1 x i )+( v i +1, j + v i +1, j +1 )( y j +1 y j ) ( u i +1, j +1 + u i j +1 )( x i +1 x i ) ( v i j +1 + v i j )( y j +1 y j ) .(3–2) Here, i and j arethenodeindicesalongthe x and y axesrespectively.Theindex i +0.5 denotesthe x location x i +0.5 =( x i x i +1 ) = 2. j +0.5similarlymeansthelocationhalf waybetween y i and y i +1 alongthe y axis. x i j =( x i y j ) T isthecoordinateofnode( i j ), 53

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and u i j =( u i j v i j ) T isthetwo-dimensionalowvelocityatthatnode. i +0.5, j +0.5 isthe circulationaroundthecellwithitscenterat x i +0.5, j +0.5 Theaveragevorticityinsidethecellisapproximated ( z ) i +0.5, j +0.5 = i +0.5, j +0.5 A 1 2 u i j + u i +1, j y j +1 y j + v i +1, j + v i +1, j +1 x i +1 x i u i +1, j +1 + u i j +1 y j +1 y j v i j +1 + v i j x i +1 x i # ,(3–3) where A istheareaofthecell. Usingbilinearinterpolation,thevorticityonanodecanbe estimatedas ( z ) i j x i +1 x i x i +1 x i 1 y j +1 y j y j +1 y j 1 ( z ) i 0.5, j 0.5 + x i x i 1 x i +1 x i 1 y j +1 y j y j +1 y j 1 ( z ) i +0.5, j 0.5 + x i +1 x i x i +1 x i 1 y j y j 1 y j +1 y j 1 ( z ) i 0.5, j +0.5 + x i x i 1 x i +1 x i 1 y j y j 1 y j +1 y j 1 ( z ) i +0.5, j +0.5 .(3–4) Equation 3–4 isusedtocomputevorticitythroughoutthisstudy. Often,thenodesonagridareequidistantalongeachaxis,i. e. x i x i 1 = x 8 i ,(3–5) y j y j 1 = y 8 j .(3–6) Infact,thisisthecaseforallexperimentaldatainthisstu dyinatleasttwoofthespatial dimensions.Whenthisisthecase,Equations 3–3 and 3–4 combinedreducesto ( z ) i j 1 8 y u i 1, j 1 +2 u i j 1 + u i +1, j 1 u i +1, j +1 2 u i j +1 u i 1, j +1 + 1 8 x v i +1, j 1 +2 v i +1, j + v i +1, j +1 v i 1, j +1 2 v i 1, j v i 1, j 1 .(3–7) Equation 3–7 isoneoftheestimationsforvorticitysuggestedbyRaffele tal.[ 46 ].The advantageofthisestimationisthatituses12valuestoesti matethevorticityofapoint, insteadof4ifusingacentraldifferencescheme,yetitdoes n'trequirevaluesfaraway fromthepointofinteresttobeknownashigherorderone-dim ensionalapproximations do. 54

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3.1.2Strain Theshearstraininthe x y planeofauidisgivenby S xy = 1 2 @ u @ y + @ v @ x .(3–8) Inthecenterofacell,itcanbeestimatedas S xy i +0.5, j +0.5 1 4 u i j +1 u i j y j +1 y j + u i +1, j +1 u i +1, j y j +1 y j + v i +1, j v i j x i +1 x i + v i +1, j +1 v i j +1 x i +1 x i # .(3–9) Thenormalstrainis S zz = @ w @ z .(3–10) Assumingthattheowisincompressible,andbyusingthecont inuityequation,itcanbe written S zz = @ u @ x + @ v @ y .(3–11) Itcanbeapproximated,atthecellcenter,as ( S zz ) i +0.5, j +0.5 1 2 u i +1, j u i j x i +1 x i + u i +1, j +1 u i j +1 x i +1 x i + v i j +1 v i j y j +1 y j + v i +1, j +1 v i +1, j y j +1 y j # .(3–12) Bilinearinterpolationisusedtoapproximatecellcentered values S xy i j x i +1 x i x i +1 x i 1 y j +1 y j y j +1 y j 1 S xy i 0.5, j 0.5 + x i x i 1 x i +1 x i 1 y j +1 y j y j +1 y j 1 S xy i +0.5, j 0.5 + x i +1 x i x i +1 x i 1 y j y j 1 y j +1 y j 1 S xy i 0.5, j +0.5 + x i x i 1 x i +1 x i 1 y j y j 1 y j +1 y j 1 S xy i +0.5, j +0.5 (3–13) andsimilarlyfor S zz .Theseapproximationsforstrainwillbeusedthroughoutth isstudy. 55

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Ifthenodesareequidistantalongeachaxis,theresultagai nreducestoapproximations suggestedbyRaffeletal.[ 46 ]: S xy i j 1 16 y u i 1, j +1 +2 u i j +1 + u i +1, j +1 u i 1, j 1 2 u i j 1 u i +1, j 1 + 1 16 x v i +1, j 1 +2 v i +1, j + v i +1, j +1 v i 1, j 1 2 v i 1, j v i 1, j +1 ,(3–14) ( S zz ) i j 1 8 x u i +1, j 1 +2 u i +1, j + u i +1, j +1 u i 1, j 1 2 u i 1, j u i 1, j +1 1 8 y v i 1, j +1 +2 v i j +1 + v i +1, j +1 v i 1, j 1 2 v i j 1 v i +1, j 1 .(3–15) Throughoutthisworktheuidstrainwillbeusedinordertoc ompute Q accordingtothe Qcriterion,whichwillbedescribedinSection 3.2.1 3.2VortexIdentication Thereisnoexact,generallyaccepteddenitionofavortexi nuidmechanics research.However,therehavebeenmanytechniquesdevelop edtoidentifythe dominantvorticalfeaturesintheow.The Q criteriondevelopedbyHuntetal.[ 68 ] willbeusedtoidentifyvortexcoresinthiswork.Severaloth ermethodsforidentifying vortexcoresexistsuchasthe criterion[ 69 ],the 2 criterion[ 70 ],andswirlingstrength ci [ 71 ].Chakrabortyetal.[ 72 ]comparedallfourofthesemethodsandfoundthat theyusuallyresultinsimilarvorticalstructuresifappro priatethresholdsofthedifferent quantitiesarechosenforwhatisconsideredtobeavortexco reregion. 3.2.1QCriterion OnemethodforidentifyingvortexcoresistheQcriterion,w hichwasdeveloped byHuntetal.[ 68 ].TheworkbeingpresentedherewillusetheQcriteriontovi sualize vorticesinthemeasuredoweldinordertounderstandowg eneratedbywings. Q is denedhereasthesecondinvariantof r u ,whichcanbewrittenusingEinsteintensor notationas Q = 1 2 @ u i @ x i @ u j @ x j @ u i @ x j @ u j @ x i ,(3–16) 56

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where u i istheuidvelocitytensor,and x i isthecoordinatetensorinCartesian coordinates. Inincompressibleow,thersttermofEquation 3–16 vanishesduetocontinuity, andthesecondcanbewritten, Q = 1 2 n ij n ji S ij S ij .(3–17) Here, S ij and n ij arethesymmetricandanti-symmetriccomponentsof @ u i @ x j : S ij = 1 2 @ u i @ x j + @ u j @ x i ,(3–18) n ij = 1 2 @ u i @ x j @ u j @ x i = 1 2 ijk k .(3–19) ijk istheLevi-Civitasymbol.Byusingthelatterformof n ij ,bothquantitiescanbe computedusingthemethodsdescribedpreviouslyinSection 3.1 TheQcriterionidentiesvortexcoresasregionswith Q > 0,whichareregions withalargerrotationratethanstrainrate.Intheresultss ections,plotsofmax( Q ,0) willbeusedtovisualizetwo-dimensionalcross-sectionso fvorticalowfeatures,and iso-surfacesof Q willbeusedtovisualizevorticalowfeaturesinthreedime nsions. 3.3LineIntegralConvolution Lineintegralconvolution(LIC)wasproposedbyCabralandL eedom[ 73 ]as amethodtovisualizevectorelds.Themethodcanreplaceth euseofordinary streamlines,withtheadvantagethatthedensityofthestre amlinescreatedwiththis methodismoreorlessthesameeverywhere.Hence,itprovide sasimplewayof visuallyidentifyingowstructuresthatmightbemissedif usingstreamlinesoriginating fromaxedsetofpoints. Themethodworksbymappingthevectoreldontopofanimage R ( x )withrandom intensitypixels.Theoutputimageiscalculatedfromthera ndomimageandthevector eld.Apixelvalueontheoutputimageatalocation x 0 isevaluatedbyrstcalculating 57

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thestreamlinegoingthroughthepixel,denedby ,where d ~ ( s ) d s = u ( ~ ( s )) j u ( ~ ( s )) j ( s 0 )= x 0 ,(3–20) s isthelocationonthestreamline,and u isthelocalowvelocityvector.Thenthepixel valueisgivenbythelineintegralconvolutionalongthestr eamline,adistance L both forwardandbackwardfromthestartingpoint: I ( x 0 )= Z s 0 + L s 0 L k ( s s 0 ) R ( ~ ( s ))d s ,(3–21) where k isalterkernelnormalizedtounity.Thealgorithmcausesp ixelsclosetoeach otheronthesamestreamlinetobehighlycorrelated,whilep ixelsthesamedistance apartbutperpendiculartothestreamlinearemuchlessorno tatallcorrelated. 3.3.1FastImplementation AfastalgorithmforcalculatingapproximateLICisdescrib edbyStallingandHege [ 74 ].Thealgorithmstartsbyfollowingthestreamlinefromapi xel,butamuchlonger distancethan L .ThestreamlinesareintegratedusingthefourthorderRung e-Kutta method.Velocitydataisobtainedthroughbilinearinterpo lation.Inbetweenthe streamlinepointsobtainedusingRunge-Kuttasteps,Hermi te-interpolationisused, i.e. p ( u )= a u 3 + b u 2 + c u + d ,(3–22) where u = s s n s n +1 s n (3–23) a =2 p (0) 2 p (1)+ p `(0)+ p `(1),(3–24) b = 3 p (0)+3 p (1) 2 p `(0) p `(1),(3–25) c = p `(0), (3–26) d = p (0). (3–27) 58

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Valuesfromtherandomimagearesampledalongthestreamlin eonevenlyspaced locations R ( ~ ( s n )),where s n = n s ,and n 2Z .Sincethestepsizeisxed,the Hermite-interpolationcanbefurtheroptimizedbydening 1 p ( u )= p ( u + ) p ( u )=3 a u 2 +(3 a 2 +2 b ) u + a 3 + b 2 + c ,(3–28) 2 p ( u )= 1 p ( u + ) 1 p ( u )=6 a 2 u +6 a 3 +2 b 2 ,(3–29) 3 p ( u )= 2 p ( u + ) 2 p ( u )=6 a 3 ,(3–30) andsteppingforwardwithconstantincrements = s = ( s n +1 s n )usingtherelationships p ( u k +1 )= p ( u k )+ 1 p ( u k ),(3–31) 1 p ( u k +1 )= 1 p ( u k )+ 2 p ( u k ),(3–32) 2 p ( u k +1 )= 2 p ( u k )+ 3 p ( u k ).(3–33) Byusingaboxlterfor k ,theresultingimageintensitiesalongthestreamlinecanb e calculated: I ( ~ ( s 0 + m s ))= 1 2 N +1 m + N X n = m N R ( ~ ( s 0 + n s )),(3–34) where N = L = s ThefastLICalgorithmcanthenbeimplementedaccordingtot hefollowing scheme: forall Pixels as Pixel do I ( Pixel ) 0 PixelHits ( Pixel ) 0 endforforall Pixels as Pixel do if PixelHits ( Pixel ) < NHits then FollowstreamlineusingRunge-Kutta 59

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InterpolatebetweenstreamlinepointsusingHermite-inte rpolation Samplepixelsin R alongstreamline Sumanddividesampledpixelstoget I forall X forwhich I isknown do I ( X ) I ( X )+ I ( X ) PixelHits ( X ) PixelHits ( X )+1 endforPixelHits ( X ) PixelHits ( X )+1 endif endforforall Pixels as Pixel do I ( Pixel ) I ( Pixel ) = PixelHits ( Pixel ) endfor Itshouldbenotedthatthiswaythemostpixelswillnotgetth evaluethatwouldbefound bydoinglineintegralconvolutionthatgoesthroughthecen tereachpixel,hencethis methodyieldsanapproximation.3.3.2ModicationstoFastImplementation Whenthesamepixelissampledseveraltimes,thevarianceoft hesumislarger thanwhenindependentpixelsaresampled.Thiscausesextre mevaluestoappearin somecases,forexamplewhenstreamlinescreateaclosedloo p.Toavoidthisproblem, thenumberoftimesthesameexactvalue(notpixel)issample discounted.Thevalue thatisstoredtotheimageisthennormalized,sothat I ( Pixel )= P N unique i =1 v i c i P N unique i =1 c 2 i 1 = 2 ,(3–35) where v i isthe i th uniquesampledvalueonthesamestreamlineand c i isthenumberof timesthevaluehasbeensampled.Assumingthat v i areGaussiandistributedrandom numberswithaunitvarianceandzeromean,thisnormalizati oninsuresthatimage 60

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valuesalsoarerandomnumberswithaGaussiandistribution ,unitvariance,andzero mean.3.3.3Example LICwillbedemonstratedusingasimpletwo-dimensionalinv iscidstagnation ow.Theowiscreatedbyfouridealvortices.Asinglevorte xat x 0 createstheow describedby: u ( x 0 ) vort ( x )= 2 y y 0 j x x 0 j v ( x 0 ) vort ( x )=+ 2 x x 0 j x x 0 j .(3–36) Theoweldiscreatedbysuperimposingfourofthese,sotha t u ( x )= u ( x 1 ) vort ( x ) u ( x 2 ) vort ( x )+ u ( x 3 ) vort ( x ) u ( x 4 ) vort ( x ),(3–37) where x 1 = ( +1,+1 ) T x 2 = ( 1,+1 ) T x 3 = ( 1, 1 ) T x 4 = ( +1, 1 ) T .(3–38) TheresultingvectoreldisshowninFigure 3-1 .Figure 3-2 showstheresultingimage usingtheLICtechniqueontheoweld.3.3.4Renements Figure 3-2 isquitebusy,whichcanbeimprovedbyapplyingaspatiallow -pass lterontherandomimageassuggestedbyCabralandLeedom[ 73 ].However,that solutionwillcausethemodicationdescribedinSection 3.3.2 tonotworkasintended. Instead,thesamerandomvaluecanbeusedbyeach N N pixelneighborhood. Figure 3-3 isobtainedbyusingan8 8neighborhoodsofthesamevalue.Byusing atwo-dimensionalcolorscale,ascalarquantity canbeshowninthesameplot.By usingthecolorscaleinFigure 3-4 and = j u j ,theowcanbeplottedasinFigure 3-5 Inthisanalyticalexample,thestreamlinescorrespondtot heowverywell everywhere.However,whenworkingwithexperimentalPIVdat a,atlowvelocities, theuncertaintynormallyincreases,asdoesthedirections ofthestreamlines.TheLIC 61

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image, I ( x ),canbemultipliedwithavariableproportionaltotheabso lutelocalow velocityforsmallvalues,butneverlargerthan1,i.e. I ( x )= I ( x ) max( C j u ( x ) j ,1).(3–39) Here, C isaconstant.Thiscausesthestreamlinestovanishatlowve locities,asin Figure 3-6 .Thisalsohelpstheobservertofocusonthepartsoftheoww heretheow velocityishigh. 3.4AerodynamicForces TobeabletovalidatethePIVdata,aerodynamicforcescanbec omputedfrom velocityelddata,andcomparedtodirectlymeasuredforce s.Thissectionwilldescribe howtime-averagedforcescanbecomputedfromPIVdata. TheincompressiblemomentumequationforaNewtonianuidw ithconstantdensity anddynamicviscosity canbewritteninEinsteintensorformas[ 75 ]: @ u i @ t + u j @ u i @ x j = @ p @ x i + @ 2 u i @ x j @ x j + f i .(3–40) where x i istheCartesiancoordinatetensor, u i isthevelocitytensor, p ispressure,and f isforceperunitvolume.Assumingthataowisperiodic,thec omponentscanbesplit intophase-averageandinstantaneousparts: u i = u i + u i ,(3–41) p i = p i + p i ,(3–42) f i = f i + f i .(3–43) Abardenotesphase-averageandatildedenotesinstantaneo usuctuations.A phase-averagedquantityisdenedhereas: ( x t )= ( x t + m T )=lim N !1 1 N N X n =0 ( x t + n T ),(3–44) where m 2Z N 2N t isthetime,and T istheperiodoftheow. 62

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Thephase-averageofEquation 3–40 is: @ u i @ t + u j @ u i @ x j = @ p @ x i + @ 2 u i @ x j @ x j + f i .(3–45) UsingtheincompressiblecontinuityEquation @ u i =@ x i =0[ 75 ],thesecondtermcanbe rewrittensothat @ u i @ t + @ u i u j @ x j = @ p @ x i + @ 2 u i @ x j @ x j + f i .(3–46) FromEquation 3–46 ,expressionsforsolvingforbothrelativepressureandfor cewillbe derivedbelow.3.4.1PressureEstimation Equation 3–46 canbewritteninaformtosolveforthepressureas @ p @ x i = @ u i @ t + @ u i u j @ x j + @ 2 u i @ x j @ x j + f i .(3–47) Sincetherearenonetbodyforcesontheowwheretheowveloc ityisknownby PIVmeasurements,theforcetermwillbeneglected.Furtherm ore,viscousforcesare signicantclosetothewingwheretheowvelocitiesareunk nown,andrelativelysmall awayfromthewing.Therefore,bothbodyandviscousforcesw illbeneglectedwhen computingrelativepressureinthemeasuredvelocityeld. Equation 3–47 thenbecomes @ p @ x i @ u i @ t + @ u i u j @ x j .(3–48) ThetermsontherighthandssidecanbeevaluatedfromPIVdata .Hence,by integratingEquation 3–48 ,the relative pressureacrosstheknowndomainisobtainedfor eachphase-averagedvelocityrepresentation. 63

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3.4.2MomentumBalance Knowingtherelativepressure,Equation 3–46 canbeintegratedoveracontrol volume V ,sothat y V @ u i @ t + @ u i u j @ x j d V = y V @ p @ x i + @ @ x j @ u i @ x j + f i d V .(3–49) Usingthedivergencetheorem,sometermscanbewrittenassu rfaceintegrals: y V @ u i @ t d V + x S u i u j n j d S = y V @ p @ x i d V + x S @ u i @ x j n j d S + y V f i d V .(3–50) Itcannowbeseenthattheviscousterm(secondtermontherig hthandsside)only mattersonthesurfaceofthevolumeinEquation 3–50 ,andifthesurfaceisnottooclose tothewingsurfacethetermshouldberelativelysmall.Itwi llthereforebeneglected infurtheranalysis.Theforcetermrepresentstheforcesac tingontheuidinsidethe controlvolume.Inthiscase,theforcetermintegratesthef orcesexertedontheowby thewingandcanbeexpressedas: y V f i d V = y V @ u i @ t d V + x S u i u j n j d S + y V @ p @ x i d V .(3–51) Thespeciccaseofmomentumalongthe y axiswillnowbeconsidered,toobtainan expressionforforcealongasingleaxis.Integratingarect angularcuboid,boundedby x 1 x x 2 y 1 y y 2 ,and z 1 z z 2 ,yieldsaforcealongthe y axisexertedonthe owinsidethecontrolvolume F y = Z x 2 x 1 Z y 2 y 1 Z z 2 z 1 @ v @ t d x d y d z + Z y 2 y 1 Z z 2 z 1 uv d y d z x 2 x 1 + Z x 2 x 1 Z z 2 z 1 p + v 2 d x d z y 2 y 1 + Z x 2 x 1 Z y 2 y 1 vw d x d y z 2 z 1 .(3–52) TherstterminEquation 3–52 correspondstothechangeofmomentuminthe y directionoftheuidinsidethecontrolvolume.Theremaini ngtermsaresurfaceintegrals correspondtothemomentumthatisleavingthecontrolvolum eandtheseareillustrated 64

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inFigure 3-7 .Usingthecontinuityequation,thersttermcanbewritten Z x 2 x 1 Z y 2 y 1 Z z 2 z 1 @ v @ t d x d y d z = Z x 2 x 1 Z y 2 y 1 Z z 2 z 1 @ @ t v j y = y 1 + Z y y 1 @ v @ y d y d x d y d z = Z x 2 x 1 Z y 2 y 1 Z z 2 z 1 @ @ t v j y = y 1 Z y y 1 @ u @ x + @ w @ z d y d x d y d z = @ @ t Z x 2 x 1 Z z 2 z 1 v j y = y 1 ( y 2 y 1 )d x d z Z y 2 y 1 Z y y 1 Z z 2 z 1 [ u ] x 2 x 1 d z + Z x 2 x 1 [ w ] z 2 z 1 d x d y d y .(3–53) ThisformallowsthersttermontherighthandssideofEquati on 3–52 tobeevaluated usingonlyowdataonthesurfaceoftheowvolume.Hence, F y canbeevaluated fromPIVdataeventhoughtherearemaskedvaluesinsidetheco ntrolvolume.Note alsothatonlyrelativepressureisnecessary,sinceanyoff setaddedtothepressurewill cancel.Therefore,thepressurecanbeobtainedusingEquati on 3–48 .Thedensitywill beassumedtobe 1.205kg/m 3 yx 0.4 0.200.20.4 0 0.2 0.4 Figure3-1.Stagnationoweldvectors. 65

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yx 0.4 0.200.2 0.4 0 0.2 0.4 Figure3-2.Lineintegralconvolutionof stagnationoweld. yx 0.4 0.200.2 0.4 0 0.2 0.4 Figure3-3.Lineintegralconvolutionof stagnationoweldand8 8samepixel neighborhoodsintherandomimageforlowerstreamlinedensity. LIC Figure3-4.Two-dimensionalcolorscaleforlineintegralc onvolutions,withoneaxisfor LICstreamlinevaluesandtheotherforanyscalarvalue oftheuidow. yx 0.4 0.200.2 0.4 0 0.2 0.4 Figure3-5.Lineintegralconvolutionof stagnationoweldwithcolorindicatingvelocitymagnitude. yx 0.4 0.200.2 0.4 0 0.2 0.4 Figure3-6.Lineintegralconvolutionof stagnationoweldwithmorepronouncedstreamlinesathighvelocities. 66

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s vw d x d y s vw d x d y s ( p + v 2 )d x d z s ( p + v 2 )d x d z s uv d y d z s uv d y d z ( x 1 y 2 z 2 ) ( x 2 y 1 z 2 ) ( x 2 y 2 z 2 ) ( x 1 y 1 z 2 ) ( x 2 y 2 z 1 ) ( x 2 y 1 z 1 ) ( x 1 y 2 z 1 ) ( x 1 y 1 z 1 ) Figure3-7. y momentumleavingacontrolvolume. 67

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CHAPTER4 EXPERIMENTALEQUIPMENT Adescriptionofthehardwareusedthroughoutthisstudywil lbepresentedin thischapter.Thisincludestwodifferentparticleimageve locimetrysystems,multiple mechanicaldevicesforappingdifferentwingsandaloadce llforthemeasurementof aerodynamicforces. 4.1PIVEquipment ThestereoscopicPIVsystemsusedinthisstudyconsistsofal ightsource,two cameras,acomputerforobtainingandstoringdata,andhard wareforcommunicating andtimingthevariouscomponents.Softwareforprocessingt hePIVimagestogenerate velocitydatausingtechniquespresentedinChapter 2 isalsonecessary.Thatcapability isavailableinthesoftwareusedforcapturingdatainthiss tudy. TwoseparatePIVsystemsareusedinthisstudy.Onesystemisa highframerate systemfromDantec.Itsadvantagecomparedtotheothersyst emusedinthisstudyis mainlyitshighframerate.Althoughtheframerateisnothigh enoughtogivewhatcould beconsideredtobetime-resolveddatafortypicalappingf requenciesusedhere,it doesallowtakingmanysnapshotsoftheowrapidly,signic antlyreducingboththetime neededforanexperimentandthewearandtearonequipment. Theothersystem,fromLaVision,hasahighercameraresoluti onanddynamic range,andhigherenergylaserpulses.However,itcanonlya cquireimagepairsat 15Hz.Hence,itiscapableofproducinghigherqualityimage sandusingsmaller seedingparticles,whilekeepingtheinvestigatedwindowa tasimilarsize.Thedownside isthatexperimentsthatareconductedinthisstudytypical lyneedsatleastanorderof magnitudelongerruntimetocapturethesamenumberofPIVsna pshots. 4.1.1DantecSystem TheDantecsystemisahighframeratePIVsystem.It'spropert ieswillbedescribed inwhatfollows. 68

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4.1.1.1Software ThesoftwarecontrollingtheDantecsystemisDantecDynami csFlowManager 4.71.ThesamesoftwareiscapableofprocessingPIVsnapshot saswellascontrolling thevariouscomponentstoobtaintheparticleimages.Whenit wasusedforprocessing PIVdata,itwassettouseaniterativecorrelationalgorithm ,asdescribedinSection 2.5 startingwitha256 256pixelinterrogationareaanditeratingdowntoa32 32pixel window. Calibration .ThePIVsystemwascalibratedusingatwo-dimensionaltarge twith equidistantblackdotsonawhitebackground.Snapshotsofth etargetwereneeded atseveraldepth-wisepositions,forthesoftwaretobeable tocalibratethephysical coordinatesforstereoscopicPIV.Forthisprocess,thecali brationtargetwasmountedon asmalltraverseforaccuratedepth-wisepositioning.Thet raversealsohasadjustments forrotatingthetargetsmallanglesintwodegreesoffreedo m,toaidaligningitwith thelasersheet.Whenthistargetwasusedinthisstudy,6snap shotsweretakenfor calibration,withthetargettraversed1mmbetweeneachsna pshot. 4.1.1.2IDTMotionProX3cameras TheDantecsystemhastwoIDTMotionProX3cameras.Thesecamer asare capableofcapturingimagepairsataresolutionof1280 1024withaminimum integrationtimeof100nsatmorethan500Hz.Thepixelsizei s12 12 mandthe sensorhasa59dBdynamicrange.ThecamerashaveScheimpugm ounts,allowing thecamerahousetoberotatedindependentlyfromtherestof thestructurewithone degreeoffreedom.4.1.1.3Leelaser ThissystemhasaLeeLaser800-PIV/40GNd:YAG.Thelaserhastw oresonators andeachoneiscapableofproducingapproximately0.1msand 15mJpulsesata 1kHzrateandmaximumlamppower. 69

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4.1.2LaVisionSystem TheotherPIVsystemusedinthisstudywasfurnishedbyLaVisio n.Thesystemhas higherresolutioncamerasandhigherpowerlaserpulsesbut amuchlowermaximum frameratecomparedtotheDantecsystemintroducedabove.4.1.2.1LaVisionDaVissoftware ThesystemiscontrolledbytheDaVis7.2imagingsoftwarewit hLaVision's FlowMasterextensionforPIVprocessingcapabilities.Whenp rocessingthePIV datainthisstudy,thesoftwarehasbeensettouseaniterati vecorrelationalgorithm, asdescribedinSection 2.5 ,doingonepassusinga128 128pixelinterrogation areaandtwopassesat32 32pixels.Finally,Whittakerreconstruction[ 76 ]hasbeen usedtoobtainhighsub-pixelaccuracy.Thealgorithmusedb yDaVisforcomputing three-componentvelocitydataisdescribedbyCalluaudand David[ 63 ]. Calibration .TocalibratetheLaVisionPIVsystem,acalibrationtargetsh ownin withequidistantmarkersintwoseparateplanesisused.Thi sallowscalibrationtobe calculatedusingasinglephoto.ThetargetisshowninFigur e 4-1 .Thecalibrationis performedusingthepinholemodel,describedinSection 2.8.1 Radialdistortioncorrection .TheDaVissoftwarehasanalgorithmtocorrectfor radialdistortion.Thealgorithmisaslightvariationofth eonedescribedinSection 2.8.2 wheretheprincipalpointisusedasthecenterofradialdist ortion,physicalcoordinates areusedinsteadofpixelcoordinates,and ˆ L ( ˆ r ) ˆx I = x I ,(4–1) ˆ r = j ˆx I j .(4–2) Thecalibrationfunctionissecondorder: ˆ L ( ˆ r )=1+ˆ 1 ˆ r +ˆ 2 ˆ r 2 .(4–3) Thisyieldsaslightlydifferent,yetstillsecondorderrad ialdistortioncorrection. 70

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Self-calibration .Whenphotographingatargetforcalibration,itisveryhard to placethetargetexactlyatthelocationwherethelasershee twillbe.Tosolvethis problem,DaVishasaself-calibrationalgorithmbuiltin,wh ichisdescribedbyWieneke [ 77 ].Thesystemmuststillbecalibrated,buttheself-calibra tionalgorithmmakesit possibletocorrectforsmallalignmenterrors.Thealgorit hmworksbymappingPIV imagesfrombothcamerastophysicalcoordinates,andthenc orrelatingthemapped imagestocomputethemisalignmentbetweenthemappedimage s.Thecomputed misalignmentisthenusedtocorrectthemapping. Timing .Themaximumcapturerateofthesystemislimitedto7Hzbyth eframe rateofthecameras.However,inpracticethecapturerateis usuallylimitedbythe transferratebetweenthecameraandthecomputer.Whenthesy stemdoesnothave timetocaptureasnapshot,thesnapshotisomitted.If T isthetimebetweentwo snapshotsthatthesystemissetuptouse,theactualtimebet weentwosnapshotswill alwaysbeanintegermultipleof T 4.1.2.2ImagerproX4Mcameras TheLaVisionsystemusestwoImagerproX4Mcameras,witha14b itdynamic range,2048 2048pixelresolution,anda7.4 7.4 m 2 sensorpixelsize.Thecameras cancapturefullresolutionimagesat14Hz,ordoubleexposu resat7Hz,withdownto a115nsinterframingtime.ThecamerashaveScheimpugmount s,allowingthelensto beindependentlyrotatedwithtwodegreesoffreedom.4.1.2.3LitronNanoLlaser ALitronNanoL135-15PIVNd:YAGlasersystemisusedtocreate thelightsheet withtheLaVisionsystem.Thelaseriscapableofa15Hzrepeti tionrate,andthelaser pulsescanhaveenergylevelsupto135mJ.4.1.3Seeding Throughouttheexperimentsconductedhere,severaldiffer entseedinggenerators withdifferentseedparticleswereused.Thesewillbediscu ssedbelow. 71

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4.1.3.1LaVisionDSaerosolgenerator TheLaVisionDSAerosolGeneratorisdrivenbyhighpressureai r,andusesfour atomizernozzlestobreakuptheseedingliquidintosmallpa rticles.Themodalparticle sizequotedbythemanufactureris d p 0.25 m.Ithasbeenusedwitholiveoilas seedingliquidinthisstudy. Theterminalvelocityoftheparticles,usingthecondition sdescribedinSection 2.4.4 and p 0.92 10 3 kg/m 3 asthedensityoftheoliveoilparticles,isestimatedtobe U t 1.7 m/s.Thisterminalvelocityisseveralordersofmagnitudel owerthanany velocitiesthatwillbeinvestigated,whicharetypicallyi ntheorderof1m/s.Furthermore, therelaxationtimefortheparticlescanbeestimated,asde scribedinSection 2.4.3 tobe s 0.18 s,whichagainisseveralordersofmagnitudelowerthantypi calow times,whichareontheorderof10 2 s. 4.1.3.2Dantechighvolumeliquiddropletseedinggenerato r AnotherseederusedinthisstudyisaDantec10F03HighVolume LiquidDroplet SeedingGenerator.Itisalsodrivenisdrivenbyhighpressur eairandhasanatomizer nozzle.Themeandiameteroftheseedingdropletsis2-5 maccordingtothe manufacturer.Forthisstudy,oliveoilisusedastheseedin gliquid.Thisleavesthe terminalvelocityintherangebetween0.1and0.7mm/s,andt heparticlerelaxationtime between10and70 s. 4.1.3.3Expancel TM microspheres Theoliveoilseedingparticlesaresmallandcanthusbehard toseesincethey donotscattermuchlight.Therefore,insomeexperiments,Ex pancel TM 461DE20d70 microsphereswillbeusedfortheseeding.Thesearegaslle dplasticmicrospheres withanaverageradiusof20micronsandanaveragedensityof 70kg/m 3 .The advantageofthesespheresistheirlowdensity,whichallow stheuseoflargeandclearly visibleseedingparticlesbutwithmuchlessdisturbancedu etobuoyancycomparedto heavierseedingsuchasoliveoilwiththesameparticlesize 72

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Therelaxationtimeoftheseparticlesisestimatedto0.3ms andtheterminal velocityto3.4mm/s.Thesevaluesarebothmorethananorder ofmagnitudebelow typicalvaluesoftheoweldsthatwillbemeasured.Thedow nsideofusingthese spheresisthattheyareverydifculttocontaininthetests ectionandtocleanup. 4.2FlappingDevices Theabilitytohaveareliabledevicetocreaterepeatablewi ngmotionsrepresents animportantchallengeinthisstudy.Notonlyistheability torepeatanexperimentat asinglelocationsought,itisalsonecessarytobeabletome asuretheowatseveral locationsofthewinginseparatemeasurementsinordertobu ildupathree-dimensional representationoftheoweld.Hence,theseseparatemeasu rementsmustgive consistentdata.Inthissection,severaliterationsofap pingdevicesthatwere conductedwithstudentsfromFlappingMAVLabintheMAEdepar tmentwillbe discussed.4.2.1Type1 Thetype1appingdeviceisshowninFigure 4-2 .Thetransmissionoftheapping deviceisbasedononethatwasdesignedandusedbyWuetal.[ 24 ],whichgivesa 9-to-1gearratio.ThemotorisaFaulhaber1628T012BK1155,ca pableofrunningwith atorqueof2.6mNmat40,000rpm.Thistranslatestoanideali zedmaximumapping frequencyof74Hz,whichwasenvisionedtobesignicantlym orethannecessaryfor thisstudy.ThemotoriscontrolledbyaFaulhaberMCBL3006Sm otioncontrollerwhich isresponsibleforkeepingthemotorataconstantspeedandc anbesetwithaprecision of1rpmupto30,000rpm.Thecontrollerusesanencodercoupl edtothemotorto keeptrackofitspositionbymeasuringthepositionwithini tsrevolutiontoaprecision of1/3600 th oftherevolution.Theaccuracyofthecontroller'sreporti ngofthemotors positionisnotexpectedtoberelevanthere,sincetheerror intimedelaybetweenthe controllersreadingandthecontrollingcomputerscompari sontoitsinternalclockis 73

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likelytobealargersourceoferror.Communicationwiththe controllerisaccomplished throughanRS232port.4.2.2Type2 Thetype2appingdevice,showninFigure 4-3 ,isanimprovedappingdevice comparedtotype1.Theappingmotionisclosertoasinusoid ,andthegearboxis morerobust.Themainstructureismadefromaluminum.Movin gpartsaremadefrom carbonberandaluminum,andtheslidersfromstainlessste el.Thereciprocatorison aslidertogetherwith8ballbearings,whichensurethatthe appingamplitudevariation isveryclosetosinusoidal.Theappingdeviceisdescribed inmoredetailbyWuetal. [ 42 ]. 4.2.3MaxonEC1615Wbrushlessmotor Theappingdeviceisdrivenbyan18VMaxonEC16brushlessDCm otor.Its nominalspeedis31,700rpmat4.76mNmtorque.Themotorisco nnectedtoaGP16 Aplanetarygearhead,witha57/13reductionratio,whichre ducesthenominalspeed oftheoutputshaftto7,230rpmor120Hz.Therecommendedinp utspeedofthegear headislessthan8,000rpm,whichcorrespondstoanoutputof approximately30Hz. Encoder .ThemotorisassembledwithanMRencoderwithHallsensors. The outputfromtheencoderisatwochannelquadraturesignalas showninFigure 4-4 .The signalisrepeated256timesperturn.Thephaselagbetweent hechannelsallowsthe directionofthemotortobereadfromthesignalandtheposit ioncanbetrackedwitha 1/1024turnresolution.Withtheeffectofthegearheadadded ,theprecisionwithwhich theappingphasecanbereadis1/4490thofaturn.Theoutput isusedforthemotor controller,butwillalsobesampledtogetherwithasynchro nizationsignaltosynchronize theappingphase,i.e.themotorturncountfromaknownoffs et,withPIVdata. Controller .ThemotoriscontrolledbyaMaxonEPOS24/1controller.Itisc apable ofreadingthespeedofthemotorupto25,000rpm,whichcorre spondstoa95Hz appingfrequency.Thecontrollerissetupusingacomputer andserialcommunication 74

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throughanRS232port.Itcanbeusedsimplybyrstspecifying amaximumrateof accelerationandthensettingacommandedspeed.Thenthecu rrentspeedcanberead tomonitorwhenthedesiredspeedisreached.Thecontroller alsokeepstrackofhow manyrevolutionsithasdone.However,thepositionofthemo torcannotbereadfrom thecontrollerwiththetemporalaccuracythatisneededtod eterminethephaseinthe appingcyclewhenthemotorisrunningattypicalspeeds.4.2.4Type3 Animprovedversionoftype2wascreatedtoaddresssomerepea tabilityissues. Thesignicantchangeisthateveryallmovingjointsarecon nectedusingballbearings. Thedesignallowedtheappertorunthroughoutallexperime ntsincludedinthispaper withoutanyadjustmentssuchaslubricatingortightenings crews.Theappercanbe seeninFigure 4-5 .Theapperusesthesamemotor,controller,andencoderas apper type2.4.2.5Type4 Muchoftheresearchconductedonappingightassumesasin usoidalapping motion.Thisinturnleadstoasinusoidalaccelerationonth ewing.InSection 5.4 ,it willbeshownthatthismayresultintheangleofattackrelat ivetothedirectionofthe wingmotionbeingcloseto90 shortlyaftermid-stroke,ifthewingsareexibleand passivelydeformed.However,asalreadynotedinSection 1.3.7 ,naturaliersoften createafavorableangleofattackalongtheentirespanofth ewinginthebeginningof eachhalf-strokeandmaintainsitthroughoutmostofthehal f-stroke. Toaddresstheproblemswithsinusoidalappingmotion,ane wapperwas designed.Theapperhasanangularvelocitythatisnearlyc onstantthroughhalfof eachhalf-stroke.Figure 4-6 showsthedesign.Byusingachaintomovetheshaft,the appercommandsanearlyconstantappingvelocityhalfoft hetimeofeachhalf-stroke andtheotherhalfisspentreversingtheappingdirection. Asforappingdevicetype3, 75

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thereisaballbearingateveryjointtoensurethatthedevic ehastherepeatabilitythatis necessaryforhighqualitylongdurationPIVmeasurements. ThepredictedmotionofthisapperisshowninFigure 4-7 .Thegoaloftheapper istoletwingswithenoughexibilitymaintainafavorablea ngleofattackthroughamuch largerfractionofeachhalf-strokethanwhathasbeenpossi blewithpreviousapper generations. Theapperdesigncausesasmallasymmetrybetweenthedurat ionofthetwo half-strokes.Byreversingtherotationofthemotor,thesid eoftheappingstrokewhich hasalongerdurationisswitched.Forhoveringightwithah orizontalstrokeplane,this maycausethethrusttonotbecompletelyalignedwiththever ticalaxis. 4.2.6MaxonEC1640Wbrushlessmotor Thisapperisdrivenbyan18VMaxon40WEC16brushlessDCmoto r.Itis similartotheMaxonmotordescribedinSection 4.2.3 ,butmorepowerful.Itsnominal speedis36,700rpmat13.4mNmtorque. Encoder .Themotorisassembledwithathree-channelencoder,witht woofthe channelscreatingaquadraturesignalallowingforthemoto rtobetrackedwitha1/1024 turnresolution,asfortheotherMaxonmotor.Thethirdchan nelonlyemitsonesignal everyrevolutionofthemotor.Thatsignalwillbeusedtoali gnthephaseofdifferent measurementsexactly(with1/1024turnprecision).Themot orisnotconnectedtoany gearbox. Controller .ThemotoriscontrolledbyaMaxonEPOS224/5controller,which contrarytotheMaxonEPOS24/1isabletosupplyenoughpowerto takeadvantageof theadditionalpowerofthe40Wmotor. 4.3Wings Thewingsusedinthisstudyallsharethesameplanform,butn otthesamematerial andreinforcement.Thissectiondescribestheplanformand outlinesthewingdesign. Specicwingdesignswillbepresentedtogetherwithexperim entalresultsinChapter 5 76

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4.3.1ZimmermanPlanform ThewingsinvestigatedinthisstudyhavetheshapeofaZimme rmanplanform. Thisshapeisdenedbytwoellipsesmeetingatthequarterch ordpoint,asshownin Figure 4-8 .Thesemispanofthewingsis75mm,witharootchordof25mm,a ndthe aspectratio(AR),asdenedbelow,is7.65. AR= 4 r 2 S .(4–4) Inthisequation, r isthesemispanofthewing,and S isthesurfaceareaofawingpair. ThisplanformisonethatiscommonlyusedforlowReynoldsnu mberMicroAirVehicle congurationsasreportedinMuelleretal.[ 78 ].Theaspectratio7.65ischosento maketheplanformaneasilydenedapproximationofthecomp ositehummingbirdwing presentedbyAltshuleretal.[ 12 ]. 4.3.1.1Aluminumwings Totestwingswithuniformandisotropicbendingcharacteri stics,wingsmadefrom 6061aluminumwereused.ThesearecutusingaCNCwaterjetto maintainaclean edge.4.3.1.2Anisotropicwings Wingsmadeforanisotropicbendingcharacteristicsaremanu facturedbylaying aCapran R r membraneonacarbonberskeleton.Thewingisattachedtoth eapper wheretheleadingedgemeetsthewingroot.Theskeletonhasa relativelythickcarbon bertriangleatthislocationasshowninFigure 4-8 ,soitcanbeapproximatedasrigid. Whenthewingdeformationismeasuredusingdigitalimagecor relation(DIC),the triangleisusedtoextractcommandedangleofthewing. Thewingroot,leadingedge,battens,andtheapproximately rigidtriangleare reinforcedbycarbonber,asshowninFigure 4-8 .Thisdesigncreatesnon-uniform andnon-isotropicexibilitycharacteristicsforimprove dthrust.Detailsofthewing 77

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manufacturingprocessisdescribedinbyWuetal.[ 25 ].Thespecicwingdesignswill bediscussedindetailwheretheexperimentalresultsarepr esented. 4.4ForceMeasurements Thissectiondescribestheforcetransducerusedtomeasure forcesfromthe appingwings,andtheadditionalequipmentnecessarytore adforcedata. 4.4.1ATINano17ForceTransducer AnATINano17ForceTransducerisusedforforcemeasurements .Itisa6axis force/torquetransducer,whichmeasuresbothforceandtor queinthethreespatial dimensions.Theweightoftheentireloadcellisonly10.1ga ndithaspreviouslybeen usedforexperimentsonappingwings[ 79 ].Itiscalibratedtosensetherangegivenin Table 4-1 .Calibratedtothissensingrange,itcanbereadwiththeres olutiongivenin Table 4-2 Theanalogforcetransducersignalissampledthrough6chan nelsofaNI-DAQ 6220card(seebelow).Thedigitalsignalisthenmultiplied witha6 6element calibrationmatrixtogiveforceandtorqueinphysicalunit s. 4.4.2ATIGammaForceTransducer AnATIGammaForceTransducerisusedforforcemeasurementsw heretherange oftheNano17transduceristoosmall.Itissimilarinfuncti ontotheNano17,butismuch largerandcanmeasuremuchlargerforces.Itiscalibratedt osensetherangegivenin Table 4-3 .Calibratedtothissensingrange,itcanbereadwiththeres olutiongivenin Table 4-4 4.4.3NI-DAQ6220DataAcquisitionCard Forsamplingofforcemeasurements,aNI-DAQ6220cardwillb eused.Itcan captureupto8differentialor16singleendedanaloginputs simultaneouslyatan aggregatesamplingrateof250kHzanda16bitresolution.Th einputrangecanbeset to 10V, 5V, 1V,or 0.2V. 78

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4.5FlappingPhaseofPIVDataSnapshots ItisnecessarytoknowthephaseofPIVdatatobeabletocomput ephase-averaged ofowquantities.Theappersoftype2,3,and4aredrivenby Maxonmotorsthathave encodersattachedtothem.Theseencodersemitatwochannel quadraturesignalas showninFigure 4-4 .Therelativeappingphaseoftheseapperscanbecomputed fromtheencodersignal.Therefore,thequadraturesignali ssampledontwoofthe channelsofaNI-DAQ6220cardandthePIVtriggersignalonath irdchannel.Anoffset mustbeknowntoobtaintheabsoluteappingphase.Theoffse tforthemotorposition isfoundbymovingtheappertoaknownappingphasewhereth ephaseposition counterisresetwhenthemotorisnotmoving. Toaccuratelykeeptrackofthemotor'spositionsynchronou slywiththesynchronization signal,thesignalmustbecapturedfromthemomentthemotor startsmovingfroma knownposition.Forthatpurpose,asoftwarewasdevelopedt ocontrolthemotor.The softwarecansimultaneouslysampleforcedataononeDAQcar dandmotorencoder dataonanotherDAQcard.ThedatafromtheDAQcardsamplingt hemotorencoder signaltogetherwiththePIVtriggersignalisprocessedbyth esoftwareinrealtimeto keeptrackofthemotorpositionatalltimes.Thatwaythepos itioncanberecordedat thetimeofeachPIVtriggersignal.Tointerprettheencoders ignalcorrectly,including thedirectionofthemotor'smotion,morethan1,024samples perrevolutionofthemotor isnecessary.FortheMaxonEC1615Wbrushlessmotordescribe dinSection 4.2.3 thatmeansrecordingmorethan4,490samplesperappingcyc leduetoits57:13ratio gearhead.Therefore,theencodersignalcanonlybereadcor rectlyuptoapproximately an18.5Hzappingfrequency.However,ifthedirectionofth emotorisalreadyknown, whichcanberememberedfromwhenthemotorisstillatlowspe edandaccelerating, onlytwosamplesperencodersignalcycleisnecessaryandth eencodersignalcan theoreticallybecorrectlyinterpreteduptoa37Hzapping frequency.Thismethod hasbeentestedbyreadingthemotorpositionwhenit'satres tandthenrunningthe 79

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motorforawhilebeforestoppingit.Whenthemotorisatresta gain,thecountfromthe motorencodercanbecomparedwiththecountobtainedbyinte rpretingthesampled encodersignal.Themethodhasbeenshowntoworkupto25Hzwi thnodifferencein count.At15Hzithasbeentriedextensivelywithoutdiscrepa nciesevenaftermorethan abillionencodersignalcycleshasbeenrecorded.Thatcorr espondstoseveralhours ofcontinuouslyrunningthemotor.However,at30Hzthecoun tratediffersslightly.By observingthephysicallocationofthemotorafterithassto ppedithasbeendetermined thatitisthecountobtainedfromthesampledsignalthatise rroneous. TheMaxonEC1640WbrushlessmotordescribedinSection 4.2.6 hasbeen trackedthesameway.Thatmotorhasonlybeenusedtogetherw iththetype4apper inthiswork.Thatapperhasagearratiosuchthattworevolu tionsofthemotor correspondtooneappingcycle.Sincethemotordoesnothave agearhead,that meansmorethan2,048samplesperappingcyclearerequired torecordallfourstates ofeveryperiodoftheencodersignal.Sincethatismuchlesst hanwhatisrequiredfor the15Wmotorusedtogetherwithappersoftype2and3,the40 Wmotorcanbe accuratelytrackedbythesoftwareuptoahigherfrequencyt hanthe15Wmotor. 4.6HoveringTestingEnvironment Measurementssimulatinghoveraremadeinthetestsectiono fanopenjetwind tunnel,whichisacubewith8footsides,toensurethatthea ppingdoesnotinducea globalcirculationpattern.Theinletandoutletofthewind tunnelarecoveredtoeliminate effectsofowinandoutofthetestsection.Table4-1.ATINano17ratedsensingrange. AxisUS(English)SI(Metric) F x F y 3lbf 13.3N F z 4.25lbf 18.9N T x T y T z 1lbf in 113Nmm 80

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Table4-2.ATINano17typicalresolutionfora16-bitDAQ. AxisUS(English)SI(Metric) F x F y F z 1 = 1280lbf1 = 288N T x T y T z 1 = 8000lbf in1 = 71Nmm Table4-3.ATIGammaratedsensingrange. AxisUS(English)SI(Metric) F x F y 7.5lbf 33.4N F z 25lbf 111N T x T y T z 25lbf in 2.82Nm Table4-4.ATIGammatypicalresolutionfora16-bitDAQ. AxisUS(English)SI(Metric) F x F y 1 = 640lbf1 = 144N F z 1 = 320lbf1 = 72N T x T y 3 = 640lbf in0.53Nmm T z 1 = 320lbf in0.35Nmm Figure4-1.LaVisioncalibrationtarget. 81

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Figure4-2.Flappingdevicetype1. Figure4-3.Flappingdevicetype2. Phase 6 Amplitude 0 90 180 270 360 ChannelA ChannelB Figure4-4.Encoderoutput. 82

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Figure4-5.Flappingdevicetype3. Figure4-6.Flappingdevicetype4. 83

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( )00.51 50 0 50 AAngle. d = d ( )00.51 200 0 200 BAngularvelocity. d 2 = d 2 ( )00.51 5000 0 5000 CAngularacceleration. Figure4-7.Flappertype4angle ,angularvelocity,andangularaccelerationasa functionofphase ,where = t = T t isthetimefromthebeginningofthe appingcycleand T istheperiodofaappingcycle.Theappingamplitude is 35 triangle rigid9.375 18.75mm 18.75mm 18.75mm 9.375 75mm25mmtrailingedgeleadingedge Figure4-8.Zimmermanprolewing.Theverticallinesonthe wingindicatebatten positions. 84

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CHAPTER5 EXPERIMENTALRESULTS Thischaptercontainsexperimentalresultsonseveraldiff erentwingsusingsimple appingmotions.TherstsetofwingsexaminedinSection 5.2 wereconstructedout ofaluminumandcanbeassumedtoberigid.Themeasurementso nthosewingsare followedbythediscussionofexperimentsonwingswithvary ingspanwiseexibility andanisotropicbendingcharacteristicsinSection 5.3 ,wherewingdeformation andaverageforcedataprovidedbyWuetal.[ 24 ]arecombinedwithaerodynamic measurements.Section 5.4 containsaerodynamicmeasurementsonwingswithvarying chordwisereinforcements.Finally,Section 5.5 showsresultsfromforceandoweld measurementsonwingswithdifferentmembranematerialsat tachedtoreinforced leadingedgesandwingroots.Howeverbeforetheresultsare presented,Section 5.1 will presentthenormalizationschemesandnon-dimensionalpar ametersthatwillbeusedto presentthedata. 5.1NormalizedQuantities Spatialaxesintheplotsthroughoutthissectionarenormali zedusingthesemispan lengthofthewing, R ,i.e. x i = x i R .(5–1) Thevelocityscalechosentonormalizetheplotsisatypical velocityforthewingmotion is U mean = hj u wingtip ji =4 ARf ,(5–2) where A istheabsoluteanglefrommid-stroketooneoftheextremes. Using U mean ,the owvelocitiesarenormalized u i = u i U mean .(5–3) Similarly,normalizedvorticityisdened i = i U mean = c 85

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where c isthewingrootchord.Thequantity Q ,whichisusedforidentifyingvortex cores,isnormalized Q = Q ( U mean = c ) 2 .(5–4) ThehoveringReynoldsnumberRe hover willbeheredenedas Re hover = U mean c = 4 ARfc ,(5–5) inlinewiththedenitionusedbyHongandAltman[ 34 ].Time t isnormalizedwith respecttothedurationofaappingcycle, T ,sothat = t T (mod1).(5–6) Theoffsetof variesbetweentheexperimentsandwillbedenedwhereitis used. Denitionsofthrustcoefcient C T andeffectivestiffness 1 aretakenfromShyy etal.[ 80 ],whodenesthemas 1 = Eh 3 12(1 2 ) U 2 mean c 3 m ,(5–7) C T = T 1 2 U 2 ref S (5–8) where E isYoung'smodulus, h ismembranethickness, isPoisson'sratio, c m isthe meanchord, T isthrust,and S isthewingplanformarea.Inthecaseswhereforce measurementsandestimationswillbepresented,liftandth rustwillbothmeanthesame thing.Thisisbecausealldatameasuresthehoveringcases, sothatthethrustisaligned withlift. 5.2RigidAluminumWings Therstaerodynamicmeasurementsconductedonthisplanfo rmweredoneusing wingsstiffenoughtobeapproximatedasrigidandthatwerea ctuatedusingasimple appingmotion.Thiswasdonetoestablishabaselinecase,t owhichwingswithfor exampledifferentcamber,exibility,angle,andamplitud ecanbecompared.Arigidwing 86

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isnotexpectedtoproduceanynetforcesaveragedoverthesy mmetricappingcycle beingusedhere.Furthermore,theweightofthewingandforc esrequiredtoapan approximatelyrigidwingsetalimitsothatthewingwasonly testedupto10Hztoavoid damagetothegearboxoftheapper.Thislimitstheapplicab ilitytothetruebiological systemthisstudyisinspiredbyandlaterresultsonexible wings,butwillnonetheless provideanexperimentaldescriptionofthevortexstructur einthissimpleappingmotion ofarigidwing. Inwhatfollows,theexperimentalsetupthatallowedforthe captureofthevelocity eldinmany y x planesalongthewingwillbediscussed.Afterthat,themetho dsused toanalyzethedatawillbedescribed,followedbyapresenta tionoftheexperimental resultswhereaphase-averagedrepresentationofthethree -dimensionalvelocityeld wasusedtocalculatethevorticityeldthroughoutthesimp leappingcycle. 5.2.1ExperimentalSetup Thissectionwilldescribetheequipmentandphysicalsetup thatwereusedfor testingtheappingrigidwings.5.2.1.1Wing Thewingusedforthisexperimentwasa0.04”thickaluminumw ing,asdescribed inSection 4.3.1.1 .Thethicknessofthewingwaschosenbyperformingadeecti on analysissothatitcanbeassumedtoberigidinthisstudy.Th edeectionanalysiswas performedbycalculatingtheone-dimensionalbendingofth ewing,inthesteady-state casewhereitiscontinuouslyexperiencingthemaximuminst antaneousangular accelerationoftheappingcycle,whileignoringaerodyna miceffects.At10Hz, thedeectionofthewingtipforthiswingcomparedtoacompl etelyrigidwingwas computedtobelessthan2.5 h ofthewingsemispan.Theedgesofthewingwere roundedtoallowforeasiercomparisonswithnumericalsimu lationswhichwere discussedin[ 3 ].Thewingwaspaintedinatblacktoreducereectionsfrom the laserlightsheetusedforthePIV. 87

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5.2.1.2Flappingdevice Thetype1appingdevice,describedin 4.2.1 ,wasused.Duetothedesignofthe appingdevice,thewingwasmountedapproximately1cmfrom thecenterofrotation. Hence,thesemispan R ofthewing,willbeconsideredtobe85mm,whichisthe distancefromthecenterofrotationtothewingtip.5.2.1.3Particleimagevelocimetrysetup ThestereoscopicDantecPIVsystem,describedinSection 4.1.1 ,wasusedforPIV measurements.Thecamerasweremountedatapproximately45 degreeanglestothe imageplane,asshowninFigure 5-1 .TheDantecseeder,describedinSection 4.1.3.2 wasusedforseeding.Theseedingwasdispensedfromabove,w hichinducedsome downwardsvelocityastheparticlessettledout.However,t hisvelocityisestimatedtobe anorderofmagnitudelowerthan U mean .Themeasurementsweremadeinthehovering environment,describedinSection 4.6 TheDantecFlowManagerSoftwarewassetuptousethesettings inTable 5-1 whilecapturingPIVsnapshots.d T listedinthistableisthetimebetweentherstand secondlaserpulseoftheimagepair.5.2.2DataAnalysisMethods Thissectiondetailstheparametersofmethodsdescribedin Chapters 2 and 3 that wereappliedtothedatacapturedforthecurrentexperiment 5.2.2.1Particleimagevelocimetry ThesnapshotswereprocessedusingDantec'sFlowManagerso ftware.Aniterative correlationalgorithmwasused(seeSection 2.5 ),setupfor3iterations,startingwitha 256 256pixelinterrogationarea,iteratingdownto32 32pixels. Theresultingwindowconsistsof68 63datapointsandcorrespondsto approximately124 81mminphysicalspace.Acamerapixelcorrespondstobetwee n 0.9 10 4 0.7 10 4 mand1.2 10 4 0.8 10 4 minphysicalspace.Thegrid, togetherwiththewing,isshowninFigure 5-2 88

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5.2.2.2Synchronization Toextractthephaseinformationforthecomputedvelocity elds,rawimages wereanalyzedtodeterminethewingapangleineachsnapsho t.Theimagesfrom thecamerainwhichthelaserreectionfromthewingwasmost clearlyvisiblewere processedtoobtaintheangleofthewingateachinstanceint ime.Therststepofthe processwastodeterminethephysicalcoordinatesoftheima geswhichwerecalculated fromacalibrationimage.Thenthepixelsofthesnapshotswe regroupeddepending ontheiranglefromthecenterofrotationofthewing,into19 9equallyspacedintervals between1 and88 .Pixelstoofarfromthecenterofrotationtopossiblycontai nthe wingwereomittedfromtheanalysis.Thentheaverageintens ityofthepixelsineachof theseintervalswasevaluated,resultinginasharpmaximum intheslicecontainingthe wing.Toavoiderroneousdatapoints,aminimumintensityth resholdtoomittheresult whenthewingwaslocatedoutsidespeciedangleswasapplie d. Sincethecapturefrequencywasknownandthemotorappedthe wingsatavery stableconstantfrequency,asinewavecouldeasilybettot hewingangleasafunction oftime.Fromthet,thephasewithintheappingcycleofeac hsnapshotcouldbe extracted.5.2.3Results ThecoordinatesystembeingusedcanbeviewedinFigure 5-3 where x isthe spanwisedirectionand z isthechordwisedirection.Theoriginisattheintersectio nof thecenterofrotationofthewingandtheleadingedge. Snapshotsofthevelocityeldwereacquiredat12differentc hordwiselocations, showninFigure 5-4 ,withthewinglocatedasshownintheplotsofFigure 5-5 .Ateach chordwiselocation,2or3seriesof815imagepairswereacqu iredforeachcamera. Whenthewingwasappingat5Hz,snapshotswerecapturedatar ateof101Hz, whileataappingfrequencyof10Hz,snapshotswereacquire dat199Hz,asshown inTable 5-1 .Thechordwiseposition z ,alongwiththenumberofsamplescollectedfor 89

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eachposition,arelistedinTable 5-2 .Theappingfrequenciesbeingexaminedhere,5 and10Hz,correspondstohoveringReynoldsnumbersRe hover ofapproximately2,130 and4,260respectively. Additionalmeasurementswereacquiredwhentheapperhadbe enmovedup 50mmverticallyinanattempttobeabletocapturethevortic esbeingejectedaround thewingreversalbeforetheupstrokeoftheappingcycle.D etailsoftheseadditional measurementstakenarelistedinTable 5-3 Thevelocitysnapshotsarecollectedinto“bins”ofthesame appingcyclephase (withinthesame1/50 th ofaappingcycle)andaveraged.At5Hzandattherstvertica l locationoftheappingdevice,resultshavebeenobtaineda t12differentchordwise locations.Iso-surfacesofthevorticitycalculatedfromt hesephase-averagedvelocity eldsarepresentedinFigures 5-5 and 5-6 .Figure 5-5 showsthechordwisevorticity, andFigure 5-6 showsvorticityintheinstantaneousspanwisedirection,i .e.alignedwith thecurrentdirectionofthewing.Theappingamplitude A isapproximatedtobe45 Figures 5-5 and 5-6 showhowvorticityappearsaroundthewingwhenit'smoving throughouttheappingcycle. =0.00occurswhenthewingisatpeakangle.Asthe wingstopsatthetopofthecyclebeforereversing,mostofth evorticityisdisappears. Asthewingapsdownagain,itleavesastreakofoppositesign vorticitybehind.This processcanbeviewedintheiso-surfacesofchordwisevorti cityplottedinFigure 5-5 Ontheupstroke,clockwiserotatingvorticityistrailingt hewinguntilapproximately = 0.16wherethewingbeginstodecelerateandfromtherethevo rtexstructure appearstobeshedawayfromthewingtodriftupwardanddecay .Additionally,as thewingisgoingthroughtheprocessofstoppingtochangedi rection,theformation ofcounterclockwiserotatingvorticitycanbeviewed.Thev orticitythenbeginstobe draggeddownasthewinggoesthroughthedownstrokepartoft hecycle.Overallthe processisquitesimilarforthespanwisevorticityviewedi nFigure 5-6 .Althoughfor thiscomponentthereisafronttobackasymmetryinthesigno fthevorticity.Onthe 90

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upstrokethereispositivevorticityonthefrontofthewing andnegativevorticityinthe backofthewing.Inamannersimilartothatofthestreamwise vorticity,vorticityisshed asthetopofthestrokeisreached,andonthedownstrokevort icityofoppositesignsis formedalongtheleadingandtrailingedgesofthewing. Inordertoextendtheviewingareaatthechordwiselocation swhereasecondset ofdatawastakenatadifferentverticallocation,theavera gevelocityeldsfromthe twodifferentverticallocationsaremergedtoformalarger eld.Theoverlapregionis calculatedbylinearlychangingtheweightbetweenthetwov elocityeldssothat: u ( x y )= u (1) ( x y ) 1 y y (2) min ( x ) y (1) max ( x ) y (2) min ( x ) + u (2) ( x y ) y y (2) min ( x ) y (1) max ( x ) y (2) min ( x ) ,(5–9) where(1)and(2)denotestheloweranduppervelocityelds, respectively. y (1) max ( x ) denotesthehighest y forwhichavelocityisdenedforthelowervelocityeldata given x location,andthecorrespondingistruefor y (2) min ( x ).Itshouldbenotedthatalthough thedatabeingpresentedismergedthroughthisweightingme thod,theoverlaplooked quitegoodimplyingtherewereminimalalignmentissuesbet weenthetwoacquisition domains.ThecombineddomaincanbeseeninFigure 5-7 Theresultsfromthecombineddomainwillnowbepresentedat z = 6.25mm, whichisthequarterchordplane,wherethespanofthewingis atitsmaximumlength. Themeansquare w velocityintheentireeldisapproximatelyanorderofmagn itude smallerthanthe u and v components,andisthereforeomittedfromtheseplots. Normalizedvorticityplotsoftheextendedregionareshown inFigures 5-8 and 5-9 ,for5 and10Hzrespectively,andwithstreamlinesvisualizedusi nglineintegralconvolutions superimposed. Figure 5-8 shows,inmoredetailthanFigure 5-5 ,howpositivechordwisevorticity diffusesoutfromunderthewingassoonasitstartsdecelera ting.Thepositivevorticity theninteractswiththenegativevorticitycreatedbytheac celerationofthewingbefore thewingwasvisibleinourwindowproducingavelocityupwar dsfromthewing. 91

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Interestingly,thisupwardsstreamisvisibleatthesamelo cationalreadyat = 0.20,so itappearsthattheowisconstantlyowingupwardsatthisl ocation. At10Hz,ascanbeseeninFigure 5-10B ,thewingdoesnotslowdownnoticeably untilitreachesitsmaximumangle.Figure 5-9 showsthatpositivevorticitydoesnot showupbeforeveryshortlyaheadoftheturningpointofthew ing.Thevortexshedding inthiscasecreatesavelocitytowardsthecenterlinebetwe enthewings,inadirection closetoperpendiculartothewingdirectionwhenthewingst ops.Sincetheapper issymmetrical,theowcannotowfarinthisdirectionbefo rereachingtheplaneof symmetry.Thismaybethereasonwhythenormalizedupwards owespeciallyat = 0.20ismuchweakerthaninthe5Hzcase. Unfortunately,duetopoorseedingqualityinthetopleftco rner,itcannotbe quantitativelydeterminediftheamountofaircominginfro mthetopleftcorner,i.e. thespacebetweenthewings,correspondstotheairthrownaw aybythewing,ormost ofthisaircomesinthe z -direction. Figure 5-10 showshowthewinganglevarieswithtimeintheappingcycle forthe twoappingfrequenciesinvestigated.Clearly,thecyclei sverydifferentat10Hzandis morerepresentativeofatriangularwavethanthesinusoida lmotionobservedat5Hz. Thewingangledetectionalsorevealsthatat10Hz,eitherth egearsareslippingor theappingdevicefailstomaintainthecommandedappingf requency,astheactual appingfrequencyisapproximately9.5Hz.Thisproblemdid notoccurat5Hz.Asa result,theoriginalphasesynchronizationmethoddoesnot workproperlyatthehigher appingfrequency.Instead,thephaseangledetectionmeth odwasusedtolocateone angularpositionofthewingforeachupstroke,andoneeachd ownstroke,andthenthe phaseprogresswasevenlyspacedintimebetweentheknownwi ngangles. 5.2.4Summary AthinZimmermanplanformwingwasappedinasimplemotioni nstillairat frequenciesof5and10Hz.Three-componentPIVmeasurements wereacquired 92

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inseveralplanessuchthatthephase-averagedvelocityel dcouldbecalculated throughouttheappingcycle.Examinationofthevorticity eldshowedtheelimination ofvorticityonthewingasitwasapproachingtheturningpoi ntintheappingcycle.Just beforethetopofthecyclethepositivechordwisevorticity isformedatthewingtipwhich isthenstretchedthroughoutthereturningdownwardstroke .Thevorticitythroughout thecycleappearstoconsistentwiththemeasurementsofHon gandAltman[ 34 ],who relatedthistotheforcesgeneratedonthewing. Three-dimensionalvelocityeldswereaveragedoverappro ximately30-50velocity snapshotsforeachlocationandphase.Itshouldbenotedtha tthereweresome differencesinthebehaviorofeachappingcycle,hencethe needforphaseaveraging. However,duetothehighnumberoflocationsatwhichsnapsho tsaretaken,thenumber ofsnapshotsateachlocationisnotashighasonewouldwish. Theconvergenceof thephase-averagedvelocityhasbeenanalyzedandthesampl eusedforthisstudywas sufcientinsomelocationsbutnotatall. Thisexperimentshowsthatitwouldbeadvantageoustobeabl etousemore pointsinthephaseaverages.Italsoshowstheneedforahigh erqualityapper,to haveidenticalappingmotionatdifferentappingfrequen cies,andtobeableto directlycomparetheeffectsoftheappingfrequency.Becau seofimperfectionsinthe manufacturingoftheappingdevice,theangletoappingph aserelationshipisquite differentwhenappingat10Hz.Whiletherelationshipisclo setosinusoidalat5Hz, itisshapedmorelikeatriangularwaveat10Hz.At5Hz,positi vechordwisevorticity canbeseentocomefromunderthewingtipsoonafterthewings tartsslowingdown. At10Hz,duetothelateandsuddenchangeofangularvelocityo fthewing,almostno positivevorticityseemstobeproducedunderthewinguntil itreachesthemaximum angle. 93

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5.3L m B1Wings TheL m B1wingsarelightweightmembranewingswithanisotropicex ibility.The introductionofexibilityallowsthewingstoproducethru st,andthelowweightofthe wingspermitsahigherappingfrequencythanthepreviousl ytestedrigidwings.It wouldbeidealtotestthewingsunderexactlythesamecondit ions.However,ahigher appingfrequencyisneededtoproducesignicantwingdefo rmationandthrust.The non-sinusoidalbehavioroftheapperathighloadsalsocal ledforabetterbehaving appingdevice.5.3.1ExperimentalSetup Theexperimentsconductedhereconsistedofmeasuringthe oweldaround threedifferentappingwingswithanisotropicexibility .Thissectionwilldiscussthe experimentalsetupusedtomeasuretheowaroundthewings. Theexperimentswere conductedinthehoveringenvironment(seeSection 4.6 ). 5.3.1.1Wing TheL m B1wingsareexiblemembranewings,manufacturedasdescrib edin Section 4.3.1.2 .ThewingsarenamedL m B n ,afterthenumberoflayersofunidirectional carbonberthatisusedtoreinforcetheleadingedge( m )andthebattens( n ).Awing isshowninFigure 5-12 ,wherethemembranehasbeenpreparedfordigitalimage correlationmeasurementsandthecarbonberreinforcemen tsareclearlyvisible.For allofthewingstestedhere,thebattenreinforcementwashe ldthesamewithone layerofthecarbonber,i.e., n =1.Therootisalwaysreinforcedwithtwolayers.The leadingedgeandrootreinforcementareconnectedbyatrian glemadebythreelayersof bidirectionalcarbonber,aswasdiscussedinSection 4.3.1.2 Wuetal.[ 42 ]hasinvestigatedthestructuralpropertiesofthesewings ,including stiffnesscomparisons,dynamicresponsemodes,andwingde formation.Digitalimage correlation(DIC)wasusedtomeasurethewingdeformationw hileappingthesewings at25Hzinbothairandvacuum.Thewingdeformationissigni cantlylargerinairforall 94

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wingsat25Hz,indicatingthataerodynamicforcescontribu temoretowingdeformation thaninertiaofthewing.Figure 5-11 showshowthemovementofthefarendofthe winglagsthewingrootinaircomparedtovacuum,anddemonst ratesthataerodynamic forcesclearlyincreasethislag. Theappingfrequency,25Hz,waschosenasitrepresentsani nterestingpoint inthethrustcurvesforallthreewings,asshowninFigure 5-13 .Atthisfrequency, L1B1'sthrusthasalreadyplateauedandL2B1generatesmoreth rustthanL3B1.At higherfrequencies,L3B1producesmorethrustthanL2B1.Since theprimarydifference betweenthesewingsisthenumberofcarbonberlayersinthe leadingedge,these resultscanbeinterpretedinthecontextofleadingedgesti ffness.Hence,theL2B1wing seemstohavealeadingedgestiffnessthatisclosetooptima lformaximizingthrustat thisappingfrequencyandtheotherwingsareeithertoosof tortoostiff.Interestingly enoughforhigherfrequencyappingandhigherReynoldsnum ber,thewingswiththe stifferleadingedgeendupout-performingbothoftheother wings.ItwasshowninWu etal.[ 42 ]thatthetrendsarenotasmuchaffectedwhenthechordwised irectionwas stiffenedaswhenthespanwisewas.Thispartofthestudywil lonlyconcentrateonthe differencesassociatedwithalteringthespanwisereinfor cement. 5.3.1.2Flappingdevice Thetype2appingdevice(seeSection 4.2.2 )wasusedtoapthewingsforallof theexperimentsonthesewings.Theappingamplitudewasse tto 35 5.3.1.3Particleimagevelocimetrysetup ThestereoscopicDantecPIVsystem,describedinsection 4.1.1 ,wasusedforPIV measurements.Thecamerasweremountedatnearly45 anglestotheimageplane toproducesnapshotsforstereoscopicPIV,assketchedinFig ure 5-15 .Thelaserwas directedatthetargetfrombelow. 95

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Expancel TM microsphereswereusedforseeding.Theseedingwasdispers ed throughatubegoingthroughthewallofthetestsectionconn ectedtoafaninsidethe testsection. TheDantecFlowManagersoftwarewassetuptousethesetting sinTable 5-4 .The wingswereappingat25Hz,andthesnapshotswereacquireda tarateof102Hz,to ensurethatanadequatenumberofsampleswereacquiredinal loftheappingphases. 5.3.2DataAnalysisMethods ThissectiondescribeshowmethodsdescribedinChapters 2 and 3 wereappliedto thedatacapturedforthecurrentexperiment.5.3.2.1Particleimagevelocimetry ThesnapshotswereprocessedusingLaVisionFlowMaster.The multi-pass correlationalgorithmwasused,withonepassusinga128 128pixelinterrogation area,andtwopassesusing32 32pixels.50%overlapwasusedbetweenneighboring interrogationareas. Atwo-dimensionaltargetwascapturedat6differentdepthwiselocationsto calibratethecoordinatesystem.Theself-calibrationalg orithmavailableinDaViswas usedtoimprovetheaccuracyofthecalibration. Theresultingwindowconsistsof84 60datapointsfortheL1B1andL2B1cases, and76 54datapointsfortheL3B1case.Inallcases,thewindowcorre spondsto approximately116 83mminphysicalspace,withsome“dead”datapointscloseto theedges.Acamerapixelcorrespondstobetween0.83 10 4 0.70 10 4 mand 1.23 10 4 0.88 10 4 minphysicalspace. 5.3.2.2Synchronization Forproperinterpretationofthevelocityeldthroughoutt heappingcycleitis crucialtoknowwhereintheappingcyclethePIVmeasurement sareacquired.Since thefeedbackfromthemotorwasnotaccurateenoughfordetai lingthephaseofthe appingcycle,aprocesswasdevelopedbasedontherawPIVima ges.Theprocess 96

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wasdifferentfromtheoneusedinthemeasurementsontherig idwingsincetheview angleandwingshape(duetodeformation)changedbetweenth eexperiments. Forsynchronizingtheoweldandwingdata,rsttherawima gesfromboth camerasweretransformedtothesamecoordinatesystemandm appedontothesame grid.Thetransformedimageswerethenmultipliedtogether pixel-wise.Thisnewimage wastheninterrogatedtondthelargestcontiguousbrighta rea.Thisareawasassumed toberepresentativeofthelocationwherethelasersheethi tsthewing.Ifnolarge enoughareaisfound,theimageisignoredforsynchronizati on.Otherwisethecenterof thebrightareawasdocumented.Asinewavewasthenttother ecordedvalues.Since theimagesofthesameseriesaretakenwithaconstantdistan ceintime,thismethod givesusthephaseofallthesnapshotsincludingthosethatw erenotusedtodothet. FortheDICwingdata,acontiguoussequenceofsnapshotswas chosen,andthe average y locationofthewingateachofthemeasuredspanwiseplanesi scalculated foreachsnapshot.Asinewavewasthenttothe y locationofthewingateach z ,asa functionoftime. =0.00wasselectedtobewhentheangleofthewingrootis0 and increasing.Thephaseoftheowelddatawaschosensothatt hephaseofthesine wavesttothePIVandDICdatacoincide.5.3.2.3Masking Duetolargeamountsofscatteredlightfromthesurfacesoft hewingwhich wouldcorruptthePIVdata,anautomatedprocedureofremovin gtheseareasbefore processingwasdevelopedasdescribedhere.Eachrawimagewa stransformedtothe physicalcoordinatesystem.Then,contiguousareasbright erthanthedarkbackground werelocated.Thenodesthatweredetectedasbrightinatlea st10%ofthesnapshots ofthesamephaseweremaskedout.Unfortunatelypartsofthe wingoutsidethelaser sheetareoftenclearlyvisiblecausinglargesectionstobe maskedout.However,aswill beseenbelow,therewasstillenoughdatatoobtaininformat ionabouttheow. 97

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5.3.3Results Thissectionwillconcentrateondiscussingthemeasuredve locityandvorticityelds aroundtheexibleappingwings.Asindicatedabove,alongw iththemeasuredvelocity elds,theactualmeasuredwingshapes[ 44 ]willbeincludedwiththeowsuperimposed onthem,tofacilitateabetterunderstandingoftheuid-st ructureinteractions. Snapshotsofvelocityeldswereacquiredforeachwingat7di fferent z locations, showninFigure 5-16 ,at z =20,30,40,50,60,70,and77mm.Ateach z location, 3seriesof815imagepairsforeachcamerawereacquired.Hen ce,2445snapshots areavailableforeachwingandspanwiselocation.However, fortheL3B1wingat z =50mm,thequalityofoneofthethreedataserieswassopoori twasomitted, leavingonly1630snapshotsforthatparticularcase.There sultinggridafterprocessing datato3dimensionalvelocityvectorsisshowninFigure 5-17 .Thecoordinatesystem beingusedcanbeviewedinFigure 5-18 where x isthechordwisedirection,and z is thespanwisedirection. y and z arezeroatthecenterofrotationofthewingand x =0 attheleadingedgeoftherootofthewing.Inordertorecreat ethephasesthroughout theappingcycle,velocitysnapshotswerecollectedinto“ bins”ofthesameapping cyclephase(withinthesame1/50 th ofacycle)andaveraged.Onaverage,eachbinthus representsthemeanof49snapshots. Theappingfrequencybeingexaminedhere,25Hz,correspon dstoahovering ReynoldsnumberRe hover ofapproximately7,300.Itshouldbenotedthatthevelocity scaleisdeterminedassumingatheoreticalrigidwingandhe ncetheyallhavethesame hoveringReynoldsnumber.Thecenterofrotationofthewing islocatedat( y z )=(0,0). =0iswhenthewingangleiszeroandincreasing. Figures 5-19 5-20 ,and 5-21 showthe z vorticityaroundtheL1B1,L2B1,and L3B1wingsrespectivelyforseveralphasesintheappingcyc le.Thearrowsshowthe velocityvectorsinthe x y plane.ThedeformedwingshapesfromDICmeasurements areincorporatedintheplots,withtheexceptionofathinbo rderaroundthewing. 98

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Additionally,thereissmallgureintheupperlefthandcorn erwhichshowstheshape ofthequarterchordlineofthewing,andthedirectionofits localvelocityattwopoints alongthespanofthewing. Figure 5-19 showstheowaroundtheL1B1wing,i.e.thewingwithlowest spanwisestiffness.At = 0.09thewingtipisonitswaydown,i.e.hasanegative v velocitycomponent,trailedbyapairofregionswithopposi tesignvorticity,mostly visiblefortheouter2/3'softhewing.Noticethatthewingr ootisalreadyonitswayup andclosetoa0 angle.At =0.01,thewingrootisclosetoa0 angle,butthewing tipisclosetoitsmaximumangle,showingthelargephaselag betweenthewingroot andthewingtip,aswasdiscussedinWuetal.[ 44 ].Thevorticitypairtrailingthewingis mostlyinthemaskedregion.At =0.11,thepreviouslytrailingvorticityisclearlyvisibl e again,asthepositivevorticityfromthetrailingedgeofth ewingispushedawayfromthe wingbynewlycreatednegativevorticity,seeminglysplitt ingupthepositivevorticityinto smallervortices.Therapidtwistingofthewingaroundthel eadingedgeseemstocause muchstrongervorticityatthetrailingedgethanatthelead ingedge.Therestoftheplots showfurtherdevelopmentoftheowthroughouttheappingc ycle.Itisnoticeablethat thereisneitherstrongvorticitynorowinthefardownstre amendoftheplots.Mostof theworkthewingdoesontheowseemstoaddmomentumtotheo walongthe y and z axis,whichisnotcontributingtoanythrust.Thisisconsis tentwiththethrustdatain Figure 5-13 .However,intheplanesat z =70mmand z =77mm,i.e.theouterpartof thewing,thereisclearlyapositive u velocitycomponentthroughouttheappingcycle. Figure 5-20 showsthevorticityaroundwingL2B1.Itshouldbenotedthatw iththis stifferwing,thephaselagbetweenthemotionoftherootand thetipofthewinghas signicantlyreduced.At = 0.37,atrailofvorticityisshownbehindthewingapping down,muchlikeintherstplotinFigure 5-19 oftheL1B1wing.ForthestifferL2B1 wing,thisoccursmuchearlierintheappingcyclesincethe phaselagbetweenthe wingrootandwingtipissignicantlysmaller.Thevorticit ycreatedthroughthestroke 99

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isnowstronger,whichcanberelatedtothefactthatthisles scompliantwingmoves throughalargervolume.At = 0.27and = 0.17,thepositivevorticityfromthe trailingedgecanbeseentobreakupintotworegionsofvorti city.Oneofthesethen seemstopairupwithnegativevorticitycreatedinthebacks trokecreatingowinthe x direction,butatadownwardsangle.Thisvorticitystructu removesdownstreaminthe subsequenttimes,inamannerwhichismuchdifferentfromwh atwasseenintheow aroundtheL1B1wing.Thedifferencesbetweenthesewingsare alsoquiteapparentin theoweldclosertotherootofthewing.Oneobservesthevo rticitygenerationinall oftheplaneswherethevelocitywasmeasured.Additionallyt hereisamuchstronger u velocitycomponentgeneratedwhichnowstretchescloserto thewingroot. ThevorticityeldaroundtheL3B1wingisshowninFigure 5-21 .At = 0.41,the wingisveryclosetothesamepositionastheL2B1wingat = 0.37,withasimilar trailingvorticitypattern.Thephaselagbetweentheroota ndthewingtipisevensmaller forthisstifferwingasexpected,butwithamuchlessdramat icchangecomparedtothe L1B1wing.Thedevelopmentoftheowthroughthecycleisvery similarfortheL2B1 andL3B1wing. TheexibilityoftheL2B1wingcausesitswingtiptomoveboth fasterandalonger distance,creatingstrongervorticity.Thelongerpathoft heL2B1wingcanbeseen inFigure 5-24 andthismaybelinkedtogreaterthrustgenerationofthiswi ngatthis appingfrequency. Figure 5-22 isrepresentativeofthewingexibilityandshowsalineoft hewing shapetakenatthequarterchordlocation.Theseplotsareus efultohelpunderstand thelocationofthewingthroughouttheappingcycletocont rastthedifferencesseenin theoweldsdiscussedabove.InFigure 5-22A ,theprescribedmotionofallthewings isonitswayup.Therigidtriangleatthecornerofthewingsi satthesamepositionfor allthreewingsatthistime.Thetrianglehasananglejustsl ightlyover0 ,butthetipof theL1B1wingisstillnearitslowestposition.InFigure 5-22B ,at =0.31,thetwostiffer 100

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wingsareveryclosetotheirpeakpositionwhiletheL1B1wing isstillonitswayupand hasalongwaytogo.Figures 5-22C and 5-22D showaverysimilarypatternforthe secondhalfoftheappingcycle. Figure 5-23 showsthevorticityaroundtheL2B1wingat z =50mmfromanend onperspective.Intheseplots,thePIVdatahasbeenmirrored at y =0mmandthe mirroredparthavebeenshiftedhalfaappingcycle.Theint ersectionoftheoriginal andmirroreddatahasbeeninterpolated.At = 0.37,itisclearlyseenhowclockwise vorticitytrailsthewinginthepathofthewingstroke,wher eascounterclockwisevorticity staysonthedownstreamsideofthewing.Thisdiffersgreatl yfromtherigidwingcase withnotwistanglewherethepositiveandnegativevorticit yismoresymmetrically trailingthewing.Inthefollowingtwoplots,at = 0.27and 0.17,thevorticityin thetophalfoftheplotskeepmovingdownstream,andthecloc kwisevorticitykeeps trailingthewing.At = 0.03,thewinghasturnedandtheclockwisevorticitythatwa s trailingthewinghasbeenshed.Thevorticityonthebottomh alfoftheplotsthenmoves downstream.Theclockwiseandcounterclockwisevorticity linesupcreatingaowata diagonaldownward/downstreamangle. Asameanstovalidatingthedata,thestreamwisemomentumper unittimeand area, u 2 ,isintegratedover y and z oftheknownPIVdomainforwhich y < 0.The valueshasthenbeendoubledtwice,oncetotakeintoaccount boththeassumed symmetrylineat y =0,andthenagaintotakeasecondwingintoaccount,sothat thevaluesarecomparabletheforcemeasurementsinFigure 5-13 .Theintegrated streamwisemomentumisplottedinFigure 5-25 Thestreamwisemomentumcorrespondstooneoftheintegrand termsinEquation 3–52 butalongthe x axisratherthanthe y axis.Datahasbeenmaskedwherethewingis disturbingtheviewandhenceitisnotpossibletouseEquatio n 3–48 torelatethe relativepressurebetweenthetwosidesofthemaskedregion .Theviscoustermsof Equation 3–49 ,thatEquation 3–52 wasderivedfrom,areestimatedbydimensional 101

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analysistobethreeordersofmagnitudeslowerthanthemome ntumtermswhere momentumissignicant: @ u i @ x i O (10 5 ) O (1) O (10 2 ) = O (10 3 ),(5–10) u 2 O (1) O (1 2 )= O (1).(5–11) Hence,theviscoustermswillbeignored.Theincreaseinstr eamwisemomentum overthemaskedintervalisapproximately5,46,and40mNfor theL1B1,L2B1,and L3B1respectively,whichcanbecomparedwith11,34,and25mN fromtheforce measurements.ThepressureterminEquation 3–52 willmostlikelyhaveanegative contributionontheforceoverthegap,sincehighervelocit yuidisexpectedtohavea lowerpressure.Theuxesofmomentumthroughthesidesnorm altothestreamwise directionhavenotbeentakenintoaccount,sincetheseareu nknownovertheinterval ofinterestduetothemasking.TheL1B1wingaffectstheowe ldmoreindirections normaltotheowdirectionrelativetotheitsinuenceinth estreamwisedirection, comparedtotheothertwowings.Thismayexplainwhythestre amwisemomentum analysisunderestimatestheforce,ratherthanoverestima tingitasfortheothertwo wings.Duetothelackofdatainthemaskedregion,adirectco mparisoncannotbe madetoseeifthePIVandforcedatagivethesameresult.Howev er,thiscomparison atleastshowsthatthemeasuredthrustandthereducedsetof componentsofforce computedfromPIVdatagivesvaluesthatdonotnecessarilyin validateeachother. 5.3.4Summary Theairowaroundthreewingsappinginasimulatedhoveren vironmentwith varyingexibilityhasbeeninvestigated.Theexperiments showhowawingthatistoo compliantinthespanwisedirectionsuffersfrompoorperfo rmanceproducingthrust.On theotherhand,whentheleadingedgeistoorigidthewingdoe snotreachthesame velocityorcoveraslargepathasawingthatislessexible. Intheseexperiments,the twoleastexiblewingsshowedsimilarowpatternsinthesp anwisedirection,butthe 102

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moreexiblewinghashigherthrustproduction.Hencethere mustbesmalldifferences intheoweldsandwingdeformationsrelatedtothis. ThemostexibleL1B1wingmostlycreatedowalongthe y axis,whichdoesnot accountforanythrustproductionatall,assumingthesyste missymmetricaround y =0. Thetwostifferwingsbothproducedowmainlyinthediagona ldownstream/downwards directionlookingattheouterportionofthewing,andassum ingsymmetry,alsothe downstream/upwardsdirection.Possibly,amoreefcientw ingwouldproduceowmore inastraightbackwardsdirection. 5.4AxxxxWings Thissectiondescribesexperimentsconductedontwosetsof wingswithdifferent chordwisereinforcements.Thespecicsofthewingswillbe describedin 5.4.1.1 ,while theresultsfromtheexperimentswillbediscussedinSection 5.4.3 .Theairowaround thewingswasmeasuredusingPIV.Themeanthrustproductiono ftheseexiblewings hasbeenreportedbyWu[ 4 ]. 5.4.1ExperimentalSetup Theexperimentsconductedhereconsistedofmeasuringthe oweldaroundtwo differentexibleappingwings.Thissectionwilldiscuss thewingsandtheequipment usedtomovethewingsandmeasuretheowandforces.5.4.1.1Wing TheAxxxxwingsareanisotropicmembranewingsmuchlikethos ediscussedin theprevioussection.Thewingsweremanufacturedasdescri bedinSection 4.3.1.2 andthewingrootofbothwingsisreinforcedbytwolayersofc arbonber,asisthe leadingedge.Thebattensarereinforcedbyasinglelayerof carbonber.xxxxinthe wingnamesindicateswhichbattenlocationsonthewing,ass howninFigure 4-8 ,that arereinforcedwithcarbonber.A1meansthatabattenispre sent,anda0meansitis not.ThetwowingsthatwereinvestigatedusingPIVaretheA000 1andA1101wings, showninFigure 5-26 .Thewingswerechosenoutofthesetofall16possibleAxxxx 103

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wings,andasetof16Bxxxxwings,whereBmeansthattheleadin gedgeisreinforced byanadditionallayerofcarbonberfromtherootchordtoth ethirdbattenposition.The selectedwingsweretwoofthemostthrustproducingwingsou tofthealltestedonesat thechosenfrequency,15Hz.Theaveragethrustforboththes ewingsisapproximately 11mNat15Hz[ 4 ],whichcorrespondstoathrustcoefcientof0.82. 5.4.1.2Flappingdevice ThewingswereappedusingtheType2appingdevicedescrib edinSection 4.2.2 Theappingfrequencywaschosentobe15Hz,asacompromiseb etweendurabilityof theapperandwingsandtheproducedthrust.5.4.1.3Particleimagevelocimetrysetup Thissectiondescribestheequipmentandphysicalsetupuse dformeasuringthe airow.Detailsofhowthedatawasprocessedwillbediscuss edinSection 5.4.2.1 ThestereoscopicLaVisionPIVsystem,describedinsection 4.1.2 ,wasusedfor PIVmeasurements.Thecamerasweremountedatindependentan glestotheimage plane,assketchedinFigure 5-27 ,toproducesnapshotsforstereoscopicPIVandfor generationofthree-componentvelocityvectors.Themeasu rementsweretakeninthe hoveringenvironment,describedinSection 4.6 .TheairwasseededbyaLaVisionDS AerosolGenerator. Pixellockingconsiderations Thesetupfortheseexperimentshadanaveragemagnication ofapproximately 0.15,andan f = #of8.Thediffraction-limitedspotdiameter,asdescribed in 2.4.2 ,ofa seedingparticlewiththecurrentsetupisontheaverageapp roximately12.0 matthe imageplane,oraround62%largerthanthesideofapixelofth ecameras'CCDs. 5.4.1.4Synchronization TorelatethetimebetweenthePIVmeasurements,theforcemea surements,and themotorlocation,aprogramwasdevelopedaccordingtothe methoddescribedin Section 4.5 .Theprogramcouldcontinuouslyprocesstheencoderoutput signaland 104

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trackthemotionafteritsmotioncounterwasresetatstands till.Theprogramstarted samplingdataonbothchannelswhenitreceivedaPIVtriggers ignal.ThePIVtrigger signalcouldthenbeusedtosynchronizethetimeofthedatas ampledonthetwoDAQ cards.Hence,themotorposition/appingphase,andthefor cetransduceroutputcould bothberecordedatthetimeofeachPIVsnapshot.However,due torelativelylarge inuencefromvibrationsandsmallaerodynamicforces,the instantaneousaerodynamic forcescouldnotbedeterminedfromtheforcemeasurements.5.4.1.5Forcemeasurements ForcemeasurementsweretakenusinganATITitaniumNano17f orcetransducer, introducedinSection 4.4.1 .Theapperwasmountedontheforcetransducersothat the x -axisofthetransducerwasalignedwiththestreamwisedire ction,andthe z -axis wasalignedwiththespanwisedirection.Theforcetransduc ersignalwassampledat 35kHz.5.4.2DataAnalysisMethods ThissectiondescribeshowmethodsdescribedinChapters 2 and 3 wereappliedto thedatacapturedforthecurrentexperiment.5.4.2.1Particleimagevelocimetry TherawPIVimageswereprocessedusingLaVisionDaVis7.2witht heLaVision FlowMastermodule.Theimagewereprocessedbyemployingan iterativecross-correlation algorithmwhichusesFFT,withonepassusinga64 64pixelwindowsizeandtwo passesusinga32 32pixelwindowsize.Inthenalpass,Whittakerreconstruct ion isusedtoachievesub-pixelaccuracy.50%overlapisusedbe tweenneighboring interrogationareas.Thesoftwareusesacalibrationgener atedfromatwolayertarget, toconvertthetwo-dimensionalpixeldisplacementvectors tothree-dimensionalvelocity vectors.Theself-calibrationalgorithmavailableinDaVis wasusedtoimprovethe accuracyofthecalibration. 105

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Pre-processing .Duetoreectionsfromthewingsandtheapper,some pre-processingwasperformedtotherawimagestoimproveth eabilityofthePIV algorithmtotrackparticles.Thiswasaccomplishedbyusin ganorder-statisticlterthat foreachpixelintheoriginalimagetakestheclosest5 5pixelregion,sortsthese pixelsvalues,andplacesthe5 th smallestvalueofthesortedvaluesinalteredimage. Thelteredimageisthensubtractedfromtheoriginalimage tocreateapre-processed image.Thisturnedouttobeamoreeffectivewayofremovingb ackgroundillumination, originatingmostlyfromreectiveout-of-planesurfaces, thanforexamplesubtractingthe meanofthesameregion.Ameanvaluecan,contrarytocharact eristicbackground intensity,increasesignicantlyinthevicinityofpartic lesastheparticleintensity increasesorevenduetothehighintensityofthescatterfro masingleparticle. Masking .Amaskwascomputedfortheprocessedvectoreldsfromther awPIV images.Thiswasdonebyrstaddingtogetherallrawimagesf romthesamecamera, exposurenumber,andphaseinterval.Then,thepartoftheim ageswherereections mayoccurwascutoutforfurtherprocessing.Ablurredimage wascreatedbyapplyinga two-dimensionalGaussianlterwitha6pixelstandarddevi ationonthecutoutsection. Then,tondedgesintheblurredimagethe3 3derivativeoperatorgivenbyJ ¨ ahneet al.[ 81 ]wasused: D = 1 32 0BBBB@ 30 3 100 10 30 3 1CCCCA .(5–12) Theabsolutegradientoftheblurredimagewasfoundbytakin gtheconvolutions betweentheblurredimageandboth D and D T ,andusingtheroot-mean-squareof eachpixellocationfromthetwoconvolutionsasthegradien t.Atthispoint,theabsolute gradientimagesfromtherstandsecondframefromthesamec amerawereadded together.Next,pixelsweremarkedasedgepixelsbyrstmar kingapixelasanedge pixelifitwasoveracertainthreshold,thenneighborsofpi xelsmarkedasedgeswere 106

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alsomarkedasedgesiftheywereaboveadifferent,lowerthr eshold,inaoodll fashion.Thenalstepwastoaddeverypixelbetweentherst andlastverticalpixel locationthatwasmarkedasedgepixelsofeachhorizontalpi xellocationtothemask.If ahorizontallineofpixelsdidnotcontainanyedges,nothin gwasmaskedontheline. Datastraightaboveandclosetostraightabovethewingshap edetectedas describedinSection 5.4.2.2 wasalsomaskedout. ScopeofPIVdata .2500PIVsnapshotsweretakenat30spanwiselocationsof theA0001wing,and52locationsontheA1101wing.However,the apperneeded occasionalmaintenance,meaningoilandtighteningofscre ws,tonotbreak.Thelatter sometimesslightlychangedthebehavioroftheapper,whic hresultedinthedatanot beingcoherentthroughallthemeasuredlocations.Therefo re,onlyasubsetofthe obtaineddatawillbepresentedhere. Thesubsetofdatathatwillbeinvestigatedhereconsistsof 16spanwiselocations at28.6,30.6,32.6,34.6,36.6,38.6,40.6,42.6,44.6,46.6 ,48.6,50.6,52.6,54.6, 56.6,and58.6mmfromthewingrootoftheA0001wing,and14spa nwiselocations at20.6,22.6,24.6,26.6,28.6,30.6,32.6,34.6,36.6,38.6 ,40.6,42.6,44.6,and46.6 oftheA1101wing.Thetwoexposuresofeachsnapshotweresepa ratedby109 s, andtwosubsequentsnapshotswereseparatedby N T ,where N isanintegerand T =1 = 3.5s.Thewingswerealwaysappingat15Hz.Thethrustwasco ntinuously measuredwhiletakingPIVdata,butithasnotbeenpossibleto separatethewinginertia andvibrationsoftheapperfromtherelativelysmallaerod ynamicforces.Hence,only thetime-averagedforceshavebeenanalyzed. Theresultinggridafterprocessingdatato3dimensionalve locityvectorsforthe twowingsareshowninFigures 5-28A and 5-28B .Theonlydifferencebetweenthe twoisthewing'slocationinsidethegridalthoughbothmeas urementsallowedfor visualizationofthewingswake.Thegridhas121 126nodes,spanningapproximately 121 116mminphysicalspace.However,alargeportionofthetoph alfofthegridwas 107

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maskedtoavoidinterferencefromout-of-planepartsofthe wingbeingvisibleinthePIV images.ThecoordinatesystembeingusedcanbeviewedinFig ure 5-29 where x isthe chordwisedirection,and z isthespanwisedirection. y and z arezeroatthecenterof rotation,and x =0attheleadingedgeoftherootofthewing.Inordertorecre atethe phasesthroughouttheappingcycle,velocitysnapshotsar ecollectedinto“bins”ofthe sameappingcyclephase(withinthesame1/50thofacycle)a ndaveraged.Eachbin thusrepresentsanaverageof50snapshots.5.4.2.2Wingtracking Inordertodeterminethewingsshapeandlocation,theparts ofthewingilluminated intherawPIVimageswereexamined.Tolocatetheintersectio nbetweenthewingand thelasersheetinasnapshot,therawimagesfrombothcamera saremappedontoa uniformgridinphysicalspaceattheplaneofthelasersheet .Thesetransformedimages arethenmultipliedtogetherandthehorizontalpositionof thebrightestpixelateach discreteverticalpositiononthegridisstored.Next,asec ondorderpolynomialistted tothelocationsofthestoredpixels,usingthepixelintens itiesasweights.Astandard deviationisthencomputedofthedifferencesbetweentheco ordinatesusedforthet andthettedpolynomialcurve.Anylocationmorethan3stand arddeviationsawayfrom thepolynomialisomittedandthenthetisrepeateduntilno morelocationsareomitted. Finallya4 th orderpolynomialisttedtotheremainingpoints,andoutli ersmorethan 2.5standarddeviationsawayfromthepolynomialisiterati velyomitted.Thisallowsfora roughtrackingofthewing. Afterttingthepolynomialasdescribed,onlyasubsetofpoi ntsremains.These pointsareusedforasecondpolynomialt,bettersuitedfor averagingresultsfrom severalsnapshots.Arotationangle, ,andanoffset, x 0 arebothinitializedto0.Then, thetisperformedbyiteratingthefollowing.Theoffsetis subtractedfromallpoints, afterwhichtheyarerotatedanangle .Afourthorderpolynomialisttedtothepoints, with x asthedependentvariable. and x 0 areiterativelychanged,untilthetted x 108

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valuesatsmallestandlargest y usedforthet, y min and y max ,areboth0,and y min = y max .Thisallowsforaroughtrackingofthewing,anditalsomake sitpossibleto interpolatethettedpolynomial,theoffset x 0 ,therotation ,andthevisiblechord, 2 y max ,overseveralsnapshotsatalmostthesamephaseintheappi ngcycle. Partsofthewingareshadedinsomeoftherecordedplanesdur ingsomeof thephases.Whenthisoccurs,thealgorithmcanonlytrackpar tofthechord.Some unintendedbehavioralsooccursforthisreasonwhenwingsh apesareaveragedovera rangewithvaryinglengthofthechordbeingshaded,whichha ppensatphaseintervals wherethewingisclosetoleavingtheinvestigatedlasershe et. 5.4.3Results Theappingfrequencybeingexaminedhereis15Hzandcorres pondstoa hoveringReynoldsnumberRe hover ofapproximately4,400.Theappingamplitude A is 35 5.4.3.1Flowelds Figures 5-31 and 5-32 showthenormalized Q andvorticity Z fortheA0001 wingforvariousplanesandphases.Figures 5-33 and 5-34 showthesamequantities fortheA1101wing.Thetwo-dimensionalplotsareshadedbyst reamlinesthatare computedusinglineintegralconvolution,andwhoseshadin gintensityisscaledwith absolutevelocityinthe x y plane. =0.00iswhenthewing'sappingangleis zeroandincreasing.Thephase-averagedwingshapeandloca tion,asdescribedin Section 5.4.2.2 ,isplottedasablacklinetogetherwiththeowdata. Lift,whichinthecaseofhoveristhesameasthrust,iscreat edwhencirculation isbuiltuparoundthewing.Thevorticityplotsshowvortici tybeingshedatthetrailing edge,asisrequiredbytheKuttacondition.Inatwo-dimensi onalow,thesameamount ofcirculationwillbeboundbythewingasisshed,according toKelvinscirculation theorem[ 17 ].Takingintoaccountthethirddimensionandtherotationo fthewing, circulationwillstillbeboundtothewing,withthechanget hatitcantravelalongthespan 109

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ofthewing.Whenthecirculationbecomestoolarge,thewings talls,andthecirculation becomesunsustainableandwillbeshed.Theguresdisplaye dinthissectionwill describethemotionofthevorticesastheyleavethewing. Asthewingchangesdirection,thecirculationmustalsochan gedirectiontocreate liftagain[ 17 ].Alargeangularaccelerationattheendofeachhalf-strok emayshedthe vorticityboundbehindthewingmorerapidly,thusallowing forcirculationtobebuiltup fasterinthebeginningofeachhalfstroke. ItcanbeseeninFigure 5-33 ,thattheA1101wingshedsonelargevortex downwardsforeachhalf-stroke.At z =33.6mm,Figures 5-33 and 5-34 showa strongnegativevortexbeingconvecteddownwardsespecial lyfor between0.04and 0.44,andastrongpositivevortexfortheremainingphases. Attheendofthestrokes,at =0.44and =0.94thereisastartingvortexshedfromthebottomsideoft hewing,at theturninglocationofthewing. TheA0001wingshowsamorecomplicatedbehavior,withsevera lvorticesbeing shedduringeachhalf-stroke.InFigure 5-31 ,avortexbeingshedcanbeseenat =0.06and0.08.Thevortexismoreseparatedfromthewingint heplotscloserto thewingroot,showingthatthevortexstartspeelingfromcl osertothewingrootand leavesatlatertimesclosertothewingtip.Thisshowsthatt hewingrootstallsearlier. Anothervortexcanbeseentobeshedat =0.18,at z =49.6and z =53.6mm, andat =0.24thevortexhasalsoshedat z =41.6mm.Thistimethevortexstarts peelingfromclosertothewingtip.Itisexpectedthatcircu lationbuildsfastercloser tothewingtip,andthereforestallsfaster,sincecirculat ionmovesspanwisetowards thewingtipduetotherotationofthewing,andbecausethewi ngmovesfastercloser tothewingtip.Thefactthattherstvortexisshedstarting fromthewingrootshould berelatedtoeffectsthatoccursattheendofthehalf-strok e,whentheangleofattack rapidlychangescausingcirculationtoberapidlyshed.Wing tipeffectscombinedwith circulationmovingtowardsthewingtipshouldalsocontrib ute,butthiseffectcannot 110

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easilybeobservedintheseplots.Thetotaleffect,however ,seemstobethatthewing hasmorecirculationclosertothewingrootrightafterthet urnofthewing,causingan earlierstallthereinthebeginningofthestroke. TheA0001wingshedsmultiplevorticeseachhalf-stroke.Two orthreevorticescan beseentrailingthewinginmostoftheplotsinFigure 5-31 ,especiallyonthereturn strokebetween =0.24and =0.74.TheA1101winggenerallyhasahigherangleof attackthantheA0001wing,andcouldthereforebeexpectedto stallearlier,butinstead theoppositeseemstobeobserved.Apossibleexplanationfo rthisbehaviorisifthe A0001wingmoreefcientlyshedsitsboundvorticityattheen dofeachhalf-stroke,and thereforemorerapidlybuildsupanunsustainableamountof vorticity. Oneofthegoalswiththesewingsisthattheyshoulddeformin suchawaythat thrustiscreatedefcientlyeventhoughtheyareonlyappe dusingasimpleonedegree offreedomappingmotion.Thisincludescreatingafavorab leangleofattack.The wingsdoindeedcreateanangleofattackthatcouldbefavora bleinthebeginningof eachhalfstroke,butthewingsareatclosetoa90 anglerelativetothestrokeplanefor alargefractionofthetimeattheendofeachhalfstroke.Thi sismostlikelybecausethe appingmotionissinusoidal,sothatthecommandedkinemat icsdeceleratesthewing throughoutthelatterhalfofeachwingstroke.Thisproblem couldmaybeberesolvedby keepingtheangularvelocityofthewingrootclosetoconsta ntformostofthehalf-stroke, andhavingalargerangularaccelerationattheendofeachha lf-stroke,suchasmany speciesinnature.This,incombinationwithawingexibili tythatcreatesafavorable angleofattackthroughoutthehalf-stroke,couldpossibly improveliftsignicantly. 5.4.3.2Forceestimations InS ¨ allstr ¨ ometal.[ 82 ]thevorticityeldsandvortexcoresgeneratedbyapping thesewingswereinvestigated.Thestudydemonstratedthat theA1101wingshedsone largevortexpairattheendofeachhalf-stroke,whereasthe A0001wingshedsmultiple vorticeseveryhalf-stroke.Thisisthoughttoberelatedto theA0001wing'shigher 111

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compliance,thatcausesthewingtoundergoamoreaggressiv eandrapidchange inwingtwistangleattheendofeachhalf-stroke.Thisresul tsinitmoreeffectively sheddingboundcirculationattheendofthehalf-stroke.Th isallowsthewingtomore rapidlybuildupboundcirculationfromthebeginningofaha lf-stroke,tothepointwhere itreachesstallandvorticesareshed.Furthermore,thestu dyconcludedthatitmaybe advantageoustoreplacethesinusoidalappingmotionused foractuatingthewingwith kinematicsinvolvingaconstantangularvelocitythroughm uchofeachhalf-stroke.Since thesinusoidalappingmotioncausesbothactuatedangular velocityandaccelerationto becontinuouslychanging,thetwistangleandthereforethe angleofattackofthewingis alsocontinuouslyexperiencingsignicantchange.Inpart icular,thewingreachesazero twistanglefarbeforeitreachestheendofeachhalf-stroke .Alinearangularvelocity, togetherwithoptimizedwingexibility,shouldhelpthewi ngkeepamoreadvantageous twistanglethroughmuchoftheappingcyclebyreducingthe largevariationsofow conditionsaroundthewing. Sincetheoriginalwork,inwhichmostofthedataupstreamoft hetrailingedgewas notused,theautomaticmaskingalgorithmhasbeenimproved signicantlyusingthe techniquedescribedinSection 5.4.2.1 .Thisallowsfordataupstreamofthetrailingedge tobeused,withoutincludingmanyerroneousowvectors.Asa result,calculationscan bemadethatwerenotpossiblebefore,andthatarenecessary fortheforceestimations thatwillbepresentedbelow. Theextentofthecontrolvolumethatisintegratedtoestima tetheforceisgivenby 0B@ x 1 x 2 1CA = 0B@ 0.8051 0.7475 1CA 0B@ y 1 y 2 1CA = 0B@ 0.1557 0.3748 1CA 0B@ z 1 z 2 1CA = 0B@ 0.38130.7813 1CA ,(5–13) fortheA0001wing,whilefortheA1101wingthecontrolvolumei s 0B@ x 1 x 2 1CA = 0B@ 0.8051 0.7475 1CA 0B@ y 1 y 2 1CA = 0B@ 0.1109 0.4195 1CA 0B@ z 1 z 2 1CA = 0B@ 0.27470.6213 1CA .(5–14) 112

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Ingure 5-35 thecontrolvolumeisgivenby 5–13 and 5–14 withtheexceptionthat thecontrolvolumeendsat y 2 atthedownstreamendofthemeasureddomaininstead ofat y 2 .Thisgureshowsthecontributionstooverallthrustbycom ponentsintegrated overthesurfaceofacontrolvolumearoundtheinvestigated wingsectionsaccordingto Equation 3–52 .Inthisgurethetime-averageofthefourtermsontheright handside oftheequationisplottedindependentlyalongwiththeirsu mmation.Thecontrolvolume startsat y 1 upstreamofthewing,whichisthespanwiselocationintheg uresupstream ofthewingwhereallvaluescollapse.Theextentofthecontr olvolumeisvariedinthe plotswith y 2 ,thenormalizedchordwisecoordinatewherethecontrolvol umeends downstreamofthewing. y 2 decreasesinthedownstreamdirection. < > denotes time-averagedvalues.Thepressuretermwassolvedforbyin tegratingEquation 3–48 Forthisanalysis,datahadtobeinterpolatedinthemaskedo utregionabovethewing wherethelasersheetwasshadedoutbythewing.Intheregion justdownstreamofthe wing,themaincontributionsconsistsofalargevalueofmom entuminthestreamwise direction,i.e.the < v 2 > term,andacounteractingpressuredroprelatedtotheow accelerationpastthewing.Whenmovingfurtherdownstream, thepressurecontribution changesasthepressureincreasestowardsthesurroundingp ressurelevel.The contributionfromthe < v 2 > componentreducesastheowfromthewingexpands. Thiscanalsobeunderstoodfromtheincreaseinthecontribu tionsfromthe < uv > and < vw > componentsas y 2 changesinthe(negative)downstreamdirection,which showsthatstreamwisemomentumisleavingthecontrolvolum ethroughthecontrol volumefacesnormaltothestreamwisedirection. Figure 5-36 showsthephase-averagedcontributionstotheforcefromac ontrol volumearoundthewingasafunctionofpositionwithinthea ppingcycle.Since thisdataistimedependent,anewtermdependingon @ v =@ t appears.Thetermis calculatedbyintegratingtherighthandssideofEquation 3–53 ,startingfromthe downstreamsideofthewing.ThedoubleintegralofEquation 3–53 inthestreamwise 113

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directionmeansthaterrorsinthedataareaccumulatedinth erstintegralandthen integratedoveragain,henceraisingsomeconcernastothea ccuracyofthisterm. Inadditiontothis,thevariablesintegratedoverarenotkn owninthevicinityofthe wingwhichincreasesthepotentialerrorfurther.Thisprob lemcanbeavoidedinfuture experimentsifthewingpathandthemaskedregionadjacentt oitnevercrossthe surfaceofthecontrolvolume.Duetothenoisynatureofthe @ v =@ t term,itwaslow-pass lteredusingaboxlterinthefrequencydomain,suchthatf requencieshigherthan 10timestheappingfrequencywereremoved.Thetime-avera geofthetermovera appingcycleiszero,andisrepresentativeofthenetmomen tumaddedtotheow bythewinginsidethecontrolvolumethathasnotyetleftthe controlvolume.The behaviorseeninFigure 5-36B isthereforeexpected:Thetotalforce, F y ,andthe @ v =@ t componentpeakssignicantlytwiceinaappingcycle.Onbo thoccations,apeakin streamwisemomentumshownbythe v 2 contributionfollows.Itcanalsobeseenthat thetotalforcewhenignoringthetimederivativeterm, F y t V @ v =@ t d V ,lagsthetotal force F y .Thisisconsistentwiththewingrstaddingmomentumtothe owinsidethe controlvolume,thenaftersometimedelaytheaddedmomentu mleavesthecontrol volume. Figure 5-36A ontheotherhand,showsthattheforcesfromtheA0001wingdoe s notfollowthesameexpectedbehavior.The v 2 contributionshowstwopeaksevery appingcycle,althoughthesearenotaspronouncedasforth eA1101wing.These peaksareaccompaniedbydropsinpressure.However,the @ v =@ t contributiondoesnot behaveasexpected.Especiallythepeaksin F y t V @ v =@ t d V arenottrailingthetotal force F y ,whichisaneffectthatisexpectedfromthe @ v =@ t contribution.Itispossible thattheerrorsin @ v =@ t aretoolarge. The uv and vw termsarebothrelativelysmallforbothwings,andtheirave rages areclosetozerorelativetothedominantterms,whenthecon trolvolumeendsat y 2 sincethatisnotfardownstreamofthewing.Thesetermscorr espondtostreamwise 114

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y momentumleavingthecontrolvolumethroughthefacesofthe controlvolumethat arenormaltothe y axis.The vw termfortheA1101wingdoes,however,havesome noticeablephase-averagedvalues.ThisisrelatedtotheA11 01winggeneratingstrong forcesclosetotheendofthecontrolvolumeinthespanwised irection,aswillbeseen later,whereastheA0001winggeneratesmostofitsforcearou ndthespanwisecenterof thecontrolvolume. Figure 5-37 showsthepressureand < v 2 > contributionsalongthespanwise planesforthetwowings.Unfortunately,thespanwisepeaks eemstobeoutsidethe measureddomainfortheA1101wingshowninFigure 5-37B .FortheA0001wing, itlookslikemostoftheforcehasbeencapturedinFigure 5-37A .Fromthegures, itcanbeseenthattheforcepeakswhen z 0.5,whereastheforcefortheA1101 wingisstillclimbingalongthe z axisattheendofthedomainfor z > 0.6.Comparing Figures 5-37A and 5-37B seemstoindicatethatthepeakforceoccursclosertothewin g tipontheA1101wing,andthatthepeakforceisstronger.Inco ntrast,theA0001wing showsitsstrongestforcevaluesclosetothemiddleofthewi ngalongthespanwhere theA1101wingexertsrelativelysmallforces.TheA0001wingo nlyhaschordwise reinforcementsatthewingrootandat0.875ofthewingspan, whereastheA1101wing alsohasreinforcementsat0.125and0.375ofthewingspan.By relatingtheforces withthebattenstructuresofthewings,itseemsthattherei nforcementsat0.125and 0.375ofthewinginhibitforcegenerationclosetothewingr oot.Atthesametime,since themembranebetween0.375and0.875isnotasconstrainedas therestofthewing, butmoreconstrainedthanontheA0001wing,theexibilityof theA1101wingismore favorableinthisregionofthewingandpeakforcesarefound inthisregion. Figures 5-38 and 5-39 showsvorticityand Q inaspanwiseplaneclosetothe planethatshowsthehighestforcefortherespectivewingin Figure 5-37 ,andclose topeakforcecontributionbythe v 2 componentinFigure 5-36 .Theguresalso containstreamlinesthatarevisualizedusinglineintegra lconvolution,[ 73 ]withan 115

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implementationmostlyfollowingthatdescribedbyStalling andHege.[ 74 ]Figures 5-38A 5-38C 5-39A ,and 5-39C showstheowbeforethepeakisreached,andFigures 5-38B 5-38D 5-39B ,and 5-39D showstheowafterthepeakwasreached.Thelattergroup showaprobablereasonforwhytheforcepeaksaremuchmorepr onouncedforthe A1101wing.Figure 5-38B showastreakofvorticityafterthewing,andFigure 5-38D indicatethattheseareseveralsmallvortexcores.Infact, studying Q forthiswing showsthatthiswingshedsseveralsmallvortexcoreseachha lf-stroke,andthevorticity plotshowsastreakofspanwise z vorticitytrailingthewing.Figures 5-39B and 5-39D ontheotherhandshowsonelargevortexcorebeingshedwhere thewingreverses. Studyingthe Q forthiswingrevealsthatforeachhalfstroke,onelargevor texcore,that breaksapartintotwovortexcores,iscreatedwhenthewingc hangesdirection.Oneof theresultingvortexcoresdissipatesorisconvectedoutof thedomain,andtheother continuesatadownstreamdiagonalangle.Thevortexcorere mainingfromtheprevious wingreversalisseeninthecenterofFigures 5-39A and 5-39C Theaverageforcesfroma pair ofthesewingsinthestreamwise y directionare approximately11mNaccordingtothemeasurementsconducte dbyWu[ 4 ].Figure 5-35 showsaforceofapproximately7mNand5mNarounda single wingfortheA0001 andA1101wingsrespectively.Figure 5-37 indicatesthatmost,butnotallofthe forcegenerationbytheA0001wingfoundintheinvestigateds ection,whereasfor theA1101wing,byextrapolation,itlookslikeasignicantp ortionoftheforceis lostduetothelimitedextentoftheinvestigateddomain.He nce,forawingpair,the aerodynamicdatasuggeststheforcearoundapairofA0001win gsisasomewhat morethan14mN,andaroundapairofA1101wings,itmaybeconsi derablymore than10mN.Fromthesenumbers,theaerodynamicdataindicat esahigherforce thandirectforcemeasurements.Onepossibleexplanationf orthediscrepancyis thediscrepanciesdiscoveredwhiletakingPIVdata,whenthe screwsontheapper hadtobetightenedwhichcauseddiscontinuitiesbetweenpl anesmeasuredbefore 116

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andafteradjustments.Hence,theforcegenerationfromthe samewingmaydifferin subsequentmeasurements.Anotherexplanationmaybenoisei ntheaerodynamicdata. Considerthemostsignicantterm,init'stime-averagefor m,contributingtoforcein Equation 3–52 : < v 2 > = < v > 2 + < v 2 > .(5–15) NoiseinthePIVdatacausesthesecondtermontherighthandss ideofEquation 5–15 tobeoverestimated.Futureexperimentshavetobedesigned tosuppresssucherrorsto beabletovalidatetheusabilityofmethodusedhereforesti matingforces. 5.4.4Summary Aerodynamicmeasurementswereconductedaroundtwodiffere ntexible membranewingswithdifferentcarbonberreinforcementst ructures.Thevelocity measurementswerephase-averagedandusedtocomputevorti cityandandvelocity eldsalongwithrelativepressureandothercomponentscon tributingtoforcederived fromamomentumbalance.Vorticityandvorticesinthewakeo fthewingswere investigated.Thetwostudiedwingscreatesimilarthrust, buthavedifferentchordwise reinforcement.Thisresultsinamuchdifferentwakestruct ure,wherethewingwithfewer chordwisereinforcements,A0001,seemstostallmorerapidl yatthecurrentapping frequency,eventhoughitgenerallyhasaloweraverageangl eofattack.TheA0001 wingshedseveralsmallvortexcoresthroughouttheapping cycle,whichcanbeseen inthetermscontributingtotheforcesincethereisalackof pronouncedpeakvaluesas wasseenfortheA1101wing.TheA1101wingmainlyshedvortexco resattheendof eachhalf-stroke.Atthesetimeslargeforcepeakscanbeseen Furthermore,thedifferentwingstructurealsochangeswhe reforceisgenerated.By comparingwheretheforceisgeneratedtothestructureofth ewings,itseemsthatboth wingsaretoostiffclosetothebattensreinforcingthewing s.Ontheotherhand,lack 117

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ofreinforcementalsoappearstoreducethethrust,atleast closetothewingtipwhere velocitiesarehigher,sincetheA1101wingproduceshighert hrustclosetothewingtip. Theseresultsindicatethatabetterwingwouldlikelyprovi delesschordwise reinforcementaroundthespanwisecenterofthewingthanth eA1101wing,andmore reinforcementclosetothewingtipthantheA0001wing.Infut urewingdesigns,it shouldbeconsideredifthetorsionalrigiditycanbeapprop riatelyadjustedeitherwitha higherdensityofweakerreinforcements,orbythechanging thepropertiesofthewing membraneitself.Thegoalwouldbetomakethetorsionalstif fnesschangeinamore continuousmanneralongthespanofthewing.Themethodofst udyingpressureand thesquaredstreamwisevelocityaroundeachwingsectionma yconstituteapossible sourceofdataforiterativelyoptimizingsuchwingreinfor cements. Itisalsonotedthatthesinusoidalappingmotionseemstob efarfromoptimal.A logicalstepforwardforimprovingthethrustmightbetoadj usttheappingmotionto haveaconstantangularvelocity,toattempttocreateanear constantexibilitythrough mostofeachhalf-stroke,asiscommoninnature. Toimprovefuturemeasurements,themostimportanttaskist omanufacturea apperwithimprovedrepeatability,sothatthesamekindof measurementscanbe takenallalongthespanofthewing,withoutanyinconsisten cies.Thiswouldallowa controlvolumetobecreatedaroundthewingwithnogaps,whi chwouldallowformore accurateforceestimations. 5.5VaryingMembraneWings Thissectiondescribesmeasurementsonwingsusingacarbon berskeleton, correspondingtotheskeletonofanL3B0winginSection 5.3 withdifferentmembrane materialsandthicknesses.5.5.1ExperimentalSetup Theseexperimentsincludedmeasuringthrustforcesgenera tedfromvewings usingforcetransducersandeightsetsofaerodynamicowme asurementsonanother 118

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setofvedifferentwingsontwodifferentappingdevices. Thissectionwilldiscuss somespecicsofthewingsalongwiththeequipmentusedtomo vethewingsand measuretheowandforces.5.5.1.1Wing Thewinghasthreelayersofunidirectionalcarbonberrein forcingtheleading edge,twolayersreinforcingthewingroot,andthreelayers ofbidirectionalcarbon berreinforcingwheretheleadingedgemeetsthewingroot. Thewinghasnobattens followingthesuggestionsinthesummaryoftheprevioussec tion.Hence,insteadof varyingtheskeletonstructure,thissectionpresentswing swithdifferentmembrane materialsandthicknesses.5.5.1.2Wingmembranematerials Differentwingmembranematerialsandthicknessesareused tovarytheexibilityof thewingstestedinthissection.Theseare0.003”polyether etherketone(PEEK),0.001” and0.002”thickpolyethyleneterephtalate(PET),0.001”and 0.002”thickperuorinated alkoxy(PFA),and0.010”thicknaturallatex.Young'smodulus anddensityforthe membranematerialsexceptlatexarelistedinTable 5-5 .Thelatexwingsdifferfrom theotherwingssincetheyweighsignicantlymorethanther estofthewingsdueto theirthicknessandalsoresonateatthreetimesthe12Hzap pingfrequencyinthe experimentsmeasuringoweld.Hence,thelatexwing'sine rtiashouldbeexpectedto contributemuchmoretothewingdeformationcomparedtothe otherwings.Figure 5-40 showsapairofwingswitha0.010”thicklatexmembrane.5.5.1.3Particleimagevelocimetrysetup ThePIVsetupusedforthissetofexperimentsmostlyresemble stheonedescribed inSection 5.4.1.3 .ThePIVsystemandmethodforsynchronizationisthesame. However,themagnicationhaschangedtoapproximately0.1 .Hence,thediffraction-limited spotdiameter,asdescribedin 2.4.2 ,ofaseedingparticlewiththecurrentsetupison 119

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theaverageapproximately11.4 mattheimageplane,oraround50%largerthanthe sideofapixelofthecameras'CCDs.5.5.1.4Flappingdevices Measurementswereacquiredusingthetype3appingdevice( Section 4.2.4 ), whichwillbereferredtoasthesinusoidalapperandthetyp e4appingdevice (Section 4.2.5 ),whichwillbereferredtoasthesemi-linearapper. 5.5.1.5Forcemeasurements Forcemeasurementsonthesinusoidalapperwereacquiredu singtheNano17 forcetransducer(seeSection 4.4.1 ).Duetoit'slargerweightandvibrations,force measurementsonthesemi-linearapperweretakenusingthe largerlesssensitive Gammaforcetransducer,describedinSection 4.4.2 5.5.2DataAnalysisMethods ThemethodsfromChapters 2 and 3 wereappliedtocaptureandprocessthedata thatwillbepresentedonthewingswithvaryingmembranes.T hissectiondescribedthe detailsofhowthemethodswereusedfortheexperimentsthos ewings. 5.5.2.1Particleimagevelocimetry ThePIVimageswereprocessedasdescribedinSection 2.9 .Thevectoreld snapshotswerepost-processedtoreducetheeffectofbadve ctorsasisexplainedin Section 2.10.1 .Finally,phase-averagedvalueswerecreatedbylinearly ttingeach vectortothecorrectphaseintheappingcyclefromallsnap shotsclosetothatphase asisdescribedinSection 2.10.2 Masking .ThemaskingofthePIVvectoreldsfortheseexperimentswer e computedmuchlikefortheexperimentsontheAxxxxwings,whi chisdescribedin Section 5.4.2.1 .Thedifferenceisthatthemaskingwascomputedfortheenti reimage, andnotjustaselectsub-region.Additionally,eachconnect edsetofmaskedpixelswas groupedtogether.Allpixelsinsidetheconvexhullformedby suchagroupofpixelswere masked. 120

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ScopeofPIVdata .Table 5-6 listsallwingsandlocationsonthewingswherePIV datawasacquired.2500PIVsnapshotsweretakenateachofloc ationslisted.The spacingbetweenthespanwisemeasurementplaneswas2.5mmf orthePET0.001”on bothappers,andtheLatex0.010”wingonthesemi-lineara pper.Fortheremaining wings,thespanwisespacingbetweenplaneswas10mm.ThePFAm embranesdidnot attachaswelltothewingskeletonastheothermembranemate rialswhenbeingglued. Therefore,notasmanyplanescouldbemeasuredwithoutthem embranecomingoffof theskeleton.Ifthemembranewouldhavebeenreattached,sm allchangesintheow eldswouldlikelyoccurandthereforemakederivativesof oweldpropertieshighly inaccurate.Inthecaseofthelatexmembranewing,highdens itymeasurementswere chosentobetakenonlyonthesemi-linearappertolimitthe amountofdatathathadto becapturedandprocessed,andbecausethewingwas(wrongly )expectedtoperform slightlybetteronthesemi-linearapper. InthemeasurementsonthePET0.001”wingonthesinusoidalap per,therewas issueswiththemeasuringdeviceonthetraversecausingthe spacingtobeslightly largerthan2.5mmintheplanesfurtherfromthewingroottha n45.0mm.Theexact locationsoftheseplaneshavethereforebeendeterminedby triangulatingaknown pointontheapperfromthecameraimages.Intotal,96plane swerecapturedoneight differentwing/appercombinations.5.5.2.2Wingtracking Thewingshapesthatarepresentedforthewingsinthissecti onwereextracted byttingsplinestomanuallyselectedpointsinrawPIVimage smappedtoaphysical coordinatesystem.5.5.3Results Inwhatfollows,forceandoweldmeasurementsonappingw ingswithvarying membraneswillbepresented.Theappingfrequencybeingex aminedhere,12Hz, 121

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correspondstohoveringReynoldsnumberRe hover ofapproximately3500.Theapping amplitude A is35 5.5.3.1Forcemeasurements Table 5-7 showstheresultsofforcemeasurementsfromthesemi-linea rapper. Thethrustvaluespresentedinthetablearecomputedbysubt ractingthethrust componentfromthetarecase(i.e.theapperrunningwithno wings)followingeach wingcase,sinceasignicant“thrust”componentwaspresen tevenwithnowing.This canbeexplainedbyadifferenceintheaveragecenterofmass oftheapperwhenitis runningandthecenterofmasswhenitisatthepositionwhere theforcevalueswere resettozero.Theresultsarelistedintheordertheywereme asured.Eachcombination ofwingandfrequencywasmeasuredbymovingtheappertoapr edeterminedposition intheappingcyclewhereabiasloadwasrecordedfromthefo rcetransducerto subtractfromthefollowingmeasurements.Theapperwasth enbroughtuptospeed andtheaverageforceoverapproximately200appingcycles wasmeasured.Thiswas repeated10timesforeachcase.Thisprocesswascompletely automatedsothatall subsequenttestsonthesamewingweredoneinsequencewithn ointermediatehuman interaction.The0.002”thickPETmembranewingisanexceptio nforwhichonlyfour measurementsarepresent. Table 5-8 showstheforcesmeasuredusingthesinusoidalapper.Since thevalues measuredofthetarecasesweresoclosetozero,theywerenot subtractedfromcases withwingsontheapperaswasdonewiththesemi-linearapp er.Again,theresults arelistedintheordertheyweremeasured.However,sincefo rcesofthesamewing weremeasuredatseveralappingfrequencies,theorderinw hichthemeasurements weretakenwaschanged.Theneworderwastotakeonesetof200 appingcyclesatat 10Hz,onesetat13Hz,thenoneat15Hz.Thatsequencewasauto maticallyrepeated 10timesforeachwingwithouthumaninteractioninbetween. 122

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Kangetal.[ 83 ]presentednumericalresultsofforcecoefcient C T asafunctionof effectivewingstiffness 1 onthesamewingplanform.Theirresultsindicatethatwithi n thetestedrange,lowereffectivewingstiffnessincreases thethrustcoefcient.However, therangeof 1 intheirdataisseveralordersofmagnitudelargerthanther ange investigatedhere.Figure 5-41 showstheexperimentallymeasuredthrustcoefcientof thewingsasafunctionofeffectivewingstiffness.Thethru stcoefcientsappeartobe higherforlowereffectivestiffnesswithinthecurrentlym easuredrangedemonstratinga similarrelationshipofthatpresentedbyKangetal.However ,if 1 isagoodchoicefor non-dimensionalstiffnessandthrustincreaseswithreduc edstiffnessinthemeasured range,the0.001”thickPETmembranewingshouldnotbeexpecte dtoexertsignicantly lessthrustthanbothPFAwings,as 1 forthePETmembranewingisinbetweenthe valuesof 1 forthePFAmembranewings. Toinvestigatetheeffectofwingstiffnessasafunctionoft hicknessandYoung's modulusmoreclosely,anumericalstudyhasbeenperformed. ANSYSsoftware version13wasusedtosolveforwingdeformationsusingani teelementanalysis. Thegridusedforthisanalysishas5043quadrilateraleleme ntsandisshownin Figure 5-42 .Shellelementsareusedtomodelthemembrane.Theelementty pe (ANSYSSHELL181)isbasedonMindin-Reissnershelltheoryandhas 3translational and3rotationaldegreesoffreedomateachofitsfournodes[ 84 ]. Inthisanalysisthetrailingedgewasmodeledasunconstrai nedandtosimplify theanalysis,theleadingedgeandrootofthewingweremodel edasrigidsincethey areordersofmagnitudestifferthanthemembrane.Fourdiff erentmembranematerial propertiesweretested,whicharelistedinTable 5-9 .Material1isasoftmaterialfor whichYoung'smodulusisapproximatelyonthesameorderofm agnitudeasforlatex andthethicknessisthesameasthelatexmembranethatwasus edintheexperiments. Material2hasthepropertiesofa0.001”thickPFAmembrane.M aterial3hasthesame sameeffectivestiffnessasmaterial2giventhat U ref andtheuiddensity arethesame, 123

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butitsthicknessis10timeslargerandYoung'smodulus1,00 0timessmaller.Material4 hasthepropertiesofa0.001”thickPETmembrane.Poisson'sra tio wasassumedto be0.4forallmaterials.Uniformpressurevaluesintherang efrom10 6 to10 2 Pawere appliedonthebottomsurfacethewingsandthesteadynon-li neardeformedsolution wasfoundforeachwingandpressurecombination.Anexampleo fthesolutiontoa deformedwingshapeisshowninFigure 5-43 Anormalizedmembranethicknesscanbedenedas h = h c m ,(5–16) withanormalizedpressureforcingthewingas P = P E ,(5–17) andthemaximumabsolutewingdeectionas u max,abs = u max,abs c m .(5–18) Here u max,abs isthemaximumabsolutewingdeection.Atlowpressurethewi ng deectionisapproximately u max,abs ( P )= c 1 c 2 P h 3 (5–19) ascanbeseeninFigure 5-44A .Theconstantvalues c 1 =4.65and c 2 =0.170was usedinFigure 5-44 .Equation 5–19 isinlinewithHooke'slawandwithusing 1 asa non-dimensionalvariableforeffectivewingstiffness.Fr omthegureitappearsthat therelationshipisapproximatelyvalidfor P = h 4 < 10 2 .Theconstantvalues c 1 and c 2 weretusingonlythesamplesforwhich P = h 4 < 10 2 andmembranes1,2,and4.The deectionformembrane3deviatesfromthatoftheothermemb ranesusingthesame constants,butthereisstillalinearrelationshipbetween u max,abs and P for P = h 4 < 0.2. Sincemembrane3deviatesandYoung'smodulusforthemembran eliesoutsidethe 124

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intervalofthemembranematerialsthatareusedinthephysi calexperimentsinthis section,itwasnotusedtottheconstants c 1 and c 2 .Athighpressures,thedeformation followstherelationship u max,abs ( P )= c 1 P h 1 = 3 .(5–20) ThisisdemonstratedinFigure 5-44B ,thatshowsthattherelationshipisapproximately validfor P > 1.Since u max,abs isnotproportionalto P = h 3 intherelationshipgivenby Equation 5–20 1 isnotsuitableasanon-dimensionalvariableforwingstiff nesswhen P > 10 2 .Thenon-linearrelationshipbetween P and u max,abs alsomeansthatlinear analysiscannotbeusedtodeterminethedeformationorreso nantfrequencyofthewing andthattheresonantfrequencyisalsoadependentonamplit ude. Areferencepressurefortheappingightcanbedenedas P ref = U 2 ref = (4 RAf ) 2 .(5–21) Atafrequencyof12Hz,thereferencepressureisapproximate ly5.8Pa. Thecaseswithmembrane2andpressuresof50and100Pawereom ittedin Figure 5-44 sinceitdeformedinamannerinconsistentwithalltheother casesandwas thereforenotcomparable.Athighpressure,thetrailingedg eofthiswingcurlsuparound thelineconnectingthetrailingedgeofthewingrootwithth ewingtipascanbeseenin Figure 5-45 .Thewingwithmembranematerial3whichhasthesame 1 asthewing withmembranematerial2deformsinlinewithalltheotherwi ngsatthesamepressure asisapparentinFigure 5-46 Whenthewingdeformationsareinnitesimal,thewingmembra nedoesnothaveto stretchtodeform.Therefore,therelationshipbetweenpre ssureanddeformationgiven byEquation 5–19 onlydependsontheamountofforcethatisnecessarytobendt he wing.Whenthedeformationsarelargerthemembranemuststre tchconsiderablyto beabletobend,sincethewingisconstrainedatboththelead ingedgeandwingroot. 125

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(Anexceptiontothatisifthewingbendsatthetrailingedgea roundandbehindthe straightlineconnectingthetrailingedgeofthewingroott othewingtipwasthecase inFigure 5-45 .)Equation 5–20 isvalidwhentheforcenecessarytostretchthewing dominatestheforcenecessarytobendthemembrane.Theforc enecessarytostretch themembraneisproportionaltothethicknessofthemembran e.Thisisconsistentwith that P / h if u max,abs isheldconstant. SubstitutingthereferencepressureinEquation 5–21 intoEquation 5–17 yields anon-dimensionalreferencepressure P ref .Thevalueof P ref = h 4 ismorethan1for allcasesplottedinFigure 5-41 .Hence,anon-dimensionalreferencedeection u ref,max,abs canbeestimatedusingEquation 5–20 .Themeasuredthrustcoefcient C T isplottedagainst u ref,max,abs inFigure 5-47 .Theplotshowsthatthethrustcoefcient isstrictlyincreasingwithincreasingreferencewingdefo rmationunderthesameapping kinematicswithonlyoneexception.Thisindicatesthat 1 islikelynotanappropriate non-dimensionalquantityforthewingstiffnessifboththe leadingedgeandthewingroot areconstrainedunlessthedeformationisverysmall. u ref,max,abs appearstobeabetter choiceforpredicting C T 5.5.3.2Floweldmeasurements Theowelddataaroundallwingsintheseexperimentshasbe enintegrated asdescribedinSection 3.4 toestimateforcesandwheretheseareexertedonthe uidsurroundingthewings.Itshouldbenotedthattheestim ationisverycrudefor themeasurementswherethespanwiseplanesareseparatedwi th10mmduetothe sparsityofthedata. Figure 5-48 showsthenetcontributiontoforcefromtheaveragestreamw ise momentumandaveragepressurebetweenaplaneinfrontofthe leadingedgeanda planebehindthetrailingedgeofthewing.Theseplaneswere chosenastheplanes closesttothewingoneachsideofthewinginthestreamwised irectionbeforethe numberofmaskeddatapointsstartstoincreaseduetothepro ximitytothewingand 126

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varyslightlyforeachcase.The0.010”latexwingonthesemi -linearappershowsa sharppeakinforceat z 0.6.Theothertwocaseswithhighdensitydataalongthe spanwiseaxisshownosuchpeak.Forthelatexwingthespanwi selocationofpeak forcegenerationappearstobeatthislocationregardlesso ftheapper.Theplots indicatethattheremainingwingshavetheirforcepeakslig htlyfurtherawayfromthe wingroot. Figure 5-49 showsthetime-averagedcontributionfromeachterminEquat ion 3–52 integratedoverthesurfaceofacontrolvolumestartingata planelocatedat y 1 upstream ofthewingtoaplaneat y 2 .Thegureshowshowthedifferentcontributionsvary withthelocation y 2 oftheplanewherethecontrolvolumeendsdownstreamofthe wing.Thegapintheguresbetweentheupstreamlocation y 1 andwherethecurve for y 2 startsconcideswiththeextentofthecontrolvolumesusedt ointegratethe valuesinFigure 5-48 .Itshouldbenotedthatforthecaseswherethespacingbetwe en thespanwiseplanesissmall,inFigures 5-49A 5-49B ,and 5-49C thetotalforce contributionfromallcomponentremainsroughlyconstanta sshouldbethecase.The othercasesshowsomevariationinthetotalforcecontribut ionwith y 2 indicatingsome discrepanciesascanbeexpectedforcoursedata.Thecasesa realsointegratedover adifferentspanwiseextent,whichmustbetakenintoconsid erationwhencomparing thecases.Anotherconsiderationisthatsomeforcemayalsob egeneratedoutside themeasureddomain,especiallyforthePFA0.001”wingonthe semi-linearapper. Allcasespresentedinthisgureshowthatthestreamwisemom entumandpressure arethetwomaincomponentstoforceclosetothetrailingedg eofthewing.These componentsdropoffasmomentumleavesthecontrolvolumeth roughsurfacesnormal tothestreamwisedirectionas y 2 increases.Themomentummostlyleavesthrough aplanenormaltothespanwise z axiswhichisshownbyariseinthe < vw > component,indicatingthattheowisgeneratedwithanoutw ard(orpossiblyinward) spanwisecomponent.Theexceptionisthelatexwingonthese mi-linearapperfor 127

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whichthe < vw > componentremainsatandmomentuminsteadleavesthecontr ol volumemostlythroughaplanenormaltothe x axisascanbeseeninFigure 5-49A Thelatexwingonthesinusoidalappershowsthestreamwise momentum < v 2 > droppingmuchslowerthanthatoftheothercases.Thiscould indicatethattheow generatedbythewingismorecloselyalignedwiththe y axis.Italsoshowsahighernet thrustthananyotherwing.Thegureindicatesathrustof3. 8mN,whichcorresponds to C T 0.89.Thesamewingonthesemi-linearapperresultsinathr ustcoefcient ofonly0.49.Allcasesexceptforthoseusingthelatexwingca sesshowaslightly negativestreamwisemomentumcontributionat y 2 = 1.5,whichmightindicatesome recirculatingow.ThethrustcoefcientsforthePETwingare 0.58(2.5mN)and0.49 (2.1mN)onthesemi-linearandsinusoidalappers,respect ively.Thesevaluestwell betweenthe10Hzand13HzresultsreportedinFigure 5-47 Thepeakforceproductionseemstobeintherange y 2 (0.6,0.8).Thevorticityfor allcaseswillthereforebeinvestigatedat y =50mm,whichcorrespondsto y 0.67. Figure 5-50 showstheowaroundthe0.010”latexwingonthesemi-linear apper. Thewingisappingfromlefttorightat =0.96,whereowovertheleadingedgeof thewingshouldbebuildinguppositivecirculationbehindt hewing.Atthesametime thewingseemstobedragginganegativevortexcorealong,th atwasshedduetothe reversalofappingdirectionattheendofthewingstroke.At =0.04,theedgeofa regionwithnegativevorticitycanbespottedbehindthewin gbutmostofitisstillhidden wherethedataismasked.Atthistime,thetwistangleofthewi nghaschangedsothat thetrailingedgeistravelingaheadoftheleadingedgeinth eappingdirection.Thisis mostlikelyduetothenaturalfrequencyofthewing.Sincethe trailingedgeisattraveling rst,owislikelygoingoverthetrailingedgefromrightto left,creatingmorenegative vorticityunderthewing.At =0.12,acoreofnegativevorticityisclearlyseentobe comingdown.Thisisalsoclosetothetimewhenthewingstart sdecelerating,ascanbe seeninFigure 4-7 .Thedecelerationcausesthewingtoshedapositivevortex, resulting 128

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inthestrongvortexpairseenat =0.24. =0.38isapproximatelythebeginningof constantcommandedangularvelocityfromlefttoright,and =0.64theend.Even thoughthecommandedangularvelocityisapproximatelycon stantduringthisperiodthe wingtwistanglechangessothatthetrailingedgeisaheadof theleadingedgeagain. Duetotheasymmetryoftheappingcycle,thevorticityshed duringthestrokefrom righttoleftismuchweakeranddoesnotseemtoresultinanys ignicantowinthe streamwisedirection. Figure 5-51 showstheowaroundthesamelatexwingonthesinusoidalap per. Sincethedurationofthetwohalf-strokesoftheappingcycl eisnowidentical,itis muchmoreobviousthatthewingresonatesatthreetimesthe appingfrequency.At =0.84,thewingisinthebeginningofthehalf-strokefromle fttoright.Justlikewhen thewingwasonthesemi-linearapper,thewingseemstobedr aggingthenegative vorticityshedattheturningpointbehind.Asthetrailinged geovertakestheleading edgebetween =0.84and =0.94,thenegativevortexcoreshouldbestrengthened. Thetrailingedgethentwistsbacktotrailtheleadingedgew hiledeceleratingascanbe seenat =0.16.Theshednegativevortexcoreisalsoclearlyseen.Byl ookingatthe wingat =0.36and =0.42,itisclearthatthetrailingedgetwistsaheadofthel eading edgeduringthesecondhalf-stroketoo.At =0.58,apositivevortexisseentohave beenshedinasimilarfashiontothenegativevortexinther sthalfstroke.Periodically repeatingthissheddingpatterncreatesareversevonK arm anstreetassketchedin Figure 5-57A .Thisowwasseentobebyfarthemostthrustproducingaccor dingto theearliercomparisonofestimatedcontributionstothefo rce.Theowshouldalsobe veryefcientcomparedtotheothercasesasitseemstoactas asinglejet,butthis cannotbeveriedfromthecurrentdata.Mostoftheothercas eslookliketheyactas therearetwoormorejetsforminginthewakewithdifferentd irectionsassketchedin Figure 5-57B .Thecaseswithtwoormorejetsthatarenotalignedinthesam edirection 129

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willcreateowforwhichcomponentsofthejetsindirection sotherthanthemeanthrust directionisawasteofenergy,astheircontributionscance leachotherout. Thevorticityaroundthe0.001”PETmembranewingonthesemi-l inearapper canbeseeninFigure 5-52 .At =0.90,theremainderofthestoppingandstarting vortexfromtheprevioushalf-strokecanbeseenasthewingi sacceleratingfromright toleft.At =0.06,astrongvortexfromthemostrecentwingreversalcan beseen topairupwiththesmallervortexfromthepreviouswingreve rsal.Thiscreatesaow thatisdirecteddownwardsandtotheleft.Thestrongnegati vevortexcoreweakens asittravelsrightoverthepositivevorticitybeforeitpai rsupwiththevortexfromthe nextwingreversalascanbeseenat =0.42.Thewingseemstocreatetwojet-like burstsofowintwodifferentdirectionsratherthanthenic ereversevonK arm anstreet withacontinuousowinthestreamwisedirectionthatcould beseeninthepreviously discussedcase. ThesamewingonthesinusoidalappercanbeseeninFigure 5-53 andshowsa behaviorthatcloselyresemblesresultsfromtheexperimen tsdiscussedinSection 5.4 Forthiscaseastoppingandstartingvortexisshedatthetur ningpointofthewingas canbeseenat =0.42and =0.84.Thatvortexpairsupwithvorticityshedduring theprecedinghalf-stroketocreateowtothedownwardslef tanddownwardsrightfor thetwohalf-strokes,respectively,justlikethesamewing onthesemi-linearapper. However,duetothesinusoidalappingmotion,weakervorte xcoresareshedatthe turningpointofthewing.Thehighangleofattackcausesvor ticitytobecreatedatboth theleadingandtrailingedgeresultinginstalloccurringr apidlyforthiscase.Thestall resultsinmanysmallvortexcoresbeingshedthroughoutthe half-stroke.Someofthese canbeseenat =0.96. Vorticityaroundthewingwitha0.002”PFAmembraneonthesin usoidalapperis showninFigure 5-54 .At =0.90whenthewingisreversingdirection,somevorticity remainingfromtheprevioushalf-strokecanbeseen.At =0.06,negativevorticity 130

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hasbeenshedduringthetravelfromlefttoright.Thetraili ngedgeofthewinghas actuallybentaheadoftheleadingedge.Thiscausesthewing tostall,justlikethePET membranewingonthesinusoidalapper.Thisresultsinmany smallvortexcoresbeing shedduringeachhalf-strokeascanbeseenat =0.06.Thedifferencebetween thiscaseandthelatexwing,wherethetrailingedgealsotra velsaheadoftheleading edge,isthatthetrailingedgeofthelatexwingrapidlytrav eledaheadoftheleading edgeandthenrapidlyreturnedtotrailingagain,causingas trongvortextobeshed.In thiscase,thewingdoesnotexbacktotrailtheleadingedge andtheowconditions remainssuchthatthewingisstillstalled.Thesamemembran eonthesinusoidalapper behavesmuchlikethePETmembranewing,withtwojet-likestru cturesbeingcreated attwodifferentinstancesduringafullappingcycle,asca nbeseenat =0.90and =0.44inFigure 5-55 .Thewingstallsonthesinusoidalappertooandshedsmany smallvortexcoresduringeachhalf-strokeascanbeseenat =0.16. Theoweldaroundthe0.001”PFAmembranewingonthesemi-li nearapperis verysimilartoowaroundthe0.002”PFAmembranewing.Compa ringFigures 5-54 and 5-56 at =0.90showsvorticityleftbehindfromtheprevioushalf-st roke,butthe thinnerwinghasshedmorevorticityinoneconcentratedand strongvortexcore.At =0.06,thenewlycreatedvortexcorecreatesajet-likestru cturewiththevorticity fromtheprevioushalf-stroke.Forthethickermembranewin g,thereisaclearimbalance instrengthbetweenthevortexcorescausingtheowtoturns ignicantly.Inthecase ofthethinnerwingtherearetwovortexcoresmoreequalinst rength.At =0.34, bothwingsaresheddingpositivevorticity.Itappearsthat duetoitshigherexibility,the thinnerwinghasbeenrotatingfromahighertwistangleatth eendofthewingstroke andthereforeitcreatesstrongervortexcoresattheendoft hehalf-strokes.Exceptforat theendofeachhalf-stroke,thewingtwistangleisremarkab lysimilarforthetwowings. 131

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5.5.4Summary Forcemeasurementswereconductedonwingswiththesamecar bonberskeleton designbutvaryingmembranematerialsandthicknessesatth reedifferentapping frequencies.Twodifferentapperswereused;onewithasin usoidalappingmotion andonewithasemi-linearappingmotion.Theforcemeasure mentssuggestedthat materialswithhigherexibilityandlessthicknessperfor mthebestintermsofforce generationintherangetested.Thesemi-linearapperoutp erformedthesinusoidal apperslightlywiththebestperformingwings.Itcannotbe determinedwithcertainty thatthesemi-linearapperisperformingbettersincethed ifferencesinforceweresmall andthenumberofwingstestedfew. Thethreewingsfromtheforcemeasurementsthatcreatedthe mostforcewere selectedformeasuringtheoweldaround.Onemoremembran ematerial,0.010” latex,wasaddedtothesetoftestedwings.Thelatexmembran ewingdiffered signicantlyfromtheotherwingsnotonlythroughthicknes sandexibility,butalso byresonatingatafrequencythreetimestheappingfrequen cy.Theoweldaround thePFAmembranewingsandthelatexwingonthesinusoidalap perwasmeasured withaspanwiseplanespacingof10.0mm,andtheremainingwi ngswithaspacingof 2.5mm. Totalforceandindividualcomponentscontributingtoforc ewereestimatedfrom theoweldmeasurementsforallwings.Thelatexmembranew ingonthesinusoidal apperappearstobethecongurationthatproducedthemost thrust.Theowelds aroundthewingshowthatitcreatesareversevonK arm anvortexstreet.Aqualitative analysisoftheowalsosuggeststhattheowislikelytobet hethemostefcient too,asmostofthemomentumseemstobeaddedinapproximatel yasingledirection. Thesamewingonthesemi-linearapperappearstoonlyaddas ignicantamountof momentumduringoneofthetwohalf-strokes.Theowmightha veproducedastronger forceifthetwohalf-strokesofthesemi-linearapperwere equalinduration,asthe 132

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effectivenessonthesinusoidalapperseemstostemfromre sonanceinthewing membrane. Theremainingwingsallhaveafairlysimilarbehavior.Thes toppingandstarting vortexofeachhalf-stroketogetherwithoppositesignvort icityshedintheprevious half-strokepairsuptocreateadownwardsdiagonaljet-lik estructureeachhalf-stroke. Thesemi-linearappercreatesstrongerstoppingandstart ingvortexcoresthanthe sinusoidalapperduetoamorerapidwingreversal.Intheca seofthePETmembrane wing,thatseemstocausethewingtostallslowerasitshedsl essvorticityduringeach half-stroke.ThesamedoesnotapplytothePFAmembranewings ,whichisprobably becausethetrailingedgeofthewingtravelsfasterthanthe leadingedgecausingthe wingtostall.Higherwingexibilitycausesthewingtotwis tmoreattheendofeach half-stroke,creatingstrongerstoppingandstartingcore sandingeneralcontributesto higherforcesintherangetested.Table5-1.PIVcapturesettingsforrigidwingoweldmeasur ements. Flappingfrequency f (Hz)Capturefrequency d T ( s)Shutterspeed( s) 51015003000 101992503000 Table5-2.PIVsnapshotstakenattheoriginalverticallocat ionoftherigidwing. Frequency(Hz) z (mm)snapshots Frequency(Hz) z (mm)snapshots 5 30.252445 5 6.251630 5 27.252445 5 3.252445 5 24.252445 5 0.252445 5 21.251630 52.751630 5 18.252445 10 9.251630 5 15.252445 10 6.251630 5 12.252445 10 0.251630 5 9.252445 133

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Table5-3.PIVsnapshotstakenoftherigidwingwithdomainsh iftedby50mmvertically. Frequency(Hz) z (mm)snapshots Frequency(Hz) z (mm)snapshots 5 31630 10 31630 501630 1001630 531630 1031630 Table5-4.PIVcapturesettingsfortheL m B1wings. Flappingfrequency f (Hz)Capturefrequency(Hz) d T ( s)Pulselength( s) 251021600.01 Table5-5.Membranematerialproperties[ 85 ]. MaterialYoung'smodulusDensity PEEK(Polyetheretherketone)3.7-4.0GPa1320kg/m 3 PET(Polyethyleneterephtalate)2.0-4.0GPa1560kg/m 3 PFA(Peruorinatedalkoxy)0.66GPa2150kg/m 3 Table5-6.ScopeofPIVdataforwingswithdifferentmembranem aterials. MaterialThicknessFlapper#ofplanesSpanwiselocations PET0.001”Semi-linear2617.5 80.0mm PET0.001”Sinusoidal2220.0 78.4mm PFA0.001”Semi-linear440.0 70.0mm PFA0.002”Semi-linear630.0 80.0mm PFA0.002”Sinusoidal620.0 70.0mm Latex0.010”Semi-linear2617.5 80.0mm Latex0.010”Sinusoidal620.0 70.0mm Table5-7.Forcesfromwingsonthesemi-linearapper. MaterialThicknessFrequencyThrust99.5%c.i. Tare-13Hz14.07mN 0.44mN PEEK0.003”13Hz4.75mN 1.02mN Tare-13Hz14.11mN 0.65mN PET0.002”13Hz2.81mN 1.60mN Tare-13Hz14.05mN 0.89mN PET0.001”13Hz6.95mN 1.03mN Tare-13Hz14.12mN 0.74mN PFA0.001”13Hz9.15mN 2.73mN Tare-13Hz14.39mN 0.79mN 134

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Table5-8.Forcesfromwingsonthesinusoidalapper. MaterialThicknessFrequencyThrust99.5%c.i. Tare-10Hz0.00mN 0.05mN Tare-13Hz0.06mN 0.04mN Tare-15Hz-0.09mN 0.16mN PET0.002”10Hz1.16mN 0.08mN PET0.002”13Hz3.05mN 0.08mN PET0.002”15Hz4.54mN 0.11mN PEEK0.003”10Hz1.13mN 0.13mN PEEK0.003”13Hz2.83mN 0.08mN PEEK0.003”15Hz3.91mN 0.28mN PET0.001”10Hz2.15mN 0.08mN PET0.001”13Hz5.79mN 0.11mN PET0.001”15Hz8.30mN 0.04mN PFA0.002”10Hz2.78mN 0.09mN PFA0.002”13Hz6.89mN 0.07mN PFA0.002”15Hz10.71mN 0.10mN PFA0.001”10Hz4.06mN 0.07mN PFA0.001”13Hz8.46mN 0.20mN PFA0.001”15Hz11.81mN 0.29mN Tare-10Hz0.07mN 0.08mN Tare-13Hz0.03mN 0.05mN Tare-15Hz-0.16mN 0.08mN Table5-9.Membranematerialpropertiesinthenumericalst udy. MaterialThicknessYoung'smodulusPoisson'sratio 10.010”0.010GPa0.4 20.001”0.660GPa0.4 30.010”0.660MPa0.4 40.001”3.000GPa0.4 135

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Figure5-1.PIVsetupschematicforrigidwingoweldmeasur ements. y (mm) x (mm) wing 00.511.5 0 0.5 1 Figure5-2.PIVdatagridforrigidwingoweldmeasurements (0,0,0)trailingedge leadingedgey z x Figure5-3.Coordinatesystemforrigidwingowelddata. 136

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Figure5-4.Chordwiselasersheetlocationsforrigidwingm easurements. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 A = 0.20. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 B = 0.16. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 C = 0.12. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 D = 0.08. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 E = 0.04. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 F =0.00. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 G =0.04. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 H =0.08. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 I =0.12. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 J =0.16. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 K =0.20. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 L =0.24. Figure5-5.Iso-surfacesofphase-averagedchordwisevort icityaroundtherigidwingat z = 2,whereredispositiveandblueisnegative. 137

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z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 A = 0.20. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 B = 0.16. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 C = 0.12. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 D = 0.08. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 E = 0.04. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 F =0.00. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 G =0.04. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 H =0.08. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 I =0.12. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 J =0.16. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 K =0.20. z y x 0 0.5 1 0.25 0.5 0.75 0.25 0 L =0.24. Figure5-6.Iso-surfacesofphase-averagedvorticityarou ndtherigidwinginthe instantaneousspanwisedirectionat span = 2,whereredispositiveand blueisnegative. secondeldofview rsteldofview wing x (mm)y (mm)Interpolatedintersection 050100 0 20 40 60 80 100 120 Figure5-7.CombinedPIVdomainforrigidwingowelddata. 138

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y0.2 0.4 0.6 0.8 1 1.2 1.4 A = 0.20. B = 0.16. C = 0.12. D = 0.08. y0.2 0.4 0.6 0.8 1 1.2 1.4 E = 0.04. F =0.00. G =0.04. H =0.08. y x 00.51 0 0.2 0.4 0.6 0.8 1 1.2 1.4 I =0.12. x 0.51 J =0.16. x 0.51 K =0.20. x 0.51 L =0.24. 2 1012 z Figure5-8.Phase-averagedchordwisevorticityandstreaml inesusingLICatthequarter chordpointoftherigidwingat5Hz.Resultsattwodifferent locationsalong the y axishasbeencombinedtoenlargetheeldofview. 139

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y0.2 0.4 0.6 0.8 1 1.2 1.4 A = 0.20. B = 0.16. C = 0.12. D = 0.08. y0.2 0.4 0.6 0.8 1 1.2 1.4 E = 0.04. F =0.00. G =0.04. H =0.08. y x 00.51 0 0.2 0.4 0.6 0.8 1 1.2 1.4 I =0.12. x 0.51 J =0.16. x 0.51 K =0.20. x 0.51 L =0.24. 2 1012 z Figure5-9.Phase-averagedchordwisevorticityandstreaml inesusingLICatthequarter chordpointoftherigidwingat10Hz.Resultsattwodifferen tlocationsalong the y axishasbeencombinedtoenlargetheeldofview. 140

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wingangle( ) 0.2 0.100.10.2 0 10 20 30 40 50 A5Hz. wingangle( ) 0.2 0.100.10.2 0 10 20 30 40 50 B10Hz. Figure5-10.Timeversuswingangleofrigidwing,measureda t z =0. angle65mmfromwingroot( )wingrootangle( ) 40 20 02040 40 20 0 20 40 Figure5-11.Approximatewingrootangleversusangleattheq uarterchordpoint65mm fromwingrootfortheL2B1wingin air and vacuum Figure5-12.AnL m B1wing. 141

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L1B1 L2B1 L3B1averagethrust(mN)appingfrequency(Hz) 0 10203040 0 10 20 30 40 50 60 70 80 90 Figure5-13.ThrustproducedbytheL m B1wingsatdifferentappingfrequencies. Figure5-14.L m B1wingappinginlasersheet. Figure5-15.PIVsetupschematicforoweldmeasurementson theL m B1wings. 142

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Figure5-16.Spanwiselasersheetlocations. y (mm) x (mm) wing 00.51 0.8 0.6 0.4 0.2 0 0.2 Figure5-17.PIVdatagrid. (0,0,0)trailingedge leadingedgey x z Figure5-18.Coordinatesystem. 143

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replacements 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 A = 0.09. 0 0.5 1 0.8 0.4 0 0 0.5 1 B =0.01. 0 0.5 1 0.8 0.4 0 0 0.5 1 C =0.11. 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 D =0.19. 0 0.5 1 0.8 0.4 0 0 0.5 1 E =0.29. 0 0.5 1 0.8 0.4 0 0 0.5 1 F =0.41. 2 1012 Figure5-19.VorticityandvelocityeldsaroundtheL1B1win g.Colorsrepresent z and arrowsshowthevelocityinthe x y plane.Thetopplotofeachplotpair showstheplanes z =30,50,and70mm.Thebottomplotsshow z =20, 40,60,and77mm.Thewinglocationinthe z y planeisshownabovethe plots.Thesmallarrowsonthewingshapeshowthedirectiono fthelocal velocityofthewing. 144

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replacements 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 A = 0.37. 0 0.5 1 0.8 0.4 0 0 0.5 1 B = 0.27. 0 0.5 1 0.8 0.4 0 0 0.5 1 C = 0.17. 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 D = 0.03. 0 0.5 1 0.8 0.4 0 0 0.5 1 E =0.11. 0 0.5 1 0.8 0.4 0 0 0.5 1 F =0.23. 2 10 12 Figure5-20.VorticityandvelocityeldsaroundtheL2B1win g.Colorsrepresent z and arrowsshowthevelocityinthe x y plane.Thetopplotofeachplotpair showstheplanes z =30,50,and70mm.Thebottomplotsshow z =20, 40,60,and77mm.Thewinglocationinthe z y planeisshownabovethe plots.Thesmallarrowsonthewingshapeshowthedirectiono fthelocal velocityofthewing. 145

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replacements 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 A = 0.41. 0 0.5 1 0.8 0.4 0 0 0.5 1 B = 0.15. 0 0.5 1 0.8 0.4 0 0 0.5 1 C = 0.07. 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 0 0.5 1 0.8 0.4 0 0 0.5 1 D = 0.01. 0 0.5 1 0.8 0.4 0 0 0.5 1 E =0.11. 0 0.5 1 0.8 0.4 0 0 0.5 1 F =0.23. 2 101 2 Figure5-21.VorticityandvelocityeldsaroundtheL3B1win g.Colorsrepresent z and arrowsshowthevelocityinthe x y plane.Thetopplotofeachplotpair showstheplanes z =30,50,and70mm.Thebottomplotsshow z =20, 40,60,and77mm.Thewinglocationinthe z y planeisshownabovethe plots.Thesmallarrowsonthewingshapeshowthedirectiono fthelocal velocityofthewing. 146

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replacements yz 0204060 40 20 0 20 40 A =0.01. z 02040 60 B =0.31. z 02040 60 C =0.51. L3B1 L2B1 L1B1 z 02040 60 D =0.81. Figure5-22.WingdeectionsatthequarterchordoftheL m B1wings. y x 00.51 0.5 0 0.5 A = 0.37. y x 00.51 0.5 0 0.5 B = 0.27. y x 00.51 0.5 0 0.5 C = 0.17. y x 00.51 0.5 0 0.5 D = 0.03. y x 00.51 0.5 0 0.5 E =0.11. y x 00.51 0.5 0 0.5 F =0.23. 2 1012 Figure5-23.VorticityaroundtheL2B1wingat z =50mm.Colorsrepresent z andLIC indicatesstreamlinesinthe x y plane. 147

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y (mm)z (mm) 40 55 70 45 30 15 0 15 30 45 Figure5-24.Approximatemotionofthequarterchordpoint65 mmfromthewingrootfor L1B1 L2B1 ,and L3B1 wings. L3B1 L2B1 L1B1streamwisemomentum(mN)x (mm) 02550 75 100 0 20 40 60 80 Figure5-25.Time-averagedstreamwisemomentumperunitti meandareaintegrated over y and z AA0001. BA1101. Figure5-26.TestedAxxxxwings,wherethex'sinthenamesare 1whereabattenis presentand0whereitisnot. 148

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Figure5-27.PIVsetupschematicforoweldmeasurementson theAxxxxwings,with thewingsatzeroappingangle.Theleadingedgeisfacingup wards, hencethestreamwisedirectionisdownwards. y (mm) x (mm) wing 0.500.5 1 0.8 0.6 0.4 0.2 0 0.2 0.4 AA0001PIVdatagrid. y (mm) x (mm) wing 0.500.5 1.2 1 0.8 0.6 0.4 0.2 0 0.2 BA1101PIVdatagrid. Figure5-28.ValidPIVdatagridsaftercalibrationformeasu rementsontheAxxxxwings, containedwithina121 126nodegrid,withwingsatzeroappingangle, andcoordinatesystemfortheA0001andA1101wings. 149

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(0,0,0)trailingedge leadingedgez y x Figure5-29.CoordinatesystemforAxxxxwingowelddata,w iththeleadingedgeat x =0. AA0001. BA1101. Figure5-30.SpanwiselasersheetlocationsforAxxxxwingow eldmeasurements. Sheetsthatareomittedinthisanalysisaregray. 150

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=0.06 1 0.5 0 =0.08 1 0.5 0 =0.18 1 0.5 0 =0.24 1 0.5 0 =0.50 1 0.5 0 =0.56 1 0.5 0 =0.62 1 0.5 0 =0.66 1 0.5 0 =0.70 1 0.5 0 =0.74 0.50 0.5 1 0.5 0 0.50 0.5 0.50 0.5 0.50 0.5 0.50 0.5 A z =29.6mm.B z =33.6mm.C z =41.6mm.D z =49.6mm.E z =53.6mm. 0 5 1015 2025 Figure5-31. Q aroundtheA0001wing. 151

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=0.06 1 0.5 0 =0.08 1 0.5 0 =0.18 1 0.5 0 =0.24 1 0.5 0 =0.50 1 0.5 0 =0.56 1 0.5 0 =0.62 1 0.5 0 =0.66 1 0.5 0 =0.70 1 0.5 0 =0.74 0.5 0 0.5 1 0.5 0 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 A z =29.6mm.B z =37.6mm.C z =41.6mm.D z =49.6mm.E z =53.6mm. 4 2024 Figure5-32.Vorticity( z )eldsaroundtheA0001wing. 152

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=0.04 1 0.5 0 =0.14 1 0.5 0 =0.24 1 0.5 0 =0.34 1 0.5 0 =0.44 1 0.5 0 =0.54 1 0.5 0 =0.64 1 0.5 0 =0.74 1 0.5 0 =0.84 1 0.5 0 =0.94 0.5 00.5 1 0.5 0 0.5 00.5 0.5 00.5 0.5 00.5 0.5 00.5 A z =29.6mm.B z =33.6mm.C z =37.6mm.D z =41.6mm.E z =45.6mm. 0 5 1015 2025 Figure5-33. Q aroundtheA1101wing. 153

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=0.04 1 0.5 0 =0.14 1 0.5 0 =0.24 1 0.5 0 =0.34 1 0.5 0 =0.44 1 0.5 0 =0.54 1 0.5 0 =0.64 1 0.5 0 =0.74 1 0.5 0 =0.84 1 0.5 0 =0.94 0.500.5 1 0.5 0 0.500.5 0.500.5 0.500.5 0.500.5 A z =29.6mm.B z =33.6mm.C z =37.6mm.D z =41.6mm.E z =45.6mm. 4 2024 Figure5-34.Vorticity( z )eldsaroundtheA1101wing. 154

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y 2forcecontribution(mN) 1.5 1 0.500.5 2 0 2 4 6 8 10 AA0001. y 2forcecontribution(mN) 1.5 1 0.500.5 2 0 2 4 6 8 10 BA1101. < F y > R x 2 x 1 R y 2 y 1 [ < vw > ] z 2 z 1 d x d y R x 2 x 1 R z 2 z 1 [ < v 2 > ] y 2 y 1 d x d z R y 2 y 1 R z 2 z 1 [ < uv > ] x 2 x 1 d y d z R x 2 x 1 R z 2 z 1 [ < P > ] y 2 y 1 d x d z Figure5-35.Time-averagedcontributionstoforcefortheAx xxxwingsoveracontrol volumestartingatthechordwisenon-dimensionalcoordina te y 1 upstream ofthewing,whereallofthecontributionsarezerointheplo ts,andending at y 2 downstreamofthewing. y 2 decreasesinthedownstreamdirection. forcecontribution(mN) 00.20.40.60.81 8 6 4 2 0 2 4 6 8 10 12 AA0001. forcecontribution(mN) 00.20.40.60.81 8 6 4 2 0 2 4 6 8 10 12 BA1101. < F y > F y F y t V @ v @ t d V R x 2 x 1 R y 2 y 1 R z 2 z 1 @ v @ t d x d y d z R x 2 x 1 R y 2 y 1 [ vw ] z 2 z 1 d x d y R x 2 x 1 R z 2 z 1 [ v 2 ] y 2 y 1 d x d z R y 2 y 1 R z 2 z 1 [ uv ] x 2 x 1 d y d z R x 2 x 1 R z 2 z 1 [ P ] y 2 y 1 d x d z Figure5-36.Phase-averagedcontributionstoforcethrough outtheappingcycleforthe Axxxxwings. 155

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zforceperspan,N/m0.30.4 0.5 0.60.70.8 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 AA0001. zforceperspan,N/m0.30.4 0.5 0.60.70.8 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 BA1101. R x 2 x 1 [ < P > + < v 2 > ] y 2 y 1 d x R x 2 x 1 [ < v 2 > ] y 2 y 1 d x R x 2 x 1 [ < P > ] y 2 y 1 d x Figure5-37.SpanwisecontributionstoforcefortheAxxxxwin gs. y x 0.500.5 1 0.5 0 A z at =0.50. y x 0.500.5 1 0.5 0 B z at =0.56. y x 0.500.5 1 0.5 0 C Q at =0.50. y x 0.500.5 1 0.5 0 D Q at =0.56. 10 5 0 5 10 0102030 40 50 z Q Figure5-38. z and Q fortheA0001wingattwophaseinstancesinthespanwiseplane z =37.6mm,or z 0.5. y x 0.50 0.5 1 0.5 0 A z at =0.44. y x 0.50 0.5 1 0.5 0 B z at =0.54. y x 0.50 0.5 1 0.5 0 C Q at =0.44. y x 0.50 0.5 1 0.5 0 D Q at =0.54. 10 5 0 5 10 010203040 50 z Q Figure5-39. z and Q fortheA1101wingattwophaseinstancesinthespanwiseplane z =43.6mm,or z 0.58. 156

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Figure5-40.Wingswitha0.010inchthicklatexmembranes. Sinusoidalapperat15Hz Sinusoidalapperat13Hz Sinusoidalapperat10Hz Semi-linearapperat13HzC T 1 5 4 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 10 2 10 1 10 0 10 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure5-41.Thrustcoefcient C T asafunctionofeffectivewingstiffness 1 forthe followingwingmembranes:1.0.003”PEEK,2.0.002”PET,3.0.002”PF A, 4.0.001”PET,5.0.001”PFA.Inthecaseswhereamaterialpropert yhasa rangeinTable 5-5 ,thevalueinthecenteroftheintervalistaken. 157

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y x 0 0.20.4 0.60.81 0 0.1 0.2 0.3 Figure5-42.Winggridwith5043quadrilateralelements. y x 0 0.2 0.40.60.81 0 0.1 0.2 0.3 00.05 0.1 z Figure5-43.Exampleofanumericallydetermineddeformedwi ngshape.Thisisthe solutionforasteadystatedeformationofthewingwithmemb ranematerial 1andwith P =100Pa. Membrane4 Membrane3 Membrane2 Membrane1 u max,abs = [( c 1 = c 2 ) P = h 3 ] P = h 4 10 6 10 4 10 2 10 0 10 2 10 4 0 0.2 0.4 0.6 0.8 1 1.2 Membrane4 Membrane3 Membrane2 Membrane1 u max,abs = [ c 1 ( P = h ) 1 = 3 ] P = h 4 10 6 10 4 10 2 10 0 10 2 10 4 0 0.2 0.4 0.6 0.8 1 1.2 Figure5-44.Relationshipbetweennormalizedpressure P andabsolutemaximumwing deection u max,abs 158

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y x 00.20.40.6 0.8 1 0 0.1 0.2 0.3 00.05 0.1 z Figure5-45.Deformedwingwithmembranematerial2andwith P =50Pa. y x 00.20.40.60.81 0 0.1 0.2 0.3 00.10.2 z Figure5-46.Deformedwingwithmembranematerial3andwith P =50Pa. 159

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Sinusoidalapperat15Hz Sinusoidalapperat13Hz Sinusoidalapperat10Hz Semi-linearapperat13HzC T u ref,max,abs 5 4 2 1 5 4 3 2 1 5 4 3 2 1 5 4 3 2 1 0.02 0.030.040.050.060.070.080.090.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure5-47.Thrustcoefcient C T asafunctionofreferencemaximumabsolutewing deformation u ref,max,abs forthefollowingwingmembranes:1.0.003”PEEK, 2.0.002”PET,3.0.002”PFA,4.0.001”PET,5.0.001”PFA.InthecaseswhereamaterialpropertyhasarangeinTable 5-5 ,thevalueinthemiddle oftheintervalistaken.Poisson'sratioisassumedtobe0.4 forall materials. 160

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zforceperspan,N/m0.20.4 0.6 0.81 0.1 0 0.1 0.2 ASemi-linearapperand0.010”Latexmem-brane. zforceperspan,N/m0.20.40.6 0.8 1 0.1 0 0.1 0.2 BSinusoidalapperand0.010”Latexmembrane. zforceperspan,N/m0.20.40.60.81 0.1 0 0.1 0.2 CSemi-linearapperand0.001”PETmembrane. zforceperspan,N/m0.20.40.60.81 0.1 0 0.1 0.2 DSinusoidalapperand0.001”PETmembrane. zforceperspan,N/m0.20.4 0.6 0.81 0.1 0 0.1 0.2 ESemi-linearapperand0.002”PFAmembrane. zforceperspan,N/m0.20.4 0.6 0.81 0.1 0 0.1 0.2 FSinusoidalapperand0.002”PFAmembrane. zforceperspan,N/m0.20.40.60.81 0.1 0 0.1 0.2 GSemi-linearapperand0.001”PFAmembrane. R x 2 x 1 [ < P > + < v 2 > ] y 2 y 1 d x R x 2 x 1 [ < v 2 > ] y 2 y 1 d x R x 2 x 1 [ < P > ] y 2 y 1 d x Figure5-48.Spanwisecontributionstoforceforwingswithv aryingmembranes. 161

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y 2forcecontribution(mN) 1.5 1 0.5 0 5 2.5 0 2.5 5 7.5 ASemi-linearapperand0.010”Latexmembrane. y 2forcecontribution(mN) 1.5 1 0.50 5 2.5 0 2.5 5 7.5 BSinusoidalapperand0.010”Latexmembrane. y 2forcecontribution(mN) 1.5 1 0.5 0 5 2.5 0 2.5 5 7.5 CSemi-linearapperand0.001”PETmembrane. y 2forcecontribution(mN) 1.5 1 0.50 5 2.5 0 2.5 5 7.5 DSinusoidalapperand0.001”PETmembrane. y 2forcecontribution(mN) 1.5 1 0.50 5 2.5 0 2.5 5 7.5 ESemi-linearapperand0.002”PFAmembrane. y 2forcecontribution(mN) 1.5 1 0.50 5 2.5 0 2.5 5 7.5 FSinusoidalapperand0.002”PFAmembrane. y 2forcecontribution(mN) 1.5 1 0.50 5 2.5 0 2.5 5 7.5 GSemi-linearapperand0.001”PFAmembrane. < F y > R x 2 x 1 R y 2 y 1 [ < vw > ] z 2 z 1 d x d y R x 2 x 1 R z 2 z 1 [ < v 2 > ] y 2 y 1 d x d z R y 2 y 1 R z 2 z 1 [ < uv > ] x 2 x 1 d y d z R x 2 x 1 R z 2 z 1 [ < P > ] y 2 y 1 d x d z Figure5-49.Time-averagedcontributionstoforceforwing swithvaryingmembranes, overacontrolvolumestartingatthechordwisenon-dimensi onalcoordinate y 1 upstreamofthewing,whereallofthecontributionsarezero intheplots, andendingat y 2 downstreamofthewing. y 2 decreasesinthedownstream direction. 162

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1 0.500.5 1.5 1 0.5 0 A =0.96. 1 0.500.5 1.5 1 0.5 0 B =0.04. 1 0.500.5 1.5 1 0.5 0 C =0.12. 1 0.500.5 1.5 1 0.5 0 D =0.24. 1 0.500.5 1.5 1 0.5 0 E =0.38. 1 0.500.5 1.5 1 0.5 0 F =0.64. 10 5 0 5 10 Figure5-50.Vorticity( z )eldsaroundthe0.010”latexmembranewingonthe semi-linearapperat z =50.0mm. 163

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1 0.5 0 0.5 1.5 1 0.5 0 A =0.84. 1 0.5 0 0.5 1.5 1 0.5 0 B =0.94. 1 0.5 0 0.5 1.5 1 0.5 0 C =0.16. 1 0.500.5 1.5 1 0.5 0 D =0.36. 1 0.500.5 1.5 1 0.5 0 E =0.42. 1 0.500.5 1.5 1 0.5 0 F =0.58. 10 5 0 5 10 Figure5-51.Vorticity( z )eldsaroundthe0.010”latexmembranewingonthe sinusoidalapperat z =50.0mm. 1 0.50 0.5 1.5 1 0.5 0 A =0.90. 1 0.50 0.5 1.5 1 0.5 0 B =0.06. 1 0.50 0.5 1.5 1 0.5 0 C =0.42. 10 5 0 5 10 Figure5-52.Vorticity( z )eldsaroundthe0.001”PETmembranewingonthe semi-linearapperat z =50.0mm. 164

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1 0.5 0 0.5 1.5 1 0.5 0 A =0.12. 1 0.5 0 0.5 1.5 1 0.5 0 B =0.30. 1 0.5 0 0.5 1.5 1 0.5 0 C =0.42. 1 0.500.5 1.5 1 0.5 0 D =0.64. 1 0.500.5 1.5 1 0.5 0 E =0.84. 1 0.500.5 1.5 1 0.5 0 F =0.96. 10 5 0 5 10 Figure5-53.Vorticity( z )eldsaroundthe0.001”PETmembranewingonthe sinusoidalapperat z =50.5mm. 1 0.50 0.5 1.5 1 0.5 0 A =0.90. 1 0.50 0.5 1.5 1 0.5 0 B =0.06. 1 0.50 0.5 1.5 1 0.5 0 C =0.34. 10 5 0 5 10 Figure5-54.Vorticity( z )eldsaroundthe0.002”PFAmembranewingonthe semi-linearapperat z =50.0mm. 165

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1 0.500.5 1.5 1 0.5 0 A =0.90. 1 0.500.5 1.5 1 0.5 0 B =0.16. 1 0.500.5 1.5 1 0.5 0 C =0.44. 10 5 0 5 10 Figure5-55.Vorticity( z )eldsaroundthe0.002”PFAmembranewingonthe sinusoidalapperat z =50.0mm. 1 0.500.5 1.5 1 0.5 0 A =0.90. 1 0.500.5 1.5 1 0.5 0 B =0.06. 1 0.500.5 1.5 1 0.5 0 C =0.34. 10 5 0 5 10 Figure5-56.Vorticity( z )eldsaroundthe0.001”PFAmembranewingonthe semi-linearapperat z =50.0mm. AReversevonK arm anstreet. BDoublejet. Figure5-57.Sketchoftrustproducingvortexstructures. 166

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CHAPTER6 SUMMARY Thisstudypresentedexperimentalresultsonoweldsarou ndappingwingsina hoveringenvironment.Themaingoalofthestudywastodevel opmethodstoacquire accurateowelddataaroundappingwingsandtorelatesuc hdatatogetherwithwing deformationstoforcesgeneratedbythewings. Therstexperimentalresultspresentedutilizedarigida ppingwing.Thiswas donetocreateabaselinecasetodevelopanunderstandingof howthenearwing vorticityisformed.Therigidityandweightofthewingtoge therwiththedesignofthe appingdevicelimitedtheexperimentstolowerthandesira bleappingfrequencies. Sincetheangleofattackwasnormaltothestrokeplaneduetot hewingsbeingand hadasymmetricalappingmotion,nosignicantthrustwasc reated,butatrailingvortex structurewasgeneratedatthewingtip.Theexperimentsals oclearlydemonstrated howtheonsetofdecelerationofthewingrapidlycausesshed dingofvorticityduetothe momentumoftheuidbehindthewing. ThenextsetofexperimentswereperformedontheL m B1wings,whichintroduced anisotropicexibilitythroughtheuseofmembranewings.T hreedifferentwingswith varyingspanwiseexibilitywereexaminedinaseriesoftes ts.Thesewingshada reinforcedleadingedgeandaexiblefreetrailingedge,al lowingwingdeformation necessarytocreatethrustfromthesymmetricappingmotio nused.Resultsfrom Wuetal.[ 42 ]showedthatthemediumleadingedgestiffnesswing,L2B1,ha dthe mostfavorableleadingedgestiffnessofthethreewingstes tedattheinvestigated appingfrequency.Theothertwowingswereeithertoosofto rtoostiffincomparison. Fromthevortexstructureitcouldbeobservedthatthesoftw ingcouldnotsustainthe aerodynamicforcesactingonthewing.Hence,itwasnotable toefcientlycreate forcesinthedownstreamdirection.Theothertwowingsboth showedabehavior wherepositiveandnegativevorticitypaireduptocreateo winadownstreamdiagonal 167

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direction,althoughthemediumstiffnesswingcreatedslig htlystrongerstructures.Thisis postulatedtobeduetobothamoreadvantageousangleofatta ckonaverageandthat theincreasedexibilitycausedthewingtoapslightlyfur ther.Thewingappingfurther allowsboththesweptareaandaveragevelocityofthewingto increase. Thenextsetofexperimentsconductedinthisstudyinvestig atedtheanisotropic A0001andA1101wings.ThesewingsdifferedfromtheL m B1wingsinthatthe locationsofthechordwisebattenswasvariedwhilethelead ingedgestiffnesswasheld constant.Thevortexstructuresshedintothewakeofthesew ingswereinvestigated byusingtheQcriteriontoidentifyvortices.Thevisualiza tionsindicatedthattheA0001 wingbecamesaturatedwithboundvorticitymuchfasterthan theA1101wing,even thoughithadaloweraverageangleofattackthroughoutthe appingcycle.Thiswas mostlikelyduetothewayvorticitywasshedandbuiltupatth eendofeachwing strokewhenthewingrotatedandchangeddirection.Theexpe rimentsclearlyshowed thatamorerepeatableappingdevicewasneededsinceonlya subsetofthetotal measurementlocationscouldbeusedduetoinconsistencyin themechanism.Thedata alsoindicatedthatasinusoidalappingmotionisnotfavor able,asthetwistanglerapidly decreasesaftermid-strokewhenthedecelerationofthewin gsetsin. Inthelastsetofexperimentsonwingswithvaryingmembrane materials,wingswith severaldifferentmembranematerialswereusedwiththegoa lofdesigningawingthat wouldpassivelydeformtohaveanadvantageoustwistangle. Itwasenvisionedthatair resistanceshouldbeabletokeepthewingatafavorabletwis tanglethroughmuchof theappingcycleifthewingiskeptataconstantspeed.Thew ingshouldthenhavea exibilitythatcausesahightwistangleatthewingtipthat isdecreasingclosertothe wingroot,sinceitwouldbeadvantageousifitcouldkeepthe effectiveangleofattack nearlyconstantalongthespanofthewing.Suchadeformation canalsobeobserved inmanynaturaliers.Therefore,asemi-linearappingdev icewasmanufacturedto beabletokeepthewingataconstantcommandedangularveloc ity.Anewapping 168

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devicewithsinusoidalkinematicswasalsodesignedtobeab lecomparethekinematics ofthetwonewappingdevicesandtoresolverepeatabilityi ssuespresentinapping devicesusedinpreviousexperiments.Thekeytoimprovemen twastheuseoflinear andcircularballbearingsateverypowertransmissionjoin tandattachingthesebearings tightlywithscrewswhenpossibleandgluewhennecessary.Af tersomeinitialtesting andadjustments,boththeseappingdevicesprovedtoresol vepreviousproblemsand neitheroneneededanyadjustmentbetweentherstandlasts etofmeasurements presentedinthiswork. Forcemeasurementswereconductedonseveralapper-wingc ombinationsto determinewhichmembranestoinvestigatetheoweldon.Th eresultsshowedthatthe morecompliantofthetestedwingsproducedmorethrust.The mostcompliantwings werethereforechosenforPIVmeasurementsandanaturallate xwingwasaddedtothe setoftestedwings. TheresultsfromthePIVexperimentsshowedthatofthewingst estedintheforce measurement,thetwowingswhichproducedthemostthrustwe realsothetwowings fromthatexperiencedthemostwingdeformation. Thedataindicatedthatwingsonthesemi-linearappercrea tedslightlystronger forcesingeneral,whencomparedtothesinusoidalapper.T hisisinlinewithwhat wasreportedbySaneandDickinson[ 30 ],whoseresultsindicatedthattheliftandthe lift-to-dragratioishigherwhenthedurationofthewingi pattheendofeachhalf-stroke isshort.However,strongerforcesfromthesemi-linearap percannotbeattributedto thewingmaintainingaroughlyconstanttwistanglethrough muchofeachhalf-stroke, aswasanticipatedbasedonpreliminarythoughts.Itmayhow everbeattributedto thecommandeddecelerationoccurringlaterineachhalf-st rokeandtothewingmore effectivelysheddingvorticityattheendofeachhalf-stro keduetoamorerapidchange ofcommandedangularvelocity.Aappingdevicewithsemi-l inearangularvelocity kinematicsandsymmetrichalf-strokesmaybebetterthanth easymmetrickinematics 169

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usedinthisstudy,asthewingcouldthenbeoptimizedforone setofconditionsthatare thesameundertheforwardandbackwardstroke,ratherthano ptimizingitforthebest performanceovertwoasymmetricalhalf-strokeswithdiffe rentowconditions. Thewingswithvaryingmembranematerialsonthesinusoidal appercreateda doublejet-likevortexstructure,withonejet-likestruct ureappearingeachhalfstroke assketchedinFigure 5-57B .Theowstructureissimilartowhatwasseeninearlier measurements.Thelatexwingwasanexceptionthatinsteadc reatedareversevon K arm anvortexstreetillustratedbyFigure 5-57A .Theestimatedforcegeneratedbythis wing-appercombinationishigherthananyoftheothercomb inationsinthesameset ofexperiments.Oneofthekeysseemstobethatthewingreson atesatthreetimes theappingfrequency.Thispropertyhelpstosignicantly alterthevortexshedding behaviorinafavorableway.However,thisstudyisnotsufc ienttodetermineunder whatconditionssuchaoweldwillappear,astherearemany propertiesthatcould bealteredwhilemaintainingtheratiobetweentheappingf requencyandtheresonant frequencyofthewing.ResultsbyVanellaetal.[ 23 ]showasignicantincreaseoflift whenthewingexperiencednon-linearresonanceat3timesth eappingfrequency underverydifferentconditions.Atwo-dimensionalnumeri calstudyatlowReynolds numberwherethewingconsistedofaexibletwo-linkmodelw asthebasisfortheir results. Alloftheexiblemembranewingsinthisstudydemonstratedt heeffectofdelayed stall.Sincetheanglebetweenthestrokeplaneandthechordo fthedeformedwingis typicallyverylarge,theresultinganglebetweenthedirec tionoftheoncomingowand thechordismorethenenoughforstallconditionstooccurin steadyconditionsatalmost allinstancesintimeforthesewings.DuetotheWagnereffec tanddelayedstall,the vorticitycreatedfromowpassingovertheedgeofawingsta ysboundforsometime asalowpressurecorebehindthewing.Thislowpressureinco mbinationwithhigh pressureinfrontofthewingcausesthewingtodeform.Dueto thedeformationofthe 170

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wing,thepressuredifferenceresultsinforcecomponentsn ormaltothestrokeplane, whichinturnresultsinanaveragenetforce.Thehistoryoft hisprocessisseeninthe oweldasvortexsystemsshedbythewing. 6.1WingPlanform Theplanformthatisusedinthisstudyisinspiredbytheshap eofhummingbird wings.Thewinghasitslargestchordlengthatthewingroot. Thewingplanformis arguablybetterdesignedforcruisingightoratleastnots pecicallyoptimizedfor hovering.Thisisapparentsincetheoweldsaroundthewin gsshowthatthrustis mainlyexertedontheowfarawayfromthewingroot.Awingde signedsolelyfor hoveringshouldthereforehaveasmallchordlengthatthewi ngrootasiscommonfor insects.Thisresearchcouldthereforebeconsideredtobea studyofawingdesigned asacompromisebetweentheneedsofforwardightandhoveri nglikeahummingbird. Studiesontheplanform'sperformanceinforwardightareex pectedtofollowafterthis one. 6.2ParticleImageVelocimetry CapturingPIVdataunderhoveringconditionsincludesauniq uesetofchallenges differentfromthosepresentwhenowswithanon-zerofrees treamvelocityarebeing measured.Theexperimentsontherigidwingpresentedchall engessinceitwas difculttoobtainconditionsunderwhichtheseedingparti cleswereclearlyvisible. TheexperimentsontheL m B1wingswereconductedwithadifferenttypeofseeding particlesthatwerelargerandmorereectiveandthusprodu cedbetterresults,butthey weredifculttoworkwith.Theseedingproblemwassolvedin amoresatisfactoryway intheexperimentsontheAxxxxwingsandthewingswithvaryin gmembranesbyusing adifferentPIVsystemwithhigherresolutioncamerasandahi gherintensitylaser.There werestilltworemainingissueswiththissolutioncompared withseedingafreestream ow.Therstonewasthatalltheuidinthetestsectionhada roughlyequalseeding density.Sincetherewerealotofparticlesoutsidethemeasu redplanethatreectlight, 171

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thebackgroundintensityandnoiseinthePIVimagesbecamesi gnicantlyhigher.The meanerroroftheprocesseddatashouldincreaseasaresult, butthereisnoreason toexpectbiaserrorstoincrease.Ahighbackgroundintensi tyandnoisemimicking realPIVdatawereassumedinthestudyoftheaccuracyofthePIV algorithminthe Appendix.Theotherremainingissuewasthattheseedingdens ityreduceswithtime andnewseedingparticlescannotbeaddedtotheowwithoutd isturbingit.However, theseedingdensityusingoliveoilparticlesdidnotreduce toomuchasahighenough seedingdensitywasmaintainedeveninthelastsnapshotsof asequencecapturingall thedataofinterestinasingleplane.ThelargerExpancelpar ticlesthatwereusedin theexperimentsontheL m B1wingsfalltothegroundmuchfasterthanoliveoil,butthat wasnotanissuesinceahighspeedPIVsystemwasusedsothatth edurationofeach measurementwasonlyontheorderoftenseconds. Thisstudyshowssomeofthepossibilitiesanddifcultiesi nobtainingthree-dimensional phase-averagedPIVdataonappingwingsinhoveringconditi onsinair.Therehave beenseveralstudiesofphase-averagedPIVdataonappingwi ngsinhovering conditionsinwater,buttothebestoftheauthor'sknowledg etherehasbeennoother suchstudiesconductedinair.Thereareadvantagestocondu ctingmeasurementsin water.ThemainadvantageisthattheReynoldsnumber,which determinesthebehavior ofincompressibleowintheabsenceofeffectsoftemperatu reandbodyforces,scales sothattoreplicatetheconditionsinair,velocitiescanbe greatlyreducedandthesize ofwingmodelsenlargedatthesametime.Thismakesmeasurem entsconsiderably easiertoconduct.Themostimportantlimitationisthatfor cesdonotscaleaccordingly. Thereforesuchexperimentsarenotveryusefulforstudying uidowthatishighly dependentonuid-structureinteraction. APIValgorithmwasdevelopedthatsignicantlycutthetimen eededtoprocessthe datainthelastsetofexperiments,andatthesametimeallow edformoreinformation tobeextracted.TheaccuracyoftheresultsfromthePIValgor ithmwasthoroughly 172

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investigatedbyusingthespecicsoftwaresettingsandgeo metriccalibrationthatwas obtainedfromtheexperiments,astheaccuracyofthemethod variesdependingonthe geometricsetupaswellasseedingandlightingconditions. Theactualseedingdensity, averageapparentparticlesize,particlescatteringinten sity,andlaserintensitywas mimickedinthevalidation.Somefactorswereignored,sucha sthatparticlesizesand thereforealsolightscatteringintensitiesvarybetweenp articlesandthatlightintensity seenbythecamerasvarybothduetonon-uniformlasersheetl ightintensityanddueto varyingdistancesbetweenthecameraanddifferentlocatio nsinthemeasuredplane. SomebiaserrorsshouldalwaysbeexpectedtobepresentinPIVd ata,butthese arenotroutinelyinvestigatedinathoroughwayeventhough theyareuniquetothe conditionsandgeometryofeachPIVmeasurement.Ifbiaserro rsforasetupare discoveredtobeunacceptablylargeforaveragedPIVdata,th eycouldbesignicantly reducedbyiterativelyprocessingasetofarticialPIVimag escreatedwiththe prescribedmotionoftheaveragedPIVresults.Thenthepresc ribedmotionofthePIV imagescanbecorrecteduntilthedifferencebetweentheave rageprocessedvectors fromthearticialandrealPIVdataissatisfactorilysmall. Severalimprovementstothemethodsusedtopost-processthe PIVdatafromthese experimentswerealsoaccomplished.Firstamethodwasappl iedtoreducetheimpact ofoutliersandtollsmalldatagapsinthevectoreldsfori ndividualPIVsnapshots. Thenphaseaverageswerecomputedbylinearlyttingveloci tyvaluesasafunctionof phaseandrejectingoutliersinsteadofjusttakingthemean ofallsnapshotsfallinginthe samephaseintervalaswasdoneforpreviousexperiments. 6.3FutureWork Therearemanyimprovementsandvariationsthatcouldbecon ductedinorderto enhanceandexpandtheknowledgegainedfromthisstudy.The currentstudyshould befollowedbyoneinwhichtheeffectsofforwardightisinc luded.Severaldifculties thathavebeendiscussed,mainlyrelatedtoseedingtheow, couldbeavoidedinsuch 173

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studies.Ontheotherhand,inwindtunneltestsaprescribed freestreamvelocityhasto bematchedtothedeformedwingkinematicsasthrustshoulde qualtodragifrealistic steadyvelocityissimulated.Determiningtheappropriate freestreamvelocityforeach casewillgiverisetoadifferentsetofdifculties. MorerevealingdatacouldbeobtainedifvalidPIVvectorscou ldberesolvedcloser tothewing.Theproblemdependstosomeextentontheunfavor ableseedingconditions resultingfrommeasuringinahoveringenvironment.Sincese edingisequallypresent everywhereandnotlimitedtotheregionaroundthemeasured plane,somelaserlight isdispersedinalldirections.Thiscontributestomakethe entirewingvisibleinthePIV imagesandmakesitnecessarytomasklargeregionsoftheow elddataaround thewing.Thisproblemmayberesolvedifmeasuringwithafre estreamowinstead ofunderhoveringconditions.Anotherissuethatarisesisth atthewingitselfreects laserlight.Therearemanydocumentedwaystosignicantly reducetheamountof lightreectedbythewing.Forexample,objectsinthelaser sheetcanbepaintedwith Rhodamine6Gpaint.TheRhodaminepaintabsorbslaserlight with532nmwavelength andemitslightatotherwavelengths.Thelaserlightscatte redbytheparticlescanthen beseparatedfromthelightemittedbytheRhodaminepaintby mountingaband-pass lteronthecamera.Unfortunately,thepaintislikelytoch angethestructuralproperties ofthewingnoticeably,sincethewingsofinteresthavesuch thinmembranes.Wing deformationmayalsocausethepainttocomeoffduringappi ng.Theidealsolution wouldbetocreateorndamembranematerialthatabsorbslig htofthewavelengthof thelaserandthatatthesametimehasappropriatemechanica lproperties. Itcouldbeofgreatvaluetostudytheowaroundaappingwin ginacylindrical coordinatesystemwiththecoordinatesystemcenteratthew ingroot.Studyingthe owataxedradialcoordinateratherthanaspanwisecoordi nateofthewingata zeroappinganglewouldtracktheowaroundapproximately thesamewingsegment throughouttheappingcycle.Thatwouldshowmoreaccurate lywhereonthewing 174

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thrustisproducedsothateffortscouldbespentinvestigat ingandoptimizingasmaller regionofthewing.Suchacoordinatetransformationcouldal readybedoneonthehigh densitydataonthewingswithvaryingmembranes. Theefciencyofappingightshouldbestudiedtoo.Itmayb ehardtoextractthe powerinputtothewingaslossesinthepowertransmissionto thewingmayaccountfor alargepartofthetotalpowerinput.Numericalstudiescoul dthereforebeofgreatvalue forestimatingpowerinput. Deeperstudiesintothesubjectofappingightwillinevit ablyresultinaneedto learnhowtomaneuveravehiclewithappingwings.Thereare manypotentialwaysto changetheightkinematicstocontroltheightpath.Flapp ingamplitudeandfrequency naturallycometomind.Itmaybesufcienttohaveindividua lcontrolovertheapping amplitudeofthetwowingstocontroltheightpath,butmore elaboratechangesin kinematicsmaybenecessarytoachievedesiredmaneuverabi lity. Wingsthatareeasiertoworkwithwouldbebenecialtofuture studies.The currentwingsaremanuallyattachedtotheappingdeviceus ingglue.Adebonder hastobeusedtodetachthewinganditissometimesdifculta ndtimeconsumingto makeitcomeloose,especiallyifithasbeenattachedforalo ngtime.Theattachment betweentheappingdeviceandthewingmakesithardorimpos sibletoattachthe wingatexactlythesamepositionandwithanidenticalorien tation.Hence,thereisa repeatabilityproblemifthesamewingistestedtwiceincas ethewingisdetachedand reattachedbetweentests. Theowcreatedaroundthelatexwingonthesinusoidalappe rappearstobethe mostpromisinginthisstudyatcreatingstrongthrustandis likelytobeefcienttoo, astheowisareversevonK arm anvortexstreet.Akeytotheowisthatthewing resonatesatthreetimestheappingfrequency.However,it isnotfullyunderstoodwhat conditionsarerequiredtocreatesuchaow.Itshouldatlea stbeinvestigatedhow 175

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sensitivetheowistothemagnitudeofwingdeformationsan difthemagnitudeofthe thrustcanbeeasilycontrolledsimplybychangingtheappi ngamplitude. Aresonantsystemdoesn'thavetobecreatedbyusingaresona ntmembrane.It couldbecreatedbyallowingthewingtotwistatthewingroot insteadifthewingroot twistingiscoupledwithaspringsystem.Thewingmembranec ouldthenberigidoronly slightlyexiblebutwouldstillbeabletoachievehightwis tangles.Aspringsystemthat doesnotresonatewiththeappingfrequencycouldbeusedto trytondawingthat maintainsanearlyconstanttwistanglethroughmostofahal f-strokeonasemi-linear apper,aswasattemptedwithexiblewingsinthisstudy.Re sultspresentedbySane andDickinson[ 30 ]indicatethatanangleofattackrelativetothestrokeplan earound 20 or30 maybeideal.Thewingsinthisstudymusttwist60 or70 fromtheiroriginal shapetoachievesuchanangleofattack.Suchhightwistangle swouldalsobeeasierto achieveifthewingcanrotatearoundtheleadingedgeatthew ingroot. Thewingtwistangleandcurvatureofthechordrelativetoth edirection,speedand accelerationoftheoncomingowarethemostimportantfact orstomaximizethrustor efciencyofaappingsystem.Thesevariablescanbetarget edinasimpliedanalysis suchasusingquasi-steadyanalysistogetherwithstructur alniteelementanalysisfora crudeoptimizationofthekinematicsandwingproperties.H owever,tofurtheroptimize thekinematicsandthestructuralpropertiesofthewing,de layedstallmustbetaken intoaccount.Bydoingso,itispossibletomaximizethedurat ionthatlowpressure structuresconnectedtodelayedstallstayattachedtothew ingandhencehigherthrust canbecreatedthanwouldbepredictedifnottakingtheeffec tintoaccount.Thisshould makeitpossibleforappingighttoresultinhigherforces thanwouldbepredictedby quasi-steadyanalysis. 176

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APPENDIX:PIVALGORITHMACCURACY TheaccuracyofthePIVAlgorithmdescribedinSection 2.9 willbeanalyzedbelow. Theaccuracydependsonmanydifferentfactors,suchashowt hecamerasaresetup comparedtothelasersheetplane,thelaserlightintensity ,backgroundlightintensity, cameranoise,seedingparticledensity,apparentseedingp articlesize,andseeding particlescatterintensity.Hence,fortheanalysistobepo ssiblesomeassumptionshave tobemade.Usingtheseassumptions,articialPIVimageswit hparticlesandnoise comparabletorealcapturedimagesbutknownvelocitiesare createdtoassessthe algorithm.Theimagesarethenprocessedinthesamewayasth edatainSection 5.5 andtheerrorscomparedtotheknownvelocityeldsusedtoge neratethearticial images. A.1Assumptions Afewassumptionshavebeenmadetoperformtheaccuracyanal ysisofthe in-housegeneratedPIValgorithm.Thecameracalibrationwi llbeassumedtobe perfect.Thisassumptionshouldbereasonablewhencompari ngtodatathatis processedusingaLaVisionDaViscalibration,ifthesoftware 's“self-calibration” algorithmhasbeenusedtorenethecalibration.Suchacalib rationshouldbeaccurate andalignedwiththelasersheettowithinacamerapixelsize projectedfromtheonthe lasersheetplane. Furthermore,itisassumedthattheapparentsizeofapartic leisgovernedbythe diffraction-limitedspotdiameter,asdiscussedinSection 2.4.2 .Usingthisassumption, aparticle'spointlocationinspacecanbeprojectedontoth ecamerasensorplane. Onthesensorplane,theparticlewillhavetheapparentsize givenbyEquation 2–3 Inagivensetup,wherethef-numberandthelaserlightwavel engthsarexed,only themagnicationvariesinEquation 2–3 .Sincethemagnication M isontheorder of0.1,andthevariationsinmagnicationbetweendifferen tlocationsintheimages isapproximatelyanorderofmagnitudesmaller,itwillbeas sumedthat1+ M in 177

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Equation 2–3 canbeconsideredtobeconstant,sothatallparticleswillh avethe sameapparentsize. Itwillalsobeassumedthatthelaserlightintensityisunif orm.Thisassumptionis knownnottobetrueforthelasersystemactuallyusedinthis study.However,sincethe laserlightintensityacrossaninterrogationareadoesnot varymucheveninrealcases, thisassumptionshouldhavelittleimpactsincethePIValgor ithmisinvarianttoifthe imageintensityofasingleimageorinterrogationareaissc aledorifauniformvalueis addedtoit. Theseedingdensitywillbeassumedtobeuniform.Inreality ,theseedingdensity canvarybothindifferentpartsofasingleimage,andbetwee ndifferentsnapshotsinthe samesequence.Theanalysiswillthereforebelimitedtoadj ustingtheseedingdensity toarandomlychosenrepresentativesnapshot.Sincebothsee dingdensityandlaser lightintensityareconstant,backgroundlightintensitya ndnoisewillalsobeassumedto beconstant. Itwillbeassumedthatthescatteredlightintensityfromas eedingparticleisonly afunctionofitsdistancefromthecenterofthelasersheet. Thismeansthatitwillnot betakenintoaccountthatlightintensityscatteredfromal ocationinthelasersheet thatisfurtherfromthecamerawillbelower.Itwillalsobei gnoredthatthescattered lightintensityisafunctionoftheanglebetweenalinefrom theparticletothelaserlight source,andalinefromtheparticletothecamera.Thesefact orsshouldnotmatteras theyvaryverylittleoverthesizeofaninterrogationarea. Itfollowsfromthepreviousassumptionthattheseedingpar ticlesizemustbe uniform.However,inrealitytheparticlesizesvary.Sincet heapparentparticlesizeis determinedbythediffraction-limitedspotdiameter,thes eedingparticlesizematters onlyfortheamountoflightscattered.Assuminguniformpart iclesizesshould,if anything,makeitslightlyhardertomatchtwointerrogatio nareasoftwoarticialimages 178

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comparedtorealimagesinceonefactorthatcontributestom akingacorrectmatch morecorrelatedthanapotentialfalsematchistakenaway. A.2GeneratingArticialPIVImages TogeneratearticialPIVimages,particlesaregeneratedat randomlocationsina volumearoundthelasersheet.Eachcomponentofeachparticl e'slocationisgenerated asauniformlydistributedrandomvaluewithinthedomainof interest,whichisthe volumewhereparticlescanbeseenbythecamerasplusthehal fthemaximumdistance aparticlecanbeconvectedbetweentwoimages.Then,thepar ticlesareconvected bothbackwardsandforwardswithaRunge-Kuttastephalfati mestepineachdirection usingapredeterminedvelocityeld,creatinga“before”an dan“after”setofparticle locations.Theparticlesarethenprojectedontotheirresp ectivelocationsonthecamera sensorplanes,usingthesamecalibrationasfortheexperim entsinSection 5.5 .The lightforeachparticleisthenintegratedforeachpixel.Th econtributionoflighttoapixel ( i j )fromalltheparticlesis: I i j = N particles X p =1 I 0 Z i +0.5 i 0.5 Z j +0.5 j 0.5 1 p 2 p exp ( i 0 i p ) 2 +( j 0 j p ) 2 2 2 p F ( z p )d i 0 d j 0 ,(A–1) where( i p j p )isthenon-integerprojectedlocationsofparticles, p istheparticlesize, I o isthescatteredlightintensityofaparticle,and z p isthelocationofaparticleinphysical coordinates,where z p =0istheplaneofthelasersheet.Thefunction F reducesthe scatteredlightintensitywhenaparticleismovedfurthera wayfromthecenterofthe lasersheet,andisgivenby: F ( z p )= 1 p 2 laser exp z 2 p 2 2 laser .(A–2) laser isproportionaltothewidthofthelasersheet,andissetto2 mm. APIVimagedoesnotonlycontainlightscatteredfromparticl esintheimageplane, butalsounwantedbackgroundlightandnoise.Totakethatin toaccount,thecenter 512 512pixelsofarealPIVimageshowninFigure A-1A isanalyzed.Tomakethe 179

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noiseinthearticialPIVimagesrealistic,thedistributio nofthemeasuredintensities inthearticialPIVimagewillbematchedtotherealimage.Sin cenormallydistributed noisedidnotmatchtheintensitydistributionoftherealPIV imagewell,thegeneralized extremevalue(GEV)distributionwasused.Theprobabilityde nsityfunctionoftheGEV isgivenby[ 86 ]: 1 exp 1+ x 1 = # 1+ x 1 1 = .(A–3) Usingthisdistribution,arandomvaluecanbeassignedtoea chpixelinamatrix I ,and acombinedimagecanbecreatedsuchthat: I = I + I .(A–4) Thelightintensitydistributionof I dependsonsixvariables: I o N particles p , ,and .Thesevariablesarettominimizethedifferencebetweent hehistogramofthe realPIVimageandanarticialimageinameansquaresense.Ma tlab's fminsearch function,whichisanimplementationoftheNelder-Meadsim plexmethod,isusedforthe minimization.Theresultingcoefcientsare: I 0 =120.4742737007416, N particles volume =1.054725901 10 9 particles/m 3 p =0.2825631893381pixels, = 0.2060210852830, =112.1678267645274, =5.505121763342960. AcomparisonbetweenhowtheimageslookcanbeseeninFigure A-1 .Anotablevisual differencebetweentheimagesisthattherearesomeparticl esintherealPIVimagethat arealotbrighterthananyparticlesinthearticialimage. Thatcanprobablyattributed 180

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tothenon-uniformparticlesizesintherealPIVimage.Figur e A-2 showsacomparison betweentheintensitydistributionsoftherealandartici alPIVimages,andshowsthat theintensitydistributionofnoiseisverywellmatched.Th eupperpartoftheintensity distributionsrepresentlightscatteredfromparticlesfo rwhichthedistributionsmatch fairlywell. A.3Results Twodifferentvelocityeldswillbeanalyzed.Therstvelo cityeldisshownin Figure A-3A andhastheform: u 1 ( x y z t )= c 1 ( y y 0 ) e x + c 2 ( x x 0 ) e y + c 3 ( y y 0 ) e z ,(A–5) wheretheconstants c 1 c 2 ,and c 3 areallpositive.Theconstants c 1 c 2 c 3 x 0 y 0 and z 0 areselectedsothattheextremevaluesforallcomponentsar eapproximately 2.3 U ref U ref referstothereferencevelocityinthecaseofvaryingmembr anewingsin Section 5.5 .Theowhasconstantavelocitygradient,sothatthereisno contribution tothebiaserrorcausedbychangesinthevelocitygradientw ithininterrogationareas. Thebiaserrorforthethreecomponents,computedfrom500sn apshots,canbeseen in A-4 .Forthein-planecomponents,thebiaserrorislessthan1%o f U ref everywhere exceptontheveryedges.Thebiaserrorfortheoutofplaneco mponentislessthan2%. Themeanabsolutedeviationfromthemeanvelocityvectorsa reshowninFigure A-5A andisbetween2.5%and3.5%of U ref everywhereexceptontheedges.Thestriped patternsindicatethatthebiaschangeswhenthedisplaceme ntisinbetweenwhole pixeldisplacements.Thepatternismorecomplicatedthanf ortwo-componentPIV, sincethecamerasmeasuredifferentdisplacementsatthesa melocation,andhence thefractionalpartofthepixeldisplacementisusuallydif ferentforthetwocameras. Theerrorsdonotseemtovarybetweenthecenterandperipher yofthesnapshots, exceptontheveryedges.Figure A-5B showsthefractionofthesnapshotsthatyield validcross-correlationsateachlocationandisabove80%e xceptontheedges.The 181

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locationswheretheerrorsaresignicantlylargerthantyp icalcannowbeseentobe wherelessthan25%ofthesnapshotsarevalid.Thesevectors areverylikelytobe removedwhenttingdataovermanysnapshotstocreatephase -averageddataas describedinSection 2.10.2 ,whereapproximately40%ofthevalueshastobevalid. ThesecondowcanbeseeninFigure A-3B ,andisaphase-averagedsnapshot withstrongshedvortices.Missingvalueshasbeenlledinu singthepost-processing methodinSection 2.10.1 ,sothatarticialparticlescanbeconvectedeverywhere. Additionally,coordinatesaremultipliedby0.9beforeextr actingtheowvectorsto convecttheparticles,sothatvelocitiesaredenedallthe wayouttotheedges.Bi-cubic interpolationisusedtointerpolatethevelocityeldbetw eengridpoints.SincethePIV algorithmisexpectedtomeasuretheaverageowvelocityin aninterrogationarea, somebiasisexpectedwherethevelocitygradientisnotcons tant,sincethecentervalue doescoincidewiththeaverage.Thebiaserrorsfortheowe ldisshowninFigure A-6 Theseare,asexpected,largerthanfortheswirlingowwith constantgradients.Inthe worstspots,thebiaserrorisaround 5%of U ref .Similarly,Figure A-7A showsthat intheworstspotsaroundthevortex,themeanabsolutedevia tionfromthemeanow vectorsisaround 10%of U ref intheworstspots,butinmostoftheowelditissimilar inmagnitudetotheswirlingow.Figure A-7B tooshowsthatallgridpointsinthebulkof thevectoreldyieldmorethan80%validcross-correlation sfromthe500PIVsnapshots thatareprocessed. 182

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ARealPIVimage. BArticialPIVimage. FigureA-1.Comparisonbetween512 512pixelsofarealandanarticiallygenerated PIVimage. RealPIVdata ArticialimageFractionofpixelsIntensity 80100120140160180200 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 FigureA-2.Particleintensitydistributionsforrealandar ticialPIVimages. 183

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y x 1 0.500.5 1.5 1 0.5 0 ASwirlingow. y x 1 0.500.5 1.5 1 0.5 0 BFlowwithvortexfromPIVdata. 2 1012 FigureA-3.FloweldsfortestingPIValgorithm.Arrowsshow u and v components,and theout-of-planecomponent w isindicatedbythebackgroundcolor. y x 1 0.5 00.5 1.5 1 0.5 0 A u y x 1 0.5 00.5 1.5 1 0.5 0 B v y x 1 0.5 00.5 1.5 1 0.5 0 C w 4 20 24 FigureA-4.Swirlingowbiaserror(%of U ref ). 184

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y x 1 0.500.5 1.5 1 0.5 0 y x 1 0.500.5 1.5 1 0.5 0 02468 10 AMeanabsolutedeviationfromaverageowvector(%of U ref ). 020406080 100 BFractionofsnapshotsyieldingacross-correlationresult(%). FigureA-5.Swirlingowmeanabsolutedeviationfromaverage owvectorandfraction ofsnapshotsyieldingacross-correlationresult. y x 1 0.500.5 1.5 1 0.5 0 A u y x 1 0.500.5 1.5 1 0.5 0 B v y x 1 0.500.5 1.5 1 0.5 0 C w 4 2 024 FigureA-6.Vortexowbiaserror(%of U ref ). 185

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y x 1 0.500.5 1.5 1 0.5 0 y x 1 0.500.5 1.5 1 0.5 0 02468 10 AMeanabsolutedeviationfromaverageowvector(%of U ref ). 020406080 100 BFractionofsnapshotsyieldingacross-correlationresult(%). FigureA-7.Vortexowmeanabsolutedeviationfromaverage owvectorandfractionof snapshotsyieldingacross-correlationresult. 186

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BIOGRAPHICALSKETCH ErikS ¨ allstr ¨ omwasbornin1982inG ¨ oteborg,Sweden.HegrewupinG ¨ oteborgas theyoungestofthreechildren. InAugust2000ErikstartedstudyingattheSchoolofBusiness,Ec onomicsand LawatUniversityofGothenburg.HeearnedaBachelorofBusine ssAdministration in2005.In2001hehadtotakepauseinhisstudiestoservefor theSwedishNavy andwasstationedonthenavalshipHMSJ ¨ agaren.In2002hestartedengineeringat ChalmersUniversityofTechnology.Hespentthefallof2006 attheUniversityofFlorida ResearchandEngineeringEducationFacility(REEF)asavisitin gstudenttoworkonhis Mastersthesis.AfterearningaMasterofScienceinEngineerin gPhysicsfromChalmers UniversityofTechnologyin2007,hegottheopportunitytor eturntoUniversityofFlorida toearnhisDoctorofPhilosophyinAerospaceEngineering. 194