On the Control of a Canonical Separated Flow

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Title:
On the Control of a Canonical Separated Flow
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1 online resource (270 p.)
Language:
english
Creator:
Griffin, John C
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:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
CATTAFESTA,LOUIS NICHOLAS,III
Committee Co-Chair:
SHEPLAK,MARK
Committee Members:
UKEILEY,LAWRENCE S
ARNOLD,DAVID P
ROWLEY,CLARENCE

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Subjects / Keywords:
canonical
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
Flow separation is generally an undesirable phenomenon that produces adverse effects to ideal aerodynamic performance. Control of flow separation is thus a popular area of research for a complex problem. A common obstacle is the lack of understanding of the complex fluid mechanics in cases of flow separation. This is evident by the substantial amount of flow control achieved through trial-and-error methods. The purpose of this work is to better understand the nature of separation for improved active control methods, which includes closed-loop control via reduced order methods. Control of a canonical separation problem, with the key features of separated flow, is proposed. Therefore, separation is created on a flat plate model, void of curvature that would otherwise include effects particular to the type of aerodynamic body. The characteristics of the imposed separation are evaluated with the intent of having a nominally two-dimensional separation, with the same essential flow characteristics of a more traditionally stalled airfoil.Results provide a reduced-order estimation technique that is used to identify global, dynamic modes through experimental measurements. Reattachment of the baseline separation is first achieved in open-loop control via an unsteady actuation. Efficient reattachment is achieved by targeting the identified characteristic flow frequencies. The baseline and control results are used to identify a reduced-order model suitable for closed-loop control, with benefits of set-point tracking and disturbance rejection.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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 John C Griffin.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: CATTAFESTA,LOUIS NICHOLAS,III.
Local:
Co-adviser: SHEPLAK,MARK.

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UFRGP
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Applicable rights reserved.
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lcc - LD1780 2013
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UFE0046219:00001


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ONTHECONTROLOFACANONICALSEPARATEDFLOW By JOHNC.GRIFFIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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2013JohnC.Grin 2

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Idedicatethistomywife,Karen. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyfacultyadvisor,Dr.LouisCattafesta.Iamverygratefulfor hisfaithfulguidancethroughoutmygraduatestudiesandhisdedicationtothisresearch project'ssuccess.Ialsoappreciatetheinputandsupportfromtherestofmycommittee, includingDrs.LawrenceUkeiley,MarkSheplak,ClarenceRowley,andDavidArnold.I wouldliketospecicallyacknowledgeDr.Ukeileyforallowingmetooccupyhislaboratory whileIcompletedmyresearch. Ialsooweadebtofgratitudetomanycurrentandformermembersofthe InterdisciplinaryMicrosystemsGroupattheUniversityofFloridaformentoringand assistingmeduringmygraduatecareer.Specically,IwouldliketothankDr.Brandon Bertolucci,Dr.MiguelPalaviccini,Dr.JessicaMeloy,Dr.NikolasZawodny,Dr.Drew Wetzel,MatiasOyarzun,AshleyJones,AdamEdstrand,andRobertReger.Thanksalso tomycollaboratorsDr.JonathanTu,Dr.AdamHart,Dr.ShawnAram,andDr.Rajat Mittal. Finally,Iwouldliketoacknowledgetheimportanceofmyfamilyspresence,love,and supportthroughoutmyeducation.Mywordscanonlyfallshortoftheimmensegratitude Ifeelfortheirguidanceandencouragementduringthisprocess.Ithankmyparents,Mike andBarbaraGrin,forshowingmewhathardworkcanachieve.Iwouldliketoexpress appreciationformybrotherBenandsister-in-lawElizabethwhosharedtheirknowledge andexperiencetohelpmesucceed.Theirdaughters,RebeccaandEleanor,havegivenme muchjoyandlaughter.Lastbutcertainlynotleast,I'dliketothankmywife,Karen.This achievementispossiblebecauseofhermanysacrices,herabilitytoshowmethebestin everysituation,andherunwaveringlove. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................9 LISTOFFIGURES....................................10 ABSTRACT........................................17 CHAPTER 1INTRODUCTION..................................19 1.1Motivation....................................20 1.1.1EcientSeparationControl......................21 1.1.2Closed-LoopSeparationControl....................22 1.2Background...................................25 1.2.1Two-DimensionalFlowSeparation...................26 1.2.1.1Boundarylayerseparation..................26 1.2.1.2Sharp-edgedblu-bodyseparation.............29 1.2.2FrequencyScalesofSeparatedAirfoilFlow..............30 1.2.3FlowControlTerminology.......................30 1.2.4Closed-LoopFlowControlStrategies.................31 1.2.4.1Model-freecontrol......................32 1.2.4.2Controlofblack-boxmodels.................33 1.2.4.3Controlofreduced-ordermodels...............33 1.2.5Reduced-OrderEstimation.......................35 1.2.5.1Stochasticestimation.....................36 1.2.5.2Model-basedestimation...................38 1.3Objectives....................................40 1.4Approach....................................40 1.5ProposalOutline................................41 2LITERATUREREVIEW:CONTROLOFSEPARATEDFLOWS........46 2.1IntroductiontoSeparationControlPlatforms................46 2.2ControlofBlu-BodyandFixed-PointSeparation..............48 2.2.1Open-LoopControl...........................49 2.2.2SingleSensorProportionalFeedbackControl.............51 2.2.3Model-FreeControl...........................53 2.2.4ControlofBlack-BoxModels......................54 2.2.5ControlofReduced-OrderModels...................57 2.2.5.1Galerkinmodels.......................57 2.2.5.2Vortexmodels.........................63 2.3ControlofBoundaryLayerSeparation....................64 5

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2.3.1Open-LoopControloftheCanonicalFlowSeparation........64 2.3.2Model-FreeControl...........................67 2.3.3ControlofBlack-BoxModels......................68 2.3.4ContolBasedofReduced-OrderStateEstimation..........70 2.4UnresolvedTechnicalIssues..........................71 3NUMERICALMETHODS..............................83 3.1ModalDecomposition..............................83 3.1.1ProperOrthogonalDecomposition...................83 3.1.2DynamicModeDecomposition.....................87 3.2Reduced-OrderEstimation...........................91 3.2.1StochasticEstimation..........................92 3.2.1.1Single-time-delaylinearstochasticestimation.......94 3.2.1.2Multi-time-delaylinearstochasticestimation........94 3.2.2Model-BasedEstimation........................96 3.2.2.1Modelidentication.....................96 3.2.2.2Kalmanlter.........................99 3.2.2.3Kalmansmoother.......................100 3.3EstimationProcedures.............................101 4EXPERIMENTALSETUP.............................104 4.1FlatPlateModel................................104 4.1.1Zero-NetMassFluxActuator.....................105 4.1.2UnsteadySurfacePressure.......................106 4.1.3SteadySurfacePressure.........................106 4.2FlowFacility..................................107 4.2.1Low-SpeedWindTunnel........................107 4.2.2SeparationSystem............................108 4.3FluidMeasurementSystems..........................109 4.3.1Hot-wireAnemometer..........................109 4.3.2PIVSystem...............................110 4.4ControlSystems.................................113 4.4.1OpenLoop................................113 4.4.2ClosedLoop...............................114 5BASELINEFLOWRESULTS............................121 5.1BaselineSeparatedFlow............................121 5.1.1MeanSeparationBubble........................122 5.1.2SeparationBubbleDynamics......................125 5.1.3MeanWakeFlow............................127 5.1.4WakeDynamics.............................128 5.2EstimationofBaselineFlow..........................131 5.2.1Reduced-OrderBasis..........................131 5.2.2StochasticEstimation..........................133 6

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5.2.3Model-BasedEstimation........................136 5.3DynamicAnalysis................................138 5.3.1DynamicModeDecomposition.....................139 5.3.2DiscreteFourierTransform.......................142 5.4SummaryofResults..............................143 6OPEN-LOOPCONTROL..............................165 6.1ActuatorCharacterization...........................165 6.1.1IndividualDiscs.............................166 6.1.2SpanwiseVariability...........................166 6.1.3ModulatedInputs............................167 6.1.4AcousticContamination........................168 6.1.4.1Burstmodulatedacoustics..................169 6.1.4.2Amplitudemodulatedacoustics...............171 6.2ControlledSeparationBubble.........................172 6.3ControlledWake................................176 6.4EstimationofControlledFlow.........................177 6.4.1Reduced-OrderBasis..........................178 6.4.2StochasticEstimation..........................179 6.5DynamicAnalysis................................180 6.6SummaryofResults..............................183 7CLOSED-LOOPCONTROL............................200 7.1Reduced-OrderModels.............................201 7.1.1ModelFormationandObserverDesign................202 7.1.2ShiftModeTracking..........................205 7.1.3OscillatoryModesOptimization....................208 7.2LinearQuadraticGaussianRegulator.....................210 7.2.1SystemIdenticationandObserverDesign..............212 7.2.2ControlResults.............................214 7.3SummaryofResults..............................216 8CONCLUSION....................................230 8.1ResearchSummary...............................231 8.1.1BaselineFlowCharacterization.....................231 8.1.2FlowControl...............................232 8.1.2.1Open-loopcontrol......................233 8.1.2.2Closed-loopcontrol......................234 8.2ResearchImpact................................235 APPENDIX 7

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ASEPARATIONCONTROLPLATFORMS.....................238 A.1Cylinder.....................................238 A.2BackwardFacingStep.............................239 A.3D-ShapedBluBody..............................240 A.4Airfoil......................................240 BPIVUNCERTAINTY................................243 B.1BiasUncertaintyTwoComponent......................243 B.1.1VelocityMagnitude...........................243 B.1.2VelocityDirection............................244 B.1.3VelocityComponents..........................245 B.1.4Time-AveragedVelocityComponents.................245 B.2SPIVBiasUncertainty.............................246 B.3RandomUncertainty..............................248 B.4MeasurementUncertainties...........................248 REFERENCES.......................................257 BIOGRAPHICALSKETCH................................270 8

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LISTOFTABLES Table page 1-1Comparisonofthereviewedcontrolmethods....................45 2-1Summaryofgeometric,ow-specic,andcontrolparametersforthediscussed separationcontrolplatforms..............................78 2-2Summaryofdirectfeedbackowcontrolstudiesforblu-bodyseparation....78 2-3Summaryofmodel-freeclosed-loopowcontrolstudiesforblu-bodyseparation.79 2-4Summaryofrobustcontrolmethodsofblack-boxmodelsforblu-body separation........................................79 2-5SummaryofowcontrolstudiesusingGalerkinreduced-ordermodelsfor blu-bodyseparationandcavities..........................80 2-6Summaryofowcontrolstudiesusingvortexmodelsforblu-bodyseparation..81 2-7SeparationbubblelocationsandsizesforcasesofZNMFforcingofthebaseline canonicalseparatedowatRe c =1 10 5 and C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 Kotapati etal. 2010. C isrecalculatedtomatchthedenitionofEq.2{9...........81 2-8DownhillsimplexoptimizationresultsforbothAMandBMsignalswithand withoutenergypenaltyforcasesofZNMFforcingofbaselineseparationfroma NACA0025airfoilatRe c =10 5 and20 angleofattackTian etal. ,2006 a ...81 2-9Summaryofadaptiveandmodel-freeclosed-loopowcontrolstudiesfor boundarylayerseparation...............................82 2-10Summaryofrobustcontrolmethodsofblack-boxmodelsforboundarylayer separation........................................82 2-11Summaryofowcontrolstudiesusingreduced-orderstateestimationfor blu-bodyseparation.................................82 3-1Descriptionsofvectornotationshorthand......................103 4-1SummaryofPIVregionsstudied...........................120 4-2ProcessedPIVsettings................................120 5-1Characteristicfrequenciesfortheseparatedshearlayerandwake.........145 B-1RandomuncertaintyintervalsforstatisticalowquantitiesBenedict&Gould, 1996..........................................252 9

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LISTOFFIGURES Figure page 1-1Schematicofcanonicalseparatedowforaatplatemodelnottoscale....42 1-2Separationofatwo-dimensionallaminarboundarylayeronaatplate......42 1-3Transitionalseparationbubble............................43 1-4Adepictionofturbulentboundarylayerseparation.................43 1-5Featuresofowoverabackward-facingstep.....................43 1-6Thewakebehindaatplatewithabluntbase...................44 1-7Schematicshowingthethreenaturalfrequenciesthatcanoccurinseparated airfoilows.......................................44 1-8Traditionalcategoriesofowcontrol.........................44 2-1Illustrationofthegeneralincreaseinowcomplexitythataccompaniesmore practicalapplications.................................73 2-2Sketchoftheplateau-likeeectoftheoutput C p forincreasing C ........73 2-3CongurationoftheD-shapedmodelwithtwoactuatorslotsandthreesensor arrays..........................................73 2-4Congurationofthebackward-facingstepwithfourloudspeakersandfour sensorarrays......................................74 2-5Congurationofthebackward-facingstepwithfourspanwiseseparated actuatorsandfoursensorarrays...........................74 2-6Schematicoftheactuation-sensingstrategybyDahan etal. 2011........74 2-7StreamlinesofthebaselinesimulatedowaroundacircularcylinderatRe d = 100...........................................75 2-8Experimentalsetupofatwo-dimensionalblubodyttedwith15dierential pressuresensorsandactuatorslotsforpulsedsuction................75 2-9Flowchartofthestochasticestimationprocessformodulationoftheactuation signal..........................................77 3-1Flowchartofmodel-basedestimatorimplementation................103 4-1Schematicoftheexperimentalsetup.........................115 4-2Photographsoftheatplatemodel.........................115 10

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4-3Actuatorcross-sectionalschematics..........................116 4-4Actuatorassembly...................................117 4-5Unsteadypressuresensorplacementontheupperatplateandatbase.....117 4-6Asectionalviewofthelowerportionoftheseparationsystemattachedtothe ceilingofthetestsection...............................118 4-7SchematicoftheSPIVsetup.............................118 4-8SchematicshowingtheapproximatelocationsofPIVmeasurementregions....119 4-9Schematicoftheclosed-loopsetup..........................119 5-1Time-averagedstreamwisevelocityprolesandskin-frictiondistributionsinthe separationandreattachmentregions.........................146 5-2Theaverageseparationandreattachmentlinesshownrelativetomodel components.......................................146 5-3AveragevelocitycomponentscomputedfromSPIVplanesatvariousslices alongthespanwisedimension.............................147 5-4AverageturbulentstatisticscomputedfromSPIVplanesatvariousslicesalong thespanwisedimension................................148 5-5PowerspectraldensitiesofunsteadypressuresensorsS1throughS6........148 5-6InstantaneousvorticityfromaPIVsnapshotofthebaselineseparationregion..148 5-7Thecross-correlationcoecientbetweenhighlightedpressuresensorsatequal spanwisebutdierentstreamwisepositions.....................149 5-8AveragevorticitycomponentscomputedfromPIVofthenearwakeregion....149 5-9AverageturbulentquantitiescomputedfromPIVofthenearwakeregion....150 5-10Unsteadypressurespectraofwakepressuretransducersforbaselineseparation andstandardblu-bodywake.............................151 5-11CPSDbetweenunsteadypressuretransducersS8andS11forthebaseline separatedow.....................................151 5-12Hot-wirespectraofasingleprobehotwireplaced0.14 c downstreamofthe trailingedge x=c =1 : 14...............................152 5-13Energycontentoftherst r PODmodesoftheseparationbubblePIVdata...152 5-14SpanwisevorticityofPODmodescomputedfromPIVeldsofthebaseline separationregion....................................152 11

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5-15Energycontentoftherst r PODmodesofthewakePIVdata..........153 5-16SpanwisevorticityofPODmodescomputedfromPIVeldsofthebaseline separationregion....................................153 5-17Schematicillustratingthedelayintervalrelativetotheestimationtime......154 5-18Errormetricforsingle-timedelaymLSEoftheseparationbubbleregion.....155 5-19ErrormetricforMTD-mLSEoftheseparationbubbleregion...........155 5-20SpanwisevorticityofanexamplePIVsnapshotoftheseparationbubbleandits correspondinglow-orderestimates..........................156 5-21Errormetricforsingle-timedelaymLSEofthewakeregion............156 5-22ErrormetricforMTD-mLSEofthewakeregion..................157 5-23SpanwisevorticityofanexamplePIVsnapshotofthewakeandits correspondinglow-orderestimates..........................157 5-24DMDeigenvaluesofMTD-mLSEestimatesandKalmansmootherestimatesfor theseparationbubbleregion.............................158 5-25RealandimaginaryportionsofselectseparationbubbleDMDmodescomputed fromMTD-mLSEestimates..............................158 5-26DMDeigenvaluesofMTD-mLSEestimatesandKalmansmootherestimatesfor thewakeregion.....................................159 5-27RealandimaginaryportionsofselectwakeDMDmodescomputedfrom MTD-mLSEestimates.................................159 5-28DMDeigenvaluesofMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregions...............................160 5-29RealandimaginaryportionsofDMDmodesfromMTD-mLSEestimates comprisedfromboththewakeandseparationbubbleregions...........161 5-30DFTmodenormsoftheMTD-mLSEestimatesandKalmansmoother estimatesfortheseparationbubbleregion......................162 5-31RealandimaginaryportionsofselectseparationbubbleDFTmodescomputed fromMTD-mLSEestimates..............................162 5-32DFTmodenormsofthewakeMTD-mLSEestimates................162 5-33RealandimaginaryportionsofselectwakeDFTmodescomputedfrom MTD-mLSEestimates.................................163 12

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5-34DFTmodenormsoftheMTD-mLSEestimatesfromboththewakeand separationbubbleregions...............................163 5-35RealandimaginaryportionsofDFTmodesfromMTD-mLSEestimates comprisedfromboththewakeandseparationbubbleregions...........164 6-1Actuatordiscs'responsesto50V pp sinusoidalinputwithfrequencyvaried between50and2600Hzinincrementsof50Hz...................184 6-2Actuatordiscs'responsestovariousamplitudesofsinusoidalinputacrossthe frequencyrangeof1600to2600Hzinincrementsof50Hz.............184 6-3Spanwisevariationoftheactuatorresponsetovariousamplitudesofsinusoidal inputwithacommonfrequencyof2050Hz.....................185 6-4Genericburstmodulatedwaveformwith6cycleswithintheburstduration....185 6-5Actuatorcharacterizationwithvariousinputlevelsandfrequenciesofburstand amplitudemodulation.................................185 6-6Genericamplitudemodulatedwaveform.......................186 6-7MeasurementsandspectraofsensorS1inresponsetoBMactuationwith f m = 90Hzand C =4 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 nofreestreamow...................186 6-8Withthewindtunnelo,theaveragecyclicresponseofunsteadypressure transducerS1toburstmodulatedinputatvariousamplitudesand90Hz modulationfrequency.................................187 6-9AttenuationoftheBMactuationpeakat f m =90HzforsensorS1andarange ofinputpeak-to-peakvoltagesnofreestreamow.................187 6-10MeasurementsandspectraofunsteadypressuretransducersS1,S3,S5,S8,and S11inresponsetoBMactuationwith F + m =2 : 68and C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 ......188 6-11MeasurementsandspectraofsensorS1inresponsetoAMactuationnoow with f m =90Hz....................................188 6-12Staticandunsteadypressurefromtheuppersurfaceofthemodelwithvarious levelsofactuationfromasinusoidalinputand f c =2050.............189 6-13Staticpressurealongtheuppersurfaceofthemodelwithburstmodulated actuationat40V pp amplitudeandtherangeofmodulationfrequencies......189 6-14Staticpressurealongtheuppersurfaceofthemodelwithburstmodulated actuationatselectamplitudeandmodulationfrequencypairings.........190 6-15Averagestreamwisevelocitycontoursoftheuppersurfaceseparationregionfor thebaselineseparatedowandselectcasesofopen-loopcontrol..........190 13

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6-16Thevariationofseparationbubbleheights H sep,1 and H sep,2 tolevelsof C ....190 6-17Phase-averagedvorticitycontoursoftheuppersurfaceseparationregionwith burstmodulationcontrol...............................191 6-18Phase-averagedvorticity,streamwisevelocity,andtransversevelocitycontours oftheuppersurfaceseparationregionwithsinusoidalcontrol...........192 6-19PSDofahotwireplacedinthewakeatapproximately x=c =1 : 14and y=c =0..192 6-20Averagespanwisevorticitycontourofthenearwakeregionforcontrolwith C =0 : 011andnomodulationofthesinusoidalinput...............193 6-21AverageturbulentquantitiescomputedfromPIVofthenearwakeregionfor controlwith C =0 : 011andnomodulationofthesinusoidalinput........194 6-22Energycontentoftherst r globalPODmodesoftheseparationbubbleand wakePIVdatafrombaselineandcontrolcases...................195 6-23SpanwisevorticityofglobalPODmodescomputedfromPIVeldsofthe open-loopcontrolledseparationbubbleregion....................195 6-24SpanwisevorticityofglobalPODmodescomputedfromPIVeldsofthe open-loopcontrolledwakeregion...........................196 6-25AverageerrormetricforMTD-mLSEoftheglobalPODmodecoecientsfor theseparationbubbleandwake............................196 6-26PowerspectraldensitiesofselectsensorsduringBMopen-loopcontrolwith f m =90Hz.......................................197 6-27DMDeigenvaluesofMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregionsinresponsetoburstmodulatedforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and1 : 9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 ,bothat f m =90Hz......................198 6-28DMDmodesfromMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregionsinresponsetoburstmodulatedforcingat C = 4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and f m =90Hz..............................198 6-29DFTmodenormsoftheMTD-mLSEestimatescomprisedfromboththewake andseparationbubbleregionsinresponsetoburstmodulatedforcingat C = 4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,bothat f m =90Hz...................199 6-30DFTmodesfromMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregionsinresponsetoBMforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and f m =90Hz.......................................199 7-1Unsteadypressureresponseoftheclosed-loopsensorsforarampincreaseofthe BMamplitude.....................................218 14

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7-2Shiftmodecoecientandinputvoltageamplitudeforthethreeowconditions includedintheglobalPODcomputation.......................218 7-3Oscillatorymodecoecientsforthethreeowconditionsincludedintheglobal PODcomputation...................................219 7-4Schematicoftheshiftmodetrackingcontrolstrategywithinputsofunsteady pressureandanoutputactuationsignal.......................219 7-5Closed-loopstepresponsesforshiftmodetrackingcontrolscheme.........220 7-6Averagestreamwisevelocityofseparationbubbleandwakeregions........221 7-7Threeinstantaneoussnapshotsfromthe b 0 ; set =0 : 030trackingcase,which convergesto C =3 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 for F + m =2 : 68.....................222 7-8Closed-looptrackingresponseoftheshiftmodecoecient b 0 ...........222 7-9Thevariationofseparationbubbleheights H sep,1 and H sep,2 toshiftmode coecientset-points..................................223 7-10ResponseofestimatedDMDcoecientstoarampincreaseoftheBMamplitude.223 7-11Schematicsoftheoscillationamplitudecontrolstrategieswithinputsof unsteadypressureandanoutputactuationsignal..................224 7-12Closed-loopoptimizationoftheoscillatoryDMDcoecientsusingBMinputat F + m =2 : 68........................................225 7-13Averagestreamwisevelocityofseparationbubbleandwakeregionsduring controlofDMD-basedstateoscillations.......................225 7-14Threeinstantaneoussnapshotsfromtheminimizationof r controlcase,which convergesto C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 for F + m =2 : 68.....................226 7-15SchematicoftheLQGcontrolstrategywithinputsofunsteadypressureandan outputactuationsignal................................226 7-16Open-looprampandmodulationchirpresponsefromtheclosed-loopsensors...227 7-17Closed-loopresponsetoLQGcontrolofanidentiedstate-spacesystem.....228 7-18AveragestreamwisevelocityofseparationbubbleandwakeregionsduringLQG control.........................................228 7-19Thevariationofseparationbubbleheights H sep,1 and H sep,2 tolevelsof C ....229 A-1Flowseparationofabackward-facingstep......................242 A-2Sketchofthewakebehindacircularcylinder....................242 15

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A-3WakeowbehindaD-shapedmodel.........................242 A-4Separatedowoveragenericairfoil.........................242 B-1Anillustrationofthebiaserrorinvectordirection,determinedbyangle ....249 B-3Biasandrandomuncertaintiesoftheseparationbubblevelocitycomponentsfor thebaselineseparatedow..............................252 B-4Biasandrandomuncertaintiesofthewakeregionvelocitycomponentsforthe baselineseparatedow................................253 B-5Randomuncertaintiesoftheseparationbubbleregionvelocitycomponentsfor thebaselineseparatedowacquiredwithSPIV...................254 B-6Biasandrandomuncertaintiesoftheseparationbubblevelocitycomponentsfor theclosed-loopcontrolcaseofminimum r ......................255 B-7Biasandrandomuncertaintiesofthewakeregionvelocitycomponentsforthe closed-loopcontrolcaseofminimum r ........................256 16

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ONTHECONTROLOFACANONICALSEPARATEDFLOW By JohnC.Grin December2013 Chair:LouisN.CattafestaIII Major:MechanicalEngineering Flowseparationisgenerallyanundesirablephenomenonthatproducesadverseeects toidealaerodynamicperformance.Controlofowseparationisacomplexproblemand thusapopularareaofresearch.Acommonobstacleisthelackofunderstandingofthe complexuidmechanicsincasesofowseparation,evidentbythesubstantialamount ofowcontrolachievedthroughtrial-and-errormethods.Thepurposeofthisworkisto betterunderstandthenatureofseparationforimprovedactivecontrolmethods,which includesclosed-loopcontrolviareducedordermethods. Controlofacanonicalseparationproblem,withthekeyfeaturesofseparatedow, isachievedatachordReynoldsnumberof10 5 .Separationiscreatedonaatplate model,voidofcurvaturethatwouldotherwiseincludeeectsparticulartothetypeof aerodynamicbody.Thecharacteristicsoftheimposedseparationareevaluatedwiththe intentofhavinganominallytwo-dimensionalseparation,withthesameessentialow characteristicsofamoretraditionallystalledairfoil.Resultsprovideareduced-order estimationtechniquethatisusedtoidentifyglobal,dynamicmodesthroughexperimental measurements.Reattachmentofthebaselineseparationisrstachievedinopen-loop controlviaZNMFactuation.Ecientreattachmentisreachedbytargetingtheidentied characteristicowfrequencies,whichisabletoreattachtheseparatedowwithless thanaquarterofthecontroleortasacomparisoncasewithhigh-frequencyforcing. Thebaselineandcontrolresultsareusedtoidentifyareduced-ordermodelsuitablefor 17

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closed-loopcontrol,withbenetsofset-pointtrackingandfullboundarylayerattachment withminimumcontroleort. 18

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CHAPTER1 INTRODUCTION Flowseparationisoneofthemostwidelystudiedowphenomenabecauseofits adverseeectsonow-relateddevices.Theperformancelossesduetounwantedow separationarewellestablishede.g.lossoflift,increaseddrag,noisegeneration,structural vibrations,andopticaldegradationforcompressibleows.Manyoftheseconditions ofseparationhavebeencontrolledtomitigatetheadverseeects,andallarepartof ongoingresearch.However,eectivereattachmentormodicationofseparatedows withminimalcontroleortisnotyetwellunderstood.Thisisdueinlargeparttothe complex,nonlinearnatureofuidmechanics,whichresultsinanoverwhelmingreliance onopen-loopstudiesthatachieveowcontrolviaparametricstudiesortrial-and-error methods.Abetterunderstandingofthenaturalinstabilitiesofseparatedowsandtheir nonlinearinteractionsisdesired. Thisresearchaimstoidentifyandtargetthenaturalowinstabilitiespresentin aseparatedow,leveragingthemtoeitherreattachtheoworsuppressseparation altogether.Ratherthananalyzeandcontroltheseparatedowfromaspecicairfoil, a canonicalseparatedow iscreatedMittal etal. ,2005;Kotapati etal. ,2010.As illustratedinFigure1-1,boundarylayerseparationfromaatplateisinducedfroma strongadversepressuregradientthatisimposedviasuction/blowingboundaryconditions ontheceiling.Thisplatformisideallysuitedforseparationcontrolbecausetheinduced separationexhibitsthetraditionalfeaturesofaseparatedowwhileremovingany curvaturedependencefromcommonlifting-typeairfoils,whichisdiculttoisolate amongotherparametersandhasbeenshowntoaectthereceptivityoftheowto actuationGreenblatt&Wygnanski,1999,2003.Theowischaracterizedbyashear layerinstability,wakevortexshedding,andapotentialoscillationassociatedwiththe separationbubble.Thecanonicalcongurationisalsoamenabletocomputationalstudies, inwhichthelocationandextentoftheseparationhavebeenprescribedbychangingthe 19

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levelofsuction/blowingaswellasthemodelplacementMittal etal. ,2005;Kotapati etal. ,2010;Aram etal. ,2010;Tu etal. ,2011.Thisisalsoadvantageouswhencompared toanairfoilstallcondition,inwhichtheadversepressuregradientislessprescribedand moredependentontheangleofattack,freestreamvelocity,andairfoilshape. Upondetectionofthecharacteristicfrequencycontentwithinthebaselineow,a zero-net-mass-uxZNMFactuatorisutilizedtoperiodicallyexcitetheowbytargeting thenaturalinstabilities.ZNMFactuators,alsocommonlyreferredtoassyntheticjets, areutilizedforthiscontrolpurposebecauseoftheirrelativelysimpledesign,abilitytouse onlytheworkinguid,andexibilityinforcingspecicfrequencycontent.Inthispaper, theeectsofvaryingtheforcingfrequencyandlevelaremeasuredwiththeobjective ofbetterunderstandingtheinteractionsoftheforcingwiththecomplexdynamicsof thebaselineow.Inadditiontotheseopen-loopresults,closed-loopcontrolalgorithms areimplementedbasedeitheronreduced-orderobservationsoftheowstateorsimpler input-outputbehavior.Theresultsfromeachmethodarecompared. Thisrstchapterintroducestheowphysics,modelingprocedures,andactivecontrol approachesforowseparation.Amoredetailedmotivationforthisresearchisinitially discussed,focusingonthebackgroundofecientseparationcontrolanditsunclear dependenceonforcingfrequency.Thisisfollowedbytechnicalbackgroundonlow-order modelingandasectiononstateestimationforreduced-ordermodeling.Finally,the objectivesandapproachofthisresearchtowardsseparationcontrolarepresented. 1.1Motivation Flowseparationoccursinmanypracticalapplicationsandhasbeenthefocusofmuch researchbecauseofitsadverseeectsonaerodynamicperformancee.g.lossofliftonan airfoil,increaseddrag,andpotentialunsteadyloads.Asaresult,owcontrolresearchis often,butnotexclusively,associatedwiththereattachmentorinuenceofseparatedows inordertomitigateorsuppresstheseunfavorableeects.Flowcontrolclearlyresidesin bothareasofuidmechanicsandcontroltheory,nottomentionitsrelevancetonumerical 20

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methodsandstabilitytheoryBewley,2001.Onafundamentallevel,however,Gad-el Hak2000denesowcontrolasanattempttoalterthecharacterordispositionof aoweldfavorably."Modernowcontrolhasrenedthisoverarchinggoaltoachieve globalorlarge-scalechangestoaoweldwithminimaleort,whichreliesonadeeper understandingoftherelevantowphysics.Intheparticularcaseofowseparation,this renedgoalhintsatleveragingthenaturalinstabilitiestoenhanceoramplifytheeects fromactuationWilliams&MacMynowski,2009.Thismotivatestheneedtoidentifythe complexinteractionofcharacteristicowdynamicsforecientseparationcontrol,aswell asdesignmoreappropriateclosed-loopcontrolmethodstoensureeectiveness. 1.1.1EcientSeparationControl Thisperspectiveofecientseparationcontrolhighlightstheshortcomingsofa majorityofcurrentlyappliedseparationcontrolmethods,whichareoftenopenloop. Open-loopcontrolmanipulatessomedevicetocontrolthesystemwithoutfeedback.For unsteadyactuation,thereareprimarilytwoparametersthatareusedtocharacterize theactuation:forcingfrequencyandoutputlevel.Theformerisoftenrepresentedasa non-dimensionalfrequency F + = f a =f n ,where f a istheforcingfrequencyand f n issome naturalfrequencyoftheseparatedowthatcanbedenedusingappropriatelengthand velocityscales.Thelatteristraditionallydescribedbythemomentumcoecient C ,a ratioofmomentumoutputbyactuationtothefreestreammomentumoverareference areaoftheplatform.Itiswellunderstoodthatforsinusoidalforcing,controlauthority increasesmonotonicallywith C Seifert etal. ,1993,1996;Seifert&Pack,1999;Glezer &Amitay,2002;Mittal&Rampunggoon,2002.However,controlauthorityvaries non-monotonicallywith F + Greenblatt&Wygnanski,1999;Seifert&Pack,2000;Glezer etal. ,2003;Raju etal. ,2007.Thereby,amoreecient C maybeachievedbyleveraging aspecicvalueorrangeofvaluesfor F + .Thisiscrucialforhigh-speedowsandother situationsthatrequireeectivecontrolviaactuationlimitedtolowforcinglevels. 21

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Unfortunately,nosingledenitionof F + appliesuniversallybecauseclassicstallcan becharacterizedbyuptothreenaturalfrequencies:thatoftheseparatedshearlayer,the wake,and,intheeventofreattachment,theseparationbubble.Eachofthesenatural frequenciesscalesbyadierentlengthassociatedwiththeseparationandacharacteristic speed,althoughfreestreamvelocity U 1 isoftenusedforthelatter.Forstudiesthatdene F + basedonthemodelchordlength,optimalvaluesof F + havebeenreportedbetween 0.55and5.5Bar-Sever,1989;Seifert etal. ,1993;Ravindran,1999;Wygnanski,2000; Margalit etal. ,2002;Darabi&Wygnanski,2002;Funk etal. ,2002.Otherstudiesthat scale F + bythelengthoftheseparationbubbleobserveoptimalvaluesbetween0.75and 2.0Seifert etal. ,1996;Pack&Seifert,2000;Pack etal. ,2002.Amitay etal. 2001 determinedthatforcingatextremesof F + > 10onanunconventionalairfoiloutperformed congurationswith F + < 4. Obviously,thereisalackofconsensusconcerningoptimalforcingfrequencies,even ifthevariousdenitionsof F + arerecognized.Animportantaspecttothesendings isthatcontrolauthorityisgovernedbybothforcingfrequencyand C ,andneither canbeneglected.Forinstance,ZNMFactuatorscanexhibitsubstantialdierencesin output C forsmallchangesinoperating F + .Alackofunderstandingofthecomplex interactionsamongthemultiplefrequencycomponentsandthecorrespondingvaluesof C mayexplainsomeofthespreadintheliterature.Amorethoroughtreatmentofthese issuesispresentedbyMittal etal. 2005.Thispaperrecognizesthediscrepanciesin optimalcontrolandaimstoinvestigatecontrolauthorityonacanonicalseparatedow conguration,whereboth C and F + arecarefullytreated. 1.1.2Closed-LoopSeparationControl Open-loopcontrolresultshelptoevaluatethereceptivityofaowtoactuation. Therequiredtasksforclosed-loopcontrol,suchasfeedingbacktheaccurateeectsof actuationwithappropriatei.e.non-intrusive,smallscale,etc.sensorsandproperly adjustingthecontrolinputduringoperation,canbedicultformanyowcontrol 22

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problems.However,open-loopcontrolsystemscanlackeciencyandarenotrobustto disturbancesorunexpectedchangesinoperatingconditions.Anecientcontrolstrategy shouldbeabletoadaptinreal-timetothechangingconditionsinordertocontinuously achieveitsobjectivewithnearminimaleort.Intheeldofowcontrol,thistaskis mademoredicultbythecomplexnatureofuiddynamics.Anidealclosed-loopcontrol methodwouldbeabletodetecttheevolutionofthehigh-dimensionalowstate,including thepreciseeectsofactuationandadjustingfortheimperfectionsandnoiseinthe sensors.Unfortunately,suchacontrolloopisnotpossibleinthenearfutureGerhard etal. ,2003;King etal. ,2005. Evenhigh-dimensionaldiscretizationofthegoverningequationsfromdirectnumerical simulationsDNSorlargeeddysimulationsLESisnotyetcapableinreal-time,which isnecessaryfordynamicfeedback.Asasimplealternative,empiricalmodelsofthelumped systembehavior,orso-calledblack-boxmodels,attempttoextractandmodelthetransfer functionsbetweenselectedinputandoutputparameters.Oeringacompromisebetween thesetwomethods,athirdapproachisalow-dimensionalmodelderivedfromdiscrete measurementsofthehigh-dimensionalsystem.These reduced-ordermodels areameansof capturingtherelevantdynamicsofacomplex,high-dimensionalsystemusingareduced numberoffeedbacksensors.Becausetheselow-ordermodelsprovidebasicphysical understandingoftheowandpermitreal-timefeedback,theyareapotentialmeansto modeltherelevantowinstabilitiesformoreecientseparationcontrolstrategies. Numerousexamplesofreduced-ordermodelinginowcontrolexist,butmostare dedicatedtothebenchmarkowofvortexsheddingbehindacircularcylindere.g. Graham etal. ,1999;Gerhard etal. ,2003;King etal. ,2005,2008;Siegel etal. ,2006.A majorityofthesearelowReynoldsnumbersimulations,usuallyRe d =100or200basedon cylinderdiameter d .Graham etal. 1999developedareduced-ordermodelusingmodes fromproperorthogonaldecompositionPODoftheuncontrolledsimulatedowover acircularcylinderatRe d =100.Acontrollawdevelopedfromoptimalcontroltheory 23

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rotatedthecylinderbasedonthemodeltoeectivelyreducethewakeunsteadiness, measuredbytheevolutionofenergyinthedominantPODmode.However,theaccuracy ofthereduced-ordermodelformedfromthebaselineowdeclined,tothedetrimentofthe controller'seectiveness,asthecontrolledowtrajectorystrayedfromtheuncontrolled oscillationsthatwereusedtogeneratethemodel.Periodicallyresettingthereduced-order modelextendedtheenvelopeofeectivenessbutatthecostofincreasedcontroller complexity.Siegel etal. 2006appliedasimilarreduced-ordermodeltocontrolsimulated owaroundacircularcylinderatRe d =100.Aproportionalanddierentialfeedback controllawbasedontherstPODmodedisplacedthecylinderinthetransversedirection andreducedthedragby15%relativetotheuncontrolledow. Amongthe experimental investigationsthathaveappliedreduced-ordermodels, Luchtenburg etal. 2010andAleksic etal. 2010conductedapairofstudiesthat controlledthewakebehindatwo-dimensionalblubodyatReynoldsnumbersbased onbodythicknessintherangeof f 2 : 69 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 : 77 g 10 4 .Variouscontrollawssynthesized aroundthemodeldecreasedthedragby15%withpulsedsuctionatthesharptrailing edgecornersofthebody.Technicalissuesremaineddespitetheserecentexperimental accomplishments.First,thereduced-ordermodelwasderivedusingmeaneld theoryNoack etal. ,2003,whichassumedfrequenciesforboththedominantwakemode andtheactuationmode.Neitherofthesewasexperimentallymotivatedfromfull-eld data,meaningthatthismethodmaynottakeadvantageofotherinuentialmodes.A modelbasedonmeaneldtheorymaynottranslatewelltoother,moresophisticated, owsystemswithmixedfrequencycontent,suchasowseparationfromatraditional lifting-bodyairfoile.g.Tian etal. ,2006 a .Second,Luchtenburg etal. andAleksic etal. actuatedwithpulsedsuction,whichwouldrequireanon-boardvacuumextrapayload infull-scaleapplications.Amoreenergyecientcontrolstrategyandsimplersetup maybepossiblebyuseofzero-net-mass-uxZNMFactuators,whichareactiveow 24

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controldevicesthatprovideunsteadyforcingusingonlytheworkinguidsupplySmith& Glezer,1997. Therefore,experimentallyderivedreduced-ordermodelsaredesiredtomitigateow separationusingZNMFactuationandtoassesstheimpactoftriggeringthenaturalow instabilities.Challengesexistforthesynthesisofareduced-ordermodelanditsusein controlofacomplexseparatedow.Understandingtheeectsofvariousinstabilitieson owseparationandapplyingexperimentaltechniquestomodelthefullelddynamics arediculttasks.Nevertheless,recentadvancementsinstateestimationbyintegration oftraditionalmeasurementsoffull-elddataandpoint"measurementscouldaidinthe formationofanexperimentallymotivatedreduced-ordermodelTu etal. ,2013.This methodhasalreadybeenappliedtoestimatethewakebehindablubody. Theproposedexperimentalplatformisaatplatemodelwithabluntbase forthestudyandcontrolofcanonicalowseparationusingZNMFactuation.The primarygoalsofthisresearcharetoidentifytheinuentialdynamicmodesfrom theboundarylayerseparationofaatplatemodelwithablunttrailingedgeandto leveragethesemodesinopen-andclosed-loopcontroloftheseparation.Themain applicationforthisowcongurationisthecontroloftwo-dimensionalairfoilstall, includingturbinebladesBalzer etal. ,2007andairfoilapsTian etal. ,2006 a ,for improvedperformance.Thisplatformisalsorelevanttoblu-bodyseparationfrom rectangularandcircularcylinderssuchaslandinggearstrutsZawodny etal. ,2009 andsubmarinefuselagesWetzel&Simpson,1992;Wetzel,1993,andhaspotentialin three-dimensionalblu-bodyseparationsuchasowsaroundaturretPalaviccini etal. 2011andautomotivevehiclesHsu&Davis,2010;Pujals etal. ,2010. 1.2Background Thissectionconsistsoffourparts.Therstintroducessomephysicalunderstanding ofowseparationfromboundarylayersandblubodies.Then,somebasicandessential owcontrolterminologyispresented.Motivatedbytheneedformorerenedclosed-loop 25

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solutions,thenextsectionoutlinesthetypesofclosed-loopcontrolstrategiesthatare commonlypracticedinowcontrol.Finally,methodsforlow-orderestimatesoftime-and spatially-resolvedvelocitydataaredescribedasenablersofanexperimentallymotivated reduced-ordermodel. 1.2.1Two-DimensionalFlowSeparation Thissectionsummarizesthephysicalbehavioroftwo-dimensionalseparatedows, whichoccurduetoadversepressuregradientsaroundstreamlinedandblubodies.The fundamentalsofbothofthesetypesofseparationaredescribed. 1.2.1.1Boundarylayerseparation Theboundarylayeristhethinregionorlayerofuidadjacenttoasurfacein whichviscousstressesaredominant.Theuidincontactwiththesurfaceobeyswhatis calledtheno-slipcondition,"inwhichthevelocityofthislocaluidmatchesthelocal surfacevelocity.Theuid'sviscosityisresponsibleforthecontinuousgradientinuid velocityfromthesurfacetothefreestream.Thedistanceawayfromthesurfacethatthe streamwisevelocitycomponentasymptotestothefreestreamvelocityistheboundary layerthickness .Thismeasuredenotestheedgeoftheboundarylayerandisdicultto deneprecisely.Theapproximateedge 0 : 99 isoftendenedbythelocationatwhichthe streamwisevelocityis99%ofthefreestreamvelocity. ThesteadyNavier-Stokesequationsaresimpliedwithintheboundarylayer byneglectingsmallterms,yieldingtheboundarylayerequations.Thesteady, two-dimensionallaminarboundarylayermomentumequationappliedatthesurface u =0, v =0ofaatplatereducesto @ 2 u @y 2 y =0 = 1 dp dx ; {1 where x isthestreamwisedirection, y isthetransversewallnormaldirection, u isthe streamwisevelocitycomponent, v isthetransversevelocitycomponent, p isthepressure, and isthedynamicviscosityoftheuid.Thisindicatesthatthevelocityprole's 26

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curvatureatthewallisdependentonthesignofthepressuregradient dp=dx .Dierent proleshapesaredepictedinFigure1-2Aforthecorrespondingsignsof dp=dx .Under theinuenceofanadversepressuregradient dp=dx> 0andskinfriction,asseenin Figure1-2B,thelowmomentumowwithintheboundarylayerdeceleratesandthe boundarylayerthickens.TheresultingproleresemblesanS-shape,inwhichachange insignoftheprolecurvatureoccursatthepointofinection.Ifthelowkineticenergy oftheinnerboundarylayeruidparticlescannotovercometheresistancefromalargeor persistentadversepressuregradient,theowwilldeceleratetothepointofseparation. Theseparationpointischaracterizedbyvanishingwallshearstress w ,where w = @u @y y =0 ; {2 andowreversaldownstream.Thisisaccompaniedbyasignicantwall-normalvelocity componentanddetachmentofthelaminarboundarylayerfromthesurface. Downstreamoftheseparationpoint,theseparatedboundarylayertransitionstoa shearlayerbetweenthehighermomentumfreestreamandthelowermomentumreverse ow.Forweakadversepressuregradientsofnitelength,theseparatedregioncan reattachtothesurfacedownstream,enclosingalaminarseparationbubble.Astronger adversepressuregradientcreatesperiodicvortexsheddingfromtheseparatedshearlayer asaresultofaKelvin{Helmholtz-typeinstabilityPauley,1988;Pauley etal. ,1990.The increasedmixingfromthevortexsheddingcaninsighttransitiontoaturbulentboundary layerreattachmentwithatime-averagedseparationbubble,asdepictedinFigure1-3.An evenstrongeradversepressuregradientseparatestheowwithoutreattachment,suchas wouldoccurforanairfoil'sstallcondition. Dependingonthenatureoftheseparation,uptothreecharacteristicfrequenciesmay bepresentMittal etal. ,2005.Iftheowdoesnotreattach,twonaturalfrequencies, thewakefrequency f wake andtheshearlayerfrequency f SL ,exist.Thelatterisaperiodic roll-upoftheshearlayercausedbytheKelvin-Helmholtzinstabilityandscaleslike 27

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f SL U 1 = ,where istheboundarylayermomentumthicknessjustupstreamof separation.Theformerscaleslike f wake U 1 =W wake Roshko,1954,where W wake isthetransversewidthofthewake.Inthecaseofreattachmentinatime-averaged sense,anadditionalfrequency f sep correspondingtotheseparationbubblescaleslike f sep U 1 =L sep ,where L sep isthecharacteristicstreamwiseortransverselengthofthe separationbubble. Laminarboundarylayersaresusceptibletoseparationgivenanadversepressure gradient.Turbulentboundarylayershavemorekineticenergynearthewallandthus betterresistancetoadversepressuregradients.Forsteadytwo-dimensionalow,the actualseparationprocessbeginswithintermittentowreversalatagivenlocation, andtheamountofinstantaneousowreversalincreasesinthedownstreamdirection. Therefore,thedetachmentofaturbulentboundarylayerhasbeendividedintofour states,"notedinFigure1-4,thataredenedbythefractionoftimetheowmoves downstream ds Simpson etal. ,1981;Simpson,1989: incipientdetachment istheinitial thresholdandquantiedbyinstantaneousowreversal1%ofthetime ds =0 : 99; intermittenttransitorydetachment requiresinstantaneousowreversal20%ofthetime ds =0 : 80; transitorydetachment requiresinstantaneousowreversal50%ofthetime ds =0 : 50;and detachment occursatthelocationwheretime-averagedsurfaceshear stress w iszeroSimpson,1989.However,theconditionsofvanishingwallshearstress andowreversalarenotsucientforthecriterionofseparation,thoughthesetendtobe satisedforsteadytwo-dimensionalow.Forunsteadytwo-dimensionalow,thesurface shearstresscanchangesignforunsteadyowreversalwithoutowseparationSimpson, 1989.Morein-depthtreatmentofturbulentboundarylayerseparationandunsteady separationarelefttomoredetailedreviews,e.g.Simpson,1989;Haller,2004. Asaresultofseparation,thewallskinfrictionandthusviscousdragarereduced. However,thedragduetoskinfrictionisinsubstantialcomparedtothepressuredrag 28

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associatedwithowseparationandthelossofliftencounteredbytraditionallifting-body airfoils.Therefore,separationisundesirableinmostaerodynamicapplications. 1.2.1.2Sharp-edgedblu-bodyseparation Manyinstancesofowoverablubodyexist,butthissectionislimitedtonominally two-dimensionalgeometrieswithsharptrailingedgesthatcausexedseparation points.Everydayexamplesincludebridges,tallbuildings,tractortrailers,andcertain automobiles.Similartoowoveracircularcylinder,Reynoldsnumbersaboveacritical valueresultinvortexsheddinginthewakeanddecreasedpressureontherearofthebody. Unliketheowoveracircularcylinder,sharpchangesinthewalldirectionorbody shapecausetheowtoimmediatelyseparate,formingafree-shearlayerdownstreamof axedseparationpoint.IftheupstreamboundarylayerislaminarandtheReynolds numberisnotsubcritical,transitiontoturbulencecanoccurafterdetachment.Inthe caseofacontinuingwall,suchasthebackward-facingstepshowninFigure1-5,theow reattachesdownstreamandaseparationbubbleisformed.Thistypeofseparationbubble hasastrongerinteractionwiththefreestreamthantheboundarylayertypeSimpson, 1989.Intheabsenceofadownstreamwall,suchasinthecaseofablunttrailingedge, theowisgovernedbyanabsolutewakeinstabilitythatproducesaperiodicKarman vortexstreete.g.Oertel,1990;Huerre&Monkewitz,1990.Anabsoluteinstabilityis anunstableinstabilitywhoseperturbationstravelupstreamanddownstream.Thisisin contrasttoadisturbancethatisonlyconvecteddownstreamfromthesource,asisthe casewithaconvectiveinstabilityHuerre&Monkewitz,1990. Inthewakeofablunttrailingedgebodywithnodownstreamwall,thecouplingof theupperandlowershearlayersisimportanttothedevelopmentofthevortexshedding thatresultsinlowmeanpressureatthebase.Referringtothewakebehindabluntbase inFigure1-6,therotationofvortex A pulls"vortex B upfromthelowershearlayerand promotescounterrotation.Therotationofvortex B pullstheuppershearlayerdownand triggersrollupoftheuppershearlayer,creatingvortex C inthenearwakePastoor etal. 29

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2008.Thisalternatingperiodicvortexcreationandsheddingatthebasecreatesaregion ofdecreasedmeanpressure,whichresultsinanincreaseinpressuredrag. Flowcontrolofblu-bodyseparationdoesnottypicallyattempttoreattachor preventseparation,especiallyforsharptrailingedges.Instead,controlofsuchowsaims toreducethepressuredragassociatedwithawidewake.Thisismorespecicallyreferred toaswakestabilization. 1.2.2FrequencyScalesofSeparatedAirfoilFlow Asalludedtointheprevioussection,Mittal etal. 2005outlinesthatasmany asthreenaturalfrequencyscalesexistinthedynamicsofseparatedowfortypical airfoils,dependingontheconditionsFigure1-7.Atextremeanglesofattack,the suction-sideboundarylayermayseparate,typicallyneartheleadingedge,andthe separatedowmayornotreattachpriortothetrailingedge.Forthemassiveseparation ofapost-stallcondition,inwhichtheowdoes not reattachinthetime-averagedsense, theseparationisatleastcharacterizedbythenaturalfrequenciesforthewake f wake and theshearlayer f SL Tian etal. ,2006 a .Thelatterisduetotheseparatedshearlayer Kelvin-Helmholtzinstabilityandscaleslikeafreeshearlayer, f SL U 1 = ,where isthe boundarylayermomentumthicknessjustupstreamofseparation.Thewakefrequency scaleslikethatofablu-bodywake, f wake U 1 =W wake Roshko,1954,where W wake is thecharacteristicwidthofthewake.Eveninthepresenceoftheglobal,absolutewake instability,theshearlayerinstabilitymechanismhasbeenshowntobeinuentialInthe eventthattheowdoesreattachinatime-averagedsense,athirdfrequencyforthe separationbubble f sep maybepresent,scalinglike f sep U 1 =L sep ,where L sep isthe characteristicsseparationlength. 1.2.3FlowControlTerminology FlowcontrolmethodologiesaregenerallycategorizedasseeninFigure1-8.The mechanismofcontrolistheprimarydivider,specifyingwhetherthetechniqueispassive oractiveinnature.Theformerclassicationachievescontrolthroughenergytransfer 30

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withinthesystem.Thisusuallyreferstocontrolfromageometricmodicationoraddition toanaerodynamicbody,suchasdimplesonagolfballorvortexgeneratorsonawing. Bothofthesearedesignedtoinduceeddiesorvorticesthatbringhighmomentumow closertothesurface,delayingseparation.Activecontrolisdistinguishedinthatitsupplies someformofenergyelectricalormechanicaltotheowsystemviaanexternalsource, oftenbymeansofsteadyblowingorunsteadyactuation.Thismethodisfurthersplitinto open-andclosed-loopcontroltypes.Open-loopcontroldoesnotdependonanysensor measurementtoalterthecontrol,sothereisnoutilizationofthecontrol'seectiveness orinuence.Closed-loopcontrolutilizesfeedbacktoaltertheactuationbasedonasome establishedcontrollaw.Finally,closed-loopowcontrolsystemscanbegroupedbased onthetemporalresponseofthecontrol.Relativetothecharacteristicfrequenciesofthe dynamics,low-bandwidthcontroladjuststhecontrolcommandsonaslowertimescale,and high-bandwidthoperatesatorabovetherelevantfrequencies.Foradetailedtreatment anddisambiguationofowcontrolterminology,thereaderisreferredtoMacMynowski& Williams2009. 1.2.4Closed-LoopFlowControlStrategies Closed-loopowcontrolmethodsareanattractivechoicecomparedtopassive andactiveopen-loopcontrolsbecausethecontrolinputcanbecontinuouslyadjusted inresponsetothemeasuredowsystem.Thissectionintroducesseveraltypesof feedbackcontrolmethods,whichcaneitherbemodel-freeormodel-basedstrategies.The lattergroupingcontainsclosed-loopcontrollawsthataddressthedynamicsoftheow system.Thesetendtofallintooneofthreetiersofcomplexity.Thehighestlevelcontains high-dimensionaldiscretizationoftheNavier-Stokesequationsorlargeeddysimulations, whicharecomputationallyunattainableforreal-timecontrolinatleasttheforeseeable futureGerhard etal. ,2003;Collis etal. ,2004;King etal. ,2005.Thelowestlevelis basedontheidenticationofso-calledblack-boxmodels,whichprovideasimplersetof equationsgoverningthebehaviorofthelumpedsystemactuators,ow,andsensors. 31

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These,inasense,identifythemostbasicinput-outputbehaviorandareusuallylinear approximationstoanonlinearuidsystem.Inthemiddletier,atractablecompromise betweenthehigh-resolutionofthenonlineargoverningequationsandthesimplicityof theblack-boxmodelsisobtainedbylow-dimensionalmodelsofthehigh-dimensional dynamicalsystem.Table1-1includessummariesandkeyfeaturesofthedierentcontrol strategies. 1.2.4.1Model-freecontrol Asthenameimplies,model-freecontrolhasnoinherentmodelassociatedwiththe control.Thecontrolisadjustedbasedonperturbationsofthesystemanddetectionof thelocalresponsetoachievesomeoptimum.Becausemodel-freecontrolisoneofthe simplerexperimentaltechniques,muchofthereviewedliteratureappliedthistechnique inadditiontoother,usuallymoresophisticated,strategiesforpurposesofcomparison. Extremum-andslope-seekingfeedbackcontrollersarethemostpopularamongmodel-free techniquesAriyur&Krstic,2003.Theseareadaptive,gradient-basedmethodsthat searchforoptimalparametersassumingasteady-statemapoftheoutputvariablesin responsetothechanginginputparameters.Theextremum-seekingcontrollerisdesigned toapproachazero-slopeinthesteady-statemapbyperturbingoneormoreoftheinput parameters,thusndingeither local minimaormaxima.Slope-seekingisavariantof theextremum-seekingmethodthat,asthenameimplies,aimstodetectaspeciedslope ratherthanalocalextreme.Likeextremumseeking,thisgenerallyassumesthatastatic mapoftheinput/outputrelationshipexists. Thesecontrollers,whicharebasedongradient-detectingalgorithms,haveslow responsetimesbecausetheinputisperturbedandtheoutputmustbeallowedto respondinsuchawayastodetectthedirectionoftheresponse.Thealgorithmisthereby continuallycomputingthecurrentresponsedirectionrelativetothedirectionoftheinput perturbation,ortheslope,andthenadjustingtheinputinordertoproceedinthedesired direction.Forowsystems,thiscontrol-drivenresponsecanbelostamidtheunforced 32

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uctuationsandnoiseofthesystemiftheperturbationisnotslowerthanthelongest characteristictime-scale. 1.2.4.2Controlofblack-boxmodels Anotherapproachappliedtoexperimentalowcontroliscontrolofanidentied system,betweentheactuationandsensingcouple.Theinput-outputbehaviorof theentireplanttobecontrolledisthenmodeled,oftenbyassumingalinearsystem orlinearizingaboutthecurrentestimatedstate.Thisisusuallyaccomplishedby measuringopen-loopresponsesfrequencyresponseand/orseriesofstep-responsesof thesystemandttingtheresponsesinordertodeterminesuitablemodelparameters. Theidentiedsystemisreferredtoasablack-boxmodel,becauseitsimplymodels theinput/outputbehaviorandneglectstomodeltheindividualcomponentsofthe process,suchasthedynamicsoftheactuators,ow,andsensors.Examplesofsystem identicationschemesincludeARMARKOVidenticationAkers&Bernstein,1997, eigensystemrealizationalgorithmLjung,1987,andMATLAB'ssystemidentication toolbox.Last,acontroller,oftenrobusttosystemdisturbances,issynthesizedbased ontheblack-boxmodel.Examplesinclude H 1 {controlBewley,2001,ARMARKOV disturbancerejectionAkers&Bernstein,1997,Smith-predictorLee etal. ,1996,and linearquadraticregulationLQR. 1.2.4.3Controlofreduced-ordermodels Reduced-ordermodelsprovidesomebasicphysicalunderstandingoftheow whilebeingcomputationallyecientenoughtopermitreal-timefeedback.Theyare obtainedfromlow-orderprojections,GalerkinmethodsHolmes etal. ,1996,orvortex modelingCottet&Koumoutsakos,2000. TheGalerkinmethod,popularintheuidscommunity,involvesaprojectionof theNavier-Stokesequationsontothesubspacespannedbyalow-orderapproximation, oftenobtainedfromproperorthogonaldecompositionPOD.Initially,adiscetenumber ofglobalmodesischosenasareduced-ordersubspaceofthefullsystem.AGalerkin 33

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approximationisthenformedbyalinearcombinationoftheseexpansionmodes,yielding anEuleriandescriptionofasimpliedoweld.Finally,aGalerkinprojectionofthe Navier-Stokesequationsontotheselectedsubspaceprovidesasetofordinarydierential equationsODEsgoverningthetemporalFouriercoecientsofthereduced-order basisRowley,2005.Theselow-ordersystemsofODEstypicallyprovidebasicphysical insightandaremorecomputationallyfeasiblethanhigh-dimensionaldiscretizationof thefullNavier-StokesequationsRempfer,2000.However,thetruncationinvolvedin reducingtheorderofthesystemsreducestheiraccuracy.Incomputationaluiddynamics, truncationoccursduetothespatiotemporaldiscretizationofthesystemandtheinability toresolvealloftheowscales,fromlargescalestructuresdowntotheKolmogorov scales.InGalerkinmodels,thechallengecouldbebothdiscretizationandselectionofthe expansionmodesNoack etal. ,2010. IncontrasttoGalerkinmethods,vortexmodelsadoptaLagrangianviewof convectivevorticalstructuresthatmergeandinteract.Galerkinmodelsarebetter suitedforcontroldesign,butvortexmodelsoerawiderbandwidthLuchtenburg etal. 2010.Vortexmodelssuperimposepoint-vorticeswithanirrotationalpotentialoweld toapproximatethedistributionofvorticityforagivenowsystemHenning etal. ,2007; Pastoor etal. ,2008.WhileHelmholtz1858andThomson1869arecreditedwiththe theoreticaldevelopmentofvortexmodels,vonKarman1911isrecognizedforapplying vortexmodelingtothestabilityanalysisofavortexstreet.Thisandotherinvestigations encouragenearlyacenturyofprogressioninwhichvortexmodelingisproposedasa simpliedalternativetoDNSGhoniem&Cagnon,1987andasamethodforcoherent structuremodelsCortelezzi,1996.Vortexmodelshavejustrecentlybeenusedasplants ofcontrolsystemsProtas,2004,2006.Stillotherinvestigationsdemonstratethat thistypeoflow-ordermodelishelpfulingivinginsightintothemechanicsandvortex interactionsofowcontrolproblemse.g.Tang&Aubry,2000;Pastoor etal. ,2008. 34

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1.2.5Reduced-OrderEstimation Hightemporalresolution,full-eldinformationishelpfulinformingaccurate reduced-ordermodelsforclosed-loopowcontrol.Thetermtime-resolved"iscommonly usedintheuidsmeasurementcommunitytodescribeadynamicmeasurementorprocess whosesamplerateisabletocapturedetailedchronologicalchangesintheowstate. Withinthiswork,thetermisusedasaconvenientdescriptorfordatathatsatisfythe Nyquistsamplingcriterion,inwhichthesamplingfrequencyexceedstwotimesthehighest systemfrequencyofinterest.Thisdescriptionispreferabletofreqeuncy-resolved,"which couldbeinterpretedasacquisitionordataprocessinginthefrequencydomain.The datamustthereforeresolvethepertinenttimeorfrequencyscalesinordertoidentify andmodelthestatedynamicsoftheow.Inadditiontocontrolmotivations,structures ofdynamicalimportancecanbeextractedusingmethodsincludingdynamicmode decompositionDMDandbalancedproperorthogonaldecompositionBPODRowley, 2005;Rowley etal. ,2009;Schmid,2010,thoughthelatterrequirestheadjointsystem. 1 Unfortunately,time-resolvedvelocityeldmeasurementsarediculttoobtain,andthe costofthenecessaryequipmentforsuchmeasurementsisoftenprohibitivelyhigh. Non-time-resolvedPIVisamoretraditionalandnanciallyfeasibletechniquefor measuringvelocityelds.Withsamplingratestypicallyontheorderoftensofsamples persecondorless,thesePIVsystemsareusuallyunabletoresolvethecharacteristic frequenciesofaow.Instead,high-ratepoint"orprobemeasurementssuchashot-wires orunsteadypressuresensorsarerelieduponforextractingsmalltemporalowscales. Suchdevicescanbesmallenoughtomitigatespatialaveragingbutobviouslyhavea lowerlimitforcapturingsmallscalestructures.Combinedwithsensorpackagingsize 1 BPODisrestrictedtocomputationaldata.Anequivalentapproach,theeigenvalue realizationalgorithmERA,appliestoexperiments. 35

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andpotentialintrusiveness,densearraysofprobesarecurrentlyintractableformost small-scaleowcontrolproblems. Asaremedy,time-resolvedestimatesareprovidedbyathree-stepmethodthat integratesnon-time-resolvedPIVsnapshotswithtime-resolvedprobemeasurements toformadynamicmodelusedtoestimatetheevolutionofthefullvelocityeldTu etal. ,2013.ProperorthogonaldecompositionPODisusedtoextractthedominant coherentstructuresforalow-orderapproximationoftheow,asonlylargescale,energetic structuresareexpectedtocorrelatebetweenthetwomeasurementtypes.Intherststep, anempiricalestimationtechniqueisusedtocomputeaninitialsetofhigh-rate,low-order estimatesofthevelocityeld.Then,theseestimateshelpgenerateamodelofthedynamic stateevolution.Finally,thismodeliscombinedwithavailablemeasurementstoforma model-basedestimator. Themodel-basedestimatorisabletorelyonboththemodelanddatameasurements, whichhelpstoltersomeofthepotentialnoisefromsensordata.Theempiricalestimator hasonlyastaticmapfromthepointmeasurementstothefull-eldestimatesanddoesn't strictlyaccountfornoise.However,empiricalmethodscanutilizemultipleinstances withintheevolutionofmeasurementstoeectivelyltertheeectsofnoiseandimprove correlationsbetweenmeasuredandestimatedvalues.Thefollowingsectionsprovide backgroundforempiricallyderivedandmodel-basedestimation. 1.2.5.1Stochasticestimation Aspreviouslydescribed,atime-resolved,full-stateestimateisdesiredfromahandful ofprobemeasurements.Thisisequivalenttoestimationofasetofconditionalvariables fromasetofunconditionalvariables.Forinstance,thevelocityatonepointinaow, u x ,canbeusedtoestimatethevelocityatanotherpoint u x 0 .Theconditionalaverage ^ u x 0 = h u x 0 j u x i {3 36

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providestheexpectedvalueof u x 0 giventhemeasurement u x ,whichisthe least-mean-squareestimateof u x 0 Papoulis,1984.Empiricalestimationisa measurement-basedtechniquethatusesaempirically-derivedmappingfunction f to computeasetofestimates^ u i fromasetofmeasurements u j ^ u i = f u j ; {4 inwhichthespatialdependence x hasbeenreplacedbyindexnotation. AstochasticestimateoftheconditionalaveragewasproposedbyAdrian1977,in whichtheaverageisapproximatedbythepowerseries h u i j u j i A ij u j + B ijk u j u k + :::; {5 wheresummationoverrepeatedindicesisimplied.LinearstochasticestimationLSE truncatestheseriesafterthelineartermsuchthatonlythecoecients A ij areretained. Stochasticestimationisaconditionaltechniqueforestimatingdynamicsthatare correlatedwiththosecapturedwithinthemeasurementsignal. Higher-orderestimatesareobtainedbyincludingmoretermsinthepowerseries. Quadraticestimation,forinstance,canbemoreaccuratewhentheestimatedquantityis velocitybasedonthemeasurementofpressureNaguib etal. ,2001;Murray&Ukeiley, 2003.Inparticular,foracavityow,Murray&Ukeiley2003foundthatwhilealinear estimatecapturedthelargescalefeaturesoftheoweld,theturbulentkineticenergyin thequadraticestimatewasmuchmoreaccurate.Thequadraticestimatewasalsoableto betterpredictfeaturesoftheowonsmallerscales. Theestimationprocesscanalsobeimprovedbyaccountingfortimedelaysbetween theconditionalandunconditionalvariablesGuezennec,1989.Ewing&Citriniti1999 incorporatedmultiplemeasurementsinthefrequencydomaintoimprovetheestimates oversingle-timeLSEofturbulentjetows.Themulti-time"formulationalsoobtained low-orderapproximationsofthejetsbyincludingPOD.Thismethodwasthenformulated 37

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inthetimedomainforpredictionsofpressurevaluesfrompastmeasurementsUkeiley etal. ,2008.Durgesh&Naughton2010foundanoptimalrangeofdelaysfortheir estimatesofPODmodecoecientsofablu-bodywake. Stochasticestimationisameasurement-basedtechniquethatdoesnotmodelthe systemdynamicsinamoretraditionalsenseofmodelingandsystemidentication. Instead,statisticalrelationshipsareutilizedtoestimatearandomvariablegivenseparate knownrandomvariables.Stochasticestimationcomputesamappingfunction,from apre-existingdataset,thatyieldsthemoststatisticallylikelyvalueofsomeunknown conditionalvariable,givensomeothermeasuredunconditionaldata.Thismethod canhelptovisualizeauidowandtosuggestwhateventsshouldbeobserved fromconditionalstatistics,butitdoesnotprovidearelationfortheunderlyingow physicsCole etal. ,1992.Furthermore,theestimatedquantitiesfromLSElieina subspaceoflimiteddimensionequaltothenumberofmeasurementsprobes,ormoreif multi-timeLSE.Thistraitisespeciallyimportantwhenconsideringasmallnumberof measurementstoestimatealargequantityofvariables,suchasthosefromavelocityeld. Dependingontheapplication,itcanbeeitheralimitationoranadvantage,unnecessarily restrictingtheestimates,orcapturingonlythefeaturesofinterest.Thespecicsof stochasticestimationaredescribedinChapter3. 1.2.5.2Model-basedestimation Model-basedestimators,orobservers,incorporateamodelofthesystem'sdynamics alongwithreal-timemeasurementupdates.Theupdatesareintendedtocorrectthe trajectoryofthemodel,whichmaydriftfromthetruetrajectoryduetoparameter uncertainty,unmodeleddynamics,andexternaldisturbances.Thismethodpermitsa weightingofimportancebetweenthemodelpredictionandmeasurements.Aproper balancedependsontheaccuracyofthemodelandthenoisecharacteristicsofthe measurements.Suchanapproachisfundamentallydierentfromanempiricalestimator likestochasticestimation,inwhichasetofmeasurementsestimatethesystemstateusing 38

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axedrelationship.Empiricalestimatorsneglectanygoverningequationsofasystem's evolution,whichcanhelptoltertheuncertaintyinthemeasurements. Themodel-basedestimatorsconsideredarethelinearKalmanlterandsmoother. Thereaderisreferredtoanystandardtextonestimation,suchasSimon2006,for in-depthdiscussionofthesetechniques.Theevolutionofalinear,dynamicsystemmodel isgivenby k = F k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + d k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 R N s k = H k k + n k 2 R N o ; {6 where isavectorof N s statevariables, isavectorof N o measurementsoutputsof thestate, d representsprocessnoise,and n representssensornoise.Giventheiteration k ,theobservedmeasurement k isdesiredtoestimatethevalueof k ,whichisotherwise unknown. Ifthesystemisobservable,aknowledgeofthesystemdynamics F andthetime evolutionof producesanestimate ^ thatconverges,intheabsenceofnoise,tothetrue valueof .Inthepresenceofnoise,thelinearKalmanlterminimizestheexpectedvalue oftheerrorcovarianceKalman,1960 k )]TJ/F15 11.9552 Tf 11.874 3.155 Td [(^ k T k )]TJ/F15 11.9552 Tf 11.874 3.155 Td [(^ k : {7 TheKalmanlterisa causal lter,meaningthatonlymeasurementsmadeupto andincludingiteration k areavailableinformingtheestimate ^ k .Incertainapplications, suchaspost-processinganalysis,measurementsoccurringafteriteration k areavailable. Thisadditionalinformationcanbeutilizedtoimprovetheestimateof k .Usingsuch informationimpliesa non-causal lter,commonlyreferredtoasa smoother ,thatcannot beusedtoestimateinreal-timeapplications.AKalmansmootherrstmakesaforward passbymeansofaKalmanlter,followedbyabackwardspassthatrenestheinitial estimates.MorespecicsoftheKalmanlterandsmoother,andhowtheyareappliedto owestimation,areprovidedinChapter3. 39

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1.3Objectives Themainobjectivesofthisresearchareasfollows.Therstisthecreationand characterizationoftheexperimentalcanonicalboundarylayerseparationfromaat platemodelwithabluntbase.Next,anensemble-basedestimatorissynthesizedfrom time-resolvedsurfacepressuremeasurementsandspatially-resolvedfull-eldvelocity measurements.Theestimatorpermitstheextractionofdynamicmodesrelevanttothe characterizationandcontrolofthebaselineow.Open-loopcontroltargetsmultiple frequencyscales,includingtheidentiedinstabilities,andcomparestheresultstothose fromhigh-frequencyforcing.Dynamicmodesareextractedfromthecontrolledow andformthebasisfunctionsforareduced-ordermodeltocontroltheseparatedshear layerabovetheatplateportionofthemodel.Simpleclosed-loopcontrolstrategies aredevelopedaroundtheexperimentallyderivedreduced-ordermodeltoreattachthe separatedow.Theresultsofthiscontrolstrategyarecomparedtoresultsfroma standardoptimalcontrolmethodappliedtoanidentiedblack-boxmodel. 1.4Approach Withtheatplatemodelfullyequippedwithsurface-basedsensors,thebaselineow forthexedtrailingedgeseparationandwakearecharacterized.Experimentsconducted inalow-speedwindtunnelemployvariousuiddynamicmeasurements,includingPIV ofthewakeandseparatingboundarylayer,steadyandunsteadypressuremeasurements beneaththeseparatedshearlayer,andhot-wireanemometryoftheunsteadysynthetic jetactuators.AsetofZNMFactuatorsisinstalledintheupperportionoftheat platemodel.Open-loopcontrolisconducted,specicallytargetingthecharacteristic frequencycomponentsofthebaselineow.Then,closed-loopowcontrolbasedon anexperimentallyderivedreduceordermodelisimplementedtoattenuatetheow separation.Thereduced-ordercontrolmethodsarecomparedtosimplerregulatorycontrol appliedtotheidentiedinput-outputbehavioroftheow.Finally,theimpactofthe closed-loopcontrolmethodsisassessed. 40

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1.5ProposalOutline Followingthisintroductorychapter,adetailedliteraturereviewofclosed-loop controlappliedtoowseparationispresented,withemphasisoncontrolusing reduced-ordermodelsandsomeoftheunresolvedtechnicalissuesthatcanbeaddressed. Chapter3providesthenumericalbackgroundandrecipeforamodel-basedestimator thathasapplicationtoclosed-loopowcontrolanddynamicalanalysisofglobal measurements.Then,theexperimentalsetupusedtoimposeandcontroltheboundary layerseparationisdescribedinChapter4.Theresultsaredividedintothreechapters: baselineuncontrolledowresults,open-loopcontrolresults,andclosed-loopcontrol results.Thereectedcontributionsbasedontheseresultsandconcludingcommentsare presentedalongwithrecommendationsforfutureresearchinChapter8. 41

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Figure1-1.Schematicofcanonicalseparatedowforaatplatemodelnottoscale. A B Figure1-2.Separationofatwo-dimensionallaminarboundarylayeronaatplate. AAverageproleshapesbasedonsignofpressuregradient.BEvolutionof proleinanadversepressuregradient.AdaptedfromWhite1991. 42

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Figure1-3.Transitionalseparationbubble.AdaptedfromHorton1968. Figure1-4.Adepictionofturbulentboundarylayerseparationthatbeginswithincipient detachment,followedbyintermittenttransitorydetachment,andnally detachmentinatime-averagedsense.Theturbulentstructuresinthedetached shearlayerprovidethesmallmeanowreversaldownstreamofthe detachmentpoint.Thedashedlinedenotesthezerotime-averagedstreamwise velocity.AdaptedfromSimpson1989. Figure1-5.Featuresofowoverabackward-facingstep.AdaptedfromDriver etal. 1987. 43

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A B Figure1-6.Thewakebehindaatplatewithabluntbaseofthickness h .Flowisleftto right.AInstantaneous2-componentvelocityeldmeasurementwithlinesof constantvorticityandvelocityvectorsTu etal. ,2013.BSketchofthe couplingbetweenthewakeandshearlayervorticesPastoor etal. ,2008. A B Figure1-7.Schematicshowingthethreenaturalfrequenciesthatcanoccurinseparated airfoilows:shearlayer f SL ,wake f wake ,andseparationbubble f sep ifow reattaches.AOpenseparation.BReattachedseparationwithclosedbubble. AdaptedfromMittal etal. 2005andKotapati etal. 2010. Figure1-8.TraditionalcategoriesofowcontrolCattafesta etal. ,2008;MacMynowski& Williams,2009. 44

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Table1-1.Comparisonofthereviewedcontrolmethods. ControlmethodAdvantagesDisadvantages Model-freecontrolNomodelrequired Optimaloperating conditions Slowadaptation Assumesstatic-map Robustcontrolof black-box models Robust Set-pointtracking Real-timecontrol Capturesdynamicsofow, sensor,andactuator Typicallylinearsystem approximation Limitedvalidity Requiresopen-loopmeasurements Controlof reduced-order models Ecientcontrol Somephysicalinsight Capturesrelevant dynamics Real-timecontrol Limitedvalidity Requiresopen-loopmeasurements 45

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CHAPTER2 LITERATUREREVIEW:CONTROLOFSEPARATEDFLOWS Thischapterprovidesanon-exhaustivebutthoroughreviewofrelevantliterature foractivecontrolofseparatedows,mainlyconcentratingonblu-bodyowseparation andboundarylayerseparation.Withinthesetwogroups,controlschemesarecategorized asdirectfeedback,model-freecontrol,controlofablack-boxmodel,orcontrolofa reduced-ordermodel,withthelevelofcomplexitygenerallyincreasinginthatorder. Emphasisisgiventostudiesoncontrolofreduced-ordermodelsofseparatedowsbecause ofitsrelevancetothiswork.Afterbriefintroductionsoftherelevantmodelplatforms, summariesofrelevantclosed-loopowcontrolactivityaregivenforblu-bodywake controlandthencontrolofboundarylayerseparation.Finally,somediscussionisgivento technicalissuesnotyetaddressedintheliterature. 2.1IntroductiontoSeparationControlPlatforms Anumberoftwo-dimensionalplatformshavebeeninvestigatedforactiveseparation control.Forxed-pointseparationandblu-bodywakes,controlstudiesmainly concentrateonafewwellestablishedbenchmarkproblems,suchascontrolofthewake behindacircularcylinder,therecirculationlengthofabackwardfacingstep,thereduction ofcavityoscillations,thepressuredragofaatplatemodelwithablunttrailingedge, and,morerecently,theinducedseparationandwakeinteractionsofacanonicalseparated ow.NumericalsimulationsaregenerallyrestrictedtolowReynoldsnumbersRe 10 5 andtendtoaddressthesimpler,morecomputationallyfeasibleplatforms.Assuch, casesofsimulatedowcontrol,whichhavesurgedduringthelastdecade,favorwake stabilizationfromcircularcylindersatReynoldsnumbersontheorderof100basedon diameterandxed-pointseparationcontrolproblems,likecavityandbackward-facing stepows.Therehasalsobeenarecentsurgeofcavityowcontrolstudies,especially thosebasedonreduced-ordermodels,whicharealsoincreasinginpopularity. 46

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Experimentalblu-bodycontrolontheotherhandspansallplatforms,butwith earlycontrolschemesfocusingonthewellestablishedcylinderwake.Experimentsare currentlyrelieduponforthemajorityofhighReynoldsnumberRe & 10 5 owcontrol problemsandpracticalgeometries,suchasowseparationfromlifting-typeairfoils.The complexityoftheowsystemgenerallyincreaseswiththepracticalapplicationofthe platformandtheReynoldsnumberoftheow.Thisrelationship,oroveralltrendinthe currentliterature,relatestotheapproachsimulationvs.experimentusedtoanalyzeand controltheowsystem.ThistrendisdepictedinFigure2-1. Thecanonicalseparatedowfromaatplatemodelisamenabletobothsimulations andexperiments,providingauniquetestcaseforfoundationalstudiesoftheuid mechanicsandinteractionsthatgovernaseparatedow.WithitsintroductionbyMittal etal. 2005,thisowcongurationisstillrelativelynew.Assuch,thecontrolmethods appliedtomanyseparationcontrolproblemsareapplicabletothecanonicalplatform. Eachseparatedowplatformhasitsowncharacteristicfeatures,instabilities,andow regimes,butthealgorithmsandapproachestowardsseparationcontrolaresimilaracross allplatformsconsideredhere.Therefore,theknowledgegainedfromcontrolofeachof theseseparatedowsandthecontrolmethodsappliedtoeachisusefultothecontrolof thecanonicalseparatedow. Abasicunderstandingofeachplatformandowcharacteristicsisessentialprior toreviewingparticularactivecontrolstrategies.However,thistreatmentismore appropriatelyplacedinAppendixA.Allsixmodelgeometriesandbaselineowpatterns includedinthischapteraredescribedthere,includingthedimensionalnomenclatureand dimensionlessquantitiesassociatedwitheachplatform.Forreferencehere,Table2-7 summarizesthosequantitiesbasedontheplatformtype.Inthesectionsthatfollow, discussionsofowcontroltechniquesappliedtotheseplatformsrefertothenomenclature andowdescriptionsprovidedinthistableandAppendixA. 47

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Thedimensionlessquantitiesimportanttomuchofthereviewedliteratureincludethe Reynoldsnumber,Strouhalnumber,andmomentumcoecient.TheReynoldsnumber istheratiooftheinertialtoviscousforcesandhelpstoquantifythepresenceofviscous eectsandturbulence.TheReynoldsnumberRe L forsomecharacteristiclength L and characteristicvelocity U iscomputedas Re L = UL ; {1 where istheuiddensity, isthedynamicviscosity,and U isoftenthefreestream speed, U 1 .TheStrouhalnumberSt L istheratioofthecharacteristicowtimetothe periodofanoscillatoryowdynamic.Thisisthedimensionlessquantityforoscillatory owdynamicsoffrequency f ,givenby St L = fL U : {2 Thisquantityisespeciallyusefulforcharacterizinginstabilitiesinowseparation.An actuation-specicdimensionlessfrequencyisindicatedby F + L ,givenby F + L = f a L U {3 foraforcingfrequencyof f a .Finally,themagnitudeofactuationfromsteadyorpulsedjet owischaracterizedbythemomentumcoecient C .Althoughmultipledenitionsare foundintheliterature,onecommonlyusedis C = A j u 2 j ,rms A ref U 2 ; {4 where u j ,rms istheroot-mean-squarermsjetvelocity, A j istheareaofthejetoriceor slot,and A ref issomereferencearea,suchasamodel'scross-sectionalorplanformarea. 2.2ControlofBlu-BodyandFixed-PointSeparation Followingabriefintroductiononthehistoryofpassiveblu-bodywakestabilization, summariesareincludedforsignicantresearchintheareaofactivecontrolappliedtoow 48

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separationofblubodies.Theseinvestigationsrelatedirectlytotheproposedblu-body separationcontrolbyuseofsyntheticjetactuators.Theincludedclosed-loopstudiesare categorizedintothecontrolstrategiesoutlinedinSection1.2.4controlusingnomodel, ablack-boxmodel,andareduced-ordermodelandanadditionalsectiononearlydirect feedbackmethods,includedforhistoricalperspective.Reduced-orderandblack-box modelsreceivemuchoftheattentionduetotheirrelevanceduetotheclosed-loopmethods appliedwithinthiswork,butbriefdiscussionisincludedformodel-freemethodsaswell. Whenspecicallyappliedtoblubodies,theseclosed-loopcontrolmethodsoften aimtodecreasetheaerodynamicdragastheprimaryobjective.Thereareatleastfour perspectivesfromwhichtoaccomplishthisobjectivewithclosed-loopcontrollersHenning &King,2005;Pastoor etal. ,2008:idirectsuppressionofthedominantvortexshedding modeinthenearwake;iihigh-frequencyforcingtomitigatethecoherenceoflarge-scale vortices;iiithree-dimensionalforcingtobreak-upthevortices;orivpreventionordelay oftheshearlayerinteractionwhichresultsinthesamemode. Thefollowingsubsectionssummarizenotablecontributionsusingthesetechniques onthepreviouslydescribedblu-bodyplatforms.Becauseaatplatemodelwitha bluntbaseistheproposedplatform,muchfocusisgiventotheowcontrolliterature dedicatedtotheD-shapedmodel.However,theowsimilaritiesbetweenotherblu-body geometriesandtheD-shapedmodeljustifyattentioninallareasofblu-bodyclosed-loop owcontrol.Withineachsubsection,theworksarepresentedinchronologicalorderexcept forrareoccasionswhengroupingbyplatformtypeismoreappropriate. 2.2.1Open-LoopControl Vortexsheddingandthecontrolofnear-wakeowhasbeenastapleofcontrol researchsinceRoshko1955installedasplitterplateonacylinderforsuppressionofthe Karmanvortexshedding,asreportedbythereviewsbyBerger&Wille1972,Bearman 1984,Oertel1990,Williamson1996,andChoi etal. 2008.Severaluniqueforms ofpassivecontroldeviceshavebeenappliedfordragreductionofablubody.Among 49

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thekeycontributorsaresurfaceroughnessFage&Warsap,1929;Achenbach,1971;Shih etal. ,1993,dimplesDavies,1949;Bearman&Harvey,1976,1993;Choi etal. ,2006, wireprotrusionsLee&Kim,1997,splitterplatesRoshko,1955;Nash etal. ,1963; Bearman,1965;Tanner,1973;Anderson&Szewczyk,1997;Hwang etal. ,2003,anda secondarycontrolcylinderDalton&Xu,2001.Passivedevicesdevelopedforblunt, orrectangular,trailingedgesareofinteresttothisworkbecauseofthexedseparation pointsatthetrailingedges.Three-dimensionalspanwisevariationsofthetrailingedge geometryhavealsobeeneectiveatinhibitingtheperiodicvortexsheddingbyintroducing aspatialphasevarianceinthenaturalsheddingcycleTanner,1972;Petrusma&Gai, 1992;Tombazis&Bearman,1997;Park etal. ,2006. Open-loopcontrolschemes,whichhaveseveraladvantagesoverpassivedevices,have alsoachieveddragreductionfromtwo-dimensionaluniforminthespanwisedirection andthree-dimensionalforcing.Oneofthemoreinterestingspecicationsistheapplication ofsteadyorunsteadyactuation,asactivedevicesoftenpermitoscillatingormodulating inputs.Furthermore,unsteadyactiveowcontrolcantargetspecicfrequenciesto leveragenaturalowinstabilities,aswellasreducetheconsumedenergyorsourceow fromsteadyactuation.Unsteadyactuationhasbeenappliedextensivelytoseparation controlofaerodynamicbodiesbutalsotoblu-bodyseparation.Someearlybackground intoopen-loopcontrolofblu-bodywakeswithsteadyforcingisintroduced,followedby examplesofunsteadyactuation. Bearman1966investigatedtheeectofbasebleed,asteadyinjectionofuidfrom thebaseofthebody,onthewakeofatwo-dimensionalmodelwithablunttrailingedge. Thebasepressurecoecientwasmeasuredagainstvariouslevelsofthedimensionless bleedcoecient C q =_ m q = m 1 ,where_ m q and_ m 1 denotedthebleedandincident massowrates,respectively.Abasepressureriseofabout70%wasachievedfor C q = 0 : 12,andvortexsheddingwascompletelysuppressedforsucientlylarge C q .Sevilla& Martnez-Bazan2004achievedsimilarsheddingattenuationfor0 : 12
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Reynoldsnumberrange4 10 3 < Re h < 1 : 2 10 4 ,whereRe h isthefreestreamReynolds numberbasedonthebasethickness.However,theeciencyofsteadyblowingwaspoor, asitrequiredasubstantialmassowrate. Unsteadyforcingoeredaseparatemeansofactiveowcontrolthatcouldmaintain controlauthoritywithreducedexpendituresofmassowrateand/orenergy.Perhaps themostpromisingofunsteadyactuationschemesisprovidedbyZNMFactuators,or syntheticjets,whichutilizeonlythesurroundinguidoftheowsysteminwhichtheyare embedded.Amitay etal. 1997andAmitay etal. 1998studiedthecontrollingeects ofZNMFactuatorsontheowoveracircularcylinderbyvaryingtheactuationangle relativetothestreamwiseowdirection.Theexperimentsrevealedthattheactuation resultedintheformationofclosedrecirculationregionsandsignicantmodicationtothe surfacepressuredistribution.Atcertainactuationangleswherethejetwasupstreamof anunforcedseparationpoint,theliftwasincreasedsignicantlyandthedragreduced. SimilarmodicationsofthesurfacepressuredistributionwereobservedbyWilliams etal. 1992withinternalacousticactuationalsoZNMF. 2.2.2SingleSensorProportionalFeedbackControl Perhapsthemostbasiccontrolmethodtobediscussed,closed-loopcontrolfrom singlesensorlinearfeedbackhasbeenshowntobeeectiveinsuppressingthevortex sheddingbehindblubodiesatlowReynoldsnumbers,typicallyontheorderof10 4 or less.Inthisregime,butstillabovethecriticalvalue,ablu-bodywakeiswellorganized withamajorityoftheenergypresentintheunstablemodemanifestedbyvonKarman vortexshedding.HigherReynoldsnumberspermitmoreecientenergytransportinto smallerowscalesviatheReynoldsshearstress,thusleadingtoincreasedenergyin turbulentuctuations.Linearfeedbackviaasinglesensoristhereforefeasibleatlow Reynoldsnumbersbecausethesensorneedonlyfeedbackthefundamentalwakemode. Albeitnotcloselyrelatedtomodel-basedowcontrol,suchmethodsareimportantfor contrastingtomoresophisticatedtechniquesrequiredforricherdynamicsand/orbetter 51

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performance.Keypointsfromeachreviewoflinearfeedbackcontrolaresummarizedin Table2-2attheendofthissection. Justtwelveyearsafterthegroundbreakingsplitter-platecontrolpassivebyRoshko 1955,Berger1967demonstratedthatobroundoval-shapedcylindervibrations inducedfromnegativefeedbackofahot-wireplacedinthecylinder'swakesuppressed thewakeforReynoldsnumbersupto80basedonthediameteralongthesemi-minor axis.Roussopoulos1993utilizedthereceptivityoftheKarmanvortexstreettoacoustic perturbationsfordirectfeedbackcontrolfromhot-wiresinthewakeofacircularcylinder. ForReynoldsnumbersupto10%greaterthantheonerequiredfortheonsetofthewake instability,completevortexsuppressionwasreportedbylocalloudspeakeractuationwith linearproportionalfeedback.Signicantattenuationwasobservedupto20%abovethe criticalReynoldsnumberbutnohigher. Huang1996foundthattheamplitudeoftheacousticexcitationcouldbe substantiallyreducedwhenthesourcelocationwasmovedclosertotheseparatedshear layerofacylinderwake.Twoloudspeakerswerefastenedtooppositeendsofahollow circularcylinderwithanarrowspanwiseslotplacedsuchthatthedisturbanceswere introducedneartheseparationpoint.Ahot-wirewasplacedintheuppershearlayerand themeasuredsignalwasfedtotheloudspeakers.Optimalphaseandgainadjustments signicantlyreducedthepresenceofvortexsheddingintheentirewakeforRe d ranging from4 10 3 upto1 : 3 10 4 ,withactuationonlyintroducedtoonesideofthewake.This demonstratedthattheentireunstablewakewasinuencedbytheperturbationsandthe interactionofthetwoshearlayers. Theseearlystudiesprovedthatsimplefeedbackcontrolmethodscouldattenuateor altervortexsheddinginblu-bodywakes.Overaspanofabout30years,thecontrollable rangeofReynoldsnumbersincreasedbythreeordersofmagnitude.Theeectivecontrol techniquesdemonstratedduringthisperiod,includingcontrolledcylindervibrationsand 52

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localforcingfromloudspeakers,continuedevenintothelastdecadee.g.Zhang etal. 2004;Henning&King,2005. 2.2.3Model-FreeControl Model-freecontroltechniquesappliedtoowsystemshavebecomepopularinthe lastdecadebecauseofthecomplexnatureofuidmechanics.Thesemethodsareonly reallyappealingforexperimentalclosed-loopstudies,inwhicheventheinput-output behaviorcanbedicultand/ortime-consumingtoobtain.Thissubsectionisdedicated toblu-bodyowcontrolusingmodel-freecontroltechniques.Thesecontrollersdetect agradientofsomechoseninput-outputbehaviorinordertodrivetheinputsuchthat theoutputisoptimized.Keyremarksandrunconditionsfortheincludedliteratureare consolidatedinTable2-3attheendofthissection. Henning&King2005conductedclosed-loopexperimentsfordragreductionona atplatemodelwithablunttrailingedgealsoreferredtoasaD-shapedmodelshown inFigure2-3.AtRe h =4 10 4 ,anextremum-seekingcontrollerwasderivedtoincrease thebasepressure,thusdecreasingthedrag.Loudspeakersconnectedtospanwiseslots alongtheupperandlowertrailingedgesprovidedharmonicZNMFactuation,angled45 towardsthestreamwiseowdirection.Ontheattrailingedge,amidspanverticalcolumn ofninelow-pressuresensorswereusedwithaKalmanltertoestimatethewall-pressure uctuation,whichwasmodeledasasinusoid.Open-loopforcingat F + h =0 : 25,the naturalvortexsheddingfrequency,actuallystrengthenedthevortexsheddingandthus increasedthedrag.Thephasedierencebetweentheactuationsignalandthebase pressurecoecientwasusedasthevaryinginputoftheclosed-loopextremum-seeking controller,withtheaveragepressurecoecientastheoutputtobemaximized.Foraxed C =2 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,theextremum-seekingmethodincreasedthebasepressurebyabout35% dragreductionofapproximately10%after10seconds.Theconvergedphasedierence was180 ,whichindicatedthatmaximumbasepressurewasachievedwithsymmetric vortexsheddingfromtheupperandlowersurfaces. 53

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Infollow-upworks,Henning etal. 2007andPastoor etal. 2008employed slope-seekingtoexploitthebasepressure'ssaturationaboveacertainactuation amplitudeandthuslimitedtheinputactuationenergy.Thesamemodeldimensions andexperimentalsetupasHenning&King2005wereused.Pastoor etal. 2008applied in-phaseactuation F + h =0 : 15ofbothupperandlowerslotsduringopen-loopforcing thatrevealedaplateau-likeeectfortheoutputpressurecoecientasafunctionof increasingtheinputmomentumcoecient.Specifyingtheslopefoundatthecuspof theplateau,exempliedinFigure2-2,theslope-seekingmethodachieveda15%drag reductionand40%meanpressureincreasewith C =5 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 .Particleimagevelocimetry PIVmeasurementsoftheRe h =2 : 3 10 4 wakewithactuationof F + h =0 : 15and C =9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 capturedsynchronizedsheddingoftheupperandlowershearlayervortices. Mostreviewedextremum-seekingresultswereconsistentwiththeopen-loopmethods. Theapplicationofmodel-freetechniqueswasthereforemostbenecialasasimple closed-loopcontrolmethodcapableofadaptingtochangingoperatingconditions,although slope-seekingwasshowntobemoreappropriatethanextremum-seekingtosteady-state mapscharacterizedbyaplateauratherthanapeak. 2.2.4ControlofBlack-BoxModels Black-boxmodelscanbefavorabletomodel-freecontrollersbecausetheidentied systemmodelcanprovideamuchfasterresponsethangradient-based,model-free techniques.Asimple,oftenlinear,modeloftheinput-outputbehaviorofaparticular systemactuators,ow,andsensorsallowsformoretraditionalcontroltechniques.The dynamic,open-loopresponseofthesystemisusedtotanassumedmodel.Acontrolleris thendesignedtomeetaclosed-loopcontrolobjective,whichisusuallytoachieveset-point trackingofthesystemoutput.KeyresultsaresummarizedinTable2-4. Therecirculationlengthoftheseparatedowregionofabackward-facingstepwas controlledbyKing etal. 2004withfourslot-loudspeakersandfourstreamwiserowsof microphonesFigure2-4.Asingle-inputsingle-outputSISOlinearmodelwasidentied 54

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frommultiplestep-responseexperimentsofspanwiseuniformactuation F + H =0 : 31 andthemeasurementofaspanwise-averagedrecirculationlength h L R i ,approximated bythermsmethod"AppendixAfromtheunsteadypressuresensors.Thewide uncertaintyintheidentiedmodelparametersnecessitatedrobustcontrolintheformof an H 1 -controllerthatguaranteedclosed-loopstabilityofeventheworstidentiedplant. Thecontrollerreducedthebaseline h L R i of6 : 5 H by23%duringatrackingcommand atRe H =4 10 3 anddemonstratedrobustnesstochangingwindtunnelspeeds.A multiple-inputmultiple-outputMIMOcontrolleroperatedthefouractuatorswith individualamplitudesandtherecirculationlengthwasestimatedforeachstreamwise arrayofsensors.Themulti-variable H 1 -controllerachievedspanwisevaryingrecirculation lengths,asdepictedinFigure2-4. Henning&King2007alsoachievedMIMOcontrolofabackward-facingstep's separationbubbleviaarobust H 1 -controllerbutatalargerReynoldsnumberofof Re H =2 : 5 10 4 Figure2-5.Actuationviaslot-hose-loudspeakerstargetedthe Kelvin-Helmholtzinstability,whichampliedtheperturbations,thusaectingthe spreadingrateoftheseparatedshearlayerandtheresultingreattachmentlocation. Open-loopcontrolresultsconcludedtheoptimalactuationfrequencyforreductionofthe recirculationlengthwasabout F + H =0 : 3. AnumericalstudybyDahan etal. 2011employedclosed-loopcontrolstrategies toincreasethebasepressureonabackward-facingstep.Thestudyusedlargeeddy simulationsLEStosimulatealaminartwo-dimensionalowatRe H =2 10 3 and aturbulentthree-dimensionalowatRe H =2 : 9 10 4 .ZNMFforcingisplacedatthe cornerofthestepandangledat45 relativetothestreamwisedirectionFigure2-6.In thetwo-dimensionalsimulations,alow-ordermodel G s inthefrequencydomainwas ttotheopen-loopowresponsefromharmonicforcing.Then,asecond-ordercontroller K s wasimplementedwithresonancesettotheshearlayerinstabilityfrequencySt H = 0 : 066St =0 : 01.Thecontrollerresultedinabasepressureincreaseof70%.Inthe 55

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three-dimensionalsimulations,asystemplant G s wasttofrequencyresponsedata usingasecond-ordermodelobtainedfromtheeigensystemrealizationalgorithm.An H 1 -loop-shapingmethodwasemployedtoreachadesiredopen-looptransferfunction L s = G s K s .Theresultingcontroller K s was,byvirtueofthismethod,robust tosmalldisturbancesandclosed-loopstable.Thecontrollerincreasedtheaveragebase pressureby20%atthecostof C =1 : 2 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 ReturningtotheworkofHenning&King2005fordragreductiononaD-shaped modelFigure2-3,linearblack-boxmodelswerealsoidentiedbetweenthecontrol input C t andthemeasuredoutputofthespanwise-averagedbasepressurecoecient h C p t i ,atRe h =4 10 4 .Withopen-loop,in-phaseZNMFforcingfromthetrailingedge corners,anoptimalactuationfrequencyfordragreductionwas F + h =0 : 17foranatural sheddingfrequencyofSt h =0 : 25.Withconstantforcingatthatoptimum,afamilyof linearrst-orderSISOmodelswasidentiedfromstepinputsoftheactuationamplitude. Quantitativefeedbacktheorywasusedtodesignarobustcontrollerbasedonthefamilyof identiedmodels.Set-pointtrackingwasachievedforaconstantreferenceof h C p i = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 4 whiletheReynoldsnumberwasvariedwithintherangeof2 10 4 Re h 6 10 4 Henning etal. 2007appliedan H 1 -controllerandafeedforwardloopfordrag reductionofaD-shapedblubody.Alinearsecond-ordermodelwasidentied betweentheinput C andtheoutput h C p i .Arobust H 1 -controllerwasdesignedfor implementationintandemwithamodel-baseddynamicfeedforwardcontroller,which wasincludedforbettertrackingperformance.Thesystemwasshowntotrackastepwise changeinthereferencecommandwhileRe h waslinearlydecreasedfrom6 10 4 to 3 : 5 10 4 .Duringthattime,thedragwasreducedbyapproximately15%. WiththeexceptionofDahan etal. 2011,alloftheblu-bodycontrolmethods basedonidentiedblack-boxmodelsachievedaset-pointtrackingobjective.Theresults fromthesestudiesindicatedthatblack-boxmodelidenticationwasfeasibleandeective 56

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forset-pointtrackingof L R forabackward-facingstepanddragreductionorbase pressureincreaseofaD-shapedmodel. 2.2.5ControlofReduced-OrderModels Reduced-ordermodelsofhigh-dimensionalowsystemstypicallyextractfundamental, orlow-order,dynamicsbasedonsomepredeterminedcriterion,e.g.energy,frequency, orvorticitycontent.Theyprovideanapproximationofthefulldynamics,usually applyingahandfulofsensorsasstateobservers.Reduced-ordermodelshaveseena surgeinpopularityinthelastdecade,withexamplesexistingforbothsimulatedand experimentalblu-bodyows.Duetothecomplexityofintegratingtheinputintothe controlmodel,thesemethodsareoftenappliedtosimulateddata.However,simulations arestillrelevanttothecontrolaccomplishedbymeansofreduced-orderdynamics.The followingsubsectionsdividetheincludedliteratureintocontrolmethodsbasedonGalerkin andvortexmodels.Techniquesapplyingtheformeraremuchmoreprevalentduetothe Galerkinmethod'spracticalapplicabilitytostandardcontroltheory.Tables2-5and2-6 summarizethekeyresultsfollowingthemoredetaileddiscussionwithineachofthese subsections. 2.2.5.1Galerkinmodels Simulatedwakestabilizationofacircularcylinderwasapopularmethodfor reduced-ordermodelingbecauseoftherelativelysimpleoscillatorydynamicsanda computationalfeasibility,atleastforlowRe d .Infact,manysimulatedstudiesshared thesameowconguratione.g.Gerhard etal. ,2003;Tadmor etal. ,2003;King etal. 2005,2008.Amongthese,theowproblemwasinitiallydevelopedbyGerhard etal. 2003indirectnumericalsimulationsDNS.Therefore,thesharedowconditions,model formation,andactuationmethodarerstdescribedandthenthedistinctcontributions fromthesereferencedworksfollow. AReynoldsnumberofRe d =100wasspeciedforthesimulation,beinglargeenough forlaminarseparationandtransitiontoavonKarmanvortexstreetbutsmallenoughto 57

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bebelowthethresholdforthethree-dimensionalinstabilityRe d 180Williamson, 1996.Alow-orderapproximationofthefulleldvelocitydataFigure2-7wasformed fromalinearcombinationoftheunstablesteadystate u s ,thersttwoPODmodes i capturing96%ofthetotalenergy,andashiftmode ,givenby u x;t = u s x | {z } steady state + 2 X i =1 a i t i x | {z } PODmodes + a t x | {z } shiftmode : {5 TheNavier-Stokesequationswereprojectedontothesemodes,yieldingaGalerkinmodel closed-loopcontrol.Theshiftmodewasnecessarytoresolvethedierencebetweenthe meanandtheunstablesteadystate.Actuationwassimulatedasavolumeforceinthe near-wakeregion,asseeninFigure2-7. InGerhard etal. 2003,adynamicobserverestimatesthestatefromaowsensor andanadjustmenttotheestimatedstatevariableviaacorrectionterm.Thecontrol objectivewastoreducethemagnitudeofthecapturedoscillations,representedbythe normofthersttwoPODcoecients,therebydelayingtheformationofthevonKarman vortexstreetinthenear-wakeregionandreducingthedragduetopressure.Rather thanusingnonlinearcontrolmethods,whichwereexpectedtolosemodelvalidityduring control,asimplecontrollawwasdeveloped.Aconditionalinputwasproposedsuchthat theinput u = 8 > < > : u 0 if g c cos )]TJ/F22 11.9552 Tf 11.955 0 Td [( > 0 )]TJ/F22 11.9552 Tf 9.299 0 Td [(u 0 otherwise ; {6 where u wasthecontrolinput, u 0 wasanamplitudecalculatedeachperiod,andthe variables g c ,,and werederivedfrommodelparametersoftheGalerkinprojection. Thiscontrollerwastermedenergy-basedcontrol"becauseitsgoalwastocontinuously reducetheuctuationenergyuntilsteadystate.Theenergywithintheentiredomainwas reducedby31%basedontheempiricalGalerkinmodelderivedfromthenaturalow. 58

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Thesubsequentcontrolleryieldeda45%reductionthroughutilizationofthePODmodes obtainedfromtheinitialcontrolledow. Thisresultdemonstratedthelimitedvalidityofthelow-dimensionalmodeltothe envelopeofoperatingconditionsthatwereusedtoderivethePODmodes.Asthecontrol alteredtheowstructureanddrovetheoscillationsawayfromthenaturallimitcycle,the stateestimationuncertaintyincreasedbecausetheidentiedmodeswereextractedfrom thenaturaluncontrolledow.Theseresultswerecomparedtoabest-case"sensing scenarioinwhichallvelocityinformationforeachinstantaneoussnapshotwasassumed known,therebybypassingtheneedforanobserverandstateestimation.Themaximum energyreductionforthisscenariowas50%. King etal. 2005expandedupontheenergy-basedcontrollerbyimplementing severalnonlinearcontrollawsdampingcontrol,input-outputlinearization,Lyapunov basedcontrol,andoppositioncontrolforcomparisontotheenergy-basedcontroller. Allmethodsachievedsimilarenergyreductionsbetween30%and33%fortheconverged post-transientuctuations.Thesevaluesmatchedtheresultobtainedfromtheoriginal energy-basedcontrollawproposedbyGerhard etal. 2003.However,severalofthe resultingcontrollaws,includingthesimpleenergy-basedcontroller,hadperiodic singularitiesbroughtaboutbytheabruptsignchangeofthecontrolinput. 1 King etal. 2008addedtothelistofcontrolmethodsappliedtothecylinderwake DNSandGalerkinmodelwithnonlinearmodelpredictivecontrolMPC.Thebasicidea ofMPCwastoupdatethecontrolinputinaniterativefashionbasedonthecalculations ofafuturetrajectorymeanttoachievesomeoptimumperformance.Thecostfunctionof 1 Suchinputsarelesspracticalwithphysicalactuatorsandwilllikelynottranslatewell toexperiments. 59

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thealgorithmwasspeciedas J = Z t + H p t e 2 d with e t = r t )]TJ/F15 11.9552 Tf 12.101 0 Td [(^ a 1 t ; {7 where r wasthereferencesignal,^ a 1 wastherstPODmode'sestimatedtime-varying coecient,and H p wasthepredictionhorizon. 2 Thereferencesignalwasspeciedas anexponentiallydecayingsinusoidsuchthattheamplitudeofthewakeoscillationswas reducedandthefrequencywasmaintained,therebyattemptingtoconnethecontrolled owwithintherealmofthelow-dimensionalGalerkinmodel.Whendirectlycomparedto theenergy-basedcontroller,thenonlinearMPCreachedapproximatelythesameenergy reduction%reductionin a 2 1 + a 2 2 inabouthalfthetimeandultimatelyconverged toalowerenergylevel.ThenonlinearMPCextendedtheaveragerecirculationlength from4 : 1 d fortheenergybasedcontrollerto5 : 2 d ,indicatingthatthevortexsheddingwas delayedfurtherdownstream. Inexperiments,Luchtenburg etal. 2010implementedanonlinearcontrolstrategy basedonalow-ordermean-eldGalerkinmodeloutlinedbyLuchtenburg etal. 2009. Areduced-ordermodelwasdevelopedtoaddressthenonlinearinteractionsbetweenthe meaneld,thecoherentstructures,andtheactuation.Whenappliedtoanexperimental D-shapedblubody,theoscillatorscorrespondingtothenaturalvortexsheddingand theactuationwereincludedinthemodel,alongwiththesteadybaseowandmean-eld deformationsbroughtaboutbyactuation.TheresultingGalerkinexpansionwas u x;t = u 0 x | {z } steady baseow + 2 X i =1 a i t i x | {z } naturalshedding + 4 X i =3 a i t i x | {z } actuation + 6 X i =5 a i t i x | {z } mean-elddeformations : {8 2 Thisperformanceindexneglectedthecontroleort,whichcouldbeimportantfor experimentalinvestigationsandeciency. 60

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Adynamicsystemfortheevolutionofthetime-varyingcoecients a i wasthenfound fromtheassumptionsofmean-eldtheory,withreferencetoNoack etal. 2003.Asliding modecontrollerwasappliedtothismodelfordragreductionoftheblubody.Thebasic premiseofslidingmodecontrolwastoforcethestatestofollowaspeciedtrajectory towardsaso-calledslidingsurface,oranattractorstate.AnextendedKalmanlter wasusedtoestimatethefullstate,whichwasrequiredforthecontrollaw.Thecontrol objectivewasset-pointtrackingofthebasepressurecoecient,whichwasmeasuredby 15pressuretapswithdierentialpressuregaugesshowninFigure2-8.Thewakewas actuatedfrombothtrailingedgesbypulsedsuction.Characterizationoftheowresponse toopen-loopforcingdemonstratedthatforcingwith C =1 : 04 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 and F + h =0 : 14 %ofthenaturalsheddingfrequencyatRe h =3 : 23 10 4 increasedthebasepressure coecientbyabout36%anddecreasedthedragbyabout15%.Theclosed-loopcontrolled pressurecoecientcloselyfollowedthereferencecommandandachievedthesamelevelof dragreductionasopenloop.However,robustnesswasalsodemonstratedbytrackingthe reference C p whileRe h wasvariedbetween2 : 69 10 4 and3 : 77 10 4 .Aleksic etal. 2010 comparedtheslidingmodecontroller,withslightmodicationtothecontrolinput,toa linearMPCscheme.Bothcontrollersachievedadragreductionofabout15%foraxed Re h =3 : 23 10 4 Theexperimentalapplicationofreduced-ordermodelsisperhapsnogreaterin oneplatformthanforsuppressionofow-inducedcavityoscillations.However,the controlobjectivesforthisproblemarelessapplicabletothecanonicalseparatedow fromaatplate,asmostaimtoreducetheowunsteadinessandthesubsequent ow-acousticcouplingthatleadstoresonanceCattafesta etal. ,2008.Thisisnot unliketheapproachforwakestabilization,however,theunforcedowstateitselfisunlike theotherseparationcontrolplatforms,involvingtheseparatedshearlayer'samplication ofitsownimpingement-generatedacousticsandtheresultingreceptivitycycle.Adetailed treatmentofthisintenseanddicultowproblemisbeyondthescopeofthisreview, 61

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andhencelefttomoreappropriatemanuscriptse.g.Rockwell&Naudascher,1978,1979; Cattafesta etal. ,2003;Rowley&Williams,2006;Cattafesta etal. ,2008.Briefattention isthereforegiventothecontrolmethodsappliedtothisproblem. Rowley&Juttijudata2005implementedcontrolschemesforthestability ofaPOD-basedGalerkinmodelandanon-physicaldynamicmodelforDNSof two-dimensionalcavityowataMachnumberof0.6.AnLQGregulatorwasimplemented basedontheempiricalGalerkinmodel,butthecontrollerwasultimatelylimitedbythe validityoftheidentiedmodel.Ontheotherhand,thedynamicphasormodelachieved completesuppression.However,suchamodelmaybelessapplicabletoanexperimental environmentwithhighturbulencelevelsandpotentialsensordisturbances.CattafestaIII etal. 1997implementedexperimentalLQGwithpoleplacementforclosed-loopcontrol thatreducedcavityoscillationswithsignicantlylessinputpowerthanwasachievedwith open-loopmethods.Samimy etal. 2007developedanexperimentalGalerkinmodelfor Mach0.3cavityowfromPODmodesofPIVdata,alongwithamethodforintroducing thecontrolinputintothelow-ordermodel.Withthebaselineowcharacterizedbya dominantresonanttone,thelinearquadraticoptimalcontrolwasabletosignicantly attenuatethetoneandredistributetheenergyintolower-frequencymodes.Theresults demonstrateaclearimprovementwhencomparedtoopen-loopforcing. Thereduced-orderGalerkinmethodsincludedhereweredividedintosimulatedand experimentalcontrolproblems.Theformerconsistedofmanycontributionsthatapplied GalerkinmodelstosimulatedowoveracircularcylinderatRe d =100.Thesesimulated controlproblemsemployedaphysicallymotivatedcontrollerenergy-basedcontrol, Lyapunovderivedcontrol,oppositioncontrol,nonlinearMPC,andlinearPDfeedback. Fromtheknowledgegained,recentexperimentalcontributionshaveappliedlinearMPC andslidingmodecontroltodecreasethedragofaD-shapedmodelby15%.Abriefreview ofreduced-ordermodelsappliedtocavityowcontrolwasalsoprovided. 62

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2.2.5.2Vortexmodels VortexmodelsareanalternativetoempiricallymotivatedPOD-basedreduced-order modelsandaresometimesusedtoapproximatevortex-dominatedowssuchaswakes. Withthemotivationofreducingwakeinducedliftforceoscillationsonacylinder,Li& Aubry2003derivedafeedbackcontrolmethodfromtheanalysisofaFopplpotential owmodelforpointvortices.Insimulation,theysuperimposeduniformincomingow aroundacylinderwithapairofpointvorticesbehindthecylinderoneithersideofthe centerline.Thecontrollawwasanalyticallyderivedfromthemodelwiththeobjective ofremovingtheliftforceoscillationsbycancelingperturbationstothesteady,unstable recirculationbubble.Thecontrolstrategywasimplementedintheimpulsivelystartedow overacylinderbasedonDNSofthetwo-dimensionalNavier-Stokesequations,usingthe globalliftmeasurementforfeedback.ForReynoldsnumbersof100and200,theoscillating liftamplitudeonthecylinderwasreducedby97%and95%,respectively. Pastoor etal. 2008derivedareduced-ordervortexmodeltomotivatethecontrol designforwake-vortexsuppressionfromaD-shapedblubodyatRe h =4 : 6 10 4 Thevortexmodelwasusedtomodelthetransientdevelopmentofthenaturalwake beginningfromthepotentialowsolution.Astagewasidentiedduringthissimplied evolution,beforethedevelopmentoftheKarmanvortexstreet,thatwascharacterized bysymmetricvortexformationin-phaseintheupperandlowershearlayersanda symmetricbasepressuredistribution.Anyperturbationofthatsymmetrywasamplied, andtheabsolutewakeinstabilityinducedalternatingvortexsheddingfromthetrailing edgeshearlayers.Thelargerwakevorticesbroughtaboutanasymmetric,alternating pressuredistributionandadecreasedaveragepressurealongthetrailingedge.Thevortex modelthusmotivatedforcingsuchthatin-phasevortexformationoftheupperandlower shearlayerswouldbeenergizedandthesymmetrymaintained,therebymakingtheshear layerslessreceptivetoperturbationsanddelayingformationofthealternatingwake vortices.Thiscontrolobjectivewasimplementedbyaslope-seekingfeedbackmethodand 63

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aphase-controllermotivatedbythesymmetricvortexsheddingintheinitialevolutionof thevortexmodel.Foraneectivemomentumcoecientof3 : 8 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 ,thedragcoecient wasreducedby15%bythephasecontroller.Thiscontroleortwas56%oftheactuation energyrequiredforsimilarresultsfromtheslope-seekingfeedback. Vortexmodelswerelessprevalentintheliteratureduetotheircomplexapplicability toclosed-loopcontrol.Themosteectivedemonstrationwastheuseofthevortexmodel byPastoor etal. 2008tomotivateasynchronizationofthesymmetricshearlayers behindaD-shapedmodel,thusdelayingtheformationoftheKarmanvortexstreetand increasingtheaveragebasepressure. 2.3ControlofBoundaryLayerSeparation BoundarylayerseparationcontrolofstalledairfoilsisoneofmotivationsforZNMF controlofthecanonicalseparatedowChapter1.Manyowcontrolprojectsare dedicatedtopracticallifting-bodyplatforms.Assuch,itisimportantforthisreview togiveattentiontostudiesthatapplyactiveowcontrolstrategiestodelayorprevent boundarylayerseparation. Thissectioncontainssummariesofrecentowcontrolstudiesforboundarylayer separationfromsymmetricorstreamlinedairfoils.Asidefromtheinitialdiscussionofthe canonicalseparatedowliterature,thesubsectionsthatfollowmatchtheclosed-control methodsoutlinedinSection1.2.4.Thekeyndingsandimportantparametersare tabulatedfollowingthediscussionineachsubsection. 2.3.1Open-LoopControloftheCanonicalFlowSeparation InresponsetothelackofconsensusamongZNMF-basedseparationcontrolliterature Section1.1.1,Mittal etal. 2005developedanovelseparatedowcongurationon aatplatemodel,describedhereasthecanonicalseparatedow"Figure1-1.The overarchinggoalofthiscongurationwastoinvestigateactivecontrolofseparatedairfoil owsinacomprehensiveandsystematicmanner."Amongthemanydiscrepanciesfrom existingliterature,thiscongurationaimedtopromotethefollowinginvestigations:ithe 64

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existenceandroleof multiple characteristicfrequencies;iitheoptimalforcingfrequency F + inrelationtothecharacteristicfrequenciesandwithregardtoconsistent C ;iiithe eectivenessofhigh-frequencyforcing F + L sep 10; 3 ivtheeectsofcurvatureor lackthereofonseparationcharacteristics;andvthepotentialbenetsofmodulation waveformsforactuation. TheinitialsetofnumericalsimulationsforthiscongurationbyMittal etal. 2005 areperformedona2%thickellipticairfoilatRe c =6 10 4 fortwoscenariosofinduced boundarylayerseparationviasuctionandblowingboundaryconditionsimposedon thetopboundaryofthecomputationaldomain.Therstcreatesaclosedseparation bubblenearmid-chordwitharecirculationbubblelengthof L sep 0 : 3 c andthesecond createsabubbleoflength L sep 0 : 35 c thatreattachesinatime-averagedsensevery nearthetrailingedge.Bothcasesexhibitdistincttemporalcontentfortheshearlayer, theseparationbubble,andthewake.ZNMFcontrolofthemid-chordseparationforces atalowfrequencyof F + L sep =0 : 6,nearthenaturalseparationbubblefrequencyof f sep L sep =U 1 0 : 42.Higher-frequencyforcingischaracterizedby F + L sep =6,whichis approximatelyequaltotherstsuperharmonicoftheuncontrolledshearlayerfrequency of f SL L sep =U 1 3whilelessthantheuncontrolledwakefrequencyof f wake L sep =U 1 8. Preliminaryresultsqualitativelydemonstratethatlow-frequencyforcingtendstoreattach theseparatedshearlayer,whereasthecaseofhigh-frequencyforcingislesseective.A limitedquantityofthree-dimensionallarge-eddysimulationsLESofasimilarunforced owcongurationindicatethattheessentialdynamicsofthisowpatternarenot signicantlyaectedKotapati etal. ,2008;Kotapati,2008. Incertainowconditions,someofthenaturalfrequenciesmaylock-on,"meaning theyassumethesamevalue.Kotapati etal. 2010simulatedthetwo-dimensional canonicalcongurationwithanaft-chordseparationbubbleandfoundlock-onstatesin 3 L sep issomecharacteristiclengthscaleoftheseparation. 65

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theunforcedow.ForRe c =1 10 5 ,whichmatchesthechordReynoldsnumberofthe containedwork,theshearlayerandseparationbubblebothlockedonatafrequencyof f lock L sep =U 1 =1 : 09.Thewakefrequencywas f wake L sep =U 1 =0 : 87.Themeanseparation bubblelengthfortheunforcedowwas0 : 286 c ,asindicatedinTable2-7,atableofmean separationbubblesizesduetoactuation.Theforcingfrequency f a targetedsubharmonics andsuperharmonicsofthelock-onfrequencywithaconstant C = dV 2 rms L sep U 2 1 =2 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 ; {9 where d isthewidthofthejetslot. 4 Therecirculationbubblelength L sep decreased monotonicallyuntilthesecondsuperharmonicof f lock F + L sep =3 : 26.Atthat frequencyandabove,theseparationbubblewasmostlyunaected.However,forthe rstsuperharmonicforcingandabove F + L sep 7 : 6,theshearlayerlockedontothe forcingfrequency,whilethewakeandseparationbubbledidnot.Therefore,therst superharmonicreducedthebubblesizethemostofalltests,butthenextsuperharmonic essentiallylostcontrolauthority. AsimilartrendforforcingfrequencywasobservedbyPostl etal. 2011,forpulsed vortexgeneratorjetsusedtocontrollaminarseparationfromaatsurfaceinDNS.A rangeofforcingfrequencieswastested,andatrendofdecreasingmeanreattachment lengthwasobserveduntiltheforcingexceededtheshearlayerfrequency,atwhichpoint theactuationwasmuchlesseective. 5 Theforcingfrequencythatmatchedthenatural instabilitymodeoftheseparatedshearlayeryieldedlarge-scale,spanwise-coherent structuresthatbetterentrainedhigh-momentumuidfromthefreestreamandreattached theow. 4 Kotapati etal. 2010dene C dierentlyandreportavalueof1 : 3 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 .Thisvalueis convertedthematchthedenitionprovided. 5 Aformaldenitionofthedimensionlessfrequencywasnotprovided. 66

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Totheauthor'sknowledge,noexperimentalworkoutsideofthisthesishasbeen conductedforthecanonicalowconguration.Assuch,thesebenchmarkexperimentsrely uponthelimitedknowledgegainedfromcomputationalmethods.However,studiesforthe separationcontrolofmoretraditionalairfoilsgeometriesaremuchmoreprevalent.Focus isgiventotheclosed-loopcontrolmethods. 2.3.2Model-FreeControl Theextremum-andslope-seekingmethodsaresimplerclosed-loopmethodsbecause nomodeloftheowsystemisrequired.Thesetranslatewelltoseparationcontrolofa stalledairfoilbecausethereisaclear,measurableeectontheliftanddragforceswhen theowisreattachedorseparationisdelayed.Anothertypeofmodel-freetechnique appliedtoowseparationfromairfoilsisdiscussedrst.Keyremarksandrunconditions areconsolidatedinTable2-9. Tian etal. 2006 a appliedamodel-freeclosed-loopcontrolalgorithmtermedthe downhillsimplextoecientlydelayseparationfromapost-stallowoveraNACA-0025 airfoilatRe c =10 5 and =20 .Thedownhillsimplexalgorithmusedfunction evaluationsfromasetofinitialconditionstoformasimplexandtheniterativelywalked" thatsimplexinthedownhilldirectiontowardsafunctionminimum.Fouron-boardZNMF syntheticjetactuatorsforcedowoscillationsthroughaspanwiseslotneartheleading edgeontheseparatingsideoftheairfoil,thoughtheslotonlycoveredthecentralone thirdoftheoverallairfoilspan.Twonaturalowinstabilities,namelytheconvectiveshear layerinstabilityandtheglobalwakeinstability,wereidentiedandtargetedforoptimal forcing.Withpotentialdemonstratedfornonlinearinteractionsbetweentheseinstabilities, theforcingparametersoftwodierentmodulationschemeswereoptimizedtoincreasethe meanlift-to-dragratio. 6 Modulatedactuation,intheformofamplitudemodulatedAM 6 Theobjectivewas,moreaccurately,todecreasethedrag-to-liftratioasthedownhill simplexisaminimizationfunction. 67

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andburstmodulatedBMsignals,targetedboththewakefrequency f wake 40Hzand shearlayerfrequency f SL 2038Hz,thoughthehighercarrierfrequencywasgenerally limitedtothenominalbandwidthofthedevice-1500Hzforsignicantoutput.The AMsignalconvergedtothelargestmean L=D of2.56.However,inclusionofanactuation penaltyreduced C byafactorofalmost30andstillachieved L=D =2 : 38.TheBMsignal furtherreducedtheenergyconsumptionbutcostofslightlyreduced L=D Benard etal. 2009utilizedtheslope-seekingtechniqueforseparationcontrol ofasymmetricNACA-0015airfoilbyapplicationofaspanwisedielectricbarrier dischargeDBDactuatoralongtheleadingedgeoftheairfoil.Theslope-seekingmethod waschosenfromopen-loopresultsshowingtheplateau-likeoptimumofthegloballiftforce inresponsetoincreasingvoltageamplitudeoftheinputsignal.Forconditionsof =15 andReynoldsnumbersofRe c =1 : 3 10 5 ,2 : 65 10 5 ,and4 10 5 ,theSISOcontrolscheme increasedliftbyapproximately107%,127%,and133%,respectively,understaticand dynamicfreestreamconditions.Thecontrollerreattachedoratleastdelayedseparation withanoptimalsupplyvoltage. Theresultsofthesemodel-freecontrolalgorithmsappliedtostalledairfoils demonstratestheeectivenessofsimplemethods.Bothexamplespanelizedcontroleort, therebyadoptingmoreecientcontrolmethodsforseparationcontrol.Theactuation typesincludedsyntheticjetsandDBDactuators. 2.3.3ControlofBlack-BoxModels Thissectionfocusesonblack-boxmodelsappliedtoairfoilseparationcontrol.Key resultsofthefollowingstudiesaresummarizedinTable2-10. King etal. 2004conductedactiveclosed-loopcontrolofseparationfroma NACA-4412apatRe c =2 : 9 10 5 and f =35 withrobust H 1 -controllers.Pulse jetswereplacedneartheleadingedgewithoptimalexcitationfrequency F + c =1,and12 dierentialpressuretransducerswereplacedonthesuctionsideoftheap.Twooutput variablesrelatedtothedegreeofseparationwereconsideredforthecontrolloop.Therst 68

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wasthelinearpressuregradient C p t = x=c computedbetweentwotransducersat about70%and10%ofthechordlength.Thesecondoutputvariablewasthechangeinlift C l t measuredbythestreamwisepressuredistributionontheuppersurelativetothe unforcedliftcontribution. H 1 -controllersweredesignedandimplementedforbothprocess variableswithresultsdemonstratingreasonablemeantrackingandresponsetimesonthe orderofaquartersecond,muchfasterthantraditionalmodel-freecontrol. TheexperimentalsetupofTian etal. 2006 a isappliedbyTian etal. 2006 b for adaptivesystemidenticationandrobustcontroltoreducethesuctionsidepressure uctuationsassociatedwithseparatedowfromaNACA-0025airfoilat12 angle ofattack.TheoverallcontrolschemeemployedsimultaneousARMARKOVsystem identicationanddisturbancerejectionalgorithmsAkers&Bernstein,1997.Thesystem identicationalgorithmrecursivelyidentiedalinearsystemmodeloftheunsteady pressureresponsefrominputactuatorvoltage,whilethecontrolleraimedtosuppress themeasuredpressureuctuations.Therecursivesystemidenticationwasshownto adequatelytrackthemeasuredunsteadypressure,evenwithlinearestimatesofthe nonlinearowdynamics.Theactiveclosed-loopcontroldecreasedthebroadbandpower ofthepressureuctuationsbyapproximately5dBandattachedtheseparatedow, asmeasuredbyPIVonthesuctionsideoweld.Theattachedowincreasedthe lift-to-dragratiobynearlyafactorof5,ascomparedtothenaturallyseparatedow. Ofthesetworobustcontrollersofblack-boxsystems,one'sobjectivewastoforce thelifttofollowacommandedreferencewhiletheother'sgoalwastousetheidentied modeltoreduceunsteadypressureuctuationsassociatedwithseparatedow.The formersynthesizedan H 1 -controllerfromopen-loopcontrolresults,whilethelatterused anonlineARMARKOVsystemidenticationanddisturbancerejectionalgorithm.An advantageofsynthesizingtheclosed-looparoundasimplemodelwasfasterresponse timesforimproveddynamiccontrol.Asisthecaseforblack-boxmodels,neitherofthese modelingapproachesaddressedthephysicalnatureoftheseparatedow. 69

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2.3.4ContolBasedofReduced-OrderStateEstimation Reduced-ordermodelinghasbeenhasnotbeentocasesofseparationcontrolfor streamlinedairfoils.However,stateestimationbasedonreduced-orderapproximationshas beenusedtodetectthepresenceofowseparation.Usingtheseapproximations,theow iscontrolledtoreducethepresenceand/ordegreeofseparation.Noteworthyndingsare summarizedinTable2-11. Glauser etal. 2004successfullydetectedandcontrolledvariousstagesofseparation onatwo-dimensionalliftingairfoilequippedwithZNMFactuators.ANACA-4412was testedunderturbulentconditionsataReynoldsnumberofRe c =1 : 7 10 5 andangles ofattackof =15 incipientlyseparatedand16 fullyseparated.Vibratingspeakers producedjetowoscillationsneartheleadingedgetodelayseparation.Pressurewas measureddownstreamoftheactuationbyelevendynamicpressuresensorsevenlyspaced alongthechordatmidspan.Thestatedetectionwasachievedbystochasticestimation, specicallymodiedlinearstochasticestimationmLSE,ofvelocityelddatafrom unsteadypressuremeasurements.TheestimationprocessisdiagrammedinFigure2-9. Inordertoestimatethestateinareduced-ordermanner,globalPODmodeswere computedfromthePIVdata. Global indicatesthatthemodesareextractedfromtheset ofallPIVmeasurementsatalltestconditionsanglesofattack.Thereby,thecondition anddegreeofseparationwasestimatedforvariousanglesofattack.Therstestimated globalPODcoecientmodulatedaxedsinusoidalsignalforspeakerexcitation.The amplitudeofthepre-modulatedsignalwasconstantanddeterminedfromopen-loop results,andthefrequencywasselectedforoptimumoperationoftheactuators.The time-seriesoftheestimatedPODcoecientwasalsolow-passlteredat100Hzinorder topreventunstablefeedback.For =15 incipientseparationandactivefeedback control,theamplitudeofthePODcoecientswasdecreased,indicatingadecreasein velocity/pressureuctuationsabovetheairfoil.Separationwasdelayeduptoandbeyond =17 70

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Thefollow-upworktothisinvestigationwasprovidedbyPinier etal. 2007,inwhich thesameessentialcontrolapproachwasappliedtothesameairfoilatRe c =1 : 35 10 5 but withchangestotheestimationtechnique.TheZNMFactuatorsweredrivenforoptimum outputat C =0 : 02 20%,unmodulated.InsteadofmLSE,thequadraticversionof modiedstochasticestimationwasappliedbecausetheinclusionofthequadraticterms yieldedmoreaccuratestateestimatesfromtheunsteadypressuremeasurementsNaguib etal. ,2001;Murray&Ukeiley,2003.Themodulationofthefeedbacksignalwasstill abletoreattachoratleastdelaytheseparatedowandthenmaintainattachedowwith decreasedlevels.Thiswasaccomplishedbeyond =18 .TheestimatedPODcoecients increasedinamplitudeinresponsetoowseparation,buttheirspectralcontentalso shiftedfromhigherfrequencies,intheattachedboundarylayer,tolowerfrequencies associatedwiththesensingoflargerscaleowstructuresduringincipientseparation. Theauthorsemphasizedthatthispracticaluseoflow-dimensionalplantestimatesfor classicairfoilseparationvalidatedfutureprogresstowardsmoresophisticatedandecient controllers. Albeitinasimplemanner,itwasshownthatalow-orderapproximationofthe dynamicoweldwassuitableforcontrollingboundarylayerseparationfromlifting airfoils.Thetestcasesdiscussedaboveallusedsomeformofstochasticestimationto estimatetheprimaryPODmodalcoecient.Thiscoecientwasusedtomodulatean actuationsignal,ultimatelydelayingseparationforanglesofattackabovethenatural thatincursseparation. 2.4UnresolvedTechnicalIssues Thischaptersummarizedstudiesspanningoverfourdecadesofclosed-loopseparation control.Forblu-bodyseparationcontrol,mostimplementedclosed-loopcontrollaws weresimplerbeforethe21 st century,atwhichtimeclosed-loopcontrolmethodsincreased inabundanceandsophistication.Thistransitionalperiodroughlyfollowedthechange infocusthatoccurredoverthelastdecadeortwo,inwhichsimplistic,oftenopen-loop 71

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methodsofseparationcontrolwereshiftedtomoreelegantclosed-loopmethodsfor ecientandadaptableactivecontroltechniques. Model-freetechniqueswereprovenvaluablefordeterminingtheoptimalcontrol parametersoradaptingtoslowlychangingoperatingconditions,butdonotprovidea meansperformreal-timeseparationcontrol.Thisshortcomingwasaddressedbysystem identicationmethodsthatwereabletospeeduptheresponsetimebasedonanopen-loop identiedmodel.Unfortunately,suchmodelsdidnotaddressthecomplexnatureofthe uidmechanics.Reduced-ordermodelswerethenextstepintheevolutionofowcontrol, usinglow-dimensionalapproximationsofhigh-dimensionalowstoimprovethemodel's abilitytoaddressdynamicowcomponentsandhowtheyaectthemeanowstructure. Althoughreduced-ordermodelsoerthemostphysicalinsightintotheseparationprocess, theyhavebeenlargelyrestrictedtosimulations. Asidefromthefewcasesofexperimentalowcontrolusingreduced-ordermodels, severalissuesstillexistthathavenotbeenaddressedwithdetailedandsystematic analyses.Forinstance,analysisofthecontrolled,globaloweldshasbeenlargely neglectedbymostclosed-loopmethodsofcontrol,onallplatforms.Experimentalresults are,forthemostpart,connedtosurfacepressureand/orglobalforcemeasurements. Furthermore,theinteractionoftheactuatedowandthenaturalinstabilitieshasnot beenadequatelyaddressedwithoweldmeasurements.Theseexamplesevidencethat theexistingliteraturetendstofocusonthecontrolaspectoftheproblemmorethan theuiddynamics,potentiallytothedetrimentofthecontroleciency.Thereisa lackofattentiontoorunderstandingoftheeectsofactuationontheowandhow thisknowledgecanbeleveragedformorecomplexowcontrolproblemswithmultiple interactinginstabilities,suchasboundarylayerseparationfromanairfoil. 72

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Figure2-1.Illustrationofthegeneralincreaseinowcomplexitythataccompaniesmore practicalapplications. Figure2-2.Sketchoftheplateau-likeeectoftheoutput C p forincreasing C .An exampleoftheoptimalslopeforslope-seekingisshownbythedashedline. Figure2-3.CongurationoftheD-shapedmodelwithtwoactuatorslotsandthreesensor arrays.AdaptedfromHenning&King2005. 73

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Figure2-4.Congurationofthebackward-facingstepwithfourloudspeakersandfour sensorarrays.Thestepheight H is20mm.AdaptedfromKing etal. 2004. Figure2-5.Congurationofthebackward-facingstepwithfourspanwiseseparated actuatorsandfoursensorarrays.AdaptedfromHenning&King2007. Figure2-6.Schematicoftheactuation-sensingstrategybyDahan etal. 2011. 74

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Figure2-7.Streamlinesofthebaselinesimulatedowaroundacircularcylinderblack circleatRe d =100.Avolumeforcecontainedwithinthegraycircleislocated 2 d downstreamandiseectiveinthe y direction.Asimulatedhot-wire anemometerisusedtomeasuretheowatapproximately6 : 5 d downstreamof and2 d abovethecylinder.AdaptedfromGerhard etal. 2003. Figure2-8.Experimentalsetupofatwo-dimensionalblubodyttedwith15dierential pressuresensorsandactuatorslotsforpulsedsuction.Straingaugesmeasure theglobalforce.AdaptedfromLuchtenburg etal. 2010andAleksic etal. 2010.

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Figure2-9.Flowchartofthestochasticestimationprocessformodulationoftheactuation signal. 77

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Table2-1.Summaryofgeometric,ow-specic,andcontrolparametersforthediscussed separationcontrolplatforms. PlatformtypeGeometricparametersFlowparametersControlparameters CylinderDiameter d ,span w St d = fd U 1 F + d = f a d U 1 Re d = dU 1 C = A j u 2 rms ;j wdU 2 1 BackwardfacingStepheight H ,span w ,St H = fH U 1 F + H = f a H U 1 steprecirculationbubble L R Re H = HU 1 C = A j u 2 rms ;j wHU 2 1 D-shapedmodelChord c ,thickness h ,St h = fh U 1 F + h = f a h U 1 span w Re h = hU 1 C = A j u 2 rms ;j whU 2 1 AirfoilChord c ,span w ,St c = fc U 1 F + c = f a c U 1 angleofattack Re c = cU 1 C = A j u 2 rms ;j wcU 2 1 Table2-2.Summaryofdirectfeedbackowcontrolstudiesforblu-bodyseparation. AuthorYear Model& Remarks conditions Berger1967Oblongcylinder, Re=80 Oblongcylindervibrationsfrom negativefeedbacksuppressedthewake Roussopoulos1993Circular cylinder, 1 : 1Re d; crit RefutedaboveclaimCompletevortex suppressionupto1 : 1Re d; crit with loudspeakerbutsensitivetohot-wire location Huang1996Circular cylinder, Re d =4 10 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 3 10 4 Movedacousticsourcelocallyto emanatefromslotinhollowcylinder, resultinginreducedvortexshedding athigherRe d 78

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Table2-3.Summaryofmodel-freeclosed-loopowcontrolstudiesforblu-body separation. AuthorYear Model& Remarks conditions Henning&King2005D-shaped, Re h =4 10 4 C =2 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 Extremum-seekingvariedphaseof actuationtoincreasebasepressureby 35%dragreductionof10% Pastoor etal. 2008D-shaped, Re h = f 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(7 g 10 4 C =9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 Slope-seekingachieved40%pressure increase%dragreduction Table2-4.Summaryofrobustcontrolmethodsofblack-boxmodelsforblu-body separation. AuthorYear Model& Remarks conditions King etal. 2004BFS, Re H =4 10 3 2 : 5 10 4 Set-pointtrackingoftherecirculation lengthvialinearblack-boxmodels andacousticactuationatthe detachingboundarylayer Henning&King2007BFS,Re H = f 1 : 7 )]TJ/F15 11.9552 Tf 11.595 0 Td [(2 : 5 g 10 4 Set-pointtrackingofspanwisevarying reattachementlengthviarobust H 1 -controller Dahan etal. 2011BFS, Re H =2 : 9 10 4 C =1 : 5 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 DLESRobustcontrollerwas synthesizedforsecond-ordermodel tthesystemplantandincreased basepressureby20% Henning&King2005D-shaped, Re h = f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 g 10 4 Set-pointtrackingof C p by identicationoflinearblack-box modelsbetweeninput C andoutput C p androbustcontrollerdesigned usingquantitativefeedbacktheory Henning etal. 2007D-shaped, Re h = f 3 : 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 g 10 4 Robust H 1 -controllerisimplemented andreduceddragbyapproximately 15% 79

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Table2-5.SummaryofowcontrolstudiesusingGalerkinreduced-ordermodelsfor blu-bodyseparationandcavities. AuthorYear Model& Remarks conditions Gerhard etal. 2003Circular cylinder, Re d =100 SimulationGalerkinmodelfromtwo PODmodesreducednaturalow domainenergyby45%reduction whenthePODmodesareobtained fromthecontrolledow King etal. 2005Circular cylinder, Re d =100 SimulationGalerkinmodelfromtwo PODmodesusedwithaphysically motivatedcontrollerandseveralother formalcontrollawsdampingcontrol, input-outputlinearization,Lyapunov basedcontrol,andoppositioncontrol obtainedenergyreductionsbetween 30%and33% King etal. 2008Circular cylinder, Re d =100 SimulationNonlinearMPCbasedon GalerkinmodeloftwoPODmodes andshiftmodeoutperforms energy-basedcontrol Luchtenburg etal. 2010D-shaped, Re h = f 2 : 69 )]TJ/F15 11.9552 Tf -64.387 -14.446 Td [(3 : 77 g 10 4 Usinglow-ordermean-eldGalerkin modelandslidingmodecontrol, pulsedsuctionatbothtrailingedges increasedbasepressure36% decreaseddrag15% Aleksic etal. 2010D-shaped, Re h = f 2 : 69 )]TJ/F15 11.9552 Tf -64.387 -14.446 Td [(3 : 77 g 10 4 Similarresultsaredemonstratedfor slidingmodecontrolandlinearMPC Rowley& Juttijudata 2005Cavity, M 1 =0 : 6, L=D =2 DNSEmpiricalGalerkinmodelwith LQGreducesoscillationsbutis limitedbythescopeoftheidentied model.Controllerdesignedfor dynamicphasormodelofoscillatory dynamicssuppressescavity oscillations. CattafestaIII etal. 1997Cavity, M 1 < 0 : 2, L=D = f 0 : 5 ; 2 : 0 g UsedLQGandpoleplacementtodrive apiezoelectricaparrayatcavity leadingedge.Suppressedoscillations withorder-of-magnitudelesspower thanopenloop. Samimy etal. 2007Cavity, M 1 =0 : 3, Re =2 10 4 L=D =4 Modelderivedfromvelocity-eldPOD modesandGalerkinprojection. Linearquadraticoptimalcontroller signicantlyreducespowerinresonant tone. 80

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Table2-6.Summaryofowcontrolstudiesusingvortexmodelsforblu-bodyseparation. AuthorYear Model& Remarks conditions Li&Aubry2003Circularcylinder, Re d =100,200 SimulationControlderivedfromthe analysisofFoppl'spotentialow modelforpointvorticesreducedwake inducedliftoscillationsby97% Re d =100and95%Re d =200 Protas2004Circularcylinder, Re d =75 SimulationLQGcompensator designedtostabilizeFopplsystem throughcylinderrotation Pastoor etal. 2008D-shaped, Re h =4 : 6 10 4 C =3 : 8 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 Vortexmodeldevelopedtomodelthe transientdevelopmentofthewake beginningfromthepotentialow solution,whichmotivatedforcing in-phasesuchthatthesymmetric upperandlowershearlayerswouldbe energizedandthesymmetry maintained,reducingthedragby15% Table2-7.SeparationbubblelocationsandsizesforcasesofZNMFforcingofthebaseline canonicalseparatedowatRe c =1 10 5 and C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 Kotapati etal. 2010. C isrecalculatedtomatchthedenitionofEq.2{9. Case F + c F + L sep Separationpoint,Separationlength,Bubbleheight, x sep =c % L sep =c % H sep =c % 0--63.028.62.35 10.950.2767.923.01.07 21.270.3667.922.60.91 31 : 900.5468.219.90.78 43 : 801.0968.815.40.55 57 : 602.1769.014.90.53 611.43.2664.228.92.44 715.24.3563.728.22.34 Table2-8.DownhillsimplexoptimizationresultsforbothAMandBMsignalswithand withoutenergypenaltyforcasesofZNMFforcingofbaselineseparationfroma NACA0025airfoilatRe c =10 5 and20 angleofattackTian etal. ,2006 a Case f m f c V pp ZL=DC %chord V rms I rms =DU 1 AM W =0781040300.392.561 : 72 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 7 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 AM W =10055124560.432.385 : 77 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(6 1 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 BM W =0801250300.422.381 : 17 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(5 7 : 8 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 BM W =100931286100.452.272 : 96 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(6 1 : 0 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 81

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Table2-9.Summaryofadaptiveandmodel-freeclosed-loopowcontrolstudiesfor boundarylayerseparation. AuthorYear Model& Remarks conditions Tian etal. 2006 a NACA-0025, Re c =1 10 5 =20 C =5 : 77 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(6 Parametersofamplitudemodulated signalsoptimizedwithdownhill simplex,whichachieved87%increase inL/DwithZNMFactuators. Becker etal. 2007NACA-4415ap, Re c =5 10 5 MIMOcontroleectivelydemonstrated andshownpracticalfor3Dwing applications. Benard etal. 2009NACA-0015, Re c = f 1 : 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 g 10 5 =15 Slope-seekingmaximizedliftwith minimuminput. Table2-10.Summaryofrobustcontrolmethodsofblack-boxmodelsforboundarylayer separation. AuthorYear Model& Remarks conditions King etal. 2004NACA-4412, Re c = 2 : 9 10 5 f =35 Reasonablemeantrackingofbothlift coecientandpressuredierentialby designof H 1 -controller. Tian etal. 2006 b NACA-0025, Re c =5 10 5 =12 OnlineARMARKOVsystem identicationanddisturbance rejectionalgorithm,controldecreased thebroadbandpressureuctuations by5dBandattachedtheseparated ow,increasingL/Dbyafactorof5. Table2-11.Summaryofowcontrolstudiesusingreduced-orderstateestimationfor blu-bodyseparation. AuthorYear Model& Remarks conditions Glauser etal. 2004NACA-4412, Re c = 1 : 7 10 5 =15 ; 16 mLSEusedforestimationofrstPOD coecient,whichmodulateda harmonicsignalanddelayed separationuptoandbeyond17 Pinier etal. 2007NACA-4412, Re c =5 10 5 C =0 : 02 Sameasabove,butreattachedbeyond 18 Hall etal. 2008NACA-4412Multi-time-LSEyieldsimproved estimateofthestate. 82

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CHAPTER3 NUMERICALMETHODS Thissectiondescribesthenumericalmethodsformodaldecompositionalgorithms andreduced-orderestimationproceduresforglobal,time-resolvedvelocity.Themodal analysistechniquesincludeproperorthogonaldecompositionPODanddynamicmode decompositionDMD.Thesemethodsaredetailed,includinganinterpretationand comparisonoftheirresultingdecompositions.Theestimationprocessesinvolvemodal decomposition,stochasticestimation,andmodel-basedestimation. 3.1ModalDecomposition Modaldecompositiontechniquesareusefulforanalysisofhigh-dimensionaluids data.Thesemethodsextractasetofmodesfromthedatathatarerepresentativeofthe characteristicfeaturesoftheow.Ideally,modalanalysiscanhelptoidentifyowfeatures thathighlightunderlyingphysicsthatisotherwisediculttodiscernfromtherawdata. However,themeaningoftheresultsisdependentonthetypeofdecompositionapplied. Inthissection,thePODandDMDalgorithmsaredescribed.Theyaregenerally,but notstrictly,appliedtouids-baseddatainordertoextractmodesbasedonenergyand frequencycontent. 3.1.1ProperOrthogonalDecomposition POD,alsocommonlyreferredtoasprincipalcomponentanalysisorthe Karhunen-Loeveexpansion,isastatisticalmethodthatidentiesstructures,ormodes, thatbestreconstructadatasetLumley,1967;Sirovich,1987;Holmes etal. ,1996. Lumley1967iscreditedwithitsintroductionintotheuidscommunitybyitsuseto extractcoherentstructuresfromturbulentows.Intheyearssince,PODhasprobably becomethemostcommonlyusedmodaldecompositionmethodappliedtouids-based data. PODisgenerallyappliedtovisualizethedistributionofthecoherentstructures withinaowortoobtainareducedsetofmodesthatbestdescribesthefull-dimensional 83

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dataset.Withreferencemorespecicallytothelattercase,adataset f n g N n =1 canbe projectedontoan r -dimensionalsubspaceobtainedfromPOD,where N isthesizeof thedatasetand n isasinglesnapshot"orrealizationofthedata.Withtheprojection operator P r ,PODprovidesthereducedorthogonalbasisthatminimizesthesum-squared projectionerrorRowley,2005 N X n =1 n )]TJ/F34 11.9552 Tf 11.955 0 Td [(P r n 2 2 ; {1 whichisequivalenttomaximizingthe L 2 innerproductontheprojectionsubspace N X n =1 P r n 2 : {2 Thismeansthatthe r PODmodescontributetothemostaccurateprojection,forall possiblevaluesof r .Thisisofteninterpretedasthemostenergetic"projection,where thetermenergyisusedtorepresentthe L 2 innerproduct.Therefore,thePODmodesare arrangedindescendingorderbasedonenergycontent,wheretherstorsmallestindex correspondstothehighestenergymode. Whenappliedtoincompressibleuidow,thedataelementsaregenerallytakentobe mean-subtractedvelocityelds,or n =u 0 n =u 0 t n ,whereu 0 encompassestheavailable velocitycomponentsi.e., u v ,and w forallprobelocations f x m g M m =1 attime t n ,andthe primenotationdenotesmean-subtractedvalues.Thetotalnumberofprobemeasurements withinasinglevelocityeld,commonlyreferredtoasasnapshot,"isdenotedby M Eachdiscretevelocityeldisarrangedintoacolumnvectorandstackedinadatamatrix X = 2 6 6 6 6 4 u 0 1 u 0 2 u 0 N 3 7 7 7 7 5 | {z } M N : Intheeventthatthenumberofprobemeasurements M isgreaterthanthenumberof snapshots N ,thenthePODmodesarebestcomputedusingthemethodofsnapshots 84

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becauseofthecomputationsavingsassociatedwithsolvingthe N N eigenvalueproblem ratherthanthe M M eigenvalueproblemSirovich,1987.ThisoftenthecaseforPIV data,where M isthenumberofgridpointstimesthenumberofvelocitycomponents. ThesingularvaluedecompositionSVDofthematrix X T WX yields X T WX = Z 2 Z T ; where W isamatrixofinnerproductweights.Fordiscretedata, W isoftenspecied asthescaledidentifymatrix Idxdydz sothatthevectornormisinterpretedasthe integratedkineticenergy,or k u 0 n k 2 =u 0 n T W u 0 n = ZZZ u 0 x;y;t n 2 + v 0 x;y;t n 2 + w 0 x;y;t n 2 dxdydz: ThePODmodes j arecomputedasthecolumnsofthematrix = XZ )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; andtheprojectionofamean-subtractedvelocityeldontothePODbasisyieldsitsPOD coecients a n = T u 0 n : Theorthonormalityofthemodesrequirestheidentity T i W j = ij {3 besatised,where ij istheKroneckerdeltafunction. BasedontheminimizationofEq.3{1,theoptimalprojectionoperatorofasnapshot u 0 n ontotherst r PODmodesisgivenby P r = r T r ; where r containsonlytherst r columnsmodesof.Thereduced-orderprojectionof snapshotu 0 t n ontotherst r PODmodesisthengivenbythepairingoftherst r POD 85

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modesandthecorrespondingvectorofPODcoecients a n = T r u 0 n ,or P r u 0 t n = r a t n = r X j =1 a j t n j : Theprojectionisthereforerepresentedbysummationofaseparableform ~u 0 x m ;t n = r X j =1 a j t n j x m ; inwhichthevaluesofthetime-dependentcoecientsdescribetheevolutionofthe spatially-dependentPODmodes. PODisanattractivetoolforbasisreductionbecauseitprovidestheoptimal projectionofthehighdimensionaldataontoalow-dimensionalsubspacespannedby themodes.Thatis,theaverageenergyintheprojectionofthevelocitydataontothe subspacespannedbythemodesismaximized.DuetotheorthogonalityofthePOD modesEq.3{3,theenergyinasinglereduced-ordermodalreconstructionisgivenby k P r u 0 n k 2 2 = a T n T r W r a n = a T n a n = k a n k 2 2 : Theenergyisthuscompletelycharacterizedbythetime-varyingmodalcoecients.Using Eq.3{2,thekineticenergycapturedbytheentireprojectioniscomputedas N X n =1 k P r u 0 n k 2 = r X j =1 j ; wherethesingularvalues j arethediagonalelementsof. ThisdecompositionisusefulforaccurateestimationsofexperimentalPIVvelocity measurementsbecausethehighdimensionalsystemcanbereducedtoamuchsmallerset ofbasisfunctions,representedby r PODmodes.Thesexedspatialmodes,whenpaired withestimatedtime-varyingPODcoecients,formalow-orderestimate.Therefore, time-resolved low-orderestimationofavelocityeldisachievedwiththeextractionof PODmodesavailablefromnon-time-resolvedmeasurementsandtheestimationof time-resolvedPODcoecients.Theselectionofmodesandmodequantitytoincludeisa 86

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decisionleftforeachapplication.Thisistypicallydictatedbytheenergydistributionor bymanualinspectionofthespatialmodalstructure. Otherdecompositionmethodsmaybemoresuitableforowcontrolapplications thanPOD.ThehighenergymodesidentiedbyPODmaynotcapturetheinput-output behaviorofthesystemIlak&Rowley,2008.Acontrolobjectivemaybebetterserved bymethodsthataddresscontrollabilityandobservability,suchasbalancedPODor theeigensystemrealizationalgorithmRowley,2005;Ma etal. ,2011.Additional extensionsofPODexistincludedoublePODSiegel etal. ,2008;Tubino&Solari,2005 andtemporalPODGordeyev&Thomas,2013,butneitherisaddressedhere. 3.1.2DynamicModeDecomposition DynamicmodedecompositionDMDisasnapshot-basedmethodthatisuseful foridentifyingmodalstructuresbasedonfrequencycontentSchmid,2010.Thisis distinguishedfromPOD,inwhichthereisnostricttemporalinterpretationofthe decomposition'seigenvaluesoreigenvectors.DMDcanbeinterpretedasameansto performKoopmanspectralanalysis,whichwasintroducedbyKoopmanin1931Schmid, 2010;Tu,2013.Rowley etal. 2009appliedtheKoopmantheorytouidmechanics becauseofitsabilitytocharacterizethedynamicsofasystem,includingnonlinear dynamicalsystems.Schmid2010introducedDMDasalinearstabilityanalysisfor thedetectionoftemporalmodes,includingattentionontheconnectionbetweenDMD andPODthatresultsincomputationofdynamicmodesusingSVD.Undertheright conditions 1 ,DMDisinterpretedasanumericalalgorithmtocalculateanapproximation oftheKoopmandecomposition,andthereforeappropriatelyextenditscapabilitytothe analysisofnonlinearsystemsTu,2013. 1 Snapshotmatrix X musthavelinearlyindependentcolumnsTu,2013. 87

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Asrstintroduced,DMDrequirestheworkingdataset X toadheretoasequential time-seriesofsnapshots,suchthat X = 2 6 6 6 6 4 u 0 u 1 u m )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 3 7 7 7 7 5 : Tu2013hascreatedamoregeneralderivationthatprovidesmoreinsightandrelaxes thisrequirement.However,themorerestrictivesequentialtime-seriesderivationis providedduetoitsapplicationtothedatawithinthiswork.Forthebriefderivationthat follows,thesnapshotsfromwithinthedatamatrix X mustbeacquiredsequentiallyat axedsamplingratethatsatisestheNyquistcriterion,basedonthehighestfrequency measured. Thestateevolutionofsomedynamicsystemisassumedtotaketheform x k +1 = f x k ,where x k isthesetofstatevariablesandindex k representsadiscretetime.Then, g x ismadetorepresentsomemeasurementofthestatee.g.,asetofvelocityprobes, suchthattheevolutionof g x isapproximatedbythemappingfunction B ,yielding g x k +1 = Bg x k .Withasetofvelocityeldmeasurements f u k g m k =0 ,whereu k =u t k thesubsequentoweldu k +1 isassumedtoberelatedtothecurrenteldbytherelation u k +1 = B u k : {4 Fornonlinearprocesses,theassumptionofalinearoperator B resultsinalineartangent approximationfromonetime-steptothenext.Atonesampleintervalahead,datamatrix X isrepresentedby X 0 X 0 = 2 6 6 6 6 4 u 1 u 1 u m 3 7 7 7 7 5 : 88

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Aslongastheassumedmappingremainsconstantfortheentiresamplinginterval, Eq.3{4canbeextendedtothemeasurementswithinthesequence,suchthat X 0 = BX: {5 Withknowledgeof X and X 0 ,thedynamicnatureoftheowisobtainedfromthe eigenvectorsandeigenvaluesoftheoperator B Schmid,2010. TherststepintheDMDalgorithmistocomputetheSVDofmatrix X as X = U Z T : {6 Thisisaccomplishedwiththemethodofsnapshots,suchthattheSVDisactually computedas X T WX = Z 2 Z T ; where W isamatrixofgridweightsSection3.1.1forfurtherexplanation.Theleft singularvectorsarethencomputedas U = XZ )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; notingtheirequivalencetothePODmodesof X .WiththecomponentsofEq.3{6 known,substitutionintoEq.3{5andrearrangingyields U T BU = U T WX 0 Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 : {7 Intheequationabove, U T BU isthematrixmultiplicationofthePODmodeswithin U byallofthesamemodesshiftedbyonetimestepintothefuture,representedby BU Therefore, U T WX 0 Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 containstemporalinformationrelatedtotheevolutionofthe PODmodes. TheeigenvalueproblemofEq.3{7is U T WX 0 Z )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 V = V ; 89

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wherethecolumnsof V anddiagonalelementsofaretheeigenvectorsandeigenvaluesof U T WX 0 Z )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ,respectively.TheDMDmode j isgivenbythe j th columnofthematrix = UV; scaledsuchthat m )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 X j =0 j =u 0 : TheDMDmodesarethuslinearcombinationsofthePODmodesandrelatedtothe originalsequenceofsnapshotsbytheequations u k = m )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 X j =0 k j j k =0 ;:::;m )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 and u m = m )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 X j =0 m j j + ? span f j g m )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 j =0 ; inwhich j isthe j th eigenvalueanddiagonalelementof.Thesequenceofeigenvalues correspondtothe j th DMDmode,wherethegrowthrateisgivenby k j k andthe frequencyofoscillationisgivenbyarg j Whenappliedtosetsofvelocityelds,thetemporalinformationcontainedwithin theDMDeigenvalueslendsitselftotheinterpretationthatDMDidentiesmodesbased onfrequencycontent.DMDanalysisofuidvelocitydatayieldsspatialstructureswhose dynamicsaredescribedbythecorrespondingeigenvalues.Thedecompositiontherefore resultsinasetofDMDmodes,eachwithacharacteristicfrequencyandgrowth/decay rateofsomeoscillatorydynamic.Forowswithoscillatorydynamics,DMDcanidentify thespatialstructuresthatcorrespondtothecharacteristicfrequenciesoftheow.The extractedDMDmodescanprovidephysicalinsightintotheunderlyingdynamicsofthe owandaprojectionforareduceddynamicalsystem.Becauseoftheseattributes,DMD 90

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isespeciallyrelevantandusefultotheidenticationandmodelingofthecharacteristic dynamicsofthecanonicalowproblem. 3.2Reduced-OrderEstimation Highlyresolved,inbothspaceandtime,velocityeldsareusefulindeterminingthe globalowstructuresandunderlyingphysicsassociatedwithaow.Forexample,POD canbeappliedtoidentifyandvisualizeowstructureswithhigh-energycontentLumley, 1967.However,thisdecompositiondoesnotprovide,norisitintendedto,dynamical informationassociatedwiththemodalcontent.First,thedatamustresolvethe temporalscalesofinterestinordertocapturethecorrespondingdynamicbehaviorsof theow.Suchmeasurementsarereferredto,atleastinthisdocument,as time-resolved measurementsofthefulloweld,whichisaconcisestatementthatthesampling frequencyexceedstwicethatofthehighestowfrequencymeasured.Iftime-resolved velocityeldsareavailable,FourieranalysisandDMDcanbeappliedtoidentifyow structuresofdynamicalimportanceRowley etal. ,2009;Schmid,2010.Theextracted modescanhelphighlightthefundamentalorcharacteristicowcontent,whichisuseful forconstructingempirically-derivedreduced-ordermodelsforclosed-loopowcontrol. Unfortunately,time-resolvedvelocityarecurrentlydiculttoobtain. ParticleimagevelocimetryPIVisthestandardtechniqueformeasuringvelocity elds,buttime-resolvedPIVTRPIVsystemsarecostlyandthusuncommon.In addition,suchsystemsareoftenrestrictedtolow-speedowsduetothelargertime intervalneededbetweensnapshotswhenusingahigh-speedlaser.Assuch,typical PIVsystemsarenottime-resolvedandasaresultareoftenincapableofresolvingthe characteristicfrequenciesofaow. Ontheotherhand,manyinstrumentsareavailablefortime-resolvedpoint" measurements,includinghot-wireprobes,unsteadypressuretransducers,and,more recently,shearstresssensorsChandrasekharan etal. ,2011;Meloy etal. ,2011.Dense arraysofsuchsmall-scalesensorscancoverlargespatialregions,butthedatamaynot 91

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resolvedthespatialscalesofinterestandtheinstrumentationmayintrudeuponthe naturalbehavioroftheow.Eachsensorislimitedbythespatialaveragingonthescale ofthesensor'slargestdimension,butthisisalsoafactorfortheimagingsensorand individualinterrogationregionsrequiredforPIVmeasurements.Pointmeasurementsare alsosensitivetoplacement,whichcanbepredeterminedand/ordiculttoadjust. Inthissection,methodsaredescribedforintegratingtime-resolvedpoint measurementsandnon-time-resolvedPIVtoestimatethetimeevolutionoflow-order velocityelds.Thereby,thehighspatialandtemporalresolutionsfromeachmeasurement areutilizedtoprovideestimatesoftheowstatewithbothcharacteristics.PODis usedtoobtainalow-orderdescriptionoftheowandlimittheestimatedowcontent todominantcoherentstructuresthataremorelikelytocorrelatebetweenthetwo measurements.Thetwoestimationtechniquesusedtoestimatethetime-varyingPOD coecientsfrompointsensormeasurementsaredescribed.Therstisavariantoflinear stochasticestimationLSE,anapproximationofaconditionalaveragethatutilizes measurementstatistics.ThesecondisaKalmanlterandsmootherthatincorporates amodelderivedfromthedynamicalanalysisoftheinitialstochasticestimates.This estimator,orstateobserver,isabletobalancetheaccuraciesandinaccuraciesofthe modelandthemeasurements,whichcanalleviatesomeofthedependenceplacedonthe sensors.Thisisimportanttotheclosed-loopcontrolmethodsappliedinthisdocument, wherethenumberoffeedbacksensorsforstateestimationislimited. 3.2.1StochasticEstimation Inthiswork,stochasticestimationutilizesthecorrelationofunsteadysurfacepressure withlow-ordervelocityelddata,andappliestheidentiedstochasticrelationship toestimatethetime-resolvedstateoftheoweldfromdynamicsurfacepressure measurements. Stochasticestimationisanempiricalestimationtechniqueforaconditionalaverage usingunconditionalstatistics.Adrian&Moin1988arecreditedfortheapplicationof 92

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stochasticestimationtouidsmeasurements,inwhichconditionaleddiesofturbulentow areapproximatedfromlocalkinematics.Astochasticestimateoftheconditionalaverage takestheformofaTaylorseriesexpansion ^ a i = h a i j p j i A ij p j + B ijk p j p k + :::; {8 where a i isthe i th PODcoecient, p j isthe j th probemeasurement, hi denotesthe expectedvalue,and^ a i istheestimateoftheconditionalaverage.Thecoecients A ij B ijk ,andsoonaredeterminedbyminimizingthemeansquareerroroftheestimate ^ a i )]TJ/F22 11.9552 Tf 11.955 0 Td [(a i 2 ; whichreducestosolvingasetoflinearequationsGuezennec,1989.Theorderofthe estimateisdeterminedbythenumberoftermsthatremainaftertruncatinghigherorder terms. Stochasticestimationofthetime-varyingPODcoecientsisusuallymuchmore computationallyecientthanestimationoftheentirevelocityeldbecausethe numberofestimatedPODcoecientsisoftenmuchlessthanthenumberofprobe measurementswithintheeld.Therefore,PODisacomplementarytechniqueto stochasticestimationBonnet etal. ,1994.WhenthePODcoecientstaketherole oftheconditionalvariables,giveninEq.3{8,thecomputationisreferredtoas modied stochasticestimationMurray&Ukeiley,2007 b .Thismethodisevenmore computationallyecientforalownumberofPODmodesandcoecientsthatcan capturethedominantbehavior,aslongasthereducedsetofPODcoecientscorrelate wellwiththeunconditionalmeasurements.Modiedstochasticestimationhasbeen appliedbyBonnet etal. 1994andTaylor&Glauser2004forlinearestimates,Naguib etal. 2001andMurray&Ukeiley2007 b forquadraticstochasticestimation,and Durgesh&Naughton2010forlinearestimateswithtime-delays. 93

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3.2.1.1Single-time-delaylinearstochasticestimation WithmodiedlinearstochasticestimationmLSE,onlythelineartermoftheseries inEq.3{8isretained.Thelinearestimateisgivenas ^ a i t = A ij p j t )]TJ/F22 11.9552 Tf 11.955 0 Td [( ; {9 whereaconstanttimedelay canaccountforpotentialleadorlagbetweenthe conditionalandunconditionalvariables,thusincreasingthecorrelationsbetween a t and p t Guezennec,1989;Cole etal. ,1992.Thecoecients A ij aredeterminedby minimizationofthemean-squareerroroftheestimates,whichsimpliesintosolvingthe matrixequation A T =[ PP ] )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 [ aP ] ; where A T = 2 6 6 6 6 6 6 6 4 A 1 ;i A 2 ;i . A N p ;i 3 7 7 7 7 7 7 7 5 ; [ PP ]= 2 6 6 6 6 6 6 6 4 p 1 p 1 p 1 p 2 p 1 p N p p 2 p 1 p 2 p 2 p 2 p N p . . . . . . p N p p 1 p N p p 2 p N p p N p 3 7 7 7 7 7 7 7 5 ; and[ aP ]= 2 6 6 6 6 6 6 6 4 a i p 1 a i p 2 . a i p N p 3 7 7 7 7 7 7 7 5 : Inthisequation, N p isthenumberofprobemeasurementsandthetimedependenceis neglectedforsimplicity. 3.2.1.2Multi-time-delaylinearstochasticestimation Thesingle-time"formofmLSEinEq.3{9canincreasethecorrelationbetween particularpairsoftheunconditionalandconditionalvariablesbutmaynotforallpairings. Multipledelayscanbeaccountedforbysummationacrossseveralvaluesof Ukeiley etal. ,2008;Durgesh&Naughton,2010: ^ a i t = A ijk p j t )]TJ/F22 11.9552 Tf 11.955 0 Td [( k : 94

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Correlationvaluescanimprovebyincorporatingmultipledelaysbecausethedelaybetween unconditionalandconditionaldatamaynotbeuniformandthedependenceonmultiple measurementseectivelylterstheeectsofnoise.Multi-time-delaymLSEMTD-mLSE hasbeenproposedforpurelynegativetimedelays,withtheestimatesbasedonlyonpast eventsUkeiley etal. ,2008,aswellasforleadingdelaysthatestimatesomeeventusing futuremeasurementsDurgesh&Naughton,2010. Thelattermethodisappliedtoestimatetheinitialtime-dependentPODcoecients a k ,andishereafterreferredtoasMTD-mLSEunlessotherwisedistinguishedasthepurely negativedelayversion. 2 Thepotentialbenetofimprovedcorrelationsfromtwo-sided MTD-mLSEcomeswiththecostofanon-causalestimator,oranestimatorthatdepends onfutureinputs.Anon-causalversioncannotbeusedforreal-timeestimationorow controlapplicationsbutisusefulwhenappliedasapost-processingtechnique.However, causalestimationcanalsobeeective.ForaderivationoftheMTD-mLSEalgorithm,the readerisreferredtotheworkofDurgesh&Naughton2010. Stochasticestimationprovidesanempiricalrelationshipbetweensomemeasurement andanother.Determiningthemappingfunctions,suchasthecoecients A ij and A ijk in thecaseoflinearestimation,requireslargedatasetsformoreaccuratecorrelations,less eectedbysensornoise.Duringestimation,however,thesefunctionsresponddirectlyto probemeasurements,whichmakesthemsensitivetonoise.Themulti-time-delayversions mayincreaserobustnesstosensornoise. 2 Whenusedtorepresentdiscretetime,theindices k and n arereservedfor time-resolvedandnon-time-resolveddata,respectively.Assumingsynchronized measurements,thenon-time-resolvedindex n isanintegermultipleof k : n = bk for k =1 ; 2 ;::: and b istheintegervaluedeterminedbytheratioofthetime-resolvedand non-time-resolvedsamplingrates. 95

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3.2.2Model-BasedEstimation Amodel-basedestimatorrequiresamodelforthetime-evolutionofthesystem states.Thisisparticularlyusefulincontrolapplicationswherethestateisunmeasurable. Rather,amodelofthestatesisobtained,andanobserverisformedthatincorporates measurementupdatestocorrectthemodel-predictedstates.Model-basedestimatescan thereforebemoreresilianttomeasurementnoisebyrelyingonsomemodeledbehaviorof thestatedynamics.Observersarecommoninclosed-loopowcontrolapplicationsbecause real-time,high-frequencymeasurementsareoftenrestrictedtosurface-basedsensors thatmaynotrepresentthesystemstatewell.Atpresentdate,high-delitynumerical discretizationoftheNavier-Stokesequationsisnotfeasibleformodelingbecauseofthe computationalexpense.Instead,empiricallyderivedmodelsaredesiredreduced-order approximationsthatdescribethepertinentowdynamics.First,thegeneralmodel identicationmethodappliedinthisworkisdescribed.Thisisfollowedbyadescriptionof thelinearKalmanltercausalandKalmansmoothernon-causal. 3.2.2.1Modelidentication Aowsystemmodelcanbeobtainedbymanymethods,suchasreduced-order projection,Galerkinprojectionaboutsomeexpansionmodes,vortexmodeling,oreven heuristicapproaches.AGalerkinprojectionispreferableforobtainingadynamicmodel becauseofitsphysics-basedrootsintheNavier-Stokesequations.However,asuitable modelforcontrolisdiculttoacquirewithexperimentaldatabecausethestandard Galerkinmodeldoesnotexplicitlyincludethecontrolinput.Inthiswork,low-order modelsareidentiedfromthedynamicanalysisoftime-resolvedstochasticestimatesof low-orderPODcoecients,whichareempiricallyderived. First,thePODmodesarecalculatedfromalargeensembleofstatistically independentPIVsnapshots.Therst r modesareretainedbasedontheenergycontent capturedbythemodesandtheabilitytoestimatethemodesfromunsteadypressure 96

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measurements.ThenumberofselectedPODmodesspeciestheorderoftheinitial estimatesandconsequentlylimitstheorderofthesubsequentlyidentiedmodel. Thenon-time-resolvedPIVmeasurementsareacquiredsynchronouslywith time-resolvedunsteadypressuremeasurements.Thevelocityeldsareprojected ontothe r PODmodestodeterminethecorrespondingsetof non-time-resolved POD coecients a n .AspreviouslydescribedinSection3.2.1,thesecoecientsareusedalong withtheprobemeasurementstodetermineaMTD-mLSEstaticmap.Themapthen transformsthetime-resolvedprobesignal f p k g into time-resolvedestimates ofthePOD coecients f ^ a k g Thestandardlinearequation ^ a k = F ^ a k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 {10 relatesthecurrentstateestimateattime t k tothepreviousoneattime t k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 bythe transferfunction F .NoticethesimilaritybetweenEq.3{10andEq.3{4,wherethe latteristheequationbasedontheKoopman/DMDoperator.Thus,thedynamicswithin thestatetransitionfunction F arewellapproximatedbyDMDofthetime-resolvedPOD estimates. 3 Giventhexedfrequencyandgrowth/decayrateDMDmodes,theestimatesforthe reduced-orderPODcoecientsareprojectedontoaselectfewofthepersistent k j k 1 andenergeticDMDmodes,representedby i ,toobtainanewsetofstatevariables b k thatrepresenttheDMDstatecoecients.Forclarity,thesearereferredtoastheDMD coecientsortheoscillatorystatesbecausethegrowth/decayratesareapproximately zeroforthisstudy.ThenumberofstatesisnowreducedtotwotimesthenumberofDMD modepairsselected.Thecomplex-valuedDMDmodesforoscillatorydynamicsappearin complexconjugatepairs.However,ifthetransformationmatrix n isconstructedsuch 3 WhenpairedwiththePODmodes,theseDMDmodescontainfull-elddynamical structuresforthelow-ordervelocityeldestimates. 97

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thattherealandimaginarycomponentsofthe n selectedoscillatory 4 modesarearranged inseparatecolumns,like n = Re n 1 o = n 1 o Re n 2 o = n 2 o Re n n o = n n o ; {11 thentheprojectionof a k onto b k isgivenby b k = + n a k ; {12 where + n representstheMoore-Penrosepseudoinverseofmatrix n iftherankof n is lessthan r ,andtheprojectionistheleast-squaressolution.Otherwise, + n isthestandard matrixinverseifallDMDmodesareretained. Thediscrete-timeevolutionoftheoscillatorystateequationisgivenby b k = F d b k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; {13 where F d = e F t isthematrixexponentialofthecontinuous-timematrix F multipliedby thediscretesamplinginterval t = t k )]TJ/F22 11.9552 Tf 12.085 0 Td [(t k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 .TheformofgiveninEq.3{11ischosen suchthatmatrix F isablock-diagonalmatrix F = 2 6 6 6 6 6 6 6 4 F 1 0 0 0 F 2 0 . . . . . . 00 F n 3 7 7 7 7 7 7 7 5 {14 formedfromsquarematrices f F i g n i =1 .Eachmatrix F i containsthegrowth/decayrate i andoscillationfrequency i forthe i th DMDmodepair,or F i = 2 6 4 i i )]TJ/F22 11.9552 Tf 9.299 0 Td [(! i i 3 7 5 : {15 4 AssumesallretainedDMDmodesarefromcomplex-conjugatepairs. 98

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Fortheestimationmodels,allgrowthratesaresettozeroandtheoscillationfrequencies areextractedfromtheDMDeigenvalues.Finally,thehomogenousequationforstate estimationisobtainedfromEq.3{13andtheinclusionofEqs.3{14and3{15. 3.2.2.2Kalmanlter Themodelobtainedbytheprocedurefromtheprecedingsectionisincorporatedinto thedynamicsystem b k = F d b k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + d k k = Hb k + n k ; where b isavectorofprojectedDMDcoecients, issomemeasuredquantity, d isprocessnoise,and n issensornoise.Matrix H ischosensuchthat k isthebest approximationoftheprobesignals.Thisrelationshipisdeterminedfromthesimultaneous valuesfortheprobemeasurementsandtheprojected b coecientsfromthetime-resolved stochasticestimatesofthePODcoecients.ThisprojectionisgivenbyEq.5{3. If d and n arewhite,zero-mean,anduncorrelatednoise,withcovariances Q and R allofwhichisassumed,thenthefollowingequationsapply: E [ d i d T j ]= Q i ij E [ n i n T j ]= R i ij E [ d i n T j ]=0 : Q and R areuser-denedmatricesusedtoweightherelativeaccuracyofthemodelversus thesensor.Themagnitudesofthesematricescanaccountfornoisysensorsand/ormodel uncertainty.Forinstance,ifthesensoriscontaminatedbynoise,themagnitudeof R is increasedtopenalizethesensornoiseandrelymoreonthemodel.Ontheotherhand, ifthemodelisknowntobeinaccurate,themagnitudeof Q isincreasedtopenalizethe processnoiseandrelymoreheavilyonthesensor. 99

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TheKalmanlterisinitializedwiththevalues ^ b + f; 0 = E [ b 0 ] P + f; 0 = E [ b 0 )]TJ/F15 11.9552 Tf 11.517 3.155 Td [(^ b + 0 b 0 )]TJ/F15 11.9552 Tf 11.517 3.155 Td [(^ b + 0 T ] ; where P isthecovarianceoftheestimationerror.Thelterisiterativelyupdated accordingtothefollowingsetofequations,for k =1 ; 2 ;::: Simon,2006: P )]TJ/F23 7.9701 Tf -0.983 -8.277 Td [(f;k = F P + f;k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 F T + Q k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 K f;k = P )]TJ/F23 7.9701 Tf -0.983 -8.278 Td [(f;k H T )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(H P )]TJ/F23 7.9701 Tf -0.983 -8.278 Td [(f;k H T + R k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ^ a )]TJ/F23 7.9701 Tf 0 -8.277 Td [(f;k = F ^ a + f;k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ^ a + f;k =^ a )]TJ/F23 7.9701 Tf 0 -8.278 Td [(f;k + K f;k k )]TJ/F22 11.9552 Tf 11.955 0 Td [(H ^ a )]TJ/F23 7.9701 Tf 0 -8.278 Td [(f;k P + f;k = I )-222(K f;k H P )]TJ/F23 7.9701 Tf -0.983 -8.278 Td [(f;k : 3.2.2.3Kalmansmoother TheKalmanlteriscausalandthereforeboundtodatauptoandincluding thelatestmeasurement.Certainsituationsallowandbenetfromnon-causal lters,commonlyreferredtoas smoothers ,becauseoftheutilizationoffuture measurementsforestimationofpastevents.AvariantoftheKalmansmoothercalled theRauch-Tung-StriebelRTSsmootherisused.Thisisaxed-intervalsmoother,in whichdataisavailableoveraxedinterval.TheRTSsmootherconsistsinitiallyofa forwardspasswithaKalmanlter,whichisfollowedbyasmoothingpassthatmarches backwardsintime. Thedataareavailablefromtimesteps0to N t .Afterperformingaforwardspasswith aKalmanlter,thesmootherisinitializedwiththevalues ^ b s;N t = ^ b + f;N t P s;N t = P + f;N t : 100

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Then,therstpassisiteratedover k = N t )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;:::; 1 ; 0Simon,2006: I )]TJ/F23 7.9701 Tf -0.882 -8.278 Td [(f;k +1 = )]TJ/F25 11.9552 Tf 5.48 -9.684 Td [(P )]TJ/F23 7.9701 Tf -0.983 -8.278 Td [(f;k +1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 K s;k = P + f;k F T I )]TJ/F23 7.9701 Tf -0.882 -8.277 Td [(f;k +1 P s;k = P + f;k )-222(K s;k )]TJ/F25 11.9552 Tf 5.479 -9.683 Td [(P )]TJ/F23 7.9701 Tf -0.983 -8.278 Td [(f;k +1 )-222(P s;k +1 K T s;k ^ b s;k = ^ b + f;k + K s;k ^ b s;k +1 )]TJ/F15 11.9552 Tf 11.517 3.155 Td [(^ b )]TJ/F23 7.9701 Tf 0 -11.258 Td [(f;k +1 : 3.3EstimationProcedures AsdepictedbytheowchartinFigure3-1,thegoalofthestochasticand model-basedestimatorsistoleveragethespatialresolutionofPIVdataandthetemporal resolutionofpointmeasurementstoobtainreduced-orderoweldestimateswithhigh spatialandtemporalresolutions.Themodel-basedestimatorisadditionallyusedasa stateobserverduringclosed-loopcontrol.Theprocedureforsynthesizingthemodel-based estimatorcanbebrokenintothreegeneraltasks: 1. Computeaninitialsetofstochasticestimates aAcquirePIVdatasynchronouslywithtime-resolvedprobemeasurements.The PIVdataneednotbetime-resolved.FromthePIVdata,computethePOD modesandselectthedominantorrelevantmodesasasetofbasisvectors j to approximatetheoweld. bProjectthePIVvelocityeldsontotheselectedPODmodes,yieldinga non-time-resolvedsetofPODcoecients a t n .UsingthePODcoecientsas theconditionaldataandtheprobemeasurementsastheunconditionaldata, computethestochasticcoecients A forMTD-mLSEUkeiley etal. ,2008; Durgesh&Naughton,2010. cUsetheMTD-mLSEcoecientsandthetime-resolvedprobedatatoestimatea time-resolvedhistoryofthePODcoecients,^ a t k 2. Identifyamodelusingtheinitialestimates 101

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PerformDMDontheinitialsetofestimatedPODcoecients.Formadynamic modelfortheevolutionofthePODcoecientsbyprojectingonthesubspace formedbytheselectionofDMDmodes.Themodelmaybeformedbyanysuitable means,suchasfromphysicalinsightorbyutilizingtheinitial,stochasticestimates. Thistaskisexibleinimplementationbecauseanadequatemodelmaybeow specic. 3. Computearenedsetofestimatesusingamodel-basedestimator UsethedynamicmodelconstructedabovetoimplementaKalmanlteror smoother.Ifusedasapost-processingestimate,applytheKalmansmoother. RecombineestimatedDMDcoecients ^ b t k withDMDmodesofPODcoecients toobtainmodel-basedPODestimates.ThetimehistoryoftheKalmansmoother's stateprovidesatime-resolved,low-orderestimateofthevelocityeld. 102

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Figure3-1.Flowchartofmodel-basedestimatorimplementation. Table3-1.Descriptionsofvectornotationshorthand. NotationEquivalentvectorDescription f u 0 x m ;t n g M m =1 u 0 t n =u 0 n Thesetofalluctuatingvelocitycomponents withinasnapshotatallgridlocations, x m for m =1 ; 2 ;:::;M ,andattime t n iscontained withinthevectoru 0 n ,representingasingle snapshotattime t n f j x m g M m =1 j ThesetofallPODmodalcomponentsatallgrid locations, x m for m =1 ; 2 ;:::;M ,formode j iscontainedwithinthevector j ,representing the j -thspatialPODmode. f a j t n g r j =1 a t n = a n ThereducedsetofPODmodalcoecientsat time t n iscontainedwithinthevector a n 103

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CHAPTER4 EXPERIMENTALSETUP Theexperimentalsetupisdesignedtoinduceboundarylayerseparationfromthe uppersurfaceofaatplatemodelwithablunttrailingedgeNa&Moin,1998.The conditionoftheseparatedowandtheboundaryconditionforimposinganadverse pressuregradientareusedinacompanioncomputationalstudybyAram etal. 2010.An adversepressuregradientiscreatedbyacceleratingandthendeceleratingtheowwith azero-netmass-uxsuctionandblowingboundaryconditionontheceilingofthetest sectionabovethemodel.AgeneralschematicoftheowconditionisvisibleinFigure4-1. Theseparatedowontheuppersurfaceoftheplateiscontrolledwithzero-netmass-ux ZNMFactuators.ParticleimagevelocimetryPIV,hot-wireanemometry,steady pressure,andunsteadypressuremeasurementsoftheseparationregionandwakeareused todiagnosetheestablishedbaselineseparatedowandthenevaluatetheeectiveness oftheopen-loopcontrol.Morespecically,thekeyobjectivesoftheexperimentsareto identifythecharacteristicfrequenciesoftheowandthentargetthosefrequencieswith unsteadyactuation.Thenaclosed-loopisformedbytheinsertionofareal-timeoperating system.Mostmeasurementsarerepeatedfortheclosed-loopcontrolexperiments.The followingsubsectionsbreakdowntheexperimentalsetupintothemodel,owfacility,and instrumentation. 4.1FlatPlateModel Thetwo-dimensionalatplatemodelgeometrybeginswitha4:1majoraxis tominoraxisellipticleadingedgeandterminateswithablunttrailingedge.The thickness-to-chordratio h=c is9.5%withchordandspanlengthsof40.2cmand29.2cm, respectively.Themodelconsistsofleadingandtrailingedgeparts,internalsupport structure,andupperandloweratplates.Photographsoftheentireassemblyareshown inFigure4-2.Theupperandloweratplatesaremachinedfromclearacrylic,which allowsthetransmissionoflightthroughthemodelandsignicantlyreducesunwanted 104

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surfacereectionsforopticalmeasurements.Theupperplatehousestheactuator assembly,sixunsteadypressuresensors,andfteenstaticpressureports,allofwhich aredescribedwithmoredetailinthesectionsthatfollow.Thetrailingedgepieceis equippedwithfteenunsteadypressuresensors. 4.1.1Zero-NetMassFluxActuator Controlisachievedviaasyntheticjetarraycomprisedoffourpiezoelectricdiscs APCInc.,PZT5J,PartNumber:P412013T-JBclampedbeneatharectangularcavity andslot.Discdeectionsresultinexpulsionfromandingestionintothecavity.Overan integernumberofcycles,thereisnonetmassuxcausedbythedeectionofthediscs. ThefourdiscsarevisiblethroughtheclearlowerplateinFigure4-2B,aphotographof themodel'slowerside.Theactuatorslotisplacedat61%ofthemodelchordlength downstreamoftheleadingedge.Thispositionisdictatedbytheseparationlineofthe baselinemeasurementsSection5.1.1. Cross-sectionaldrawingsoftheactuatorareincludedinFigure4-3.Notethatthe zy crosssectionutilizestheactuator'sgeometricsymmetryaboutcenterspan z =0. Theslotwidthanddepthareeach2.0mm,anditslengthis177.8mmsuchthattheslot coversthecentral61%oftheoverallmodelspan.Duringexperimentation,theactuator slotiscoveredforthebaselinemeasurementsinordertogaugetheslot'simpact,ifany, ontheseparationcharacteristics.Otherwise,theslotisopenforallmeasurements.This carefultreatmentoftheslotisnecessarytoevaluateitspassiveeectontheseparated ow. ThestagesofactuatorassemblyareshowninFigure4-4.Acompositediscis composedofabrassshimsandwichedbetweentwopiezoceramicdiscswithathin electrodesilveronthesurfaceofeachpiezoceramicdisc.Thediscsareclampedaround theiroutercircumferencesbytwoaluminumplatesFigure4-4,stage3.Anotherplate withalargerectangularcutoutisplacedontopofthepreviousassemblytoformthe 105

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cavityspaceabovethediscsstage4.Finally,thetopplatewiththeslotcutoutisxed upontheentireassemblystage5. 4.1.2UnsteadySurfacePressure Unsteadypressuretransducersareusedtodetectthepresenceofhydrodynamic uctuationsassociatedwiththeinducedseparationandwake.Flush-mountedPanasonic WM-61Aelectretmicrophonesmmpackagediameterenablereal-timemeasurementsof theuctuatingsurfacepressure.Theelectretmicrophoneshaveaquotedtypicalfrequency responsemagnitudeuncertaintyof 1dBforthefrequencyrangeof20Hzto20kHz. Sixsensors,namedS1throughS6,areplacedintwostreamwiserowsandush mountedintheupperatplate.Thetworowssymmetricallystraddlecenterspanby 2cm,orroughly10%chord,ineitherdirection.Theirpreciseplacementisgivenin Figure4-5.Anadditionalfteensensorsareembeddedintothetrailingedgepart,with threeplacedontheuppersurfaceandtheremainingtwelveonthebase.Amongthese fteensensors,onlythevesensorsatmidspanareusedinthispaper,andtheyare denotedbyS7toS11. Eachmicrophoneispoweredbya4mAdccurrentexcitationsuppliedbyaNational InstrumentsPXI-1042QchassisequippedwithaNIPXI-4498dataacquisitionmodule. Eachchannelofthismodulehas24-bitresolutionwith114dBdynamicrange,abuilt-in anti-aliasinglter,andaccouplingwitha0.5Hzcut-onfrequency.Dataareacquiredata rateof5kHz. 4.1.3SteadySurfacePressure Steadysurfacepressuremeasurementsaidinidentifyingandcomparingstatic pressuredistributionsexperiencedbytheuppersurfaceforvariousforcinglevelsand frequencies.Themeasurementsaremadealongthecenterspanlineoftheuppersurface atplate.Atotaloffteenstaticpressureports,withaninnerdiameterof0.711mm,are installedtomeasurethestaticpressuredistributionassociatedwiththebaselineseparation andcontrolledows.ThedistributionoftheportsisshowninFigure4-5. 106

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A16-channelEsterlinePressureSystemspressurescannerwitharangeof10inH 2 O measuresthepressuredierentialbetweenthelocalstaticpressure p s andthefreestream staticpressure p 1 .Foreachtestcase,atleast100samplesareacquiredatarateof5Hz. Afterpressuremeasurementsaretaken,surfacepressurecoecients C p arecomputedby normalizingtheaveragedierentialpressure, p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 ,byameasureofdynamicpressure, C p = p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 1 2 U 2 ; {1 where and U arethelocalfreestreamdensityandspeed,respectively. 4.2FlowFacility Theentireowfacilityconsistsoftwoow-generatingdevices:alow-speedwind tunnelandanexternalbranchthatevacuatesairowthroughthetunnel'stestsection ceilingandthenreturnsthisowdownstream.Eachportionhasitsownfan,whichis operatedindependentlyoftheother.Theexternalductingandfannecessarytocreatethe boundaryconditionsistermedtheseparationsystem. 4.2.1Low-SpeedWindTunnel TheexperimentsareconductedinanAerolabopencircuit,lowspeedwindtunnel withatestsectionthatmeasures30 : 5cm 30 : 5cm 61 : 0cminheight,width,and length,respectively.Upstream,thetestsectionisprecededbyanaluminumhoneycomb panel,twoanti-turbulencemeshscreens,anda9:1contractionsection.Theairspeedis controlledbyavariablefrequencydrivenfanaftofthediusionsection.AnexternalPID controllersetsthetestsectionvelocitybyreferencingthestaticandstagnationpressures fromaPitot-statictubeplaced6 : 5cmupstreamofthetestsectioninlet,atthemid-height ofthetestsection.Theemptytestsectionfreestreamvelocitiesrangefromapproximately 3to45m/s.Thefreestreamvelocityissetto3.9m/s,whichcorrespondstoRe c =10 5 Re h =9 : 5 10 3 107

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4.2.2SeparationSystem Theseparationsystemisdesignedtoimposeadjacentsuctionandblowingboundary conditionsthatcreatealocaladversepressuregradient.Theadversepressuregradientis imposedtoseparatetheboundarylayerontheupperatsurfaceofthemodel.Theow enterstheseparationsystemthrougharectangularcutoutintheceiling.Thecutoutspans fromonesidewallofthetestsectiontotheotherandis20.2cm : 50 c inthestreamwise direction.AsillustratedinFigure4-1,airenterstheupstreamhalfofthecutoutintothe suctionportionofthesystemandreturnstothetunnelthroughthedownstreamhalfof thecutout.Thetime-andspatially-averagedsuctionvelocityisapproximately0.20 U 1 Becausetheextractedairisreinserted,themassowratethroughthewindtunnelis conserved.ThisowconditionismonitoredbyaHeiseST-2Hpressureindicatorwitha 0 )]TJ/F15 11.9552 Tf 10.132 0 Td [(2in-H 2 Odierentialpressuretransducer.Thedierentialpressurebetweentheambient pressureandthestagnationpressureinsidethesuctionportionoftheseparationsystem ducting,about25cmabovethetestsectionceiling,ismaintainedat0 : 096 0 : 006in-H 2 O. Thisconditionishighlyrepeatablewithveryseldomnetuningofthefanspeed. Theseparationsystemconsistsofductworkthatenclosesthepathoftheextracted airow,anin-lineductfanthatdrivestheow,andinternalelementsthathelpcondition theow.ASoler&PalauPV-250xin-lineductfanwithamaximumowrateof 618CFMdrivestheowthroughtheducting.Therectangularsheetmetalductingis linedwith2.54cmthickEchoAbsorber acousticfoam. Placementofthemodelrelativetothesuctionandblowingpanelsisalsoguided bythesimulations,whichachieveaclosedseparationbubblewiththetrailingedge alignedwiththedividerplatebetweenthetwopanelsAram etal. ,2010;Tu etal. 2011.However,theexperimentsinthisthesisplacethetrailingedgeofthemodel0 : 050 c downstreamofthedividerplate.Inadditiontothis,thetransverseplacementofthe modelissetsuchthatthedistancebetweentheceilingandthemodel'suppersurfaceis 0 : 15 c 108

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4.3FluidMeasurementSystems 4.3.1Hot-wireAnemometer Thewakefrequencyisidentiedwithunsteadypressuresensorsbutalsowitha hot-wireanemometer.The5 mdiameter,1mmlongwireisalignedwiththe z axisand placedinthewakeatapositionof x=c =1 : 14and z=c =0.Measurementsarerecorded atthreelocations:parallelwiththeupperatplatesurface y = h= 2,parallelwiththe lowersurface y = )]TJ/F22 11.9552 Tf 9.298 0 Td [(h= 2,andatthecentralmodelheight y =0.Thesemeasurements arerecordedforthebaselineowandthecontrolledowcases. Thehotwireisalsoutilizedtocharacterizethesinusoidalactuatoroutput mean-squarevelocityinquiescentconditions.Thehotwireisalignedwiththelong dimensionoftheslotandplacedinthemiddleoftheslot.Thejetexitspeedsareacquired overthecenterofallfourdiscsforvariousforcingamplitudesandfrequenciesforasimple sinusoidalinput.Onceasinglesinusoidalforcingfrequencyisselectedbasedonthe resonantvaluesfromeachindividualdisc,thehotwireistraversedin2mmincrements alongthespanwisedimensionoftheslot.Theoutputspeedismeasuredwithallfourdiscs operatingatthesamefrequencyandseveralamplitudes.Oncethespanwisevariabilityis assessed,thehotwireisusedtomeasuretheactuatoroutputforaburstmodulatedBM inputwaveform.Thistime,thehotwireisstationaryinthecenteroftheslot,andthe responseismeasuredforvariousmodulationfrequencies f m andamplitudes.Thecarrier frequency f c oftheburstissettotheoptimalfrequencypreviouslydeterminedfromthe sinusoidalinputs. Allhot-wiremeasurementsareacquiredusingaNIPXI-1042Qchassiswithafour channelNIPXI-4462dataacquisitioncard.Eachchannelhas24-bitresolutionwith 118dBdynamicrangeandabuilt-inanti-aliasinglter.Forthewakemeasurements,the samplingrateis5kHz.TheresultsdemonstratethatthisfrequencysatisestheNyquist criteriontocapturetheuctuationsoftheforcingandthehighestfrequencyofinterest fromtheow. 109

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4.3.2PIVSystem APIVsystemisusedtoacquirebothtwo-componentandthree-component stereoscopicvelocityeldmeasurements.Formeasurementsoftheseparationbubble region,aNewWaveResearchSolo120-XTNd:YAGlasercoupledwithadjustableoptics generatesalaserlightsheetforPIVmeasurements.Becauseoftheplacementofthe separationsystem,thelightsheetmustenterthroughtheclearceilingatananglepointing downstreaminordertoilluminatetheowintheinducedseparationregion.Anexample ofthisisshowninthethree-dimensionalschematicofFigure4-7.Amirror,externaltothe testsection,isusedtodirectthelightsheetinthismanner.Unfortunately,theseparation systemstillcastsashadowthatpreventsilluminationoftheentireowregionbetweenthe upperatplateandtheceiling. Whentheregionofinterestisthenearwakebehindtheblunttrailingedge,aQuantel LaserEvergreen200Nd:YAGlaserislocatedunderneaththetestsection.Identicaloptics areusedtogeneratealightsheetthatentersverticallythroughtheoorofthetest section,illuminatingthewakeow. AllPIVimagesarecapturedbyLaVisionImageProX4Mcameraswith2048 2048pixelresolution.EachcamerahasaSigma105mmf/2.8macrolensattached,and onemeasurementincludes1.4xteleconverterstoincreasezoomandnarrowtheeld ofview.Onelaser,dependingonthesubjectow,andcamerasaresynchronizedbya LaVisionexternalprogrammabletimingunitcontrolledbyLaVisionDaVis8.1image acquisitionsoftware.OliveoilparticleseedingisproducedbyaATITDA-4Baerosol generatorforatypicalparticlesizeof1 m. OneofthemanyPIVcongurationsisforastereoscopicPIVSPIVmeasurement ofthebaselineseparationregionatseveralspanwisepositions.Thepurposeofthis measurementistoassessthespanwisevariabilityofallthreevelocitycomponentswithin theseparationregion.Aschematicofthesetupwiththelightsheetalignedwithcenter spanisgiveninFigure4-7.Inthissetup,amirrorisplacedwithinthemodeltoreectthe 110

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incidentlightsheetbacktowardstheceilingdownstream,thusilluminatingthepreviously mentionedshadowregion.Becausetheactuatorplacementontheuppersurfacewould alsopreventlightfromreectingooftheinternalmirror,thesymmetricmodelisipped upsidedown,andthemirrorisushmountedtotheinsidesurfaceoftheloweracrylicat plate,whichbecomestheuppersurfaceforthepurposesofthisparticularmeasurement. Theverticallyorientedlightsheetisalignedwiththemidspanline,andthentheentire setupcamerasandlaseristraversedinthespanwisedirectiontoacquiresevenevenly spacedSPIVeldsbetween z=c =[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 15 ; 0 : 15].Atotalof500instantaneousspapshotsis acquiredateachposition,exceptat z =0,where1000spanshotsareacquired. InadditiontothisSPIVmeasurementofthebaselineseparationregion,several dierenttwo-componentPIVmeasurementsareacquired.Theseincludeclose-upsof theseparationandreattachmentlocationsalongthespan,ahigh-resolutionviewof theseparationbubble,andthenearwakeregiondirectlydownstreamofthebase.All acquiredregionsarenotedinFigure4-8,separatedintomeasurementsofthebaseline andcontrolledowsforclarity.Asummaryofeachregion,includingtheopticsused, descriptionofthemeasurement,andnalvectorresolution,isincludedinTable4-1.The setupfortwo-componentvelocitymeasurementsisverysimilartothatoftheSPIV,except thatthecamerasarealignednormaltothelightsheet.Theycanbeplacedside-by-sideso thatalargereldofviewisacquired,ifdesired. Therstsetoftwo-componentPIVmeasurements,notedasregion A ,isofthe baselineseparationandreattachmentlocations.Thecamerasfocusontheselocations independently,employing1.4xteleconverterstoincreasezoomandincreasetheprocessed vectorresolutionwithintheseparatingandreattachingboundarylayers.Similartothe SPIVmeasurements,thismeasurementplaneistraversedacrosstenspanwisepositions between z=c =[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 22 ; 0 : 22]withequalspacing.Thisspanwisecoveragecorrespondsto thecentral62%ofthetotalspan.Thevelocityeldsallowcomputationofthestreamwise 111

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shearstress,andthereforedeterminationoftheseparationandreattachmentpointsforall acquired z=c planes. ThenextsetofbaselineowPIVusesasinglecameratoacquirethenearwake region,denotedasregion C .Thelightsheetentersverticallythroughthetestsection oorandisosetfromcenterspanby1cm z=c =0 : 025withnotraversal.Region D isasimilarowareaacquiredduringopen-loopcontrol.Finally,theinducedseparation region E isacquiredforboththebaselineandcontrolledows,thoughitisonlydepicted inthecontrolledportionofFigure4-8forsimplicity.Inthisconguration,thecameras areplacedside-by-sidesuchthattheirareasofinterestoverlap,whichoncetheimages areprocessed,resultsinasinglevelocityeldthatcoversasignicantportionofthe separationregion.Thisregionisxedtothespanwisepositionof z=c =0 : 025. ThelaserandcamerasaresynchronizedbyaLaVisionexternalprogrammabletiming unitcontrolledbyLaVisionDaVis8.1imageacquisitionsoftware.Theoliveoilparticle seedingisproducedbyaATITDA-4Baerosolgeneratorforatypicalparticlesizeof1 m. Theseederisplacedupstreamofthetunnelentranceandalignedsothatparticlespass throughthelightsheet. TheacquiredPIVimagesareprocessedwithLaVisionDaVis8.1software.The generalimageprocessingstepsforallregionsincludesubtractinglocalaverageintensities andmaskingoutignoringstaticportionsoftheimages,i.e.surfaces.Then,multi-grid cross-correlationscomputevelocityelds,withdecreasinginterrogationwindowsizefor thereningpasses.Inbetweenpasses,outliersarereducedbyapplyingarecursivespatial outlierdetectiontestWesterweel&Scarano,2005.Thenalvelocityeldsareagain testedforoutliers,andthedetectedoutliersareremovedGrin etal. ,2010. Formanyexperiments,unsteadypressuremeasurementsareacquiredsimultaneously withthePIVimages.Toaccomplishthis,thePIVsystemgeneratesatriggersignal halfwaybetweentherstandsecondlaserpulsesthistimechangesdependingonthe experimentafteralsoaccountingfora293 s delaybetweenthePIVacquisitionandthis 112

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triggersignal.ThetriggersignalismonitoredbytheNIPXI-4498dataacquisitioncard. Withasamplingrateof5kHz,theLabVIEWsoftwaremaintainsabuerof320samples perchanneluntilthedigitizedPIVsynchronizationsignalexceeds1Vac.Atthisevent, thebuerofsamples,thecurrentsample,andanadditional320samplesperchannel areacquiredandrecorded,suchthatthedurationoftheunsteadypressurerecordisless than0.13seconds.ThetriggerrateofthePIVsystemisapproximately4Hzbutwith irregularitiescausedbythedownloadspeedfromthecameras.Inthecaseofcontrol,the actuator'sinputsignalisalsoacquiredwiththesynchronizedblockofunsteadypressure data. 4.4ControlSystems Thebaselinetestparametersaremaintainedduringcontrol,however,additional equipmentisneededtoforcetheactuators,conditionthesignals,and,inthecaseof closed-loopcontrol,assessthereal-timestateoftheowsystem.Inthissection,the additionalhardwareisgenerallygroupedintoopen-andclosed-loopcategories,though eachsetofexperimentsutilizesnearlyallofthepreviouslymentionedinstrumentationto analyzethecontrolledowconditions.Thefollowingcontainsdetailspertainingtothe necessaryequipmentandgeneralstrategiesofthecontrolmethods.Specicsfortheopenandclosed-loopcontrolobjectivesarecontainedinChapters6and7,respectively. 4.4.1OpenLoop Aprogrammablevoltagesourceandamplierareneededtodrivethefour piezoelectricactuatorsduringopen-loopcontrol.Sinusoidalandburstmodulated sinusoidalsignalsaregeneratedwithaTektronixAFG3022BArbitrary/Function Generatorcapableofupto25MHzand12.5MHzsinewavesintheserespectivemodes. Thepeak-to-peakamplituderangeis10mV pp to10V pp givena50load.Thegenerated waveformsareampliedbyagainratioof50V/VviaTrekModelPZD350ampliers.For thesafetyofthediscs,thenalpeak-to-peakvoltageislimitedto100V pp .Thecontrolled 113

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owconditionsaremeasuredwiththeunsteadyandstaticpressuretransducers,thePIV system,andthehot-wireanemometer. 4.4.2ClosedLoop Fourelectretsareusedforunsteadypressurefeedbackduringclosed-loopcontrol. 1 A dSPACEDSPsystemModeldS1006witha3.0GHzAMDOpteronprocessorclosesthe loopbyacquiringthefoursignalswitha16-bitA/DdS2001boardwithoutaccoupling andananti-aliasinglter,performingthecontroloperations,andoutputtingthecontrol signalthrougha16-bitD/AdS2102board.Closed-loopcontrolalgorithmsarecoded inSIMULINKandcompiledviaanintegrationofMATLABanddSPACEReal-Time Workshop.Theiterationrateofthecontrolsystemis41 : 04kHz. BecausethedSPACEacquisitionchannelsdonotaccoupleorlow-pass lterforanti-aliasing,theunsteadypressuresignalsareaccoupledwithacut-on frequencyof0.1Hzandlow-passlteredwithacut-ofrequencyof500Hzbya KEMOVFBF35multi-channellterpriortothedSPACEA/Dconversion.Theelectret sensitivitiesnominally30mV/Paareappliedwithinthecontrolsoftware.Theoutput signalfromthedSPACEactuatorinputisthenpassedthroughtheKEMOVBF35 forthepurposesofanti-aliasingandaccoupling.Aowchartoftheclosed-loopcontrol systemisincludedinFigure4-9. 1 ThedSPACEA/Dboardislimitedtofourfunctioningchannels. 114

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Figure4-1.Schematicoftheexperimentalsetup. A B Figure4-2.Photographsoftheatplatemodelshowthegrayleadingedgepart,the aluminumtrailingedgepart,thefourpiezoelectricdiscshousedwithinthe actuator,pressuretapstubingnotshown,andunsteadypressuresensors. AUpperside.BLowerside. 115

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A xy crosssection B zy crosssection Figure4-3.Actuatorcross-sectionalschematics.AThe xy crosssection.BThe zy cross sectionshowshalfoftheoverallspansymmetricaboutindicatedline. 116

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Stage1 Stage2 Stage3 Stage4 Stage5 Figure4-4.Actuatorassembly. Sensor x=cy=cz=c S10.800.0480.10 S20.800.048-0.10 S30.850.0480.10 S40.850.048-0.10 S50.900.0480.10 S60.900.048-0.10 S70.980.0480.00 S81.000.0340.00 S91.000.0110.00 S101.00-0.0110.00 S111.00-0.0340.00 Figure4-5.Unsteadypressuresensorplacementontheupperatplateandatbase.The unsteadysensorsusedinthisstudyaredenotedbytheirsensornumberinthe rangeofS1-S11. 117

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Figure4-6.Asectionalviewofthelowerportionoftheseparationsystemattachedtothe ceilingofthetestsection.Bluearrowsmarktheowdirections. Figure4-7.SchematicoftheSPIVsetup. 118

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Figure4-8.SchematicshowingtheapproximatelocationsofPIVmeasurementregions. Region E isalsoacquiredinthebaselinemeasurementsbutnotshownthere forclarity. Figure4-9.Schematicoftheclosed-loopsetup. 119

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Table4-1.SummaryofPIVregionsstudied. RegionPIVTypeOptics z=c Description A 2-component,105mmlens,[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 22 ; 0 : 22]Baselineseparationand 2cameras1.4xteleconverterplanesreattachmentpoints B stereo105mmlens [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 15 ; 0 : 15]Spanwisevariationof planesbaselineseparationbubble C 2-component, 105mmlens0.025 Nearwakeowof 1camerabaselineseparation D 2-component, 105mmlens0.025 Nearwakeowof 1cameracontrolledow E 2-component, 105mmlens0.025 Baselineandcontrolled 2camerasseparationregion Table4-2.ProcessedPIVsettings. RegionVectorresolutionvec/mmFinalwindowsizepx A 4.67upstream 16 16 4.21downstream B1.5124 24 C1.8124 24 Dopenloop1.9224 24 Dclosedloop1.9824 24 Eopenloop1.9124 24 Eclosedloop1.9824 24 120

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CHAPTER5 BASELINEFLOWRESULTS TheresultsoftheunforcedexperimentsdescribedinChapter4aredividedinto threesections.First,thebaselineseparatedowisanalyzedusingPIVmeasurementsand standardspectralanalysismethodsfromunsteadypressureandhot-wireanemometry.An eortismadetoidentifykeycharacteristicsoftheseparationbubbleandwake,including theirdominantfrequenciesandanycoherentstructures.Then,thebaselineresultsare decomposedintoasetofmodes,ofwhichtheorderisreducedforstochasticestimation andKalmanlteringmethods.Thelow-orderestimatesretainthehigh-spatialresolution ofthePIVeldsandthehigh-temporalresolutionoftheunsteadypressuremeasurements. Finally,theestimatesareanalyzedforglobaldynamics,whichbetterillustratethenature anddistributionofthecharacteristicowdynamics. 5.1BaselineSeparatedFlow Thebaselineowconditionisestablishedastheinducedboundarylayerseparation fromtheupperatsurfaceofthemodel.Theseparationisduetoanadversepressure gradientimposedbyceilingboundaryconditions.Theobjectiveoftheseresultsistoassess theextentofthebaselineseparationanditspotentialinteractionswiththewakeow.The chordReynoldsnumberisdenedas Re c = U 1 c ; where isthekinematicviscosity,andRe c =10 5 forallexperiments.Thisbaseline separatedowisdistinguishedfromatraditionalblu-bodywake,whichresultsfrom removaloftheimposedadversepressuregradient.Thelattercaseisofinteresttothe controlresultsbecauseofitsattachedboundarylayersuntiltheabrupttrailing-edge cornersatthebaseofthemodel.However,thetwoowconditionsarenotanalogous inatraditionalsense.Evenintheeventofsuppressionoftheseparationinthebaseline 121

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separatedowontheuppersurface,theboundarylayerstillexperiencesandrecoversfrom theadversepressuregradientimposedbytheceilingboundaryconditions. Withthatinmind,thebaselineseparatedowresultsfollowbelow.Theseparation bubbleisinvestigatedrstbydeterminingthestreamwiseextentoftheseparation. Thenthethree-dimensionalcharacteristicsofthemeanseparationbubbleareanalyzed usingstereoscopicparticleimagevelocimetrySPIV.Spectralanalysisoftheunsteady pressurewithintheseparationbubbleisperformedtodeterminethefrequencycontent associatedwiththeseparatedshearlayer.ThemeanowresultsfromPIVandthe dynamicmeasurementsfromunsteadypressuresensorsonthebasehelptocharacterize thenearwakeow.Theseresultsarecomparedtothestandardblu-bodywakewiththe sameRe c 5.1.1MeanSeparationBubble Therststeptowardscharacterizingthebaselineowistomeasuretheextent ofboundarylayerseparationresultingfromtheimposedadversepressuregradient. Two-componentPIVisusedtoaccomplishthisobjective.Twovelocityelds,onezoomed intothesuspectedseparationpointandtheotherzoomedintothesuspecteddownstream reattachment,aremeasured.InFigures5-1Aand5-1B,time-averagedstreamwisevelocity prolesaresuperimposedontime-averagedstreamlinesforthetwoacquiredregions. Refertoregion A inFigure4-8forthelocationsofthetwoeldsofview.Thewallshear stress w ,denedas w = d u dy y =0 ; iscomputedfromthetime-averagedstreamwisevelocitycomponentsnearthewalland plottedinFigures5-1Cand5-1Dastheskinfrictioncoecient, C f = w 1 2 U 2 1 : Thereare20velocitymeasurementswithintheincomingboundarylayerheightof approximately4.8mm,withthegridpointclosesttothewallat n =0 : 2mm. 122

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Theuncertaintiesof C f nearthepointsofseparationandreattachmentaretypically 0 : 5 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 and 1 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,respectively. Theuncertaintyinthepointofseparationisdeterminedfromtheuncertaintybounds of C f ascomputedbythenear-wallcurvets.Inordertoobtainarangeof C f values, Monte-Carlosimulationswith1,000iterationsperturbthestreamwisevelocityproles basedontheuncertaintyinthePIVmeasurements.AsdescribedinAppendixB,thetotal uncertaintyinthestreamwisevelocityconsistsofrandomandbiasuncertainties.The randomuncertaintyinthestreamwisevelocityisassumedtobenormallydistributedwith 95%condenceintervalboundsof 1 : 96 u rms =N ,where u rms isthestandarddeviationof theensembleof N velocitymeasurements.Theuncertaintyinthewall-normaldirection y isauniformprobabilitydensityfunctionintheintervalof[ )]TJ/F15 11.9552 Tf 9.299 0 Td [( y= 2 ; y= 2],where y istheverticalspatialresolutionofthePIVdata.Thebiasuncertaintyintheaverage streamwisespeedisalsouniformlydistributed,butthemagnitudeisdeterminedfollowing theequationsinAppendixBColeman&Steele,2009.Finally,theuncertaintyinthe skinfrictiondistributionontheupperatplaterevealsarangeofstreamwisecoordinates forthelocationofseparation. Theguresshowthatthewallshearstressdecreasessharplypriortoseparation, crossesthe C f =0lineatthepointofaverageseparation,andmaintainsanaverageow reversal C f < 0untilthepointofreattachment,where C f =0againbutwithapositive slopethatindicatesnewboundarylayergrowth.Thisisconrmedbytheaverageproles inFigure5-1B.Atmidspan,thetime-averagedseparationpoint x sep andreattachment point x att aredeterminedtobeat0 : 656 c 0 : 010 c and0 : 951 c 0 : 009 c ,respectively.The characteristicbubblelength L sep istherefore0 : 295 c 0 : 019 c .Thisvaluesupportsthe separationbubblesizesforthesimulatedcasesofthecanonicalseparationreportedby Mittal etal. 2005andKotapati etal. 2010Section2.3.1,althoughthisisgivenasa loosecomparisonduetodierencesinthecongurations. 123

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TheaveragePIVeldsshownarefromthespanwisepositionof z=c =0 : 025and arecomputedfromatotalof100instantaneoussnapshots.Thisprocessisrepeatedfor anadditionalninePIVplanesbetween z=c =0 : 22and z=c = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 22,withanequal spacingof z=c =0 : 050betweenneighboringplanes.Thetimeaveragedseparationand reattachmentpointsarecalculatedforeachplaneandtheresultsareplottedinFigure5-2 asseparationandreattachmentlines.Theuncertaintyrangesforthestreamwiselocations ofaverageseparationandreattachmentaregivenfortheopen-slotconguration.These measurementsarerepeatedwiththeactuatorslotcoveredinordertoassesstheimpact oftheslotonthetime-averagedseparatedowtopology.Theaveragelinesindicatethat thereislittledierencebetweenthecoveredslotandtheopenslotandthatbothare typicallywithinthemeasurementuncertainty.Theaverageseparationandreattachment locationsthereforerefertotheopenslotcasehereafter. Despitethelackofsignicantinuencefromtheslot,therearesmallvariationsin theextentoftheseparationbubblealongthespanwisedirection.Attheextremes,near z=c = 0 : 2,reattachmentappearstooccurslightlyfurtherupstreamthanatthemore centralspanpositions.However,theaverageextentoftheseparationbubbleappearstobe uniformwithinthecentralrangebetweenthesetwovalues. Next,SPIVoftheseparationbubbleisusedtomorecriticallyassessthemagnitudeof three-dimensionalitywithinthisregion.Sevenmeasurementplanesareacquiredbetween z=c = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 15and z=c =0 : 15withanequalspacingof z=c =0 : 050.Time-averaged contoursofeachvelocitycomponentaredisplayedinFigure5-3.Refertoregion B in Figure4-8fortheeldofviewrelativetothemodel.Concentratingrstontheaverage streamwisevelocity,aseparationbubbleisobserved.Thestreamwisebubblelength decreasesbyamaximumof1.7%chordfortheextremecasesof z=c = 0 : 15,whichagrees withthepreviousmeasurements.Theowupstreamandabovetheseparatedshearlayer isacceleratedabovetheincomingfreestreamspeedbeforedeceleratinginthepresenceof theadversepressuregradient. 124

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Theaveragetransversevelocitycomponent v showsreasonablespanwiseagreementin bothmagnitudeanddistribution.Thereisaclearupstreamanddownstreamdivision betweenpositiveandnegativevaluesforthiscomponent.Recallthatthedividing platebetweenthesuctionandblowingportionsoftheseparationsystemislocatedat astreamwisepositionof x=c =0 : 95.Whiletheaverageverticalvelocitywithinthebubble isrelativelysmall,theouterowrisesintheacceleratedregionbeforereachinganapex v =0andchangesdirectionbacktowardsthemodelsurface. Theboundsontheaveragespanwisecontourlevels w arespecicallychosento matchthoseofthe v component.Thefreestreamislargelyvoidofanysignicant averagespanwisevelocity,butpocketsexistwithintheseparationregion.However, theirmagnitudesarerelativelysmall,especiallywithinthecentralspanboundedby z=c =[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 10 ; 0 : 10]. Figure5-4displaystheaverageturbulentquantities.Inspectionoftheseeldsshowsa uniformityinthree-dimensionaluctuationsacrossthespan.Thevelocityuctuationsare mostprominentinthedownstreamregionoftheseparationbubble,namelyfor x=c> 0 : 8. Withthemeanseparationoccurringataround x=c =0 : 66,theseuctuationsindicatethat theseparatedshearlayertransitionstoamoreturbulentmixinglayerbeforeboundary layerreattachmentpriortothetrailingedge. 1 5.1.2SeparationBubbleDynamics Sixunsteadypressuresensorsontheuppersurfacearestrategicallyplacedwithin theseparationbubbleinordertodetectthedynamicsassociatedwiththeseparatedshear layeranditsdevelopment.Forthebaselineseparation,dataareacquiredatasampling rateof5kHzfor80seconds.Unsteadypressurespectraarecomputedusingablocksize of1,000samples,resultingina5Hzbinwidth.AHanningwindowisappliedtoeach 1 Reattachmentnotcapturedbythesemeasurements. 125

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datablockwith75%overlap.Theauto-spectraldensitiesforsensorsS1toS6aregivenin Figure5-5. UnsteadypressuresensorsS1andS2arelocatedat x=c =0 : 80,whichistheclosest sensorpositiontotheseparationpoint.Thoughthereareseveralpeakswithinthespectra ofthesetwosensors,thepeaksaboveSt L sep =5 : 95havebeendeterminedtobeacoustics relatedtotheoperationofthetunnelandseparationsystem.ExaminationofsensorsS3 andS4,whicharethetwosensorsat x=c =0 : 85,revealsaprominentpeakataround St L sep 2 : 7andincreasedbroadbandlevelswhencomparedtosensorsS1andS2.Even furtherdownstream,sensorsS5andS6at x=c =0 : 90,haveevenhigherbroadbandlevels butstillshowthepeakfrequencyatSt L sep =2 : 68 f =90HzorSt =0 : 010,where St L sep = fL sep U 1 and St = f x U x x =0 : 6 c : Duetotheplacementofthesesensorsunderneaththeseparatedshearlayerandthe increasedturbulencelevelsobservedinthedownstreamportionoftheseparationbubble, thispeakishypothesizedtocorrespondtotheshearlayerfrequency. Thishypothesisissupportedbyinstantaneousvelocityeldsoftheseparationregion. Figure5-6showsthevorticityofaninstantaneoussnapshot.Thissnapshotprovides ausefulreferenceforthelocationsofthesensorsinFigure5-5 x=c =0 : 80,0.85,and 0.90relativetotheseparatedshearlayer.Refertoregion E inFigure4-8fortheshape andlocationofthefulleldofview.Downstreamfromthelaminarseparation,the Kelvin-Helmholtzinstabilityleadstoroll-upoftheshearlayer.Resultingvorticespinch" o,convectdownstream,andbreakupintolesscoherenteddiespriortoreachingthe trailingedge. Becauseofthisconvectiveinstability,crosscorrelationsbetweensensorsalonga similarowpathareusedtocalculatetheconvectivetimedelay c .Thepeakinthe cross-correlationcoecient correspondstotheaveragetimeittakesforcoherent 126

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structurestoconvectfromonesensortothenext.Thecross-correlationcoecients betweensensorpairS3andS5andalsopairS4andS6areplottedinFigure5-7.The averagedelayassociatedwithbothofthesepairingsis8 : 8 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 seconds,whichamounts toaconvectionspeedof U c =2 : 27m/s U c =0 : 58 U 1 with2cmbetweensensors. FromFigure5-6,thedistancebetweenthetwovortexcoresisestimatedasapproximately 0 : 065 c .6cm. 2 Thisconvectionspeedandvortexspacingestimatethefrequencyofthe convectingvorticestobeSt L sep =2 : 59 f =87Hz,whichagreeswellwiththeobserved spectralpeak.Therefore,theshearlayerfrequencySt L sep ; SL = f SL L sep =U 1 isidentied asapproximately2.68 f SL 90Hz.ThisfrequencyisequivalenttoSt c; SL =9 : 23when scaledbythechordlengthTable5-1. 5.1.3MeanWakeFlow Thenextregiontoinvestigateforthebaselineseparatedowisthewake.Anatural comparisonforthisregionistheblu-bodywakewithKarmanvortexshedding.In additionaltothebaselineseparatedow,measurementsaremadeforthisblu-bodywake withoutinducedseparation,whichisreferredtoasthestandardblu-bodywake. Two-componentPIVisusedtomeasuretheowimmediatelydownstreamofthe base.Refertoregion C inFigure4-8forthiseld'splacementwithintheentiretest section.Thetime-averagedvorticity ,normalizedby U 1 =c ,isplottedinFigure5-8 forboththebaselineowandtheblu-bodywake.Theaveragevorticityofthebaseline separationrevealsanasymmetricwakewithamuchthickershearlayerdetachingfrom theuppersurface.Theinducedseparationcausestherecirculationregiontodeectinthe negative y direction.Contrarytothis,thestandardblu-bodywakeissymmetricabout y =0,atleastinatime-averagedsense. Inordertoinvestigatethemagnitudeanddistributionofuctuationsforeachof theseowscenarios,themeanturbulentstatisticsarecomputedandshowninFigure5-9. 2 ThisvalueagreeswithdistancesfrommultiplesimilarinstantaneousPIVsnapshots. 127

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Again,theasymmetryofthebaselineseparatedowisimmediatelyapparentinthe uctuations.Large-scaleuctuations,especiallyinthe v component,areprominentin thelowershearlayerandmuchlesssointheuppershearlayer.Themeanturbulence quantitiesofthestandardblu-bodywake,however,exhibitclearsymmetrywithregions ofhighuctuationsindicativeofKarmanvortexshedding.Finally,themagnitudesof thewakevelocityuctuationsinthebaselineseparatedcasearenearlyhalfofthisin thestandardblu-bodywake.Thisisimportantwhenconsideringthemeasurementof unsteadypressureuctuationsfromthebaseofthemodel. 5.1.4WakeDynamics Thewakedynamicsareinvestigatednextinordertoidentifythecharacteristic frequencyassociatedwiththewake.Fourunsteadypressuresensorsonthebaseofthe modelaresampledatarateof5kHzfor80seconds.Theauto-spectraldensitiesforthese sensorsareshowninFigure5-10.Thespectralparametersare1,000FFTpointsthat resultsinafrequencybinwidthof5Hz,aHanningwindowappliedtoeachblock,and 75%overlap. Theblacklinesrepresentthebaselineseparatedowcase.ForsensorS8nearthe uppersurface,theshearlayerfrequencyofSt L sep =2 : 68persists,buttheredoesnot appeartobeanadditionalstrongpeakassociatedwithanytraditionalwake-likecoherent structures.Toverifythatthesesensorsarecapableofdetectingwakeoscillations,the blu-bodywakecaseisalsorecordedandshownasthedashedredline.Anewpeak atSt L sep =0 : 89Hzappearsforallsensors,butthepeakmagnitudeislargestfor sensorsclosesttotheupperandlowercornersofthebasesensorsS8andS11.When thisfrequencyisnormalizedbasedonplatethicknessandalocalfreestreamvelocity,its StrouhalnumberSt h is0.25,where St h = fh U W : Thespeed U W =4 : 5m/sisthefreestreamvelocityunderneaththemodelandatthe trailingedge. 128

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ThedynamicrelationshipbetweenthemeasurementsfromsensorsS8andS11is investigatedbymeansofthecrosspowerspectraldensityCPSD,showninFigure5-11. Thisanalysisisrestrictedtothebaselineseparationforclarity.Themagnitudeofthe CPSDhasmanypeaks,buttheinterestingpeaksarehighlightedbysignicantphase dierences.Nearzerophasepeaksaremorelikelytobealiatedwithcommonnoise sourcesoracousticsandless-likelytheresultofin-phasehydrodynamicuctuations. Aphaseshiftofnearly180 occursatSt L sep =0 : 92Hz,whichsuggeststhatthe wake-likevortexstreetpersistsbutatadecreasedmagnituderelativetotheblu-body wake.Interestingly,anotherphaseshiftofabout120 isdetectedatSt L sep =2 : 62Hz. Thismayindicateinteractionbetweentheuctuationsfromthelowershearlayerofthe wakewiththeconvectingstructuresfromtheseparatedshearlayerontheuppersurface. Ifso,arelationshipbetweenthestateofthebaselinewakeandtheseparationbubble dynamicshasbeenidentied. Referringbacktotheturbulentvelocitystatisticsforthebaselinewakeandthe blu-bodywakeseeFigure5-9,thedierencesinthemagnitudeanddistributionof high-energyuctuationsbetweenthesetwocasessuggeststhatthevortexsheddingforthe baselineseparatedowisprimarilyfromthelowersurfaceshearlayer.Thecombination ofthoseresultswiththeunsteadypressurespectrasuggeststhatthepressureuctuations associatedwiththissheddingarenotasimpactfulonthemodelbaseandthereforenot detectedrelativetothebroadbandpressurelevelsandpotentialdisturbances. Toremedythis,ahotwireisplaceddownstreamofthetrailingedge,atalocationof x=c =1 : 14inordertofurtherinvestigatethewakedynamics.Thehotwire'stransverse placementissettothreepositions:alignedwiththeuppermodelsurface y=h =0 : 5, alignedwiththelowermodelsurface y=h = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 5,andalignedwiththecentralheightof themodelbase y=h =0.TheselocationsareshownasdotsinFigures5-8and5-9and areselectedinordertodetectvelocityuctuationsindicativeofvortexsheddingfromthe uppershearlayer,thelowershearlayer,andpotentialinteraction,respectively. 129

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Auto-spectraldensitiesofthehotwireinthebaselineseparatedowandthe blu-bodywakeowarecomputedandshowninFigure5-12.Parametersimportantto thespectralcontentincludeasamplingfrequencyof5kHz,aseriesdurationof80seconds, 5,000FFTsamplesHzfrequencybinswith75%overlap,andapplicationofthe Hanningwindowfunction. Theresultingspectrafromthebaselineseparatedowshowanoveralldecreaseinthe powerofthefundamentalpeakatSt L sep =0 : 92Hz.Thispeak'spowerdecreasesas themeasurementlocationmovesfromthelowershearlayertowardstheuppershearlayer, reinforcingtheresultsfromtheunsteadypressuremeasurements.Thesedynamicresults andthefull-eldturbulentstatisticssuggestthatthedominantinstabilitymechanism intheasymmetricwakeofthebaselineseparatedowisvortexsheddingfromthelower shearlayerandthatitscharacteristicfrequencySt L sep ; wake =0 : 92 f wake =31Hz. However,thereisadditionalevidenceofwakeinteractionwiththeseparatedshearlayer fromtheuppersurface.ThisphenomenonisinvestigatedfurtherinSection5.3. Thisspectralanalysisisrepeatedfortheblu-bodywakeowFigure5-12B.There isclearagreementbetweenthedominantpeaksassociatedwiththeupperandlower shearlayers.However,whenthehotwireisplacedcentrallybehindthebaseofthemodel, theapparentdominantpeakdoublesinfrequency.Thiseectiscausedbythehotwire's inabilitytodetectvelocitydirection.Thehotwiresensestheoscillatorynatureofthewake velocityasonlypositiveshiftsinspeed,whichgivesrisetoadoublingoftheprimary sheddingfrequency. Duringonecycleofastandardblu-bodywake,thecenterlineofawakeexperiences twovortices,onefromthelowershearlayerandonefromtheuppershearlayer.At thewakecenterline,eachvortexproducesavelocitydecitin u whilealsoproducing anegativeminimumorpositivemaximumin v .Therefore,ifthemeasurementsare restrictedto u ,thespectracouldreecttherstharmonicofthesheddingfrequency. However,becausethehotwiredoesnotrecognizenegativevaluesorowdirection,itis 130

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possiblethatboth u and v uctuationscontributetoaperceivedfrequencydoublingofthe primarysheddingfrequencyalongthecenterline,whichdoesn'tevenincludethenatural harmonicsofwithwake.Astheprobeismovedawayfromthecenterline,bothvelocity componentsshowtheprimarysheddingfrequencyandsuper-harmonics.Referringbackto thebaselineseparatedcase,theharmonicpeakforthe y=h =0spectrumissignicantly lessthanfortheblu-bodywakeow,whichpotentiallyhighlightslessinteractionbetween theupperandshearlayersatthisfrequencyandatthislocation. 5.2EstimationofBaselineFlow ThesimultaneousunsteadypressureandPIVsnapshotsarecombinedto performstochasticandmodel-basedestimationofthecoherentdynamics.First,the full-dimensionalvelocityeldsareapproximatedbyprojectionontoareduced-orderbasis obtainedfromproperorthogonaldecompositionPOD.Then,stochasticestimationis usedtoidentifyalinearrelationshipbetweenunsteadypressureandtime-varyingPOD coecients.Thisrelationshipisappliedtoestimatethereduced-ordervelocityeldsbased ontheunsteadypressure,wheretheestimatesretainthehighspatialresolutionofthe velocityeldsplusthehightemporalresolutionoftheunsteadypressuremeasurements. Thesetime-resolvedestimatesareusedtoidentifyamodelforthestatedynamics, governedbythetime-varyingPODcoecients.ThemodelisincorporatedintoaKalman ltercausalandaKalmansmoothernon-causal.Theestimatesfromallestimatorsare usedtoidentifythedynamicstructureswithintheshearlayerandwake. 5.2.1Reduced-OrderBasis Inordertolimitthedimensionsoftheestimatedvelocityeldswhileretaining coherentinformation,PODiscomputedfromthe u -and v -velocitycomponentsof themean-subtractedvelocityelds.AsmentionedinChapter3,thePODmodesare consequentlybasedonthekineticenergycontentcontainedwithineachmodeandplaced indescendingorderofenergy,suchthattherstmodecontainsthemostenergy.The 131

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decompositionsarecompletedindependentlyfortheseparationbubbleandwakeregions becausethedataarenotacquiredsimultaneously. Thesummedenergyfractionsfortherst100PODmodesoftheseparationbubble PIVareshowninFigure5-13.TounderstandthenatureofthesePODmodes,themean eldandthersteightmodesareshowninFigure5-14,representedbythespanwise vorticityofthemodalcontributions.However,theseeightmodescaptureapproximately 50%ofthetotalenergywithinthesetofPIVsnapshots.Amongtheseeight,thesecond andthirdPODmodescontainwellorganizedcellsofvorticitythatalternatesigninthe streamwisedirection.Thepairingofthesetwoisindicativeofaconvectivevortexstreet. Therearenorecognizablepatternsamongtheremainingmodes. ThisprocessisrepeatedforthewakePIVdata.Thesummedenergyfractionsof therst100modesareshowninFigure5-15.Themeaneldandrstfteenmodesare showninFigure5-16,withnearly40%ofthetotalenergyinthersttwomodes.Inthis region,therstandsecondmodesareemblematicofstreamwiseconvectingvorticesfrom thelowershearlayerofthetrailingedge.Thereislittlecontributionfromtheupper shearlayerinthesetwomodes,whichisunlikethesymmetricPODmodaldistributions forastandardblubodywakee.g.Tu etal. ,2013.Modesthreethroughsixappearto capturemoredynamicswithinthelowershearlayer,withmodesfourandsixresembling aspatialharmonicofmodesoneandtwo.It'snotuntilmodesevenandabovethat signicantvorticalcontributionsfromtheuppershearlayerappear.Thisindicatesthat thehighestenergymodesareprimarilyrestrictedtodescribingthevelocityuctuations withinthelowershearlayer,whichisconsistentwiththehot-wireresults.Amongthe lower-energymodes,organizedpatternsfortheuppershearlayerappearinmodesnine throughfteen,withemphasisonmodeseleven,thirteen,andfourteen.Thesemodesalso exhibitstructuresfromthelowershearlayer. 132

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5.2.2StochasticEstimation Inthisimplementationofstochasticestimation,atemporalintervalisselectedfrom amongtheunsteadypressuresignals,andalinearrelationshipbetweenthemeasurements withinthatintervalandanaccompanyingsetofnon-time-resolvedPODcoecients isdetermined.Inthisrelationship,theunsteadypressureassumestheroleofthe unconditionaldatauponwhichthePODcoecientsaredependent,eveniftheintervals occur after theacquisitiontimesofthePODcoecients.Thismeansthattheinterval ofmeasurementsusedtoestimatethePODcoecientsisnotrestrictedtoasymmetric intervaloftimesaroundtheestimationofthePODcoecient,aswasthecaseinprevious endeavorsTu etal. ,2013.Instead,aninitialosettimeof 0 isincludedthatdescribes thelead 0 < 0orlag 0 > 0timeofthemeanintervalrelativetotheestimatedevent. AnexampleschematicinFigure5-17showsanintervalofmultipledelaysforestimation ofasnapshot,ormoreaccuratelyasetofestimatedPODcoecients.Thetimeofthe estimatedeventis t k .Theintervaloflagsissymmetricallydistributedabout t k )]TJ/F22 11.9552 Tf 12.221 0 Td [( 0 with ahalf-spanwidthof max ,where max > 0.Thatis,toestimatethestateattime t k ,probe datacollectedattimes t k )]TJ/F22 11.9552 Tf 11.955 0 Td [( 0 j areused,forarangeofdelays j satisfying j max U 1 =c; {1 where indicatesthattimeisnormalizedbytheratioofchordlengthtofreestream velocity. Thefollowingdescribestheprocessappliedtodeterminetheoptimaltimeinterval forestimatingthePODcoecientsfromagivensetofdata.Foreachparameterchange, asetofestimatedPODcoecientsisgeneratedandcomparedtotheprojectedtrue coecientsfromtheacquiredPIVelds.Theerrormetricusedtoquantifythedierence betweenthetruecoecients a k andtheestimatedPODcoecients^ a k istheratioofthe 133

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squarederrorintheestimatetothemeanenergycapturedbythesetof r PODmodes,or e r t k = k ^ a t k )]TJ/F22 11.9552 Tf 11.955 0 Td [(a t k k 2 2 D k a t k k 2 2 E = P r i =1 h ^ a i t k )]TJ/F22 11.9552 Tf 11.955 0 Td [(a i t k i 2 P r i =1 D a i t k 2 E : {2 Thisisequivalenttothekineticenergyoftheerrorinthelow-ordervelocityeldestimates normalizedbythemeankineticenergy,orsimply,thefractionofmeankineticenergyin theestimationerror. Theoptimaldelayisdeterminedrstfortheestimationoftheseparationbubblefrom thesensorsindicatedinFigure4-5.Initially,100PODmodesareretainedtodetermine theoptimalsingle-timedelay 0 max =0.Theminimumerrormetricoccursfor 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,asseeninFigure5-18A,whichcorrespondstoaleadrelativetothe estimationtimeof20samplingintervalsatarateof5kHz. Next,thenumberofPODmodesincludedinthelow-orderbasisforestimation isbasedonthenumberofmodesthatcanbeestimatedwithameasurabledegreeof accuracy.Withaxedleadtime,thenumberofPODmodesincludedintheestimate isincreasedfrom1to100inordertoassessconvergenceoftheestimationerror.Once convergenceisachieved,includingmoremodesinthelow-orderprojectionhaslittleto nobenecialimpactbecauseoftheinabilitytoaccuratelyestimatetheadditionalmodes. TheeectofthisincreaseisplottedinFigure5-18B.Specicallyfortheselectionofthe numberofmodestoretain,theaverageerrormetric e r iscomputedrelativetotheenergy inallmodes,hence r = 1 correspondstothefull-dimensionaldata.Inthiscase,modes notretainedhaveestimatedcoecientsofzeroandthuscontributetheirfullenergyto theerrormetric.Thisisessentialforafaircomparisonofrelativeerrorcomputationsfor variousquantiesofretainedPODmodes.Thebenetofincreasingthenumberofmodes intheestimatediminishes,andtheerrorappearstoconvergetoabout0.88byabout 20modes.Therefore,20modesareretainedforMTD-mLSEoftheseparationregionPIV. Figure5-13showsthatthese20modescaptureabout62%ofthetotalaveragekinetic energywithintheacquiredvelocitymeasurements. 134

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Multipletimedelaysareemployedbyincludingasymmetricrangeofpointsabout theinitialleadtime.Again, max describesthehalf-spanwidthoftheintervaloverwhich themultipledelaysareutilized,andthisvalueisincreasedovertherangefrom0to 97 : 5 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 .Figure5-19suggeststhattheoptimal max valueforminimum e r occurs between0.04and0.06.ThenalMTD-mLSEparametersselectedforestimationofthe separationbubblePIVare max =52 : 6 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,and r =20. AnexamplevelocitysnapshotseeFigure5-6orFigure5-20Aisprojectedontothe selectedPODbasis,yieldingtheprojectedsnapshotinFigure5-20B.Thiseldisalso estimatedwiththeestablishedstochasticrelationshipandtheunsteadypressuresignals acquiredsimultaneouslywiththePIV,andtheresultisshowninFigure5-20C. TheprocessfordeterminingtheoptimalparametersforMTD-mLSEisrepeated forthewakemeasurements.Inthiscase,thesamesetofunsteadypressuretransducers Figure4-5wereacquiredsimultaneouslywithPIVmeasurementsofthenearwake region.With100PODmodes, 0 isagainvaried,andtheresultingerrormetricvalues areshowninFigure5-21A.Alagof 0 =0 : 21resultsintheminimumvalueof e k thoughthereisevidenceofperiodicuctuationswithinthedistributionofsingletime delays.Figure5-21Bshowstheasymptoticconvergenceofestimationaccuracyfor increasingthenumberofmodes,andthenumberofmodesisagainlimitedto20dueto themarginalimprovementbeyondthatquantity.Referringbacktothecumulativemodal energydisplayedinFigure5-15,about60%oftheaveragetotalkineticenergywithinthe measuredwakevelocityeldsiscapturedbytherst20modes. Multipletimedelaysareincorporatedsurroundingtheinitiallagtime,wherethe optimalvalueof max =29 : 2 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 isdeterminedfromFigure5-22.Atthesamplingrate of5kHz,thiscorrespondstoatotalof31timelagswithinthedelayintervalsatisfying Eq.5{1.ThenalMTD-mLSEparametersforestimationofthewakePIVdataare max =29 : 2 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 0 =0 : 21,and r =20.Withthesevalues,anexamplelow-order projectionofthewakePIVisestimatedandshowninFigure5-23Ccomparedtothetrue 135

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PODprojectioninFigure5-23B.Theestimatecapturesthebulkstructurefromtheupper andlowershearlayersofthetrueprojectionbutwithdiscrepancies.Bothprojectionand estimatelackthenescalestructuresthataretruncatedbythelow-orderPODbasis. 5.2.3Model-BasedEstimation Asthenamesuggests,model-basedestimationincorporatesamodelofthesystem dynamicsthatreducessomeoftherelianceplaceduponthesensorsusedtoestimatethe state.SeveralmodelsareidentiedusingtheinitialsetofMTD-mLSEestimates,but themosteectiveutilizesthedynamicalanalysisresultsthatappearintheproceeding section.TheidenticationprocessisdetailedinSection3.2.2.1.Torefresh,asetofDMD modesisselectedfromamongthetotalnumberavailable,whichisequaltotheorderof thePODestimates.Amodelisformedfromthedynamiccharacteristicsoftheselected DMDmodes.ThemodeldescribestheDMDtime-varyingcoecients b t k ,whichcanbe projectedbackontothereduced-orderPODbasisbytherelation a k = n b k ; {3 where n isthematrixofselectedDMDmodes. Inmodel-basedestimation,themodelpredictsthesystemstatebasedonthehistory ofthecoecients,andthensensormeasurementscorrectthepredictedvalues.For estimationofthePODcoecients,aKalmanltercausalandaKalmansmoother non-causalaredesignedandimplementedforeachoftheseparationbubbleandwake ows.Theunsteadypressuresignalsareusedtocorrectthemodeleddynamics. Forthemodel-basedestimationoftheseparationbubbledynamics,a6 th -orderlinear modelisformedfromthreeDMDmodeswithcharacteristicfrequenciesofSt L sep =0 : 87, 2.37,and2.73.TheKalmanlterfortheseparationbubbledynamicsisinitializedwith 136

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values ^ b + f; 0 =0 : 1[11 ::: 1] T P + f; 0 =5 I; where I istheidentitymatrix.Withthisinmind,thenoisecovariances,denedin Section3.2.2.2,areselectedtobe Q k = I and R k =1 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 I: Thevaluesof Q and R aredeterminedbyminimizingtheerrormetricinEq.5{2.Once theordersofmagnitudeareassessed,theresultsareinsensitivetofurthernetuningof theseparameters. Finally,theimplementationsoftheKalmanlterandsmootherareassessedby computingeachusingtheblocksofmicrophonedataacquiredduringbeforeandafter eachPIVsnapshotSection4.3.2.Theestimates ^ b k areprojectedbackontothePOD basistoyieldthecorrespondingPODestimates^ a k ,andthenthesevaluesarecompared tothetruePODprojectionsusingEq.5{2.Themeanresultsof e r forboththeKalman lterandsmootherareincludedinFigure5-19.Thelow-orderestimatesofanexample PIVsnapshotarealsoincludedinFigure5-20. Theseresultsdemonstratethatthemodel-basedestimationtechniquesincreasethe amountoferrorintheestimateswhencomparedtostochasticestimation,whichisat leastpartiallyattributedtothereductioninorderfromtwentystatestosix.However, theKalmanlterallowsforasimplecausalimplementationofastateobserverthatis amenabletostandardlinearcontroltheoryandmorerobusttosystemandmeasurement disturbances. Thisprocessisrepeatedforthelow-orderestimatesofthewakeregion.Forthis region,onlytwocharacteristicmodesareretainedfromtheDMDanalysisthatfollows. 137

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TheircorrespondingfrequenciesareSt L sep =0 : 88and2.49.Thenoisecovariancesareset to Q k = I and R k =1 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 I: TheerrorassociatedwiththeresultingestimatesofthePODprojectionsisincludedin Figure5-22,andanexampleestimatedsnapshotiscomparedtothetrueprojectionand stochasticestimateinFigure5-23.Again,themodel-basedestimatesincreasetheerror relativetotheoptimalstochasticestimates. 5.3DynamicAnalysis Thehighersamplingrateofthelow-orderestimatesfromtheprevioussectionhas asubstantialimpactonthedynamicalanalysisoftheseowregionsbecauseofthe increasedNyquistfrequency.However,theincreasedtemporalresolutioncarriessacrices intheformsofreduceddimensionalityandestimateddynamicswhicharedependent oncorrelatedpressure-velocityuctuations.Theinformationlostbythetruncation ofPODmodesisgenerallyaliatedwithsmall-scalestructuresandhigh-frequency uctuations.Theestimatorsarelimitedtothemoreenergetic,coherentowelements thatcorrelatewiththepressureuctuationssensedbytheunsteadypressuresensors. Therefore,dynamicalanalysisoftheestimatesisexpectedtoreectabiastowardsthe lowerfrequencycontentofthemorecoherentstructures.Essentially,thetemporalcontent oftheestimatesislimitedtotheunsteadypressurecontentthatcorrelateswiththe measuredvelocityelds,ormoreaccuratelythereducedsetofPODcoecients.Finally, theresultsofthedynamicanalysismethodsarerestrictedtothelowdimensionalityofthe estimates. Becausethestochasticestimatesexhibitasmallererrorthanthosefromthe model-basedtechniques,emphasisinthissectionisplacedonthederivedresults fromMTD-mLSE.However,analysisoftheKalmansmootherdynamicsisincluded 138

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forcomparison.DynamicmodedecompositionsDMDanddiscreteFourier transformsDFTareappliedtotheestimatedowelds.Eachmethodisinitially appliedtotheseparationbubbleandwakeregionsindividuallyinordertoisolatethe dynamicswithineachestimatedregion.Then,thesimultaneousestimatesfromboth regionsarecombinedasanaggregate,andtheanalysisisrepeated.Thelatterapplication helpstovisualizetheidentieddynamicsaspartofthelargeroweld,wheredynamics spreadandinteractbetweenthetwocapturedregions. 5.3.1DynamicModeDecomposition AsetofPODcoecientsisestimatedfromaseparatetime-seriesof4 10 5 unsteady pressuremeasurementsacquiredatasamplingrateof f s =5kHz.DMDiscalculated forthelow-orderestimatestoextractxed-frequencymodes.BecausetheMTD-mLSE estimatesarealinearcombinationoftwentyPODmodes,theDMDresultsarelimitedto twentymodesaswell.InthecaseoftheKalmansmootherestimates,theoutputisfurther limitedtotheorderoftheidentiedmodel.Unlikemoreconventionalspectralanalysis basedonFouriertransforms,DMDdoesnotspecifythefrequenciesofinterestbutrather determinesthisinformationbasedontheeigenvaluesofthestateevolutionSchmid, 2010. Beforethedecompositionisperformedforasetofestimates,theestimatedPOD coecientsarelow-passlteredwithan8 th -orderButterworthlterwithacuto frequencyof500Hz.Thenthetimeseriesisdown-sampledtoarateof1kHztoavoid potentialjitter"orreverseowinstancesamongtheestimatedtrajectories,andDMD modesarecalculated.TheDMDeigenvaluesandspectrumfromtheMTD-mLSE estimatesoftheseparationbubbleregionareshowninFigure5-24bluedots.Recall thatthemodalfrequency f isgivenby f = arg 2 f s ; 139

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where f s isthesamplingfrequency,andthegrowthrateofthemodeisgivenby k k .In theright-handplot,persistentoscillationsareidentiedbyincorporatingthegrowthrate ofthemodesovertwelveiterations 12 intothemodalenergy,representedbythenorm oftheDMDmode.Twelveisapproximatelythenumberofsampleswithinonecycleof theshearlayeroscillationatasamplingrateof1kHz.Theprominentpeakstherefore representoscillationsthatpersistrelativetotheotherextracteddynamicsduringone averagecycleoftheshearlayer.Theseprominentpeaksarealsoeigenvaluesclosesttothe unitcircleintheleft-handplot. TheprimaryoscillationisfoundatafrequencyofSt L sep =2 : 73 f =91 : 7Hz,which agreeswellwiththepreviouslyidentiedfrequencyoftheshearlayerseeSection5.1.2. PeaksarealsoobservedatSt L sep =2 : 37and0.87,withthelattersuggestiveofan inuentialoscillationfromthecharacteristicwakefrequency.However,thisfrequencymay bepronouncedintheKalmansmootherDMDanalysisbecauseitwasmodeledfromthe DMDresultsoftheMTD-mLSEestimates.TheMTD-mLSEestimatessuggestthatthis Ontheotherhand,theKalmansmootherisabletoimprovethestabilityofitsreduced setofmodes,i.e.allofitseigenvalueslieontheunitcircleinFigure5-24.Therefore,the wakefrequencyDMDmodeisinterpretedwithcaution. ThemodaldistributionsoftwoDMDmodepairscomputedfromtheMTD-mLSE estimatesaredisplayedinFigure5-25,visualizedbythevorticityoftherealand imaginarycomponentsfromeachpair.TheDMDmodalcontributionsforSt L sep =2 : 37 areneglectedbecauseoftheirstrongsimilaritytothosefromSt L sep =2 : 73.Themodal distributionsinFigure5-25Aareclearlyidentiedwithconvectingstructuresfromthe separatedshearlayer.Themodespotentiallyassociatedwiththewakefrequencyare displayedinFigure5-25B.Thedistributionofthismodepairsuggeststhatitsmost inuentialuctuationsaremoreconcentratedintheaftportionoftheuppersurfaceow, closetothetrailingedge.Thisareaofinuenceisconsistentwithanoscillationthatis likelyattributedtothewakeinstability. 140

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Thisprocessisrepeatedseparatelyforthewakeestimates.TheDMDeigenvalues andspectrumforthewakeareshowninFigure5-26.Thepersistentfrequenciesare similartothoseextractedfromtheseparationbubbleestimates,butthedominantpeak switchestothelowerfrequencyofSt L sep =0 : 88.ThehigherfrequencyisSt L sep =2 : 49. Thecorrespondingmodalcontributionsforeachoftheseoscillationsareillustratedin Figure5-27.Figure5-27Adepictstheconvectionanddownstreamgrowthofshedding vorticesfromthelowershearlayer.Thisisidentiedasthedominantwakemode.The higherfrequencyofFigure5-27Bcontributesalternatingsignsofvorticityfromtheupper surfaceandthelowershearlayer.Theupperpatternofstructuresdonotappearto emanatefromtheuppershearlayer,similartothedominantwakemode,butrather convectintotheacquiredeldofviewfromtheowabovetheatplatemodeland interactwiththeuppershearlayer.Thissheddingalsoappearstoinuencethelower shearlayer,wheresmallerscalestructuresthatgenerallymatchthespacingoftheupper patterarevisibleinthiswakeDMDmode.Thismodalcontentisthereforeattributedto convectionofthevorticesfromtheuppersurfaceandtheirinteractionwiththewake's upperandlowershearlayers. ThesimilarityofthefrequenciesfromtheprominentDMDmodessuggeststhatthey maybeinuentialinboththeseparationbubbleandwakeregions.Inordertoassessthe dynamicsfromamoreglobalperspective,thesimultaneousPODcoecientestimatesfrom eachregionarecombinedintoonerecord.DMDiscalculatedforthisaggregateset,and theresultingeigenvaluesareshowninFigure5-28.Theprominentpeaksshowthatthe characteristicwakeandshearlayerfrequenciespersist,thoughthelatterisnotidentied precisely.Instead,theshearlayerappearstoberepresentedbytwopeaksthatstraddle theexpectedfrequencyofSt L sep 2 : 68.OneoccursatSt L sep =2 : 50andanotherat St L sep =2 : 84. TheDMDmodaldistributionsthataccompanytheseeigenvaluesandtheonefor thecharacteristicwakeareshowninFigure5-29.ThewakemodeinFigure5-29Ashows 141

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thatthevorticesshedfromthelower-wakeshearlayeraremuchstrongerthanthosein theseparatedshearlayerontheuppersurface.However,theseparatedshearlayermodes Figures5-29Band5-29Cclearlyshowthetransitionthatoccursasthevorticesappear, convectdownstream,andinteractwiththelowershearlayerinthewake.Betweenthese two,thelower-frequencydynamicappearstoprotrudeintothewakemoresubstantially, whereasthehigherfrequencyismoreconnedtotheseparationbubbleregion. 5.3.2DiscreteFourierTransform SimilartotheDMDanalysis,theestimatedPODcoecientsarelow-passltered, butthesamplingrateremainsat5kHz.Then,aonesidedDFTiscomputedfrom1 10 4 samples.ThisprocessisrepeatedforbothMTD-mLSEandKalmansmootherestimates. ThenumberofpointsintheDFTcomputationisequaltotherecordlength,whichyields afrequencybinwidthof0.5Hz. ThenormsoftheDFTmodes fortheseparationbubbleestimatesareshownas aspectrumforallspeciedfrequenciesinFigure5-30.Thedominantpeakappearsat St L sep =2 : 65.0Hz,althoughthereareseveralsmallerpeaksinthebandwidthrange ofSt L sep [2 : 3 ; 3 : 2].Theredoesnotappeartobeanysubstantialfrequencycontent atornearthecharacteristicwakefrequency.Thespectrafromeachsetofestimates overlapquitewell,demonstratingagreementbetweenfrequencycontentcapturedby boththeMTD-mLSEandKalmansmootherestimates.Therealandimaginaryportions ofthetwodominantDFTmodesareshowninFigure5-31.Bothareclearlyaliated withtheseparatedshearlayer,thoughthelowerenergyhigherfrequencymodeisless representativeoftheoutershearlayerdownstreamof x=c =0 : 9.AlongwiththeDMD resultsfromtheprevioussection,thissuggeststhattheshearlayerconsistsofasmall rangeoffrequencies. InFigure5-32,theDFTspectraofthewakeestimatesrevealasubstantialshiftin energyfromhighertolowerfrequencies,followingthetrendfromsmallerstructuresinthe separatedshearlayertolargercoherentstructuresinthewake.Thereislessagreement 142

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ofmodalenergiesinthelower-frequencyspectrabetweentheMTD-mLSEandKalman smootherestimates.However,thefrequencypeaksalignwellforthisowregionaswell, whichcouldbeevidenceofnonlinearcouplingbetweentheupperandlowershearlayers. ThreedominantDFTmodesareshowninFigure5-33.Thersttwo,Figures5-33A and5-33B,exhibitsimilarstructureandfrequencytotheDMDmodesinFigure5-27.The thirdisafrequencysubharmonicoftheseparatedshearlayerbutthemodaldistribution suggeststhatthismaybemorealiatedwiththelowershearlayerinthewake. DFTisagaincalculatedfromthecombinedrecordthatconsistsofPODestimates frombothseparationbubbleandwakeregions.Theresultingspectraareshownin Figure5-34versustheDFTmodalnorms.Therearethreeprominentpeaks:oneforthe characteristicwakefrequency,oneforthecharacteristicseparationbubblefrequency,and ahigherfrequencypeakatSt L sep =3 : 14.TheDFTmodaldistributionsaredisplayed inFigure5-35andagreewellwiththeanalogouscaseofDMDmodesfromthejoint estimates. 5.4SummaryofResults Intheseexperiments,theimposedsuctionandblowingboundaryconditionsofthe proposedcanonicalcongurationareabletoinduceboundarylayerseparationfrom theuppersurfaceofaatplatemodel.Theseparatedowregionisinvestigated,and thetime-averagedseparationandreattachmentpointsaredeterminedtobenominally two-dimensionalacrossthecentralspanwiseregionoftheatplatemodel.Thebaseline separatedowcausesthewakebehindtheblunttrailingedgetodeectandexhibit asymmetricvelocityuctuations.Thenaturalfrequenciesassociatedwiththeseparated shearlayerandthewakearedeterminedfromdynamicmeasurementstobeSt L sep ; SL 2 : 68 f SL 90HzandSt L sep ; wake 0 : 89 f wake =30Hz,respectively. Stochasticandmodel-basedestimationmethodsareimplementedforlow-orderPOD descriptionsofPIVelddata.Fromtheestimates,dynamicanalysisisconductedby DMDandDFTmethods.Theidentiedmodesconrmthecharacteristicfrequency 143

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content.Furthermore,themodalcontentfromthisanalysisdemonstratesinteraction betweentheseparatedshearlayerfromtheuppersurfaceanddynamicswithinthewake ow. 144

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Table5-1.Characteristicfrequenciesfortheseparatedshearlayerandwake. DimensionalfrequencySt L sep = fL sep U 1 St = f x U x x =0 : 6 c St c = fc U 1 St h = fh U W f SL =90Hz2.680.0109.230.76 f W =30Hz0.890.0033.080.25 145

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A B C D Figure5-1.Time-averagedstreamwisevelocityprolessuperimposedonaverage streamlinesintheseparationandreattachmentregions.Theskin-friction coecientsarecomputedintheseregions.Themeasurementplaneis z=c =0 : 025.ASeparationregionvelocityeld.BReattachmentregion velocityeld.CSeparationregionskinfriction.DReattachmentregionskin friction. Figure5-2.Theaverageseparationsolidandreattachmentdashedlinesshownrelative tomodelcomponents,includingtheactuatorslotat x=c =0 : 61,unsteady pressuresensorslargerblackcircles,andstaticpressureportssmallerblack circles. 146

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A B C Figure5-3.AveragevelocitycomponentscomputedfromSPIVplanesatvariousslices alongthespanwisedimension.AStreamwisevelocity.BTransversevelocity. CSpanwisevelocity. 147

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Figure5-4.AverageturbulentstatisticscomputedfromSPIVplanesatvariousslicesalong thespanwisedimension. A B C Figure5-5.Powerspectraldensitiesofunsteadypressuresensorsundertheseparated shearlayer.Thesensorlocationsaremarkedinthebottomleftcornerofeach plot.ASensorsS1andS2.BSensorsS3andS4.CSensorsS5andS6. Figure5-6.InstantaneousvorticityfromaPIVsnapshotofthebaselineseparationregion. 148

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Figure5-7.Thecross-correlationcoecientbetweenhighlightedpressuresensorsatequal spanwisebutdierentstreamwisepositions. A B Figure5-8.AveragevorticitycomponentscomputedfromPIVofthenearwakeregionat z=c =0 : 025.Dotsindicatethelocationofhot-wiremeasurements.ABaseline separatedow.BStandardblu-bodywake. 149

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A B Figure5-9.AverageturbulentquantitiescomputedfromPIVofthenearwakeregionat z=c =0 : 025.Dotsindicatethelocationofhot-wiremeasurements.ABaseline separatedow.BStandardblu-bodywake. 150

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A B C D Figure5-10.Unsteadypressurespectraofwakepressuretransducersforbaseline separationandstandardblu-bodywake.ASensorS8.BSensorS9. CSensorS10.DSensorS11.Thesensorlocationsaremarkedinthe southwestcornerofeachplot. Figure5-11.CPSDbetweenunsteadypressuretransducersS8andS11forthebaseline separatedow. 151

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A B Figure5-12.Hot-wirespectraofasingleprobehotwireplaced0.14 c downstreamofthe trailingedge x=c =1 : 14.ABaselineseparatedow.BStandard blu-bodywake. Figure5-13.Energycontentoftherst r PODmodesoftheseparationbubblePIVdata i.e.aPODbasis f j g r )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 j =0 Figure5-14.SpanwisevorticityofPODmodescomputedfromPIVeldsofthebaseline separationregion.Themodesarearrangedinorderofdecreasingenergy content. 152

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Figure5-15.Energycontentoftherst r PODmodesofthewakePIVdatai.e.aPOD basis f j g r )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 j =0 Figure5-16.SpanwisevorticityofPODmodescomputedfromPIVeldsofthebaseline separationregion.Themodesarearrangedinorderofdecreasingenergy content. 153

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Figure5-17.Schematicillustratingthedelayintervalrelativetotheestimationtimeof t k Inthisexample, j 0 j > max theintervaldoesnotspan t k and 0 > 0the centermostprobemeasurementwithinthedelayintervaloccursbefore t k 154

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A r =100, 0 =[ )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 58 ; 0 : 58] B r = 1 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(39 : 0 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 Figure5-18.Errormetricforsingle-timedelaymLSEoftheseparationbubbleregionwith a100PODmodesandvaryingdelaysandbaxeddelayof 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 andvaryingmodequantitiestoestimate. Figure5-19.ErrormetricforMTD-mLSEoftheseparationbubbleregionwith r =20, 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,andvaryingvaluesof max 155

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A B C D E Figure5-20.SpanwisevorticityofanexamplePIVsnapshotoftheseparationbubbleand its20 th -orderPODestimates.AAcquiredPIVsnapshot.BPOD projection.CStochasticestimate.DKalmanlterestimateEKalman smootherestimate. A r =100, 0 =[ )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 58 ; 0 : 58] B r = 1 0 =0 : 21 Figure5-21.Errormetricforsingle-timedelaymLSEofthewakeregion.A100POD modesandvaryingdelays.BFixeddelayof 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(39 : 0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 andvarying modequantitiestoestimate. 156

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Figure5-22.ErrormetricforMTD-mLSEofthewakeregionwith r =20, 0 =0 : 21,and varyingvaluesof max A B C D E Figure5-23.SpanwisevorticityofanexamplePIVsnapshotofthewakeandits20 th -order PODestimates.AAcquiredPIVsnapshot.BPODprojection. CStochasticestimate.DKalmanlterestimateEKalmansmoother estimate. 157

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Figure5-24.DMDeigenvaluesofMTD-mLSEestimatesblueandKalmansmoother estimatesredfortheseparationbubbleregion. A B Figure5-25.RealandimaginaryportionsofselectseparationbubbleDMDmodes computedfromMTD-mLSEestimates.ASt L sep =2 : 73.BSt L sep =0 : 87. 158

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Figure5-26.DMDeigenvaluesofMTD-mLSEestimatesblueandKalmansmoother estimatesredforthewakeregion. A B Figure5-27.RealandimaginaryportionsofselectwakeDMDmodescomputedfrom MTD-mLSEestimates.ASt L sep =0 : 88.BSt L sep =2 : 49. 159

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Figure5-28.DMDeigenvaluesofMTD-mLSEestimatescomprisedfromboththewake andseparationbubbleregions. 160

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A B C Figure5-29.RealandimaginaryportionsofDMDmodesfromMTD-mLSEestimates comprisedfromboththewakeandseparationbubbleregions. ASt L sep =0 : 90.BSt L sep =2 : 50.CSt L sep =2 : 84. 161

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Figure5-30.DFTmodenormsoftheMTD-mLSEestimatesblueandKalmansmoother estimatesredfortheseparationbubbleregion. A B Figure5-31.RealandimaginaryportionsofselectseparationbubbleDFTmodes computedfromMTD-mLSEestimates.ASt L sep =2 : 65.BSt L sep =3 : 14. Figure5-32.DFTmodenormsofthewakeMTD-mLSEestimates. 162

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A B C Figure5-33.RealandimaginaryportionsofselectwakeDFTmodescomputedfrom MTD-mLSEestimates.ASt L sep =0 : 95.BSt L sep =2 : 65.CSt L sep =1 : 33. Figure5-34.DFTmodenormsoftheMTD-mLSEestimatesfromboththewakeand separationbubbleregions. 163

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A B C Figure5-35.RealandimaginaryportionsofDFTmodesfromMTD-mLSEestimates comprisedfromboththewakeandseparationbubbleregions. ASt L sep =0 : 95.BSt L sep =2 : 65.CSt L sep =3 : 14. 164

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CHAPTER6 OPEN-LOOPCONTROL Open-loopcontrolresultsarepresented,includingresultsinwhichtheperiodicZNMF forcingtargetsthecharacteristicowfrequenciesdeterminedfromthepreviouschapter. Thebenetsofcontrolareevaluatedbasedonstaticpressurerecoveryandreductionin separationbubblesize.Thecostofcontrolisassociatedwiththecoecientofmomentum C ,whichisproportionaltothepowerrequiredtooperatetheZNMFactuator. First,theactuatorischaracterizedbyhot-wireanemometrymeasurementsfor theforcinglevels,frequencies,andwaveformtypesusedinopen-andclosed-loop control. 1 Theparasiticacousticsgeneratedbytheactuationandmeasuredbythe unsteadypressuresensorsaremodeledandsubsequentlyattenuatedbysubtraction ofthepredictedacousticswithinthemeasuredsignals.Flowmeasurementresultsare providedforthecontrolledowinboththeseparationbubbleandwakeregions.Initially, theresponsetohigh-frequencyforcingismeasured.Thisisfollowedbyamodulation frequencyandamplitudeparametricstudythatisrestrictedtotherangeofcharacteristic frequenciesofthebaselineseparatedow.PIV,hot-wireanemometry,andsurfacepressure areutilizedtodeterminetheeectivenessofthecontrolwithregardstoreattachment.A coupleofthecontrolcasesareusedtoperformstochasticestimationofthetime-resolved dynamicsbasedonthesynchronizedmeasurementsofPIVandunsteadypressure,anda dynamicalanalysisoftheestimatesfollows. 6.1ActuatorCharacterization Priortocontrol,theactuatorischaracterizedfortherangeoffrequenciesand amplitudestested.Thisbeginswithindependentmeasurementsoftheslotoutputvelocity overeachofthefourembeddeddiscs.Then,ahigh-frequencysinusoidaloperating conditionisselected,andthespanwisevariabilityoftheoutputfromtheactuatorslotis 1 Closed-loopresultsaregiveninChapter7. 165

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measuredforseveralforcingamplitudes.Finally,thecenterslotischaracterizedforthe burstandamplitudemodulatedwaveformsthatareappliedinbothopen-andclosed-loop controlschemes. 6.1.1IndividualDiscs Theinitialtaskinthefullcharacterizationoftheactuatoristoisolatetheforcingto eachofthefourdiscstodeterminethejetresponsetoasimplesinusoidalinput.However, becauseaclampeddisc'sresonanceisassociatedwithpeakdeection,operatingatthis conditionmakesthediscsusceptibletofailure.Therefore,operatingtheactuatorata secondresonanceisdesiredbecausetheamplitudeofthediscdeectionsistypicallymuch smaller.AsdisplayedinFigure6-1,eachdisc'sresponsetoa50V pp sinusoidalinputis measuredbyahotwirelocatedcentrallyintheslotaboveeachdisc.Thecarrierfrequency isvariedfrom50to2600Hzinincrementsof50Hz.Thoughthisfrequencyspacingisa bitcoarsetoaccuratelydeterminetheactuatoroutputresponse,maximumrmsvelocity outputgenerallyoccursbetween700and800Hz.Anotherbenetofnotoperatinginthis rangeistheavoidanceofsubstantialoutputdierencesamongdiscsifchoosingasingle inputfrequencyforallfourdiscs,whichcouldresultinnon-uniformoutputacrosstheslot. Therearesecondarypeaksintheoutputvelocitiesfromapproximately1600to 2600Hz.Thisrangeisthentestedagainforvariousamplitudes,andtheresultsare plottedinFigure6-2.ThoughtheoutputoverdiscsA1andA4thediscsontheendsof thearrayismaximumnear2200Hz,theoutputoftheentireactuatorismaximizedifthe optimalconditionsareselectedbasedondiscsA2andA3.Therefore,thecarrierfrequency isselectedas2050Hz. 6.1.2SpanwiseVariability Thenextstepinthecharacterizationistomeasuretheentirespanwiseoutputofthe actuatoroperatingat2050Hz.Theoutputspeedismeasuredin2mmincrementsforthe entireslot.Figure6-3showsv rms forfouramplitudelevels.Thedisccenterlocationsare plottedasdashedverticallinesforconvenience.Theaverageoutputappearstobeuniform 166

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acrossthespanwisedirectionoftheslot,particularlyinthespanof z=c =[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 15 ; 0 : 15] wherev rms istowithin0.2m/softhespanwiseaverage. 6.1.3ModulatedInputs TheburstmodulationBMwaveformtypeisselectedtomodulatethecarriersine waveforopen-loopcontrol.BMenablestheactuationtotargetspecicfrequencieswhile maintainingahighoutputbasedonthecarrierfrequency.TwoperiodsofanexampleBM waveformareshowninFigure6-6withsixcarriercyclesandamodulationperiodof f )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 m Anintegernumberofcarriercycleswithinasingleburstisenforcedwiththecondition thatthenumberofcyclesistheminimumamounttoensureadutycycleofatleast50%. Theactuatoroutputatthecenteroftheslotismeasuredformodulationfrequencies between0and200Hzinincrementsof10Hzandforamplitudesfrom10to80V pp in incrementsof10V pp .Basedonthemeasuredoutputsofv rms thecoecientofmomentum iscomputedas C = A j v 2 rms A sep U 2 1 ; where A j istheareaoftheactuatorslot, A sep = L sep w istheapproximatefull-spanarea oftheseparationbubble,and w isthemodelspan.Becauseallcontrolresultsthatfollow areforaconstantRe c =10 5 ,themeasuredactuatoroutputtoBMisplottedasthe momentumcoecientinFigure6-5Aassumingthecorrespondingfreestreamowspeed. Similarly,themodulationfrequencyisnormalizedas F + m = f m L sep U 1 : Consequently,thisshowsthetrendin C forarangeof F + m valuesthatareapplicable toopen-andclosed-loopcontrol,althoughtheactual U 1 iszeroduringactuator characterization.Intheplot,thesinusoidalinputofthecarriersignalisrepresented by F + m =0.Thereisanimmediatedrop-oinoutputasBMisenabled,whichis attributedprimarilytotheapproximate50%dutycycleofthemodulatedwaveforms. Anothercontributingfactoristhatthetransientsinvolvedwithstartingandstopping 167

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theactuatorinaburstbecomealargerportionofthemodulationcycleas F + m increases. Thisisapotentialreasonfortheslightlydownwardtrendingoutputwhile F + m increases. Controlresultshereafteruse C whenapplicableand F + m whenreferencingaparticular BMcontrolpair. InadditiontotheBMactuation,thecharacterizationforamplitudemodulationAM isrequiredforsomeclosed-loopcontrolresults,inwhichthestatevariablesofthe identiedsystemaresinusoidalinnatureandcanbeusedforproportionalfeedback. Thebaselineresultshavealreadydemonstratedthattheseparatedowandwakeexhibit characteristicoscillationsthatarewellrepresentedbysinusoidaloscillations.Thus, characterizationoftheactuatorwithanAMwaveformisshowninFigure6-5Bandis usefulforestimatingtherequired C levelsforsomecasesofclosed-loopcontrol. 6.1.4AcousticContamination TheunsteadypressuremeasurementsfromsensorsindicatedinFigure4-5are contaminatedbytheparasiticacousticsandpotentialvibrationsthataccompany actuation.Inordertoinvestigatehowmuchtheforcingfromtheactuatoraects thepressuresensors,theactuatorisoperatedwhilethewindtunnelandseparation systemarenot.Theacousticresponsefortheunsteadypressuretransducersismodeled rstforBMinputsignals.Thismethodmodelstheaverageresponseto f m =90Hz actuation F + m =St L sep ; SL ,whichisprevalentinbothopen-andclosed-loopresults. AnothermodelisidentiedbasedonanAMsignalcapableofawider,butnite,range offrequencies.Thissecondmodelisonlyrelevanttosomeclosed-loopresults,wherethe statefeedbackofstateoscillationsiswellrepresentedbytheAMinput. Apredictionandattenuationofthemeasuredacousticsismotivatedbythedesireto isolatethemeasuredpressureuctuationstohydrodynamics.Thesepressureuctuations arethenusedtoanalyzeandestimatethecontrolledow.However,theacousticsmay beasignicantfactorintheperturbationandcontroloftheseparatedshearlayerduring excitation.Infact,manystudieshaveutilizedthereceptivityofaseparatedshearlayerto 168

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acousticexcitationforcontrolpurposesCollins,1979;Nishioka etal. ,1990;Roussopoulos, 1993;Huang,1996.Yet,Kotapati etal. 2007andKotapati etal. 2010demonstrate eectiveforcinginincompressiblesimulationsofacanonicalseparatedow,voidof acousticpressureuctuations.Therefore,themechanismforeectivecontrolisnot attributedstrictlytotheunsteadyZNMFforcingorthegeneratedacoustics,butthe analysisofthecontrolledowandestimationofitsglobalvelocityuctuationsisrestricted tothelarge-scalehydrodynamicpressureuctuations.Futureexperimentationisnecessary toassesstheeectoflocalacousticexcitationaloneonthecanonicalseparatedow. 6.1.4.1Burstmodulatedacoustics TheactuatorisdrivenbyaBMinputsignalwith f m =90Hzand f c =2050Hz becausethedynamicresultsfromthismodulationfrequencyareimportanttobothopenandclosed-loopresults.Theamplitudeofthewaveformisincreasedinincrementsof 10V pp fromthersttestat30V pp tothelasttestat80V pp Thedataaresampledat5kHz,whichmatchestheacquisitionrateduringbaseline andcontrolmeasurements.Atotalof4 10 5 samplesisacquiredpersensor.Aportion oftherecordacquiredforsensorS1isshowninFigure6-7Aalongwithitscorresponding powerspectraldensityPSDinFigure6-7B.ThissignalistheresponsetoBMactuation at f m =90Hzand80V pp largestcalibratedamplitude.Muchofthepowerintheraw signalisconcentratedinthehigh-frequencycontentofthecarriersignal.Therearealso substantialpeaksfromthemodulationtoneanditshigher-orderharmonics. Thehigh-frequencyacousticsarediculttomodeldirectly,andanyresulting errorwouldbesubstantialrelativetothemagnitudeofthehydrodynamicuctuations. Therefore,thehighfrequencycontentassociatedwiththecarriersignalislteredwith an8 th -orderlow-passButterworthlterwithacutofrequencyof500Hz.Theltered output,alsoincludedinFigure6-7,becomestheobjectivesignaltopredictfromthe knowninputsignal. 169

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Thelowerfrequencytonesassociatedwiththemodulationoftheinputremaininthe lteredsignal.Anaveragecycleofthissignaliscomputedforeachsensorandforeach testedinputamplitude.TheaveragesforsensorS1areshowninFigure6-8Aatthetested inputamplitudesbutwitheachsignalnormalizedbyitsrmsvalue,whichissuitableto collapsethecycles.TheseaveragesareshownwithzerophaselagrelativetotheBMinput signalFigure6-6.Thus,withknowledgeoftheinputBMsignal'sphase,eachsensor's normalizedaverageacousticresponsecanbepredicted,butnotyetthemagnitudeofthe response. Thenormalizedaveragecyclesfor60V pp arechosentorepresentthenormalized responsesforallsensors.Becauseoftheagreementamongthecollapsedaverageresponses, themagnitudesofthepredictedacousticsignalsaredeterminedbytherelationship betweentheinputvoltageamplitudeandthermsvalueoftheaverageacousticresponse, whichiswelldescribedbyasecondorderpolynomial.Thisrelationshipisshownin Figure6-8B.Thisprocessisrepeatedforallrelevantsensors. Thelow-frequencyacousticresponsecanbepredictedwithknowledgeoftheBM inputsignal'smagnitudeandphase,eitherduringcontrolorasaseparatelyacquired signal.Figure6-7Ademonstratesthisbycomparingtheoriginallow-passlteredsignal withitspredictedsignalandtheresultingsubtraction.Thespectralcontentisalso comparedbeforeandaftersubtractionofthepredictedacoustics.Completesuppressionof thecontrol-generatedacousticsisnotaccomplished,butsignicantattenuationisachieved bymodelingtheaverageresponse.Thepeakattenuationofthe90Hzmodulationfor sensorS1isgiveninFigure6-9forarangeofpeak-to-peakvoltageamplitudes.Forthe 40V pp case,themaximumamplitudeoftheacousticpressureuctuationsisontheorder of0.3PaforsensorS1.Giventhecorrespondinghydrodynamicpressureuctuations duringcontrolatRe c =10 5 areontheorderof3to4PaforsensorS1Figure6-10B, anattenuationof51dBreducestheacousticlevelstolessthan2%ofthemaximum hydrodynamicuctuations.Allproceedingunsteadypressuredataareprocessedto 170

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attenuatetheportionofthemeasuredpressureuctuationsduetotheacousticsfromthe BMactuation. 6.1.4.2Amplitudemodulatedacoustics ThesensorresponsestotheacousticsfromAMforcingarealsomodeledbecause oftheirimportancetoclosed-loopcontrol.Thismodelingisapplicabletocontrolthat modulatesthecarriersignalbasedonstatefeedbackthatresultsinanear-sinusoidal modulation,asopposedtoburstmodulation.However,thefrequencyoftheoscillationsis notxed,andcanthusvarywiththeuctuationsoftheow'sstatevariables.Inorderto modelthisresponse,anAMwaveformisspeciedwitharepeatingchirpandincreasing amplituderamp.Themodulationfrequencyassumestheroleofthechirpfrequency, varyinglinearlybetween10Hzand150Hzin8secondintervals.Theamplitudeofthe AMsignalislinearlyincreasedfromabout5V pp to85V pp overatotalacquisitiontime of70seconds.Thesevariationsinwaveformparameters,andsensorS1'sresponseare includedinFigure6-11. SimilartothepreviousBMexample,thesignalislow-passltered.Anautoregressive exogeneousARXpolynomialmodelisdeterminedusingMATLAB'sSystem IdenticationToolbox.TheARXmodelhastheform p 0 t k + a 1 p 0 t k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 + + a n a p 0 t k )]TJ/F23 7.9701 Tf 6.587 0 Td [(n a = b 1 u t k )]TJ/F23 7.9701 Tf 6.587 0 Td [(n d + + b n b u t k )]TJ/F23 7.9701 Tf 6.587 0 Td [(n b )]TJ/F23 7.9701 Tf 6.587 0 Td [(n d +1 + e t k ; where p 0 isthelow-passlteredunsteadypressure, u isthemodulationinput, n a isthe numberofpoles, n b isthenumberofzeroesplusone, n d isthenumberofdelaysbetween theinputandoutput,and e isassumedtobeawhite-noisedisturbance.Throughtrial anderror,theleastresidualoutputisdeterminedwithparametersof n a =2, n b =2, and n d =1.Thispredictionandtheitsresidualaftersubtractionfromthetruesignal forsensorS1isshowninFigure6-11.Thesubtractedoutputiswithin 0 : 5Pauntil about60V pp seconds.Themodelbeginstoloseaccuracybeyondthatamplitude, 171

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butstillmanagestoattenuatethesensorresponsetothechirpedactuation.Thismodelis implementedinsomeoftheclosed-loopexperimentsdetailedinthenextchapter. 6.2ControlledSeparationBubble Open-loopexperimentssweepthroughtherangeofinputamplitudesandmodulation frequenciesgiveninFigure6-5A,aswellasthesine"caseswithoutmodulationofthe carrierwave.Surfacepressureismeasuredfromthestaticportswithrecordlengths of400samplesperportandfromtheunsteadysensorsatasamplingrateof5kHz for80seconds.Ahotwireisplacedinthewaketosupplementtheunsteadypressure measurementsfromthemodelbase.Fromthesemeasurements,aselectfewcasesare investigatedfurtherwithPIVmeasurementsoftheseparationbubbleandwakeregions. Theseresultsarecomparedtothedatafromthebaselineseparatedow. Theinitialsetofcontrolexperimentsappliesthesinusoidalinputwaveformno modulationatthevarious C levelsprovidedbytheactuatorcharacterization.The staticanddynamicsurfacepressuresaremeasured,andtheeectivenessofthecontrol isinvestigatedrstwithoutmodulation.Thecoecientofpressure C p = p s )]TJ/F22 11.9552 Tf 12.671 0 Td [(p 1 =q isdenedwiththedynamicpressure q computedusingthefreestreamspeedattherst pressuretaplocatedat x=c =0 : 27.Theseresultsalongwiththeunsteadypressure measuredatlocationS4areshowninFigure6-12.Errorbarsshow95%condence intervalsforthebaselinecase,buterrorbarsforthecontrolcasesareneglectedforclarity. Becausethedynamicpressureforthisowconditionissmall,theseerrorbarsarenot muchwiderthanthoseforthenoiseoor. Pressurerecoveryisobservedformanyofthesecontrolcases,andtheadverse pressuregradientbecomesmoregradualoverthebaselineseparationregionforincreasing valuesof C .Intheunsteadypressurespectra,increasing C leadstoincreasedbroadband turbulenceuntil C =3 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 .Furtherincreasing C decreasesthebroadbandlevels ofsensorS4untiltheshearlayerfrequencySt L sep ; SL =2 : 68issuppressedforthecontrol casesof C =7 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 and1 : 1 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 .Thereisalsoaslighttonalpeaknearthe 172

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characteristicwakefrequencyofSt L sep ; wake =0 : 89forthesetwocontrolcases,which suggeststhattheseparatedshearlayermaybefullyattachedforthesecasesandthatthe wakeinstabilityismoreinuentialupstream. Next,BMactuationisimplementedforall C and F + m valuesshowninFigure6-5A total.Themodulationfrequencyisvariedfrom10to200Hzinincrementsof 10Hzsuchthatboththecharacteristicshearlayerfrequencyandthecharacteristicwake frequencyareincluded.Thegeneraltrendof C p distributionfortherangeof F + m values isshowninFigure6-13forallthecasesof40V pp .Withinthisrange, C variesbetween 2 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and6 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 .However,thisisstillanarrowrangeandthe C p distributions areassumedtovaryprimarilybythechangein F + m .Giventhisassumption,theoptimal rangeof F + m iscenteredaroundtheshearlayerfrequencyofSt L sep ; SL =2 : 68.Signicant recoveryalsooccursatlowfrequencyforcing,thoughtoalessdegree. Fromamongthe160totalcombinations,fourinterestingcasesareselectedforfurther investigation.Becauseofthetrendobservedbyvarying F + m andbecausetargetingthe naturalowfrequenciesisafocusofthisthesis,thecasesfor F + m =0 : 89 f m =30Hz and F + m =2 : 68 f m =90Hzareselectedforsmaller C valuesthatexhibiteective controlvia C p distributionsandunsteadypressurespectraintheseparationregion.The rangeofmodulationfrequenciessurrounding F + m =St L sep ; SL isanoptimumintermsof lowerlevelsof C basedonpressurerecoveryinthe C p distributions.Becauseofthis,a caseat F + m =St L sep ; SL butwithahigher C isadded.Finally,acaseofsinusoidalforcing withthelargestcharacterized C isincluded.The C valuesforthesecasesarelistedin Figures6-14and6-15. The C p distributionsintheseparatedregionaftoftheactuatorareplotted inFigure6-14forthefourselectedcontrolcasesandtwoadditionalsinecasesfor comparison,oneat C =5 : 1 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 andanotherat C =1 : 5 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 .Thesefew casesaredividedintothreegroupsbasedonthevalueof C .Therstgroupcontains threecasesoflow-levelforcing,characterizedby C 5 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 .Thebenetofmodulation 173

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isclearwhencomparingthe C p distributionsfromtheBMcasestothesinusoidalinput case.Thepressurerecoveryisgreatestforthemodulationfrequencyof F + m =2 : 68while thesinusoidalforcingschemehasanegligibleeectonthebaselinepressuredistribution. ThenextgroupcomparesBMforcingwith F + m =2 : 68andagreater C valueof1 : 9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 toacomparablesinusoidalforcinglevel.Again,theBMhasamuchgreaterimpacton thestaticpressurerecoverythanthesinusoidalforcingscheme.Theincreasedlevelresults inincreasedpressurefurtherupstreamthanthelow-levelcases.Thenalgroupconsists ofonlythehigh-levelsinusoidalforcingcase C =0 : 011becauseBMforcingdoesnot reachasimilar C value.Thismaximumoutputcaseproducesthemostgradualpressure recovery,whichmaybeindicativeofamorereducedoreliminatedseparationbubble. PIVdataareacquiredphaselockedateightequallyspacedphasesofthemodulation cyclefortheseparationregionabovetheatplate.Thetime-averaged u -velocityfor theentiremodulationcycleisshowninFigure6-15foreachofthecontrolcases.The baselineseparatedowisalsoincludedforcomparison.Inordertoobservethecontrol eectiveness,theseparationbubbleheightiscalculatedintwowaysforeachofthesecases. First,theseparatedstreamlineisestimatedastheminimumheightstreamlinethatdoes notshowrecirculationeects,andisthereforenotintherecirculationbubble.Thenthe height H sep,1 isthemaximumvalueofwallnormalposition n reachedbytheseparated streamline.Aseconddenitionisthemaximumheightforwhichthe u -velocityiszero, or H sep,2 = f max n j u x;n =0 g .Thislinefor u =0doesnotenclosetherecirculation regionbutcontainstheareaofaveragereverseow.BothlinesareincludedinFigure6-15 andthecomputedheights H sep,1 and H sep,2 areplottedinFigure6-16forthesecontrol cases. Allofthecontrolcasesdelayseparationandpromoteearlierreattachmentbut withslightlyvaryingdegreesofeectiveness.Theredoesnotappeartobeasubstantial dierenceinthedistributionof u -velocityamongthetwolowest C values.Thelarger modulationfrequenciesthatexcitetheshearlayerfrequencyachieveaslightlysmaller 174

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separationbubble.Thenexthighestoutputcaseof C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 and F + m =2 : 68reduces theaverageseparationbubblesizeevenfurther.Thehighestoutputof C =0 : 011with nomodulationappearstosuppressseparationaltogetherbutattheexpenseofan8.5x increasein C .Thesemeasurementsareconsistentwiththe C p prolesinFigure6-14. Theseparatedshearlayerisunstableandsensitivetoperturbations,describedby theKelvin-Helmholtzinstability.Thisinstabilityyieldsperiodicformationandrelease ofvorticesthatconvectandgrowalongtheshearlayer,witheventualdissipationand potentialvortexmergingdownstream.Itiswellestablishedthatperiodicexcitationat thisnaturalfrequencyincreasesthecoherenceofthespanwisevortices,whichinturn enhancesmixingandreducesaperiodicuctuationstypicaloftheunforcedow.This eectisutilizedbyZNMFactuationthattargetsmultiplefrequencyscalesbymodulating thehigh-frequencyforcingnecessarytodrivetheactuatornearresonance. Thedynamicsoftheselectedmodulatedforcingschemesareanalyzedbyinspectionof thephase-averagedvelocityelds.Eightequallyspacedphase-averagedvorticitycontours areplottedinFigure6-17fortwocases: C =5 : 7 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and F + m =0 : 89Hzand then C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and F + m =2 : 68Hz.Theshearlayervortexsheddinglockson tothemodulationfrequencyforbothcases.Inthephasesofthe F + m =0 : 89modulation inFigure6-17A,successivevortexpairsappeartomergeintoalargervortexbefore detachingandconvectingdownstream.Thiscausesspacingbetweenconvectingvortices toincreasebecauseofthedecreasedrateofformationwhencomparedtotheshearlayer ofthebaselineseparatedow.However,theaveragemagnitudeofthevorticitywithinthe shedvorticesislargerthantheothercaseofsimilar C .The F + m =2 : 68forcingoccurs atornearthenaturalsheddingfrequencyoftheshearlayer.Thishigherrateofvortex roll-upresultsintighterspacingbetweenconvectingvortices.Theseparatedshearlayer isreceptivetothisburstrateatthenaturalfrequency,whichenhancesmixingofthe high-momentumfreestreamwiththelow-momentumownearthesurface.Theresultisa slightlyshorteraverageseparationbubblewithcomparablevaluesof C 175

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Thesecasesofmodulationarecontrastedtotheeectivesinusoidalforcingstrategy. Phase-averagedresultsareshownforthiscaseinFigure6-18,wherethephasesarerelative tothecycleofthehigh-frequencysinewave.Theseaverageoweldsshowthatthe high-levelforcingpiercestheboundarylayerduringexpulsion,acceleratingthestreamwise owbeforereductionoftheboundarylayerduringtheingestionportionofthecycle.This forcesoscillationsofthefreestreamspeedandperiodicmomentumchangeswithinthe boundarylayerthatpreventseparationfromtheimposedboundaryconditions. Therefore,high-amplitudesinusoidalforcingwitha C of0.011isabletosuppress boundarylayerseparationaltogetherforthisow.However,amodulated,orpulsed, actuationisshowntoatleastpartiallyreattachtheowwithamuchsmalleroutput of C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 bytargetingthenaturalshearlayerinstability.Theshearlayer locksontothepulsing,whichresultsinstrong,periodicvortexsheddingthatlowersthe separatedshearlayer,enhancesmixingfromthefreestreamintotherecirculationregion, andreattachestheowfurtherupstreaminatime-averagedsense.Becauseoftheshear layer'sreceptivitytoforcingnearitscharacteristicsheddingfrequency,theactuatoroutput canbereducedbyasmuchasanorderofmagnitudetoyieldsimilarpressurerecovery. 6.3ControlledWake Ahotwireplacedat y=c =0and x=c =1 : 14markedinFigure6-20Bismeasured duringcontrolinordertodetecttheuctuationsandcomparetheselectedcontrolled resultstothebaselineseparatedowandtheblu-bodywake.Figure6-19conrms thattheselectedcontrolcasesalterthewakedynamicstobehavemoresimilarlytothe standardblu-bodywake.Theagreementofthespectrasuggeststhatthecontrolcases causethewakeowtoresumeKarman-likevortexshedding,thoughtheresultsatthis measurementlocationdonotdiergreatlyfromcasetocase.Thenearwakeowdoes notappeartodiersignicantlyduetotheforcingthattargetsthecharacteristicwake frequencywith F + m =0 : 89.Therefore,targetingthisfrequencydoesnotappeartoenhance oralterthewakedynamicscomparedtotheotherselectedcontrolcases. 176

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PIVisalsoacquiredinthenearwakeregion.Forconciseness,theresultsare restrictedtothehighest C casewithasinusoidalinput.Allothercasesshowsimilar averageanductuatingowpatterns.Figure6-20Bshowstheaveragespanwisevorticity forthecontrolledoweld,whichiscomparedtothebaselineseparatedwakeow inFigure6-20A.Thecontrolled-owwakeisstillasymmetricsimilartothebaseline separatedow,thoughtoalesserdegree.Thedierenceismoreapparentintheturbulent statistics,showninFigure6-21alongwiththecorrespondingeldsfromthebaseline separatedowFigure6-21Aandtheblu-bodywakeFigure6-21B.Forthecontrolled owFigure6-21C,regionsofhighlyuctuatingvelocityaremoretop-bottomsymmetric likethestandardblu-bodywake.Thisimprovedsymmetryinvelocityuctuations supportshot-wireresultsfromFigure6-19. 6.4EstimationofControlledFlow Aswiththebaselineow,thehydrodynamicuctuationsmeasuredbytheunsteady pressuresensorsareusedinconjunctionwithsynchronousPIVmeasurementsofthe separationbubbleandwakefortime-resolvedestimation,althoughtheseregionsare acquiredseparately.Unliketheestimationofthebaselineow,thereduced-orderbasisis determinedbyglobal"PODGlauser etal. ,2004,encompassingdatafromtwocontrol casesandthebaselineseparatedow.The F + m =2 : 68HzBMcontrolcasesarechosen becauseoftheirlowoutputlevelsrelativetothesinecaseandbecausetheyshouldhave similarmodalstructures. PODmodesarethusobtainedfromarichervarietyofsnapshotsthatcanincludethe eectsofactuationonthebaselineowGerhard etal. ,2003;Glauser etal. ,2004;Rowley &Juttijudata,2005.Inthecaseoftheseparationbubble,200PIVsnapshotsareincluded fromabaselinePIVmeasurement,1600snapshotsareincludedfromthe C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and F + m =2 : 68controlcase,andanother1600areincludedfromthe C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 and F + m =2 : 68controlcase.Similarly,200snapshotsofthebaselinewakeand500snapshots fromeachofthecontrolcasesisincludedinthewakeglobalPODcomputations.Then,a 177

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linearstochasticrelationshipisidentiedbetweentheunsteadypressureandtime-varying PODcoecients.Thismappingfunctionisusedtoestimatethetime-resolvedPOD coecientsfromeachofthecontrolcases. 2 Theestimatesallowapplicationofdynamic modedecompositionDMDanddiscreteFouriertransformDFTanalyses. 6.4.1Reduced-OrderBasis ThoughthePODbasisiscomputedfrommultipleowconditions,theactual decompositionmustaddressthewakeandseparationbubbleregionsseparatelybecause thesnapshotsfromthetworegionsarenotacquiredsimultaneously.Thesummedenergy fractionsfortherst100globalPODmodesoftheseparationbubblePIVareshown inFigure6-22A.Theequivalentplotforthewakeregion'sglobalPODisdisplayed Figure6-22B.Theseenergydistributionscannotbedirectlycomparedtothebaseline PODdistributionsbecauseofdieringrecordlengths,butit'snotableatleastthat theseenergyfractionssuggestmoreenergyisconcentratedintherstfewmodes.For instance,thewake'scumulativeglobalPODenergyexceeds60%ofthetotalwithjustfour modes.ThebaselinewakePODrequiresapproximately20modestoreachthatenergy fractionFigure5-15.Physically,thissuggeststhatmoreowenergyisconcentratedin thelarge-scale,coherentstructuresofthecontrolledwakeowthanthereisinthebaseline separatedwakeow. TheglobalPODmodalstructuresoftheseparationbubbleandwakeregionsare showninFigure6-23andFigure6-24,respectively.Mode1,theglobalPODmodewith thelargestenergycontent,fromtheseparationbubblePIVdatarepresentsashift" modethatcanalterthemeanowtoaccountforchangesinthesteadystate.Thismode eectivelyshiftsthemeanshearlayerupordowndependingonthemagnitudeandsign ofthecorrespondingmodalcoecient.However,ashiftmodeistypicallyreservedto 2 Themodel-basedestimatorisestablishedinthenextchapter,whereitisappliedto closed-loopcontrol. 178

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representamean-eldcorrection,accountingforthetransientprocessbetweenanunstable steadystateandthemeanowNoack etal. ,2003.TheremainingmodesinFigure6-23 appeartorepresentconvectionoftheshearlayervortices,eithernaturallyoccurringor enhancedbyactuation. Thewake'shigher-energyglobalPODmodesFigure6-24appeartobesimilarto thebaselinemodesFigure5-16withthepossibleexceptionofmode3,whichmaybe morerepresentativeofashiftmodefortheupperandlowershearlayers.Globalmodes1 and2aremoresymmetricaboutthe y=c =0butstillslightlyfavorthewakeshedding fromthelowershearlayer,asopposedtotheclassicallysymmetricKarmanvortex sheddingpattern.Modes4through7appeartobespatialharmonicsofthefundamental sheddingmodes.Coherentstructuresaregenerallyabsentfromthelower-energymodes notshown,includingtheconvectingvorticesfromtheuppersurfacethatmergeintothe wakebaselinewakePODmode11.Theirabsencesuggeststhattheenergywithinthese shearlayervorticeshasdissipatedpriortothewakeowregion,whichcontraststothe structuresthatpersistfromthebaselineseparatedshearlayer. 6.4.2StochasticEstimation TheparametersforMTD-mLSEoftheglobalPODmodes 0 max ,and r aredeterminedinthesamewaytheyaredeterminedfromthebaselineseparated owSection5.2.2.Theoptimumsingledelay 0 forestimatingthe100highestenergy modesisdeterminedtobe )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 and1 : 6 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 fortheseparationbubbleand wakeregions,respectively.Then,theconvergenceoftheminimumerrormetricEq.5{2 isassessedforincreasingthenumberofglobalPODmodesusedtoformtheestimate. Theseparationbubbleestimatesconvergewithonly9modes,whichcapture58.0%ofthe totalkineticenergyinthePIVdata.Theaccuracyoftheestimationforthewakeregion's globalPODcoecientsconvergesbytherst20modes,capturing73.5%oftheenergy. Thesemodequantitiesarethenusedtoevaluatetheminimumspanofdelaysappropriate toreducetheestimationerrorofMTD-mLSE.Figure6-25showsthattheminimumerror 179

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metricisachievedwithsmalldelayintervalsof max =1 : 95 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 and11 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 forthe separationbubbleandwakeestimates,respectively. 6.5DynamicAnalysis Theprocedureandparametersfromtheprevioussectionareusedtoestimate eachoftheseparationbubbleandwakeregionsusingalongtimeseriesofunsteady pressuremeasurements,therebyprovidingalongtime-seriesofglobalPODcoecient estimates.Beforeproceedingtodecomposetheseestimatesfortemporalcontentandthe correspondingmodalstructure,thedynamicsofthepressurefromwhichtheestimatesare obtainedareinvestigatedinresponsetocontrol.Theopen-loopresultsfromSection6.2 revealthatareducedseparationbubbleisobtainedfromBMactuationtargetingthe naturalshearlayerfrequency.Therefore,theunsteadypressureresponseto F + m = St L sep ; SL =2 : 68andarangeof C valuesisinvestigatedthroughspectralanalysisofsix selectsensorsinFigure6-26. ForsensorsS1andS3Figures6-26Aand6-26B,whichareclosetotheseparation line,thebroadbandlevelsgenerallyincreasealongwithincreased C .Thebaseline separatedshearlayerlocks-ontoandincreasesthepowerinthemodulationfrequency. Downstreamofthosesensors,sensorS5Figure6-26Cexperiencesadecreaseinpower forfrequenciesbelowSt L sep =3 : 4andaslightincreaseforfrequenciesaboveSt L sep =3 : 4. SensorS7Figure6-26D,nearthetrailingedge,hasitsbroadbandlevelsdecreaseddueto increased C ,exceptatSt L sep ; SL =2 : 68andSt L sep ; wake =0 : 89.Thespectraofthetrailing edgesensorsS8andS11aremorediculttoanalyzeduetothepresenceofexternal disturbancesattheselowerbroadbandlevels,whichareattributedprimarilytoacoustics generatedbythemotordownstreamofthewindtunnel'stestsectionanddiuser. 3 At theuppercorner,sensorS8Figure6-26Eexhibitspowerincreaseatthewakeandshear 3 ThesedisturbancesarealsovisibleinthespectraofsensorS1Figure6-26A.Notable peaksarebetweenSt L sep =[1 : 19 ; 1 : 49]-50HzandSt L sep =[3 : 02 ; 4 : 02]-135Hz. 180

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frequenciesandallotherfrequenciesdecreaseduetoactuation.Atthelowercorner,sensor S11Figure6-26Falsorecordsanincreaseinpoweratthecharacteristicwakefrequency, butthisfrequencyisdecreasedslightly. Withabetterunderstandingofhowtheunsteadysurfacepressuredistribution changesduetoactuation,thetime-resolvedestimatesofthetwocontrolcasesusedtoform thestochasticestimationprocedurearedecomposedintofrequency-basedmodesbasedon DMDandDFTanalyses.Treatmentoftheindividualestimatedowregionsisneglected. Instead,theestimatesfromeachregionareproperlyalignedintimeandcombined intoanaggregatesetofestimatesfortheglobalPODcoecients.Thereby,twosetsof estimatesareobtainedforthecombinedregions,eachconsistingofnearly4 10 5 estimates atasamplingrateof5kHz.TherstsetistheestimatedglobalPODcoecientsforthe open-loopcontrolcasedescribedby C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and F + m =2 : 68.Thesecondisfor C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 and F + m =2 : 68. BeforecomputingtheDMDmodes,theestimatedglobalPODcoecientsare low-passlteredwithan8 th -orderButterworthlterwithacutofrequencyof500Hz. Thenthesetoflteredestimatesisdownsampledtoarateof1kHz,andtheDMD calculationisperformed.TheDMDeigenvaluesandspectrumfromtheMTD-mLSE estimatesofthejointregionsareshowninFigure6-27.The C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 caseis representedbybluedots,andthe C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 caseisrepresentedbyreddots. BothcasesidentifyeigenvaluesassociatedwiththeshearlayerSt L sep =2 : 67and wakeoscillationsSt L sep =0 : 88butwithmuchmoreenergyinthewake'smode.The correspondingDMDmodesfromthe C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 caseareshowninFigure6-28. Theeectsofthecontrolonthedynamicsisapparentwhenthesemodesare comparedtotheDMDmodesofthebaselineseparatedowFigure5-29.Thefrequency ofthecharacteristicwakemodeisslightlyreducedduringcontrol.Also,themodal distributionofthismodeismuchmoresymmetricforthecontrolledcase,indicatingthat thereisstrongerinteractionbetweentheupperandlowershearlayersatthisfrequency. 181

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Bothofthesechangesagreewellwiththechangeinpressureassensedbythesensorson thebaseofthemodel. TheseparationbubbledynamicinFigure6-28Balsochangessubstantially.Inthe baselineseparatedow,twomodeswereidentiedfortheshearlayerFigure5-29,one atSt L sep =2 : 50HzandtheotheratSt L sep =2 : 84Hz.Theshearlayermode fromthecontrolledowsplitsthesetwotoessentiallymatchtheforcingfrequency.The structureofthebaselineseparatedDMDmodesfortheshearlayeroscillationsshowsthat thehighlyorganizedelementsofvorticityappearslightlydownstreamof x=c =0 : 8and alsoslightlyabovetheatuppersurface.Thepocketsofalternatingvorticityextend wellintothewakeregionofthemodes,suggestingthatthevorticesfromtheshearlayer frequencyconvectintothewakeofthebaselineseparatedow. Duringcontrol,however,thisvorticalpatternbeginsjustdownstreamof x=c =0 : 7 andveryclosetothemodelsurface.Thehighlyorganizedpatternextendstothetrailing edgebutdoesnotappeartobepulleddownintothewake.Instead,thepatterncontinues inamorestreamwisedirectionintheupperwakeandappearstoinuencetheupper shearlayerofthewake.Thewakestructuresatthishigherfrequencydonotappearhighly organizedandthereforecannotbeconclusivelyclassiedasalocked-ondynamictothe forcingfrequency.Thereisalsonoevidenceofenhancedharmonicuctuationsmeasured bythehotwireforthiscaseoranyoftheothersFigure6-19. Thewakedynamicisalsoheavilyinuencedbythechangeinthemeanboundary layeratthetrailingedge.Theseparatedshearlayerlocks-ontotheforcing,ampliesthe modulationbursts,andbeginstoshedvorticesfurtherupstream.Theshedvorticesfrom theforcedshearlayerconvectdownstream,andtheirstrengthdissipatessubstantially priortothexedseparationpointattheuppertrailingedgecorner.Withtheaverage reattachmentpointoccurringfurtherupstreamduringcontrol,theboundarylayerisable todevelopoveralongerdistance.Thereby,thenaturalinteractionoftheupperandlower shearlayerisabletoresumeitsmoresymmetric,Karman-likevortexsheddingpattern. 182

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TheresultsoftheDFTanalysisforthecontrolledowFigures6-29and6-30agreevery wellwiththeDMDresults. 6.6SummaryofResults UnsteadyactuationfromaZNMFactuatortargetsawiderangeoffrequencies byutilizingaburstmodulatedinputsignal.Boththeshearlayerfrequencyandwake frequencyaretargetedviaopen-loopcontrol.SurfacepressureandPIVmeasurements determinethattheburst-modulatedowcanreattachwithamuchsmaller C valuethan isrequiredwithoutmodulationoftheinputsinusoidalwaveform.Theoptimalforcing frequencytoachievepressurerecoverygivensimilarvaluesof C coincideswiththeshear layerfrequencyfromthebaselineseparatedow,thoughforcingatthewakefrequencyis alsoeectivetoalesserextent.Thisisduetothemodulatedoutput'sabilitytoperturb multiplefrequencyscales,therebyutilizingtheow'snaturalshearlayerinstabilityto moreecientlyreattachtheseparatedow. 183

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Figure6-1.Actuatordiscs'responsesto50V pp sinusoidalinputwithfrequencyvaried between50and2600Hzinincrementsof50Hz.Measurementsprovidedbya hotwireplacedatthedisccenters,alignedwiththecenteroftheslot,andat theslotjetexitplane y=h =0 : 5. A B C D Figure6-2.Actuatordiscs'responsestovariousamplitudesofsinusoidalinputacrossthe frequencyrangeof1600to2600Hzinincrementsof50Hz.Measurements providedbyahotwireplacedatthedisccenters,alignedwiththecenterofthe slot,andattheslotjetexitplane y=h =0 : 5.ADiscA1.BDiscA2. CDiscA3.DDiscA4. 184

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Figure6-3.Spanwisevariationoftheactuatorresponsetovariousamplitudesofsinusoidal inputwithacommonfrequencyof2050Hz.Measurementsprovidedbya hotwirealignedwiththecenteroftheslot,attheslotjetexitplane y=h =0 : 5,andtraversedinthespanwisedirectioninincrementsof z=c =5 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 .Verticallinesmarkthecentersofthefourdiscsandthespan ofthe z=c axisisequaltothatoftheslot. Variable : Description f m : modulationfrequency f c : carrierfrequency N : numberofcarriercycles A m : modulationamplitude Figure6-4.GenericBMwaveformwith N =6carriercycleswithintheburst. A B Figure6-5.Actuatorcharacterizationwithvariousinputlevelsandfrequenciesof modulation.ABurstmodulation.BAmplitudemodulation.Thesinusoidal inputnomodulationisincludedas F + m =0butisnototherwiserepresented by F + m =0. 185

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Figure6-6.GenericAMwaveform. A B Figure6-7.MeasurementsandspectraofsensorS1inresponsetoBMactuationwith f m =90Hzand80V pp nofreestreamow.ATimeseriesofraw,low-pass ltered,predicted,andattenuatedsignals f s =5kHz.BPower-spectral densityofmeasuredandlteredsignalsHzbinwidthandHanningwindow with75%overlap. 186

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A B Figure6-8.Withthewindtunneltunnelo,theaveragecyclicresponseofunsteady pressuretransducerS1toBMinputatvariousamplitudesand90Hz modulationfrequency.AAverageunsteadypressureresponsesnormalizedby theirrespectivermsvalues.BRelationshipbetweeninputamplitudeV pp and measuredrmspressure S1 duetoacoustics. Figure6-9.AttenuationoftheBMactuationpeakat f m =90HzforsensorS1anda rangeofinputpeak-to-peakvoltagesnofreestreamow. 187

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A B Figure6-10.MeasurementsandspectraofunsteadypressuretransducersS1,S3,S5,S8, andS11inresponsetoBMactuationwith F + m =2 : 68and C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 ATimeseriesofunsteadypressuresignals f s =5kHz.BPower-spectral densityofsignalsHzbinwidthandHanningwindowwith75%overlap. A B Figure6-11.MeasurementsandspectraofsensorS1inresponsetoAMactuationno owwith f m =90Hz.ATime-traceofAMinputamplitudeand modulationfrequencyandmeasuredraw,low-passltered,predicted,and attenuatedsignals f s =5kHz.BPower-spectraldensityofmeasuredand lteredsignalsHzbinwidthandHanningwindowwith75%overlap. 188

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A B Figure6-12.AStaticpressurealongtheuppersurfaceofthemodel.Avertical dash-dot lineat x=c 0 : 66showstheaverageseparationpointforthebaselinecaseat z=c =0 : 025.BUnsteadypressurefromsensorS4withvariouslevelsof actuationfromasinusoidalinputand f c =2050.Theactivecontrolresults arecomparedtothebaselineseparatedowcondition. Figure6-13.Staticpressurealongtheuppersurfaceofthemodelwithburstmodulated actuationat40V pp amplitudeandtherangeofmodulationfrequencies.The rangeof C valuesisbetween2 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and6 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 189

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Figure6-14.Staticpressurealongtheuppersurfaceofthemodelwithburstmodulated actuationatselectamplitudeandmodulationfrequencypairings. Figure6-15.Averagestreamwisevelocitycontoursoftheuppersurfaceseparationregion forthebaselineseparatedowandselectcasesofopen-loopcontrol.Solid linesshowtheestimatedseparationstreamline.Dashedlinesshowthe u =0. Estimatedseparationbubbleheights H sep,1 and H sep,2 aredeterminedfrom theselines.Theverticalaxishasbeenstretchedtomoreclearlyvisualizethe separation. Figure6-16.Thevariationofseparationbubbleheights H sep,1 and H sep,2 tolevelsof C shownasthe p C forclarityofsmallforcinglevels. 190

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A B Figure6-17.Phase-averagedvorticitycontoursoftheuppersurfaceseparationregionwith BMcontrolatRe c =10 5 .Theverticalaxishasbeenstretchedtomoreclearly visualizetheseparation.APhasesrelativetomodulationcycleof F + m =0 : 89 and C =5 : 7 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 .BPhasesrelativeto F + m =2 : 68and C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 191

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A B C Figure6-18.Phase-averagedvorticity ,streamwisevelocity u ,andtransversevelocity v contoursoftheuppersurfaceseparationregionwithsinusoidalforcingat F + =70 : 0and C =1 : 1 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 .APhase-averagedvorticity. BPhase-averaged u velocity.CPhase-averaged v velocity. Figure6-19.PSDofahotwireplacedinthewakeatapproximately x=c =1 : 14and y=c =0.Theactivecontrolresultsarecomparedtothebaselineseparated owandthestandardblu-bodywake,bothatRe c =10 5 192

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A B Figure6-20.Averagespanwisevorticitycontourofthenearwakeregion.Measurement planelocatedat z=c =0 : 025.ABaselineuncontrolledseparatedow. BControlwith C =0 : 011andnomodulationofthesinusoidalinput.Dot indicateslocationofhot-wireprobeforcorrespondingmeasurements. 193

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A B C Figure6-21.AverageturbulentquantitiescomputedfromPIVofthenearwakeregion. Measurementplanelocatedat z=c =0 : 025.ABaselineuncontrolled separatedow.BBlu-bodywake.CControlwith C =0 : 011andno modulationofthesinusoidalinput. 194

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A B Figure6-22.Energycontentoftherst r globalPODmodesoftheseparationbubbleand wakePIVdatafrombaselineandselectcontrolcasesi.e.aPODbasis f j g r )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 j =0 .ASeparationbubble.BWake. Figure6-23.SpanwisevorticityofglobalPODmodescomputedfromPIVeldsofthe open-loopcontrolledseparationbubbleregion.Themodesarearrangedin orderofdecreasingenergycontent. 195

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Figure6-24.SpanwisevorticityofglobalPODmodescomputedfromPIVeldsofthe open-loopcontrolledwakeregion.Themodesarearrangedinorderof decreasingenergycontent. A B Figure6-25.AverageerrormetricforMTD-mLSEoftheglobalPODmodecoecients. Selected max valuesareindicatedbylledcircles.ASeparationbubblewith r =9and 0 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 .BWakewith r =20and 0 =1 : 6 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 196

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C values A B C D E F Figure6-26.PowerspectraldensitiesofselectsensorsduringBMopen-loopcontrolwith F + m =2 : 68.Dataacquiredat f s =5kHzfor80seconds.Spectracomputed with5,000FFTpoints,75%overlapwithHanningwindowfunctions. ASensorS1.BSensorS3.CSensorS5.DSensorS7.ESensorS8. FSensorS11. 197

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Figure6-27.DMDeigenvaluesofMTD-mLSEestimatescomprisedfromboththewake andseparationbubbleregionsinresponsetoBMforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 blueand1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 red,bothat F + m =2 : 68. A B Figure6-28.DMDmodesfromMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregionsinresponsetoBMforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and F + m =2 : 68. 198

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Figure6-29.DFTmodenormsoftheMTD-mLSEestimatescomprisedfromboththe wakeandseparationbubbleregionsinresponsetoBMforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 blueand1 : 9 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 red,bothat F + m =2 : 68. A B Figure6-30.DFTmodesfromMTD-mLSEestimatescomprisedfromboththewakeand separationbubbleregionsinresponsetoBMforcingat C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 and F + m =2 : 68. 199

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CHAPTER7 CLOSED-LOOPCONTROL Theideaofecientcontroliscarriedoverfromtheopen-loopresultsintothe closed-loopcontrolinitiatives.Areduced-ordermodelisformedfromthedynamicanalysis oftheglobalPODestimates,whichencompassarichersetofdynamicswiththeinclusion oftwocontrolcasesandthebaselineow.Anothermodelisdevelopedfortheshiftmode, whichdescribestheextenttowhichthebaselineseparatedshearlayerisbroughtcloser tothesurfaceduetoactuation.Controlschemesaredesignedforeachofthesemodels, utilizingtheBMactuationthattargetsthebaselineshearlayerfrequency.Byassumingan actuationinputwithaxedmodulationfrequency,thecontrollerattemptstoconnethe controlledowwithintherealmofthelow-dimensionalmodelKing etal. ,2008. Thecontrolresultsfromthereduced-ordermodelsarecomparedtoasimpleapproach thatidentiesastate-spacemodeloftheinput-outputsystembehaviorandimplements alinearquadraticregulatorLQRtoreducethestateoscillations.Becausethecontrol isgovernedbyproportionalstatefeedback,thisapproachisnotamenabletoBMdue topotentialsuddenspikesincontrolvoltage.Thestatefeedback,whichissinusoidalin nature,assumestheroleofthemodulationsignal. RecallfromChapter4thattheclosed-loopcontrolsetupisrestrictedtofourfeedback channels.Thislimitationhasconsequencesonallofthecontrolschemesbecauseof thesensorselectionforstatefeedback.Theopen-loopresultsdemonstratethatcontrol eectivenessisprimarilyrestrictedtotheseparatedshearlayer.Thewakedoesnotappear tobedirectlycontrollable,butratherindirectlyaectedbythemodicationoftheupper surfaceseparation,reattachment,andboundarylayerdevelopment.Furthermore,thewake dynamicsdonotdiersignicantlybetweenlow-andhigh-levelforcingFigure6-19, whichcouldlimitthescopeofacontrollawdesignedtoincreasethecharacteristicwake oscillations.Therefore,thefoursensorschosenforfeedbackaresensorsS1,S3,S5,andS7, 200

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whicharealllocatedontheuppersurfaceFigure4-5.Theseunsteadypressuresignals bestdescribethedynamicsoftheseparatedshearlayer. 7.1Reduced-OrderModels Beforethereduced-ordermodelsaredescribed,asimpleopen-loopexperimentis conductedtomeasurethefeedbacksensors'responsestoincreasing C .Inthisexperiment, thevoltageamplitudeofthe F + m =2 : 68HzBMforcingsignalisincreasedlinearly withtime,spanning0to80V pp .Thecorresponding C valuesandtheunsteadypressure responsesfromtheindicatedsensorsareplottedinFigure7-1. Theunsteadypressurefromthesefoursensorsisindicativeoftheshearlayer positioningontheuppersurface.SensorsS1,S3,andS5arealignedwith z=c =0 : 1 andseparatedinthestreamwisedirectionby0 : 05 c ,withsensorS1atthemostupstream positionof x=c =0 : 8Figure4-5.Downstreamofthosethreeat x=c =0 : 98,sensorS7 isalignedwithcenterspanofthemodel.At t =0seconds,thecontrolamplitudeiszero andthesensors'responsesarerepresentativeofthebaselineseparatedow.Thebaseline resultsforturbulentkineticenergyinFigure5-4demonstratethatvelocityuctuations growinthestreamwisedirectionandareconcentrateddownstreamofabout x=c =0 : 9, whichcoincideswiththepositioningofunsteadypressuretransducerS5.Forthisunforced owcondition,themagnitudeofthepressureuctuationsalsoincreaseswiththeincreased streamwiselocationofthesensors.Atthepointthecontrolinputamplitudereachesabout 20V pp t 12seconds,themagnitudesofthepressureuctuationsbegintochangefor eachofthesensors.SensorsS1andS3experiencegreateructuationsastheamplitudeis increasedbeyondthispoint,whiletheuctuationsinsensorsS5andS7begintotaper. Thisbehaviorsuggestsmovement"oftheshearlayerduetoactuation.Asthe forcinglevelincreasesfromzero,theshearlayerperturbationsareamplied,causing higherpressureuctuationsassociatedwithvortexformationandroll-uptooccurfurther upstream.TheuctuationsreachamaximumforsensorS3rst,andthensensorS1 follows.Continuingtoincreasetheforcinglevelreducestheuctuationlevelsineach 201

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sensoruntilaminimumisreached,withtheminimumoccurringforsensorS7rstand progressingsuccessivelyupstreamuntilsensorS1.Furtherincreasingtheforcinglevel causesaslightincreaseinpressureuctuationsafteraminimumisreached,butthelevels appeartoconvergeabove70V pp .Thiscouldbeasaturation-likeeectinwhichincreasing the C leveldoesnotyieldameasurableeectonthemeanow.Anunderstandingofthe unsteadysurfacepressureduetoactuationishelpfulwhenconstructingareduced-order modelandcontrollerthatrelyuponmeasurementupdatesforaccuratestateestimation. 7.1.1ModelFormationandObserverDesign WithreferencetoSection6.4.1,tworeduced-ordermodelsareformedfromtheglobal PODbasismodes.Therstisasimplemodelbetweentheinputvoltageamplitude andthecalculatedshiftmodecoecientPODmode1inFigure6-23,termed b 0 for closed-loopcontrol.Therefore,themodulationamplitude,andthustheamplitudeofthe pre-ampliedactuationsignal,isdesignatedasthecontrolinput u .Figure7-2showsthe valueoftheshiftmodecoecientforthevarioustestcasesincludedintheglobalPOD analysis.Thevaluesof b 0 rangefromapproximately-0.06to0.02withinthesetestcases, withthelowerrangecorrespondingthebaselineuncontrolledseparatedow.These valuesarepairedwiththecorrespondingvaluesfor u .Theshiftmodecoecientbehaves likeadcosetthatshiftsthemeanowbetweenthecontrolcases.Thisvalueisclearly relatedtothelevelofforcing.Alinearrelationshipoftheform b 0 = B ~ u isdetermined,where ~ u isadcshiftedvalueof u .Matrix B servesasthelinearmodelfor theshiftmodecoecient. 1 1 Analternatemodelingschemeistoassumeaunitystatetransitionmatrix A basedon thedctrendofthecoecient.However,aKalmanlterofthismodelwouldrelyheavily onmeasurementstoaccountforchangesinstate. 202

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Next,anobserverisdesignedaroundthemodelthatincorporatesmeasurements toalleviateinaccuraciesandpotentialdisturbancesassociatedwiththeinputmodel. Unfortunately,theclosed-loopsystemarchitecturereliesuponac-coupledmeasurements ofthefourunsteadypressuretransducers.Therefore,amoving-rmscalculationisdevised foreachofthefoursensorstoextractadc"quantityfromeachoftheacsignals.The unsteadypressure p 0 isrelatedtoitsrmspressure p rms by p rms = q p 0 2 ; wherethemeanisaccomplishedbya4 th -order,Butterworthlow-passlterwithacuto frequencyof1Hz.Then,alinearrelationshipiscomputedbetweenthermspressure uctuationsandthetrueshiftmodecoecients.AlinearKalmanlterisgivenby ^ b 0 ;k = B ~ u + d k y k = Hb 0 ;k + n k ; where k isthetimeindex,themeasurementupdatesareprovidedbyy= p rms )]TJ/F22 11.9552 Tf 12.918 0 Td [(p 0 and p 0 isavectorofconstantvalues.Theestimatesarethencomputedwithmodeland measurementnoisecovariancessetas Q k =10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(5 and{1 R k = 2 6 6 6 6 6 6 6 4 1000 0100 000 : 010 0000 : 01 3 7 7 7 7 7 7 7 5 : {2 Withinmatrix R k ,thevaluesarechosentopenalizethemeasurementsfromsensorsS1 andS3morebecauseoftheirnon-linearrelationshipbetween p rms andinputvoltage amplitude p 0 S1 and p 0 S3 inFigure7-1.Themodelandobserverarevalidatedby comparisonoftheKalmanlterestimatestothetrueshiftcoecientsinFigure7-2. 203

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Theestimatesagreewellwiththetruevaluesforthesecases,butthevariabilityisgreater forthebaselineowcase,wherethecontrolinputiso u =0.Forlow-amplitude forcing,theKalmanlterreliesmoreontheunsteadypressuremeasurementsduetoslight modelinaccuracy. Thesecondreduced-ordermodelisdevelopedfromtheoscillatoryDMDmodes computedfromthetime-resolvedestimatesoftheglobalPODcoecientsSection6.5. TherealandimaginaryportionsoftheDMDmodepairfortheshearlayerarearranged incolumnsofmatrix n ,suchthattheprojectionoftheglobalPODcoecients a k ontotheoscillatorystatedynamicsisgivenbyEq.5{3.ThisDMDmodepair isgeneratedfromtheopen-loopcontrolcasecharacterizedbySt L sep =2 : 67and C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 Figure6-28B.Theprojectedoscillatorystatecoecients b 1 and b 2 areshowninFigure7-3asthetrue"valuesforthethreetestcasesincludedintheglobal PODanalysis.AwakeDMDmodepairisnotincludedinthisdynamicmodelbecauseof theselectedsensors'inabilitytoproperlyestimatethewakeoscillations. AnotherlinearKalmanlterisdesignedfortheestimationoftheoscillatorystate dynamicswithcorrectionprovidedbythefourunsteadypressuremeasurements.The discrete-timeevolutionofthehomogeneousstateequationisgivenby b k = e A t b k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ; where e A t isthematrixexponentialofmatrix A multipliedbytheconstantsampling interval t = t k )]TJ/F22 11.9552 Tf 10.787 0 Td [(t k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 .Theformof n speciesthatthestatetransitionmatrix A isgiven by A = 2 6 4 SL SL )]TJ/F22 11.9552 Tf 9.299 0 Td [(! SL SL 3 7 5 ; where SL istheshearlayermode'soscillationrateinradianspersecondand SL isthegrowth/decayrate.However,thelatterissettozerointheKalmanlterfor simplicity.AlinearKalmanlterisdesignedasdescribedinSection3.2.2.2withmodel 204

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andmeasurementnoisecovariancessetas Q k =10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 I and R k = I: TheKalmanlterestimatesoftheDMDcoecientsareshowninFigure7-3against thetrueprojectedvaluesforthecasesusedtogeneratethereduced-orderbasis.The estimatescomparefavorablytothetruecoecients,thusvalidatingtheKalmanlterasa stateobserverduringcontrol. 7.1.2ShiftModeTracking Therstcontrollerisasimpleproportionalcontrolschemebasedontheerrorbetween thespeciedandestimatedshiftmodecoecient,givenby e = b 0 ; set )]TJ/F15 11.9552 Tf 11.518 3.154 Td [(^ b 0 : ThiscontrollerisrepresentedbytheschematicinFigure7-4,whichconsistsofthe operationsconductedwithinthedSPACEDSPSystemblockofFigure4-9.Aproportional gainconstantof K =0 : 6providesadequateresponsetimeofthecontrollerwhenoperating withaniterationrateof41.04kHz.Separatefromtheclosedloop,unsteadypressure dataareacquiredat5kHz,andthestaticpressureissampledat5Hzandacquired simultaneouslywiththeunsteadypressure. Intheinitialsetoftests,theshiftcoecientset-point b 0 ; set issettothreediscrete levelswithinthemodeledrangeandonevalueoutsidethisrange.Thesevaluesare )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 030, )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 006,0 : 015,and0 : 030,wherethelastset-pointisgreaterthanthemaximummodeled valueofabout0.02.Thestartingpointforthesetestsisalwaysthebaselineseparated ow,whichhasamean b 0 ofabout )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 060.Theclosed-loopstepresponsetoeachof theseset-pointsisprovidedinFigure7-5.Theestimatedshiftmodecoecienttracksthe set-pointswithamaximumdelayoftwotothreesecondsbetweenthecontrolactivation andset-pointconvergence.Thisisnotunexpectedgiventheaccuracyofthemodeland 205

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Kalmanlterestimates.Thereisevidenceofslightovershootassociatedwiththehigh gainfactorforfastresponse.TheinclusionofaderivativegainPDcontrollercouldhelp reduceovershoot,butintegralcontrolappearsunnecessarybecauseofsmallstead-state error. Thestateestimatesoftheoscillatorymodes b 1 and b 2 ,thecoecientof momentum C ,andthestaticpressuredistributionarealsoplottedalongwiththe shiftmodecoecient. 2 Amongthesefourcases, b 1 and b 2 achievemaximumlevelsfor theset-pointof b 0 ; set = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 006.Thiscorrespondstoaconverged C valueofabout 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 ,whichisveryclosetotheopen-loopcontrolcasefromwhichtheDMDmodesare derived C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 The C p distributionsillustratehowtheadversepressuregradientisaectedbythe magnitudeof C .Increasedlevelsof C tendtoenhancegradualpressurerecovery.This relationshipisalreadyestablishedbytheopen-loopresults.Interestingly,themagnitudes ofthestateoscillations b 1 and b 2 arenotdirectlycorrelatedwiththelevelofpressure recovery.Theamplitudeoftheoscillationsisnonlinearlyrelatedtothecontroleort, describedby C Alongwiththesurface-basedpressuremeasurements,PIVoftheseparationbubble andwakeregionsisacquired.TheaveragestreamwisevelocityisplottedinFigure7-6 forthefourshiftmodecontrolcasesandthebaselineseparatedow.Thesizeofthe time-averageseparationbubble,representedbytheestimatedseparationstreamlineand theowreversalboundary u =0,decreasesinstreamwiselengthandtransverseheight withtheincreasedvalueoftheshiftmodecoecient. 2 Thesquarerootof C isshowninFigure7-5inordertovisualizethelow-level forcingonthesamescaleasthehigh-levelforcing.Otherwise,thelow-levelforcingis indistinguishablefromzeroinput. 206

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Forthelargestset-pointof b 0 ; set =0 : 030,fullreattachmentinanaveragesensemay beachieved.However,thereisagapheightof1.0mmbetweenthemodelsurfaceand thevelocityeldgrid,withinwhichtheaveragevelocityisnotmeasuredandnothighly resolvedwithrespecttothenear-wallportionoftheboundarylayer.Thisaverageow eldisachievedwitha C ofabout3 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 ,whichisabout3timeslessthanthe C reportedforasimilaraverageoweldusinghigh-frequencysinusoidalforcingbottom rightofFigure6-15.Afewinstantaneoussnapshotsfromthistestcaseareshownin Figure7-7.Withinthesesnapshots,theboundarylayerappearstobeattachedinperiodic spatialintervals.Betweentheattachedportions,theboundarylayerliftsfromtheupper surfaceandrollsbacktowardstothesurface.Thisperiodicnatureappearstotransitionto moreturbulentuctuationsbetween x=c =0 : 80and0 : 85.Thus,slightunsteadyseparation remainsforthiscontrolcase,buttheperiodicseparatedshearlayersremainclosetothe surfaceandbegintransitiontoaturbulentboundarylayerwellupstreamofthetrailing edge.Thisresultsinatime-averagedreattachedboundarylayer. Twomoretestsareconductedtomeasuretheclosed-loopresponseofthesame proportionalcontrolsystemtotime-varyingset-pointconditions.Thespeciedshift modecoecientisvariedintimerstviaanincreasingrampandthenaperiodicrandom set-pointbasedonauniformrandomdistributioncoveringthepredeterminedrangeof shiftcoecientvalues.Figure7-8includesunsteadyandstaticpressuremeasurements fromthetwotests.Trackingisgenerallyachievedfortheramp,exceptintheupperrange of b 0 ; set > 0 : 015.Abovethisvalue,theestimatedcoecientdoesnottrackthedesired ramp.Theerrorislikelyduetoasaturation-eectofthecontrol,andanindicationofan upperboundaryforthevalidityofthemodel.Thiseectisrepeatable.Liketheindividual stepresponses,trackingisachievedforthetheperiodicrandomset-point,inwhichthe maximumoftherangeofset-pointsislimitedto b 0 ; set =0 : 015.Theoscillatorycoecients demonstratesuddenbutshortdurationsofhigh-amplitudeuctuationswhentheset-point 207

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ischangedbyalargeamount.Thepressurerecoveryfollowsthetrendoftheset-pointfor theshiftcoecient. Thoughtheshiftcoecienthasphysicalmeaningwhenpairedwiththecorresponding shiftmode,itsinterpretationwithrespecttoamorepracticalcontrolobjectiveparameter isdesired.Therefore,therelationshipbetweenthebubbleheightandtheshiftmode coecientisplottedinFigure7-9,forthetwoseparationbubbleheightspreviously introduced.Asareminder, H sep ; 1 isthemaximumheightoftheestimatedseparated streamlineand H sep ; 2 isthemaximumwhere u =0.Thisguresuggeststhatthe separationbubbleheightislinearlyrelatedtotheshiftcoecientset-pointandthatthe owfullyreattachesatornearthelargestset-pointof b 0 ; set =0 : 030.Again,thisset-point convergestoa C valueof3 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 7.1.3OscillatoryModesOptimization Fromthepreviousresults,themagnitudeoftheDMDoscillationcoecientsvaries nonlinearlywithcontroleort.Therefore,asimplefeedbackschemeisdevelopedbasedon themagnitudeofthestateoscillationsinordertoinvestigateinterestingextremecases. ThiscontrollerislooselyinspiredbythemodelsappliedbyNoack etal. 2003,Tadmor etal. 2003,andGerhard etal. 2003forcontrolofacylinderwakeandRowley& Juttijudata2005forcontrolofcavityoscillations,inwhichanon-lineardynamicalmodel forstateoscillationsatfrequency isassumedandgiveninageneralformby r = r )]TJ/F22 11.9552 Tf 11.955 0 Td [(r 3 = !; where > 0and areconstantsand r istheamplitudeofthestateoscillations b 1 ;b 2 = r cos ;r sin .Giventhismodel,Rowley&Juttijudata2005designasinusoidalcontrol inputwithappropriatephaseandatthesamefrequencyasthenaturalcavityshearlayer oscillation.Acontrollawisdevelopedtostabilizetheoscillationswhichisultimately 208

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governedby u = )]TJ/F22 11.9552 Tf 9.299 0 Td [(Kr cos )]TJ/F22 11.9552 Tf 12.242 0 Td [( c ,where K isaconstantcontrolparameterthatdictates thebehaviorofthesystemresponse. Becausestabilityorreductionoftheseparatedshearlayeroscillationsisnotthe objectiveoftheclosed-loopexperimentsherein,asimilarcontrollawisappliedbut withoutregardforthephaseorthesinusoidalnatureofthecontrollaw.Instead,the BMforcingthattargetstheshearlayerfrequencyismaintained,andtheamplitudeof thepre-ampliedactuationagainassumestheroleofthecontrolinput u ,governedby u = Kr .Therefore,dependingonthechoiceof K ,theamplitudeoftheDMDoscillation coecientsisincreased. Theopen-loopresponseoftheestimatedoscillationcoecients ^ b 1 and ^ b 2 toaramp increaseinBMamplitudeisshowninFigure7-10.Theamplitudeofthestateoscillations r isalsoincluded.Open-loopresultssuggestthatsignicantreattachmentoccursfor C valuesthatmeetorexceedthevaluenecessarytoachievemaximum r .Therefore, K is selectedtodrivethecontroltowardsmaximum r startingfromthebaselineseparatedow. AdetailedschematicofthecontrolstrategyisprovidedinFigure7-11A. Theclosed-loopresponseofthiscontrolschemeispresentedinFigure7-12A.The converged C valueforthemaximum r controlisapproximately4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 ,whichisthe sameastheshiftmodecontrolcasefor b 0 ; set = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 006.Thisisaninterestingconvergence becausethis C valueisoneoftheselectcasesfromopen-loopcontrolthatachievesnear thesamepressurerecoveryasthefullyattachedcaseFigure6-14,anditisalsotheow conditionfromwhichthemodel'soscillatoryDMDmodesareextracted. Theoptimizationofthestateoscillationsalsoincludesatestdesignedtominimize r Inordertoswitchtheconvergencetowardsaminimum r ,thecontrollawisrestructured as u = K r 0 )]TJ/F22 11.9552 Tf 12.114 0 Td [(r .Withtheproperlyselected K ,thecontrollershouldthenminimize r or equivalentlymaximize r 0 )]TJ/F22 11.9552 Tf 12.265 0 Td [(r aslongas r 0 isalwaysgreaterthan r withinthepotential amplituderange.Theopen-loopresponseof r 0 )]TJ/F22 11.9552 Tf 11.997 0 Td [(r isshownfortherampincreaseinBM amplitudeinFigure7-10.Therearetwolocalregionsinwhich r canbeminimized:one 209

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forlittlecontroleortandanotherforhighcontroleort.Theformerisatrivialcondition becausethecontrolinputshouldconvergetonearzero.Therefore,theotherconditionis desirableforconvergenceofaminimum r .Thisconvergencerequiresthattheinitialor minimuminputamplitudeexceedthevaluerequiredtoobtainamaximum r .Thereby,the controllercanprogresstheinputamplitudealongthepositiveslopingportionof r 0 )]TJ/F22 11.9552 Tf 12.318 0 Td [(r towardsthelargeramplitudeassociatedwiththedesiredminimum r .Figure7-11Ashows thecontrolschematicforthiscontroller. TheimplementationofthiscontrollerisshowninFigure7-12B,where C increasesto aconvergedvalueof C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 andareduced r .Theamplitude r oftheoscillatory coecientsisreducedfromitsmaximumofabout0.5toabout0.25,andamoregradual pressurerecoveryintheregionnear x=c =0 : 8isaccomplishedatthisforcinglevel. TheaveragePIVeldsforbothcontrollersareincludedinFigure7-13.These averagestreamwisevelocityeldsconrmthattheaverageowstateislargelygoverned bythemagnitudeof C .Aslighttime-averageseparationbubbleonthetheupper surfaceisvisibleforthetestthataimstomaximize r .Ontheotherhand,theseparation bubblefortheminimum r controlcaseisbarelydetected.Thiscouldsuggestthatthe magnitudeoftheoscillatorystatedynamicsisreducedtoaminimumnearthepointof fullreattachment.Instantaneoussnapshotsfromtheconvergedcontrolforminimum r areshowninFigure7-14.SimilartothemaximumshiftcoecientcontrolFigure7-7,a wave-likepatternofseparatingshearlayerstransitionstomoreturbulentuctuations.The detectedoscillationsarenearminimumforthiscondition. 7.2LinearQuadraticGaussianRegulator AlinearquadraticregulatorLQRisanoptimalcontrollerdesignedtominimizea quadraticperformanceindexDorf&Bishop,2005 J = Z 1 0 )]TJ/F43 11.9552 Tf 5.479 -9.684 Td [(x T Q lqg x + u T R lqg u dt: 210

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Thematrices Q lqg and R lag aredesignparametersandvectors x and u arethestate andinputvectorsforacontinuous-timestate-spacerepresentationgivenbythestate dierentialequationandtheoutputequation: x = A x + B u y = C x + D u : Thefeedbacklaw u = )]TJ/F22 11.9552 Tf 9.299 0 Td [(K x isdeterminedbythechoiceof Q LQG and R LQG .Theindex,or costfunction,isminimizedwhenthefeedbackgain K iscomputedas K = R )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 lqg B T P; wherethesquarematrix P iscomputedfromtheRiccatiequation, A T P + PA )]TJ/F22 11.9552 Tf 11.955 0 Td [(PBR )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 lqg B T P + Q lqg =0 : Foradiscretestate-spacemodel,liketheoneobtainedherein,theperformanceindex becomes J = 1 X k =0 )]TJ/F43 11.9552 Tf 5.48 -9.684 Td [(x T k Q lqg x k + u T k R lqg u k : {3 Thegainmatrixiscalculatedfrom K = )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(R lqg + B T d PB d )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 B T d PA d andthediscrete-timealgebraicRiccatiequation, P = A T d PA d )]TJ/F28 11.9552 Tf 11.955 9.683 Td [()]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(A T d PB d )]TJ/F22 11.9552 Tf 12.952 -9.683 Td [(B T d PB d + R lqg )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(B T d PA d + Q lqg ; {4 where A d and B d arethestatetransitionandinputmatricesforthediscrete-time state-spacerepresentation: x k +1 = A d x k + B d u k y k = C d x k + D d u k : 211

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Thenotation x k isshorthandfor x t k LQRassumesthatallstatesareknownormeasurable,butthisisrarelythecase inmostowcontrolscenarios.Inthissection,areduced-orderstate-spacemodelis identiedbasedontheinput-outputbehaviorofthesystem,wheretheinput u isthe modulationsignalpreviouslythemodulationamplitudeandtheoutputs y arethefour pressuretransducersindicatedforfeedback.Anobserverisimplementedintheform ofalinearKalmanltertoestimatethesystemstateswithknowledgeoftheoutputs. ThecombinationoftheLQRcontrollerandKalmanlterisknownasalinearquadratic gaussianLQGregulator.AschematicofthiscontrollerisshowninFigure7-15,where thisrepresentsthecomputationswithinthedSPACEDSPSystemblockofFigure4-9. 7.2.1SystemIdenticationandObserverDesign Astate-spacerepresentationoftheowsystemisidentiedfromtheinputand outputmeasurementsofanopen-looptestthatisdesignedtotargetasmallrange offrequenciessurroundingthatofthebaselineseparatedshearlayer.Anexperiment isconductedinwhichtheinputmodulationsignalofanAMwaveformischirped" periodicallyandtheamplitudeislinearlyincreased.Thechirplinearlyvariesthe modulationfrequencyofthesinusoidalactuationwaveformat f c =2050Hzbetween 70and110Hz,andthiscyclerepeatsevery5seconds.Thepeak-to-peakamplitude oftheAMwaveformisincreasedlinearlyfrom0V pp to80V pp forthedurationofthe experiment,orapproximately70seconds.Thecorresponding C values,basedonboth f m andamplitude,areshowninFigure7-16Aalongsidetheunsteadypressuremeasurements oftheclosed-loopsensors. WiththeMATLAB r SystemIdenticationToolbox ,theinput-outputbehavior isbestidentiedthroughtransferfunctionsbetweeneachoftheinputandoutput pairingsfourtotal.Then,thecompletesystemoftransferfunctionsisconvertedinto adiscrete-timestatespacesystemconsistingof22statevariables.Becausethestate-space modelisdeterminedfromasetoftransferfunctions,thestatedynamicsandmodeled 212

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outputsareindependentfromoneanother.Therefore,thissystemcanalsoberepresented byasetoffourstate-spacerepresentations. Themodelpredictions,givenzeroinitialconditions x 0 =0,arealsoshownin Figure7-16A.ThemodelpredictionsforsensorsS1andS3capturethegeneraltrendof thevaryingfrequencybutobviouslycannotaccountforthenon-linearresponseofthe unsteadypressuretothelinearincreaseoftheinputamplitude.Thepredictedoutput magnitudesforsensorsS5andS7arepoor.Contributingfactorsincludetheneglectofthe naturalstateoscillationsassociatedwiththebaselineseparatedowandtheinabilityofa lineartime-invariantmodeltoaccuratelyrepresentanon-linearsystemoverawiderange ofoperatingconditions.Still,thereissomeagreementbetweenthephaseofthetrueand modeledoutputs,oneforeachinput-outputpairing. AKalmanlterisdesignedaroundtheentiresystemtoreducerelianceuponthe modeledsystemandemphasizeaccuracyofthemeasuredsignals,particularlythosefrom sensorsS5andS7Figure4-5.Thenoisecovariancesarespeciedas Q k =10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 and{5 R k = 2 6 6 6 6 6 6 6 4 1000 00 : 100 000 : 10 0000 : 1 3 7 7 7 7 7 7 7 5 ; {6 wherethediagonalelementsof R k indicatethatthemeasurementsfromsensorsS3,S5, andS7areweightedmoretocorrectthemodeledsystemstates.TheKalmanltered estimatesoftheunsteadypressuresignalsareshowninFigure7-16B.WiththeKalman lterheavilyweightingtheunsteadypressuremeasurementsfromsensorsS3,S5,andS7, theagreementofthesemeasurementestimatestothetruemeasurementsdemonstratesthe lter'sabilitytocompensateforpoorlymodeledstatedynamics. 213

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7.2.2ControlResults ThecontrollawforLQGisstillbasedonthelinearfeedbackoftheestimatedstates. However,thehighsystemordercancausethecontrolsignaltobedistorteddueto thecontributionsfrommanysystemstates,asopposedtoamoredesirablesinusoidal modulation.Therefore,feedbackisdesignedaroundoneofthefourfeedbacksensors. Initially,acontrollerisdesignedforsensorS1becauseofitsmodelaccuracy.Atthe baselineowconditions,however,thepressureuctuationsatsensorS1'slocationare verysmallbecausetheunsteadinessoftheshearlayerismoreconcentratedtowardsthe trailingedge,abovethemoredownstreamsensors.Withthisinmind,sensorsS5andS7 areselectedforclosed-loopcontrolbecauseoftheirabilitytosensethebaselineoscillations andtheirfavorablestateresponsetoactuation. ResultsarerstpresentedforacontrolcasefromsensorS7.Oncethestate-space equationsareestablished,thecontroldesignisrestrictedtotheselectedscalingofthe Q lqg and R lqg matrices.Thematricesarespeciedas R lqg = 1 = 0 : 3 2 and{7 Q lqg = 1 = 10 6 2 I {8 inordertobalancethecontributionsfromthestatevariablesandinputwithinthecost functionEq.7{3andultimatelycalculateagain K thatconvergestoasmall C value. Increasingthenormof Q lqg and/orreducingthenormof R lqg decreasesthegain K .Here, themagnitudeof R lqg reectsaninputvaluethatcorrespondstothecontrolamplitude priortoamplicationandaresultinggainof50V/V.Thegainvectoriscalculatedafter solvingthediscrete-timealgebraicRiccatiequationEq.7{4inMATLAB r ThemodeledoutputforsensorS7isdependentonstatesx 18 tox 22 ,andthus thestate-spacemodelisreducedtothis5 th -ordersystem.Thestatevariablex 18 is representativeofthesestates,anditsresponsetotheclosed-loopcontrolisshownin Figure7-17Aalongwithstatesrepresentativeoftheotherthreesensors.Intheinitial 214

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secondsafterthecontrolisactivated,themagnitudeofthestateuctuationsforx 18 reducebutultimatelyconvergetoaslightlyhighervaluethantheinitialreduction.During thisprocess,themodulationfrequencyofthecontrollawshiftsfromaninitialfrequency ofabout86Hztoalowervalueofabout74Hz.Themodulationfrequencyisestimated basedonthepeakfrequenciesforaspectrographofthemodulationinputsignal.This shiftdoesnotappeartosignicantlyaltertheestimatedvalueof C ,whichconvergesto anaverageof C =1 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 .AnaveragePIVmeasurementforthisclosed-loopcontrol caseisacquiredandshowninFigure7-18A.Thisaverageoweldiscomparabletothe smallestset-pointtestoftheshiftmodecontrollerinFigure7-6E,whichisachievedwith C 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(5 .ThisLQGcontrollerdoesachieveashorterseparationbubblethanthat particularcase,wheninspectedclosely,butatthecostofan3.25xincreasein C ThenextcontrollerisdesignedbasedonthemeasurementsfromsensorS5,whichis modeledbya4 th -orderstate-spacesystem.Forthistest,thedesignmatricesarechosenas R lqg = 1 = 0 : 5 2 and{9 Q lqg = 1 = : 5 10 5 2 I {10 inaneorttoconvergetoagreater C .Thetimeresponseofthistestisshownin Figure7-17B.Liketheprevioustest,theinitialmodulationfrequency86Hzdropsto about74Hzafterafewsecondsofcontrol.Themodulationfrequencyisdependentonthe stateoscillations.Thistrendofafrequencyshiftappearstobealiatedwiththedecrease instateuctuationsassociatedwithsensorS3,butthereasonforthisshiftisunknown. WhencomparedtothepreviousLQGresults,thistestconvergestoaslightlygreater C of2 : 1 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 butisreducedfromit'sinitialvalueof4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 followingtheswitchto activecontrol.TheaveragestreamwisevelocityforthiscaseisshowninFigure7-18B. Theextentoftheaverageseparationbubbleisnoticeablyreducedwhencomparedtothe previouscase,butthisisattributedtotheincreasedcontroleort.Thisaverageoweld isagainconsistentwiththepreviouscontrolresults. 215

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7.3SummaryofResults Anumberofclosed-loopcontrolmethodsareperformed.Thisincludesset-point trackingoftheobservedshiftmode,optimizationoftheamplitudeforthepairofDMD modalcoecientsoscillationcoecients,andanLQGregulatorappliedtoidentied systemsofindividualinputandsensorpairings.Theresponsesofallcontrolmethodsare recordedbystaticandunsteadypressure.Theconvergedoweldsarealsorecordedwith PIV. Inordertoassesstheeectivenessofallthesecontrolschemes,thesizeofthe separationbubbleiscalculatedfromeachtime-averagedPIVeld.Theaverageseparation bubbleheights H sep,1 and H sep,1 refertotheheightsoftheseparatedstreamlineandthe boundaryforthereverseowregion,respectively.Thesevaluesarecalculatedfromall ofthetime-averagedPIVeldsfromthischapter,andthevaluesareshownagainstthe converged C valuesinFigure7-19.Actually,thesquare-rootof C isplottedtobetter depictthelow-levelforcing.Thecalculatedvaluesfromtheopen-loopresultsandthe reportedvaluesof H sep,1 fromKotapati etal. 2010arealsoincluded. ThetwocontrolcasesfromKotapati etal. 2010areopen-loopforcingwith F + L sep = 1 : 09and2.17and C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 Table2-7.However,thesesimulatedcontrolcases enforcesinusoidalactuationwithcomparable C values.Thistypeofactuationisnot plausiblegiventhepiezoelectricZNMFactuatorbecauseforcingatanactuatorresonance isrequiredinordertoreachthenecessary C values.Specicfrequenciesaretargetedin theexperimentsbasedonamodulationofthesinusoidalforcing.Thesetupisalsonotably dierent,withthemostsignicantdierencebeingthetwo-dimensionalassumptionof thelargeeddysimulations.Despitethesesubstantialdierences,thesimulatedresults comparereasonablywelltotheexperimentaltrend.Asidefromthehigh C valueforthe sinusoidalactuation,alineartrendisobservedforallcases.Futuretestcasesshouldfocus onthehigh-frequencysinusoidalforcinginordertobetterestablishthistrendandnd 216

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amoreprecise C valueforthesaturationoftheseparationbubblesizeduetoincreases forcingamplitude. 217

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Figure7-1.Unsteadypressureresponseoftheclosed-loopsensorsforarampincreaseof theBMamplitude. Figure7-2.Shiftmodecoecientandinputvoltageamplitudeforthethreeow conditionsincludedintheglobalPODcomputation:baselineseparatedow, BMcharacterizedby C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 and F + m =2 : 68,andBMwith C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 and F + m =2 : 68. 218

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Figure7-3.Oscillatorymodecoecientsforthethreeowconditionsincludedinthe globalPODcomputation:baselineseparatedow, F + m =2 : 67BM characterizedby C =4 : 3 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 ,and F + m =2 : 67BMwith C =1 : 9 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 Figure7-4.Schematicoftheshiftmodetrackingcontrolstrategywithinputsofunsteady pressureandanoutputactuationsignal. 219

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A B C D Figure7-5.Closed-loopstepresponsesforshiftmodetrackingcontrolscheme.Results includetheestimatedshiftmodecoecient,theestimatedoscillatorymode coecients,themomentumcoecient,andthestreamwisedistributionof C p The C p colorscaleisincludedinFigure7-8.A b 0 ; set = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 030. B b 0 ; set = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 006.C b 0 ; set =0 : 015.D b 0 ; set =0 : 030. 220

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A B C D E Figure7-6.Averagestreamwisevelocityofseparationbubbleandwakeregions. ABaselineseparatedow.B b 0 ; set = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 030.C b 0 ; set = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 006. D b 0 ; set =0 : 015.E b 0 ; set =0 : 030. 221

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Figure7-7.Threeinstantaneoussnapshotsfromthe b 0 ; set =0 : 030trackingcase,which convergesto C =3 : 7 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 for F + m =2 : 68. A B Figure7-8.Closed-looptrackingresponseoftheshiftmodecoecient b 0 .AIncreasing rampsetpoint.BPeriodicrandomsetpoint. 222

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Figure7-9.Thevariationofseparationbubbleheights H sep,1 and H sep,2 toshiftmode coecientset-points. Figure7-10.ResponseofestimatedDMDcoecientstoarampincreaseoftheBM amplitude. 223

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A B Figure7-11.Schematicsoftheoscillationamplitudecontrolstrategieswithinputsof unsteadypressureandanoutputactuationsignal.AMaximizeoscillatory stateoscillations.BMinimizeoscillatorystateoscillations. 224

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A B Figure7-12.Closed-loopoptimizationoftheoscillatoryDMDcoecientsusingBMinput at F + m =2 : 68.AMaximize r controller.BMinimize r controller,starting frommaximum r initialcondition. A B Figure7-13.Averagestreamwisevelocityofseparationbubbleandwakeregionsduring controlofDMD-basedstateoscillations.AMaximize r controller. BMinimize r controller,startingfrommaximum r initialcondition. 225

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Figure7-14.Threeinstantaneoussnapshotsfromtheminimizationof r controlcase,which convergesto C =2 : 4 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 for F + m =2 : 68. Figure7-15.SchematicoftheLQGcontrolstrategywithinputsofunsteadypressureand anoutputactuationsignal. 226

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A B Figure7-16.Open-looprampandmodulationchirpresponsefromtheclosed-loopsensors. AModeledoutputfromidentiedstate-spacesystem.BKalmanltered outputoftheidentiedstate-spacesystem. 227

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A B Figure7-17.Closed-loopresponsetoLQGcontrolofanidentiedstate-spacesystem. AState-spacemodelofsensorS7.BState-spacemodelofsensorS5. A B Figure7-18.Averagestreamwisevelocityofseparationbubbleandwakeregionsduring LQGcontrol.AState-spacemodelofsensorS7.BState-spacemodelof sensorS5. 228

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Figure7-19.Thevariationofseparationbubbleheights H sep,1 and H sep,2 tolevelsof C shownas p C forclarityofsmallforcinglevels. 229

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CHAPTER8 CONCLUSION Eectivereattachmentofseparatedowswithminimalcontroleortisnotyetwell understood.Thisisdueinlargeparttothecomplex,nonlinearnatureofuidmechanics. Abetterunderstandingofthenaturalinstabilitiesofseparatedowsandtheirroleinow separationwasdesired.Tothateort,Mittal etal. 2005developedaso-calledcanonical separatedowcongurationconsistingofaprescribedboundarylayerseparationfrom aatplatemodel.Insimulations,thisowhasbeenshowntoexhibitseveralofthekey characteristicsofamoretraditionalairfoilseparation.Thisthesiswasdedicatedtowards thecreationandstudyofthiscongurationforthepotentialbenetsofleveragingthe naturalinstabilitiesformoreecientanduniedseparationcontrolstrategies. Theprimarytasksofthisresearchwereasfollows.Therstwastodesignan experimentalsetupandmodelthatwererepresentativeoftheproposedcanonical conguration.Thisplatformwasthenusedtoestablishanexperimentalinstanceofthe separatedowthathasalreadybeenthesubjectofseveralcomputationalstudiesMittal etal. ,2005;Kotapati etal. ,2010;Aram etal. ,2010;Tu etal. ,2011.AtachordReynolds numberof10 5 ,thebaselineseparatedowwascharacterized,identifyingpotential instabilitymechanismsviadynamicanalysismethods.Thentheseparatedowwas controlledwithZNMFactuation,specicallytargetingthecharacteristicfrequenciesand comparingtheeectivenesstohigh-frequencyforcingthatdoesnotaimtoleveragesuch instabilities.Finally,thereduced-orderdescriptionsoftheevolvingoweldareusedfor severalclosed-loopcontroleorts. Thisnalchaptersummarizesthekeyndingsofthisthesisandoutlinesthekey contributionstotheowcontrolcommunity. 230

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8.1ResearchSummary 8.1.1BaselineFlowCharacterization Thisresearchaimedatcreatinganovelexperimentalsetupforthestudyofboundary layerseparationfromaatplatemodelwithabluntbase.Anadversepressuregradient wasimposedviasteadyexternalboundaryconditions,resultinginboundarylayer separationandnaturalreattachmentthatenclosedatime-averagedrecirculationbubble. Thekeyattributesofthisowcongurationwereitspreservationofthetraditional featuresofaseparatedow,theabilitytoprescribetheextentandlocationofthe separation,andlackofsurfacecurvaturewhichhasbeenshowntoaltertheowresponse toactuationGreenblatt&Wygnanski,1999,2003. Aspartofthebaselineowcharacterization,two-andthree-componentstereo particleimagevelocimetryPIVmeasurementsdeterminedthelocationandextentofthe inducedseparationbubble,andfurtherassesseditsspanwiseuniformitywithrespectto meananductuatingvelocity.Meaneldmeasurementswerealsoacquiredforthewake region.Theadversepressuregradientandseparationfromtheuppersurface,evenwith averagereattachmentpriortothetrailingedge,deectedthestandardwakerecirculation regiondownward,awayfromtheuppersurface.Averageturbulentquantitiesreectedthis asymmetrywithlargervelocityuctuationsinthelowerwakethanintheupper.These measurementswerecontrastedtothestandardblu-bodywakethatresultsfromthe removaloftheimposedadversepressuregradient. Thentheunsteadysurfacepressureintheseparatedshearlayerandwakeregions wasobtainedtounderstandtheuctuatingnatureoftheow.Periodicuctuations underneaththeshearlayerwereattributedtotheroll-upandconvectionofspanwise vorticeswiththeaidofcross-correlationanalysisfromneighboringsensorsaswellasthe typicallengthscalesofthepassingvorticesfromPIVmeasurements.Unsteadypressure measurementsonthemodel'sbluntbaseshowedareductionintheuctuationstypical ofaKarmanvortexstreetandveriedtheasymmetricowcontentinthecross-stream 231

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direction.Thewake'suppershearlayerdownstreamoftheuppertrailingedgecorner containedperiodicpressureuctuationsassociatedwiththeconvectingstructuresfromthe uppersurfaceshearlayer.Ahot-wireprobewasplacedinthewaketoconrmthatthe sheddingfrequencyfromthelowershearlayerwasmoreprominentthanintheupper.The hot-wiremeasurementsalsosolidiedtheidenticationoftwocharacteristicfrequencies associatedwiththebaselineseparatedow:theshearlayerandthewake. Modaldecompositionmethodsofthetime-resolvedoweldweredesiredto discerntheglobalstructureandinteractionofthecharacteristicowoscillations. Therefore,thehighspatialresolutionfromthevelocityeldsandthehightemporal resolutionfromtheunsteadypressuremeasurementswereutilizedtogetherforstochastic estimationoflow-ordervelocityeldestimatesthatexhibitbothcharacteristics.With theunderstandingthattheestimateswereatmostlimitedtothecorrelateddynamics betweenthetwomeasurementtypes,theeectiveNyquistfrequencyofthefull-elddata wasincreasedforanalysisbythedynamicmodedecompositionDMDandthediscrete FouriertransformDFT.Thesemethodsextractedmodesofxed-frequenciesforboth theshearlayerandwake,revealingthedistributionandextentoftheuctuations.This providedamorecompletepictureoftherelevantowphysicsinthecanonicalseparated ow. 8.1.2FlowControl FlowcontrolwasachievedbyaZNMFactuatorconsistingoffourpiezoelectricdiscs clampedbeneatharectangularcavityandslot.Theactuatorwasplacedsuchthattheslot isabout0 : 05 c upstreamofthebaselineseparationlocation.Theactuatorwascapableof excitingmultiplefrequencies.Acarriersinewasestablishedasahigh-frequencyforcing thatincreasedthermsvelocityoutputataconditionwithtypicallylowdiscdeections, forthebenetofdiscpreservation.Then,theparasiticacousticsgeneratedbyamplitude andburstmodulationschemesweremodeledforeachpressuretransducer.Thepredicted 232

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acousticsweresubtractedfromthemeasuredpressuresignalsorfromthereal-time measurementsinthecaseofclosed-loopcontrol. 8.1.2.1Open-loopcontrol BurstmodulationBMandsinusoidalforcingwereappliedforopen-loopcontrol, withtheformertargetingarangeoffrequenciessurroundingandincludingthe characteristicowfrequenciesidentiedfortheshearlayerandwake.Eectiveforcing wasdeterminedbyvaryingtheinputlevel C andmodulationfrequencywhilemeasuring theowresponseviastaticandunsteadysurfacepressuretransducers.Eightlevelsof sinusoidalforcingnomodulationand160combinationsofforcinglevelsandmodulation frequenciesweretestedinopenloop.Theeectivecasesweredeterminedbyrecoveryof theadversepressuregradient,indicativeofreattachedorfullyattachedow. Anoptimalrangeofforcingfrequenciesencompassedboththewakeandshearlayer frequencies.FourcasesthreeforBMandoneforasinusoidalactuationwereidentied forfurtherinvestigationviaPIVmeasurementsofthecontrolledseparationbubbleand wakeregions.Theheightofthemeanseparationbubblewascalculatedforeachcase, andtheresultsshowedthatverysmallbubbleheightswereachievedforBMwithorders ofmagnitudeless C thanwasusedtoreattachtheowcompletelywithhigh-frequency sinusoidalforcing.Phase-lockedPIVoftheBMcontrolresultsshowedthattheseparated shearlayerlockedontothemodulationfrequencyandwasabletoreattachtheowwith lowforcinglevels C 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 .Theunsteadyforcingenhancedthecross-streammixingof streamwisemomentum,therebyincreasingtheboundarylayer'sresistancetoseparation fromtheadversepressuregradient.PIVmeasurementsofthewakerevealedthatall theselectedcontrolcasesalteredthewakeinasuchawaythatincreasedtop-bottom symmetryofthevelocityuctuations.Spectraofhot-wiremeasurementsconrmed thepresenceofKarman-likevelocityuctuationsthataccompaniedthecontrolled reattachmentoftheseparationbubble. 233

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Followingtheprocedureestablishedbythebaselineresults,thecontrolledowwas alsoestimatedfromthetime-resolvedunsteadypressuremeasurements.Thedierence thistimewasthereduced-orderPODbasiswasformedfromPIVsnapshotsofdierent controlcasesandalsoasmallnumberofsnapshotsfromthebaselineow.Thetwocontrol casesweresubsequentlyestimatedandanalyzedformodalanalysiswithcorresponding frequencycontent.DMDandDFTresultsagree,showingtheshearlayermuchcloser totheuppersurfaceandstartingfurtherupstream.Thesestructuresdonotappearto interactasstronglywiththewakesheddingasthebaselineseparatedow.Therefore,the wakeexhibitedamoretraditionalvortexstreetsheddingpattern. 8.1.2.2Closed-loopcontrol Theresultsfromtheopen-loopcontroltransitionedtowardsslightlymore sophisticatedclosed-loopcontrolbasedonreduced-orderstateobservations.A model-basedstateobserverwasformedbasedontheoscillatorymodesidentiedfrom theopen-loopcontrolestimates.Inaddition,aPOD-basedshiftmodewasextractedthat describedtheadjustmentinshearlayerheightinresponsetoBMcontrol.Atracking strategywasimplementedwithproportionalfeedbackoftheobservedshiftmodeerror. PIVandstaticpressuredemonstratedtheeectsoftheset-pointtrackingonthemean bubblesize.Thestateoscillationswerealsoestimatedduringcontrol,andthesemotivated closed-loopcontrolaimedatenergyoptimizationofthestateoscillations.Control strategiesareimplementedformaximizingandminimizingtheamplitudeofoscillatory statecoecients,withtheminimumconditionyieldingaseeminglyfullyattached boundarylayerinthebaselineseparationbubbleregion.ThisresultsuggeststhatBM isabletostrengthenthenaturalshearlayeruntilthepointatwhichtime-averaged suppressionoftheseparationisachieved.Theminimumoscillationswereinterpreted asasortofsaturationpointinwhichthestateoscillationswereminimizedduetothe attachedboundarylayerandthereducedforcinglevelrequiredtoachieveattachment. Finally,theseresultsarecomparedtoanLQGregulationcontrolaimedatminimizing 234

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theidentiedstatevariablesandthecontrolinput.Thecontroldieredfromtheprevious methodsinthatproportionalfeedbackofthestateoscillationswasusedtomodulatethe actuationsignal.Attemptswithdierentstate-spacemodelsachievedtheappropriatelevel ofeectivenessbasedonallpreviouscontrolresults,butdidnotimprovetheeciencyof thecontrol. 8.2ResearchImpact Aliteraturereviewofpreviousworkforseparationcontrolfoundatleasttwoareas thatwereinsucientlyaddressed.HighlightedbyMittal etal. 2005,therstwasin regardstoZNMFforcingofstalledairfoilcongurations.Moststudieslackedadetailed treatmentoftheentirecontrolsystem,withparticularemphasisonthenaturalow instabilitiesandthesettingofcontrolparametersforcingfrequencyandamplitude. Therefore,thecanonicalowcongurationwasproposedsothatsystematicapproaches toowcontrolcouldbeaddressedbybothexperimentsandsimulations,potentiallyin tandem.Thesecondissuewasthelackofphysics-basedknowledgewithinthedevelopment ofcontrolstrategiesforairfoilseparation. Withregardstotherst,thisthesisachievedanexperimentalinvestigationonthe canonicalseparatedowconguration,onesuitableforcomparisontoexistingwork andforreferencebyfuturework,inbothexperimentalandcomputationaldomains. ResultswereparticularlyagreeabletotheworkofKotapati etal. 2010,achievinga baselineseparationbubbleofnearlyidenticallengthandheight.Furthermore,therelevant physicalmechanismspresentwereidentiedbyapplicationofwell-establishedestimation techniquesAdrian&Moin,1988;Bonnet etal. ,1994;Murray&Ukeiley,2007 b anda relativelynewapproachtomodaldecompositionSchmid,2010;Rowley etal. ,2009.Each oftheseareashasinuenceinthegreateruidmechanicscommunity. Intermsoftheimpacttowardstheseparationcontrolcommunity,thisthesis recognizedthediscrepanciesinmanyofthereportedoptimalcontrolparametersand thereforeinvestigatedcontrolauthoritywithcarefultreatmentof C and F + .Control 235

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methodswereconductedwithaneyetowardsleveragingthenaturalowinstabilitiesfor moreecientcontrol.Forinstance,acoupleofclosed-loopcontrolmethodsthattargeted thenaturalshearlayerinstabilityachievedcompleteornearcompleteboundarylayer attachmentwithlessthanaquarterofthe C valueofthatshownforahigher-frequency sinusoidalactuationFigure7-19.Theresultsoftheappliedcontrolmethodshave providedmoreunderstandingforeectivelycontrollingseparatedowcharacterizedby multipleowinstabilities,e.g.ashearlayer,aseparationbubble,andawake. Theseachievementsshouldbeimpactfultotheowcontrolcommunity,where detailedtreatmentoftheentireowsystemisoftenneglected.Thisthesisprovided acomprehensivestudyoftheseparatedow,priortoandaftercontrol.Though thecontainedcontrolmethodslackedthephysics-basedsignicanceofamore complexGalerkinprojection, 1 simpleandeectivecontrollersaredesignedaroundthe reduced-orderstateobservationswhicharederivedfromfull-eldvelocitymeasurements. Thesemethodsshouldbeparticularlyusefultoairfoilseparationcontrol,where reduced-ordermodelinghasseenlittleimpact,atleasttotheauthor'sknowledge. Althoughreduced-ordermodelsoerthemostphysicalinsightintotheseparationprocess, theyhavebeenlargelyrestrictedtosimulations. Inadditiontothesecontributions,theauthor'scollaborationtowardstheworkbyTu etal. 2013demonstratedtheabilitytoidentifyglobal,oscillatoryowstructuresbased ontime-resolved,low-orderestimates.Thisworkwasdedicatedtowardsasimpleblu bodywakeatRe h =3 : 6 10 3 ,whichservedasasortofstepping-stoneforthecomplex canonicalseparatedow.Inthisthesis,thefreestreamReynoldsnumberbasedonplate thicknesswasRe h =9 : 5 10 3 .Theowwasgovernednotonlybyawakeinstability 1 AGalerkinprojectionistheprojectionoftheNavier-Stokesequationsontoan establishedreduced-orderbasis,resultingasetofnonlinearordinarydierentialequations. AhindrancetoaGalerkinmodelisthecomplicatedtreatmentoftheactuatorinputtothe modeledoweld. 236

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butalsoaKelvin-Helmoltzinstabilitythatwasshowntointeractwithandsignicantly inuencethewakecharacteristics.Also,unsteadysurfacepressurewasusedforestimation ofthetime-resolved,low-orderstates. Allofthesefactorssuggestthatabroadaudiencewithcomplex,highReynolds numberowcouldbenetfromthisreduced-orderapproachtoidentifyingrelevantow structures.Thiswasemphasizedforamajorityofexperimentalinvestigationsthatlack thenecessaryhardwarefortime-resolvedvelocityelds.Theapproachprovidedmore physicalinsightandunderstandingthantraditionaluidmeasurementtechniquesbecause itwasabletodecomposethecomplexmeasurementsintoafewtractablemodeswith physicalsignicance. 237

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APPENDIXA SEPARATIONCONTROLPLATFORMS Sixmodelgeometriesandbaselineowpatternsaredescribed.Theplatformsfor separationcontrolincludethecircularcylinder,thebackwardfacingstep,thecavity, theD-shapedblubody,lifting-bodyairfoils,andtheatplatemodelforthecanonical separatedow,alternativelycalledthecanonicalatplatemodel.Thedimensionaland dimensionlessquantitiesassociatedwitheachplatformareestablished. A.1Cylinder AsseeninFigureA-2,acircularcylinderofdiameter d isorientedsuchthatthe centralaxisisorthogonaltothestreamwisedirection.Thelocationsoftheseparation pointsforacylinderwakearevariableandprimarilydeterminedbythefreestream Reynoldsnumber,butotherfactorscontribute,includingsurfaceroughnessandfreestream turbulencelevel.Forpost-criticalReynoldsnumbers,thatisRe d verynearandabove49, 1 thewakeischaracterizedbylaminar,periodicKarmanvortexsheddingatafrequency nearSt d 0 : 12.ThesheddingfrequencyincreaseswithincreasingRe d untilSt d 0 : 19for acorrespondingRe d 180,whichmarkstheonsetofathree-dimensionalinstabilitythat yieldssmall-scalestreamwisevorticesWilliamson,1996.TheStrouhalnumberisthen approximatelyconstantatSt d 0 : 2forReynoldsnumbersfrom100to10 5 White,1991. TheshearlayerinstabilityappearsforRe d =1 ; 000to200 ; 000.Thereaderisreferredto Williamson1996foradetailedreviewofthecylinderwakeanditsvariabilitywithRe d Themostpopularcontrolobjectiveissuppressionormitigationofthevortexstreet.The consequencesofsuchagoalincludedragreductionbyincreasingthepressureinthenear wakeofthecylinderanddecreasingthemagnitudeofliftoscillations,whichcaninduce structuralvibrations. 1 Here,thecriticalReynoldsnumberdenotesthetransitionfromthesteady,laminar cylinderwaketotheunsteady,laminarvortexsheddingratherthantransitionfrom laminartoturbulentowortransitionfromtwo-dimensionaltothree-dimensionalow. 238

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A.2BackwardFacingStep Anotherbenchmarkproblemforseparatedowsisthebackwardfacingstepow. AsdepictedinFigureA-1,thedownstreamwakecanbedividedintofourregions:the shearlayer,therecirculationbubble,thereattachmentlocation,andthedeveloping boundarylayerBecker etal. ,2005.Mostcontrolstrategiesaimtocontrolthesizeof therecirculationbubble,ormoreprecisely,thestreamwiselengthoftherecirculation bubble L R .Thisdistanceisalsocommonlyreferredtoasthereattachmentlengthbecause itextendsfromthebaseofthesteptothepointofboundarylayerreattachment.The reattachmentlocationisdenedby w x = L R =0,wherethewallshearstressinthe streamwisedirectionis w = @u @y y =0 : A{1 Withouttheavailabilityofadirectmeasurementofshearstress,therecirculation lengthisoftenestimatedfromtheempiricalobservationthatthermsofthepressure uctuationsontheoorbehindthestepisamaximumatapproximately90%ofthe time-averagedrecirculationlengthMabey,1972.However,theresponsetimeofthis so-calledrms-method"ishinderedbytherecordlengthrequiredforanaveragepressure distribution. Unliketheowseparationfromacylinder,theseparationpointforabackward facingstepowisxedatthestepcorner.Thenatureoftheseparationisgovernedby atleastthreeinstabilitymodes.Therstistheshearlayermodebroughtaboutbythe Kelvin-Helmholtzinstability.Thismodescaleswiththemomentumthickness andis characterizedby St =0 : 011forturbulentseparationHasan&Khan,1992.Thesecond isthesheddingmodeofthegrowingshearlayerasitbendstowardsthewall.Thismode istypicallycharacterizedbySt H 0 : 185,independentoftheseparatingboundarylayer, where H isthestepheightLeschziner&Lardeau,2011.Thepinching"andconvection ofvorticescausesthereattachmentlengthtobeunsteady.Thisresultsinthethirdmode, 239

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aappingmodeoftherecirculationbubble,anditsfrequencyistypicallyanorderof magnitudelessthanthesheddingmodeBecker etal. ,2005. A.3D-ShapedBluBody Theso-calledD-shapedmodelmostcloselyresemblesthecanonicalplatform,which isaatplatemodelwithahalf-ellipseattheleadingedgeandablunttrailingedge.The D-shapedblu-bodymodelshareskeyfeatureswithboththewakefromacylinderand theseparationfromabackwardfacingstep.Likethecylinder,aKarmanvortexstreet resultsfromanyperturbationtothesymmetryoftheupperandlowershearlayers.The non-dimensionalizedsheddingfrequencyissimilarlycharacterizedbySt h 0 : 2,where h istheplatethicknessorbaseheight,forroughlythesameReynoldsnumberRe h regime asthecircularcylinder.However,thepointsofseparationarexedatthetrailingedgesof thebase,muchliketheseparationfromthebackwardfacingstep. Theseparationareabehindthebluntbase,depictedinFigureA-3,isdividedinto similarregionsasthecylinderwake.Thetime-averagedrecirculationzonecontainsthe regionofmeanowreversalinthenearwakeregion.Thisregionisbetweenthetwoshear layersbehindtheupperandlowertrailingedges.Theseshearlayersinteract,yielding aKarmanvortexstreetwithlargecoherentstructuresthatdecreasethebasepressure, eectivelyincreasingthedragonthebodyHenning&King,2005. A.4Airfoil Athighanglesofattack ,owoveranairfoilcanexperiencealargeenoughadverse pressuregradientsuchthattheboundarylayerseparates.Undercertainconditions, thedownstreamowmayreattachtoformaseparationbubblebutmoreoftenremains separated.Inthelattercase,representedinFigureA-4,theoweldcontainsthe Kelvin{Helmholtz-typeinstabilityintheshearlayerandtheglobalwakeinstability causingavonKarmanvortexstreet.Theshearlayerneartheleadingedgeofthesuction sidecanrollupintosmallvorticesthatconvectdownstreamandmergeintolarger structuresduetodeceleration.Anotherseparationpointisxedatthetrailingedgeon 240

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thepressuresideoftheairfoil.Thisowrollsuptoformthecounter-rotatingvorticesin thewake.Inthecaseofowreattachmentnotshown,athirdcharacteristicfrequency mayexistthatisassociatedwiththeseparationbubblebetweenthepointsofseparation andreattachmentMittal&Kotapati,2006;Raju etal. ,2008. Thenatureofseparation,andthereforeitscontrol,fromanairfoiloraerodynamic bodyisfundamentallydierentthanthatofablubody.Boundarylayerseparation frommostblubodiesandespeciallythosewithgeometricslopediscontinuitiessharp cornersisinevitablebecauseofthestrengthandabruptnessoftheadversepressure gradient.Blu-bodycontrolschemesareconsequentlylimitedtodirectwakeorshearlayer modication.Aerodynamic,liftingbodiesarebytheirverynaturedesignedtoprovidelift withasmalldragpenalty.Separationisthereforeadetrimentalando-designcondition thatcanbe,andhasbeenshowntobe,controllableiftheadversepressuregradientis weakenoughtobeovercomebyboundarylayerperturbationsGad-elHak&Bushnell, 1991;Joslin&Miller,2009. 241

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FigureA-1.Flowseparationofabackward-facingstep.AdaptedfromBecker etal. 2005. FigureA-2.Sketchofthewakebehindacircularcylinder.Notethattheshearlayer instabilityappearsforRe d =1,000to200,000. FigureA-3.WakeowbehindaD-shapedmodel. FigureA-4.Separatedowoveragenericairfoil. 242

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APPENDIXB PIVUNCERTAINTY TheuncertaintyinPIVvelocitymeasurementsincludesbiasandrandom uncertainties.Thebiasuncertaintyfortwo-componentvelocitydataisdetermined followingtheequationsofColeman&Steele2009.Thismethodincorporatesallbias uncertaintyinthevelocitycalculations,includingthetimebetweenlaserpulses,the imagecalibrationconstantunitsofmm/px,andthemeasureddisplacements.Thebias uncertaintiesofSPIVdataareestimatedbyarelationofthein-placedisplacementsfor eachcamera,satisfactionofsub-pixelaccuracy,andaroot-sum-squareanalysisofall contributingfactors.Therandomuncertaintyinstatisticalowquantitiesissetbythe intervalsfromBenedict&Gould1996.Thisuncertaintyanalysisdoesnotdirectly addressthespatialaveragingofcorrelationwindowsizeortheoccurrenceoffalsePIV correlationsthatareundetectedbyoutlierlters.Thesefactorsmaycontributeto uncertainty,especiallyinmeasurementsveryclosetoasurface. B.1BiasUncertaintyTwoComponent ThebiasuncertaintyiscomputedfollowingthecriteriaoutlinedbyColeman&Steele 2009andimplementedwithPIVdatabyMurray&Ukeiley2007 a andWetzel2011. Theequationsforthebiasuncertaintyinthevelocitycomponentsisrstderived.Then, thisapproachisextendedtothestatisticalquantitiesofturbulenceintensityandReynolds stress. B.1.1VelocityMagnitude Thebiasuncertaintyinthemagnitudeofthevelocitymeasurementisafunctionof theindividualuncertaintiesforthecomponentsusedtodeterminethevelocity.APIV velocitymagnitudeiscomputedby j v j = DL m L P t ; 243

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where j v j isthemagnitudeofthevelocityvector, D isthemeasuredpixeldisplacement, L m isthemeasuredlengthofthecalibrationtarget, L p isthecorrespondingpixellength oftheimagedcalibrationtarget,and t isthetimebetweenlaserpulses.Thebias uncertaintyiscomputedastheroot-sum-squareoftheuncertaintiesforeachofthese values.Thetotalbiasuncertaintyinthevelocityequationisgivenby B j v j = D B D 2 + L m B L m 2 + L p B L p 2 + t B t 2 1 = 2 : B{1 Thebiasuncertaintyofthepixeldisplacement B D istakenas0.03pxfortheLaVision DaVissoftwareDaVisv8.1,2012.Thebiasuncertaintyinthephysicallength B L m usedforcalibrationisone-halftheresolutionofthedeviceusedtomeasurethedistance. Thebiasuncertaintyinthepixellengthofthecalibrationimage B L p is0.5pxateachof thetworeferencepoints,whichsumstoa1pxoverall.Thebiasuncertaintyinthetime betweenlaserpulses B t is1ns,whichisacombinationoftheuncertaintyinthelaser pulsesynchronizingunitandtheresponsetimeofthelasers.Thetermsarethepartial derivativesofthevelocitymagnitudewithrespecttothesubscriptedvariables.Theseare givenas D = L m L P t ; L m = D L p t ; L p = L m D L 2 p t ; and t = L m D L p t 2 : B.1.2VelocityDirection Becausethereisaxeduncertaintyof0.03pxforthemeasuredpixeldisplacementas opposedtouncertaintyineachvelocitycomponent,thereisalsoadependentuncertainty onthedirectionofthemeasurement.IfaPIVvelocitymeasurementisavectorquantity inthe x y plane,asseeninFigureB-1,thenthemeasuredvectorresidesinacircular subspacewhoseradiusisdeterminedbythebiasinthepixeldisplacement.Thus,thereis arangeofanglesforthevelocitymeasurementwithinthiscircleofuncertainty.Thebias uncertainty B associatedwiththeangleofthevector isdeterminedasthemaximum uncertaintyin basedonthemeasuredangleandtheuncertaintyinpixeldisplacement. 244

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ReferringtoFigureB-1,theangularuncertaintyiscalculatedas B =sin )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 0 : 03px D : Intheeventthat D< 0 : 03,thedirectionalbiasincludesallanglessuchthat B = B.1.3VelocityComponents Withtheuncertaintiesforthemagnitudeanddirectionknown,thebiasuncertainties intheindividualcomponentsofatwo-componentPIVmeasurementcanbedetermined. Inthe x y coordinatesystem,thecomponentsfor u and v arecalculatedas u = j v j cos and v = j v j sin : FollowingthesameguidelinesetbyEquationB{1,thebiasuncertaintyinthese componentsiscalculatedas B u = h )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( j v j B j v j 2 + B 2 i 1 = 2 = h )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(B j v j cos 2 + B j v j sin 2 i 1 = 2 and B v = h )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [( j v j B j v j 2 + B 2 i 1 = 2 = h )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(B j v j sin 2 + B j v j cos 2 i 1 = 2 : B.1.4Time-AveragedVelocityComponents PIVmeasurementsareoftenusedforaverageoweldvelocity.Theensemble averageofasetof u -velocitymeasurements f u k g n k =1 is u = 1 n n X k =1 u k : 245

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Thebiasuncertaintyforthisensembleaverageis B u = u 1 B u 1 2 + u 2 B u 2 2 + + u n B u n 2 1 = 2 = 1 n n X k =1 B 2 u k 1 = 2 : B.2SPIVBiasUncertainty ThebiasuncertaintiesforthethreecomponentsofSPIVdataarecomputedby relatingthein-planedisplacementsforeachcamera x; y totheactualparticle displacementinthree-dimensionalphysicalspace X; Y; Z Hu,2013.Application ofthepinholelensmodelequatesthephysicaldisplacements X; Y; and Z tothethe displacementswithintheimageplanesofthetwocameras,representedby x 1 ; y 1 and x 2 ; y 2 .Thecapitallettersdenoteaphysicallength,andthelowercaseletters arepixellengths.TheanglesofthecamerasplayakeyroleinSPIV,providingdierent perspectivesofthethinlightsheetbythetwocameras.Withthecamerasplacedon thesamesideofthelightsheet,theanglesbetweentheviewingraysandthelightsheet normaldirectionaredenotedas 1 and 2 withinthe xz -planeandas 1 and 2 inthe yz -plane.TheseanglesaredepictedinFigureB-2.Thepixellengthsineachdirectionare computedas x = x 2 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( x 1 tan 2 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 2 ; B{2 y = y 2 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 1 tan 2 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 2 = y 1 + y 2 2 + x 2 + x 1 2 tan 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 1 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 2 ; B{3 and z = x 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( x 1 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 2 = y 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 1 tan 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(tan 2 : B{4 Animagecalibrationresolution R isusedtoconvertthesedisplacementsfrompixel lengthstophysicalunits,or X = R x .Thisquantityisprovidedbythecalibration fromwithintheDaVissoftware. 246

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Aroot-sum-squareanalysisisperformedonEqs.B{2,B{3,andB{4withthe inclusionoftheresolution.ThisyieldstheanalyticalbiasuncertaintiesfortheSPIV velocitycomponents: X bias = s 1 X x 1 2 + 2 X x 2 2 + X 2 sp ; B{5 Y bias = s 1 Y x 1 2 + 2 Y x 2 2 + 1 Y y 1 2 + 2 Y y 2 2 + Y 2 sp ; B{6 and Z bias = s 1 Z x 1 2 + 2 Z x 2 2 + Z 2 sp : B{7 Theuncertaintyanalysismakesuseofasub-pixelsprmserror.Nobach etal. 2005 determinedanrmserrorofabout0.06pixelsforimagedparticlediametersofabout 1.5pixels.Thepartialderivativesofthebiasdisplacementsarecomputedusingacentral dierencemethod,givenby X x 1 = R x 2 tan 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [( x 1 + 1 tan 2 tan 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(tan 2 )]TJ/F21 7.9701 Tf 13.15 5.698 Td [( x 2 tan 1 )]TJ/F21 7.9701 Tf 6.587 0 Td [( x 1 )]TJ/F23 7.9701 Tf 6.586 0 Td [( 1 tan 2 tan 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(tan 2 x 1 + 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( x 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( 1 ; where i isthermspixelerrorinthetoftheSPIVconformalmappingfunctionforthe i th camera.ThesevaluesarecomputedwithintheDaVissoftwareforeachcamera. Finally,thebiasuncertaintyinthevelocitycomponentsiscomputedbydividing thebiasdisplacementsbythetimebetweenlaserpulses t .Thebiasassociatedwith thetemporalresolutionofthesynchronizedPIVhardwareisneglectedduetothe highaccuracyofthetimingunitabout1ns.ForallSPIVmeasurementsincludedin Section5.1.1,the t isconstantat145 s.Theresultingbiasuncertaintiesforthese measurementsare B u =0 : 081m/s ;B v =0 : 081m/s ; and B w =0 : 146m/s : 247

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B.3RandomUncertainty TherandomuncertaintyofPIVvelocitymeasurementsfollowstheequationsin Benedict&Gould1996.The95%condenceintervalsforusefulowstatisticsarelisted inTableB-1.Benedict&Gould1996recommend n 1000foreachmeasurement. Undercertainconditionsinthisinvestigation,lessthan1000measurementsaresucient forthepurposesofthemeasurement,andtheserandomuncertaintyintervalsareused. B.4MeasurementUncertainties Themethodsdescribedfromtheprevioussectionsareappliedtotwo-componentand stereo-PIVdatainordertoinvestigatethedominantregionsofpotentialerroraswell asthemagnitudesoftherandomandbiaserrorquantities.Separationbubbleandwake dataarechosenfromthebaselineseparatedowandacontrolcase.Therandomandbias uncertaintyeldsareplottedforallavailablevelocitycomponents. Therstsetofdataistwo-componentPIVfromtheseparationbubbleregionfor thebaselineseparatedowregionEinFigure4-8.Therandomandbiasuncertainty eldsareshowninFigureB-3forthe u -and v -velocitycomponents.Asexpected, theregionsofhighrandomuncertaintyareconcentratedwithinthedownstream portionoftheseparationbubble.Amaximumofabout0.05m/sforboththe u -and v -velocitycomponentsisobserved,whichisabout1.3%ofthefreestreamspeed.The biasuncertaintiesareconcentratedinregionsofhighspeed.Therefore,thefreestream portionofthestreamwisebiasuncertaintyisgreaterthanwithintheseparationbubble andanorderofmagnitudegreaterthanthebiasuncertaintyoftheverticalvelocity. Next,thewakeisinvestigatedandsimilartrendsareshowninFigureB-4.Finally,the randomuncertaintyforacaseofSPIVdataatcenterspanisplottedinFigureB-5.The magnitudesanddistributionsofrandomuncertaintyagreewellforallthreecomponents, withthemaximumoccurringatabout0.07m/sforeachvelocitycomponent. 248

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FigureB-1.Anillustrationofthebiaserrorinvectordirection,determinedbyangle 249

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A B C D FigureB-2.SchematicsofageneralSPIVsetup,withanglesnotedrelativetothecameras, lightsheet,anddeterminedthree-dimensionalvector.AIllustrationof camerapositionsprojectedontothe xz -plane.BIllustrationofcamera positionsprojectedontothe yz -plane.CThree-dimensionaldisplacement vectorprojectedontothe xz -plane.DThree-dimensionaldisplacementvector projectedontothe yz -plane.AdaptedfromHu2013. 251

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A B C D FigureB-3.Biasandrandomuncertaintiesoftheseparationbubblevelocitycomponents forthebaselineseparatedow.ARandomuncertainty95%condence intervalfor u m/s.BBiasuncertaintyfor u m/s.CRandomuncertainty 95%condenceintervalfor v m/s.DBiasuncertaintyfor v m/s. TableB-1.RandomuncertaintyintervalsforstatisticalowquantitiesBenedict&Gould, 1996. Flowstatistic95%condenceinterval u i 1 : 96 u 0 2 i n 1 = 2 u 0 2 i 1 = 2 1 : 96 u 0 4 i )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 0 2 2 n 1 = 2 u 0 i u 0 j i 6 = j 1 : 96 u 0 2 i u 0 2 j )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 0 i u 0 j 2 n 1 = 2 252

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A B C D FigureB-4.Biasandrandomuncertaintiesofthewakeregionvelocitycomponentsforthe baselineseparatedow.ARandomuncertainty95%condenceintervalfor u m/s.BBiasuncertaintyfor u m/s.CRandomuncertainty95% condenceintervalfor v m/s.DBiasuncertaintyfor v m/s. 253

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A B C FigureB-5.Randomuncertaintiesoftheseparationbubbleregionvelocitycomponentsfor thebaselineseparatedowacquiredwithSPIVat z=c =0.ARandom uncertainty95%condenceintervalfor u m/s.BRandomuncertainty95% condenceintervalfor v m/s.CRandomuncertainty95%condence intervalfor w m/s. 254

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A B C D FigureB-6.Biasandrandomuncertaintiesoftheseparationbubblevelocitycomponents fortheclosed-loopcontrolcaseofminimum r .ARandomuncertainty95% condenceintervalfor u m/s.BBiasuncertaintyfor u m/s.CRandom uncertainty95%condenceintervalfor v m/s.DBiasuncertaintyfor v m/s. 255

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A B C D FigureB-7.Biasandrandomuncertaintiesofthewakeregionvelocitycomponentsforthe closed-loopcontrolcaseofminimum r .ARandomuncertainty95% condenceintervalfor u m/s.BBiasuncertaintyfor u m/s.CRandom uncertainty95%condenceintervalfor v m/s.DBiasuncertaintyfor v m/s. 256

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BIOGRAPHICALSKETCH JohnGrinwasbornin1985inBrunswick,Maine.Afterspendinghisearlyyears travelingalongtheeastcoastduetohisfather'sprofessionintheUnitedStatesNavy, hisfamilymovedtoOrangePark,Floridain1992.Hegraduatedasvaledictorianofhis classatClayHighSchoolin2004andsoughttofurtherhiseducationattheUniversity ofFlorida.JohngraduatedsummacumlaudewithhisBachelorofSciencedegreein aerospaceengineeringinthespringof2008.Hethencontinuedonundertheguidance ofDr.LouisCattafestaattheUniversityofFloridatoearnhisMasterofSciencedegree inaerospaceengineeringinthefallof2011andhisDoctorofPhilosophydegreein mechanicalengineeringinthefallof2013. 270