The Control of Boundary Layers and Channel Flows Using Plasma Actuators

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Title:
The Control of Boundary Layers and Channel Flows Using Plasma Actuators
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1 online resource (208 p.)
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english
Creator:
Riherd, Paul M II
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University of Florida
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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:
ROY,SUBRATA
Committee Co-Chair:
MEI,RENWEI
Committee Members:
BALACHANDAR,SIVARAMAKRISHNAN
SLINN,DONALD NICHOLAS
VISBAL,MIGUEL

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Subjects / Keywords:
boundary-layer -- flow-control -- plasma-actuator
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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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:
Dielectric barrier discharge actuators can be used to control boundary layer and channel flows. These actuators introduce a body force to the flow, adding momentum and vorticity in a beneficial manner for control purposes. The key contributions of this study are an increased understanding of how plasma actuators can be used to modify the stability properties of a laminar boundary layer, characterization of how serpentine geometry plasma actuators can be used to generate boundary layer streaks, and an exploration the bulk flow properties and flow structure of plasma driven channel flows. Linear stability analyses of a plasma modified boundary layer have been performed, indicating that plasma actuators can be used to stabilize or destabilize a laminar boundary layer, depending on the magnitude and orientation of the actuator. The local and global stability properties of plasma modified boundary layers have been characterized, and evidence for additional instability modes has been discovered. It is also shown through numerical simulations that when the electrode geometry of these actuators is modified in a sinuous manner, boundary layer streaks can be generated, opening additional avenues for control using this class of actuators. The use of these actuators to drive a channel flow is demonstrated experimentally. Empirical relationships are drawn, based upon the data collected. Characterization of these channel flows is also performed using numerical simulations and experimental techniques.
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In the series University of Florida Digital Collections.
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Includes vita.
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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 Paul M II Riherd.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: ROY,SUBRATA.
Local:
Co-adviser: MEI,RENWEI.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-06-30

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THECONTROLOFBOUNDARYLAYERSANDCHANNELFLOWSUSINGPLASMAACTUATORSByPAULMARKEYRIHERDIIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013PaulMarkeyRiherdII 2

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Idedicatethistomyfamilyandmyancee,foralwaysbeingpatientwithme. 3

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ACKNOWLEDGMENTS TherearemanypeoplethatshouldbethankedfortheirassistanceandinsightasIhaveworkedandresearchedattheUniversityofFlorida.IwouldliketothankDr.SubrataRoyforhisguidanceandendlesspatiencewithmeasastudent.IwouldalsoliketothankDrs.MiguelVisbalandDonRizzettafortheirsupportduringtheseveralsummersIspentworkingwiththematAFRLinOhio.Similarly,IwouldliketothankSteveWilkinsonforthechancetolearnandworkatNASAatLaRCinVirginiaforasummer.IwouldalsoliketothankthemanyothergraduatestudentsatAPRGformakingitsuchauniqueandwelcomingplacetowork.Inparticular,IwouldliketothankRyanDurscher,Chin-ChengWang,TomasHouba,ArielBlanco,andNavyaMastanaiahforthemultitudeofconversations,lunches,andsharedproblemsolvingthatledtothispoint.ThisworkwouldalsonotbepossiblewithoutthesupportofUF'sHighPerformanceComputingcenter.ThehelpprovidedtomebyCharlesTaylorandCraigPrescotthasbeenessentialinthewritingandrunningofcodesinthisresearch.Finally,Iwouldliketothankmyfamilyandmyancee,MeganWallis,fortheirendlesspatienceasIslowlymovedthroughgraduateschool.Theirsupportinparticularhasmadethedifcultiesofgraduateschoolmucheasiertobear. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 15 CHAPTER 1INTRODUCTION ................................... 16 2OVERVIEWOFRELEVANTFLUIDMECHANICS ................ 21 2.1BoundaryLayerFlows ............................. 23 2.2ChannelFlows ................................. 27 2.3HydrodynamicStabilityandTurbulentFlows ................. 28 2.4ImplementationofDBDBodyForceIntoNavier-StokesEquations .... 43 3LOCALSTABILITYANALYSISOFAOFAPLASMAACTUATEDBOUNDARYLAYER ........................................ 46 3.1BaselineFlowModication .......................... 48 3.1.1BoundaryConditions .......................... 49 3.1.2CalibrationofthePlasmaModel .................... 50 3.1.3SimulatedBaseows .......................... 52 3.2LocalLinearStabilityTheory ......................... 56 3.2.1NumericalModeloftheEigenvalueProblem ............. 56 3.2.2Co-FlowActuation ........................... 58 3.3AModeloftheLocalBoundaryLayerProles ................ 60 3.3.1AdditionalInstabilityModes ...................... 66 3.3.2ComparisonBetweenComputedBoundaryLayerProlesandTheModel ................................ 67 3.3.3LinearStabilityUsingthe1DFlowModel ............... 68 3.3.4Co-FlowActuation(>0) ....................... 71 3.3.5CounterFlowActuation(<0) .................... 74 3.3.6ComparisonoftheOnsetofDifferentStabilityModes ........ 76 3.3.7ComparisontotheUniversalCorrelation ............... 77 3.3.8UnsteadyEffects ............................ 78 3.4Conclusions ................................... 80 4BI-GLOBALSTABILITYANALYSISOFAOFAPLASMAACTUATEDBOUNDARYLAYER ........................................ 83 4.1Bi-GlobalStabilityAnalysisoftheTollmien-SchlichtingWave ....... 84 5

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4.1.1NumericalMethodforBi-GlobalStabilityAnalysis .......... 84 4.1.1.1Numericaldiscretizationandboundaryconditions .... 85 4.1.1.2Gridresolutionstudy ..................... 86 4.1.2Results ................................. 87 4.1.3StabilizationoftheTSWave ...................... 95 4.2StabilizationofBoundaryLayerStreaks ................... 97 4.2.1GenerationofBoundaryLayerStreaks ................ 98 4.2.2GridResolutionStudies ........................ 99 4.3Results ..................................... 101 4.3.1BaselineCase ............................. 101 4.3.2WithPlasmaActuation ......................... 103 4.3.3ScalingoftheDamping ........................ 106 4.4Conclusions ................................... 109 4.4.1ConclusionsontheStabilizationoftheTSWave .......... 110 4.4.2ConclusionsontheStabilizationofBoundaryLayerStreaks .... 111 4.4.3FutureWork ............................... 111 5FLOWSTRUCTUREINABOUNDARYLAYERWITHSERPENTINEGEOMETRYPLASMAACTUATION ................................ 114 5.1SerpentineGeometryActuatorUnderQuiescentConditions ........ 117 5.1.1NumericalDetails ............................ 119 5.1.2FlowCharacterization ......................... 120 5.2BoundaryLayerModicationUsingSerpentineGeometryPlasmaActuation 121 5.2.1FlowCharacterization ......................... 123 5.3Conclusions ................................... 127 6USINGPLASMATODRIVEACHANNELFLOW ................. 134 6.1ExperimentalMeasurementsofaFiniteLengthChannelFlow ....... 135 6.1.1VelocityMeasurementsandSimulationsoftheFiniteLengthChannelFlow ................................... 138 6.1.2PressureMeasurements ........................ 141 6.1.3DeviceEfciency ............................ 144 6.2Conclusions ................................... 146 7FLOWSTRUCTUREOFPLASMADRIVENCHANNELFLOWS ........ 148 7.1ProblemDescription .............................. 149 7.22DLaminarFlowCharacterization ...................... 150 7.2.1NumericalDetails ............................ 151 7.2.2DescriptionoftheResultingFlowFields ............... 153 7.2.2.1Effectsofheightinthechannel ............... 160 7.3ExperimentalValidation ............................ 162 7.3.1CharacterizationoftheDBDPlasmaActuators ........... 163 7.3.2ConstructionoftheChannelExperiment ............... 163 7.3.3DetailsoftheExperimentalMethod .................. 165 6

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7.3.4PIVResults ............................... 167 7.3.4.1Singlegeometrychannels .................. 167 7.3.4.2Doublegeometrychannels ................. 169 7.4Conclusions ................................... 169 8SUMMARYANDRECOMMENDATIONSFORFUTUREWORK ........ 174 8.1ControlofBoundaryLayerStability ...................... 174 8.2SerpentineGeometryPlasmaActuators ................... 176 8.3PlasmaDrivenChannelFlows ........................ 177 APPENDIX AMODELOFTHELOCALTEMPORALSTABILITYANALYSIS .......... 179 A.1EigenvalueProblemFormulation ....................... 179 A.2TransientStabilityAnalysis .......................... 180 BMODELOFTHEBI-GLOBALSTABILITYANALYSIS ............... 183 B.1NumericalConcernsandImplementation .................. 184 B.1.1DifferencingSchemesUsed ...................... 186 B.1.2MemoryDistribution .......................... 186 B.1.3ArnoldiAlgorithm ............................ 187 B.2Verication ................................... 188 B.2.1ChannelFlow .............................. 188 B.2.2DuctFlow ................................ 189 CDESCRIPTIONOFFDL3DI ............................. 192 REFERENCES ....................................... 197 BIOGRAPHICALSKETCH ................................ 208 7

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LISTOFTABLES Table page 2-1Overviewofdifferentlinearstabilitymethods. ................... 32 3-1Convergenceofthemostunstableeigenvaluefortwosamplecases. ...... 57 3-2Parameterstocomparetheboundarylayervelocityproleowmodeltothesimulatedows. ................................... 68 3-3Approximatefrequenciesandfreestreamvelocitieswhereplasmaactuationcanexciteinstabilitymodes. ............................. 80 4-1DetailsofthegridresolutionstudyperformedforthelinearTSwavecalculations. 87 4-2Detailsofthegridresolutionstudyperformedforthelinearboundarylayerstreakcalculations. .................................. 100 5-1Dimensionalandnon-dimensionalvaluesusedtocomputetheowwithaserpentinegeometryplasmaactuator. ....................... 121 6-1Dimensionsoftheplasmachannelusedforvelocitymeasurements. ...... 137 7-1Meshesusedforgridresolutionstudiesofthechannelgeometry. ........ 152 B-1Memoryrequirementsforthesizeofcertainmatricesinvolvedinbi-globalstabilityanalysis ........................................ 186 B-2OperationcountsforthesizeofcertainalgorithmsinvolvedintheEVandSVDproblemsforadensematrix. ............................ 187 B-3Gridresolutionsandmostunstableeigenmodesforthechannelow. ...... 189 B-4Gridresolutionsandmostunstableeigenmodesfortheductowofaspectratio1. ......................................... 190 8

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LISTOFFIGURES Figure page 2-1Blasiusboundarylayervelocityprole ....................... 25 2-2Turbulentboundarylayervelocityproles. ..................... 26 2-3Diagramofthetransitionpathswhichtheowisreceptivetobasedonthemagnitudeandnatureoftheperturbationforboundarylayers. .......... 34 2-4TSwavecomponents. ................................ 35 2-5UniversalcorrelationofthecriticalReynoldsnumber ............... 37 2-6ExamplesofsecondaryinstabilitiesforTSwave. ................. 39 2-7Examplesofstreamwisevortexstreaksinaboundarylayer. ........... 40 2-8Exampleofby-passtransitiongeneratingaturbulentspot. ............ 40 2-9Visualizationofturbulentstructuresinaatplateboundarylayer. ........ 42 3-1Schematicofthemeshusedforcomputations. .................. 49 3-2Calibrationdatafortheplasmaactuation. ..................... 51 3-3Velocityeldsaroundtheplasmaactuators. .................... 52 3-4Pressureonandnearthesurfacearoundtheplasmaactuator. ......... 53 3-5Boundarylayerheightsandshapefactorasafunctionofthevelocityratio. ... 55 3-6Evaluationofparallelowapproximation. ..................... 57 3-7Gridconvergenceofthecriticaleigenvaluefortwodifferentboundarylayerinstabilies. ....................................... 58 3-8Computedeigenspectradownstreamofthepalsmaactuator. .......... 59 3-9Stabilitydiagramsofthemodiedboundarylayerows. ............. 60 3-10CriticalvaluesofRexfortheactuatedow. ..................... 61 3-11Componentsoftheplasmamodiedboundarylayerowmodel. ........ 62 3-12Comparisonsbetweenthesimulatedandmodeledvelocityproles. ....... 64 3-13Boundarylayerprolesusingthecalculations. .................. 65 3-14Regionwhereowreversaloccursintheboundarylayerprole. ........ 66 3-15RegionwhereFjrtoft'scriterionismet. ...................... 67 9

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3-16Comparisonanderrorbetweenthecritialeigenvaluesofboundarylayerprolesfromthesimulationsandthemodel. ........................ 69 3-17Comparisonoftheeigenmodesfromthesimulationsandthemodel. ...... 70 3-18Eigenspectraofthemodelasafunctionof()]TJ /F4 11.955 Tf 9.3 0 Td[(0.5<<0.5)for=0.3andaboundarylayerheightratioof=1.0 ...................... 71 3-19Neutralstabilityintermsofthereducedfrequencyandphasevelocityfortheco-owmodiedboundarylayers. ......................... 73 3-20EigenmodesoftheTSandouterinstabilitiesfortheco-owmodiedboundarylayers. ......................................... 74 3-21Neutralstabilitycurvesforthecounter-owmodiedboundarylayers. ..... 75 3-22NetmapsforlowReynoldsnumbercounter-owmodiedboundarylayers. .. 76 3-23AcomparisonofthecriticalReynoldsnumbersofthedifferentinstabilitiescomparedtofor=1. .................................... 77 3-24Comparisonofcurrentresultstootherboundarylayerproles. ......... 78 4-1Perturbationsenforcedattheinletofthedomainatvaryingfrequencies. .... 86 4-2ConvergenceofamplitudesandspatialgrowthratesforthegridconvergencestudyofthelinearTSwave. ............................. 88 4-3Realcomponentoftheuperturbationvelocityforthenon-dimensionalfrequencyofF106=100andvaryinglevelsofplasmaactuation. ............. 89 4-4MagnitudeoftheTSperturbationsatvaryingfrequenciesandasafunctionofthemagnitudeoftheplasmaactuation. ...................... 90 4-5SpatialgrowthratesoftheTSperturbationsatvaryingfrequenciesandasafunctionofthemagnitudeoftheplasmaactuation. ................ 91 4-6ComparisonofuR,kineticenergyproduction,andkineticenergydissipationofTSwavesinthenearplasmaregion. ...................... 93 4-7ComparisonofdifferentkineticenergyproductiontermsfortheTSwavesinthenearplasmaregion. ............................... 94 4-8u-velocityprolesoftheboundarylayerdisturbancedownstreamoftheplasmaactuatorforvaryinglevelsof0. ........................... 95 4-9ComparisonofrelativeTSwavemagnitudeasafunctionofvelocityratio. ... 96 4-10DampingratioforTSwavesdownstreamoftheplasmaactuator. ........ 97 10

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4-11Convergenceofboundarylayerstreakmagnitudesforthegridresolutionstudiesinanunforcedboundarylayer. ........................... 100 4-12Convergenceofboundarylayerstreakmagnitudesforthegridresolutionstudiesinaplasmamodiedboundarylayer. ........................ 101 4-13Growthandstructureofaboundarylayerstreak. ................. 102 4-14Growthofkineticenergycontainedinboundarylayerstreaksfor=0.6withandwithoutplasmaactuation. ............................ 103 4-15Comparisonofstreaktubulentkineticenergy,kineticenergyproduction,andviscouskineticenergydissipationaroundtheplasmaactuator. ......... 104 4-16Comparisonofboundarylayerstreakkineticenergyproductiontermsaroundtheplasmaactuator. ................................. 105 4-17Totalandnormalizedkineticenergyproductionintheboundarylayerstreaks. 106 4-18Effectofplasmaactuationonindividualstreakamplitudeandtheenvelopeofstreakenergywithandwithoutplasmaactuation. ................. 108 4-19Overallcomparisonofstreakmagnitudewithandwithplasmaactuationappliedtotheow. ...................................... 109 4-20Dampingoftheboundarylayerstreaksbytheplasmaactuatorsasafunctionof0andDc. ..................................... 109 5-1Schematicofvariousserpentineplasmaactuatorgeometries. .......... 114 5-2Comparisonofunsteadyowmeasurementsinaboundarylayerwithandwithoutboundarylayerstreaksapplied. ...................... 116 5-3Differentboundarylayerstreakbreakdownmodes. ................ 117 5-4Spanwiseslicesofthestreamwisevoriticitygeneratedbyaserpentinegeometryplasmaactuatorunderquiescentconditions. ................... 118 5-5Flowvisualizationofserpentineandstandardgeometryplasmaactuators. ... 119 5-6Computationalmeshusedtoperformserpentineplasmaactuatorowsimulations. 120 5-7Comparisonofsimulatedandexperimentalvelocityeldsinducedbyaserpentinegeometryplasmaactuatorunderquiescentconditions. .............. 122 5-8Comparisonofsimulatedandexperimentalvelocitystreamlinesinducedbyaserpentinegeometryplasmaactuatorunderquiescentconditions. ....... 122 5-9Streamwisevorticityatx=1.025oftheserpentinegeometryactuatoroperatedunderquiescentconditions. ............................. 123 11

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5-10Slicesoftheboundarylayeroweldstakenatthepinchingandspreadingpointsandstreamwisevorticitywhenaserpentineplasmaactuatorisapplied. 124 5-11Angleofthevectoredjetasthevelocityratioisvaried. .............. 125 5-12Visualizationoftheowstructurearoundanddownstreamoftheserpentineplasmaactuatorinalaminarboundarylayer. ................... 126 5-13Variousvelocityprolestakendownstreamofaserpentineplasmamodiedboundarylayerasafunctionofthevelocityratio. ................. 129 5-14Normalizedboundarylayerstreakvelocitymagnitudeandstreamwisevortexmagnitude. ...................................... 130 5-15Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.01). ................................... 130 5-16Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.025). .................................. 131 5-17Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.05). ................................... 132 5-18Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.10). ................................... 133 6-1Schematicoftheplasmachannelandofaplasmaactuator. ........... 135 6-2Circuitdiagramoftheplasmachannel. ....................... 136 6-3PowerconsumptionbyasingleDBDactuatorwiththesamedimensionsanddielectricasusedinthechannel. .......................... 138 6-4PIVSetupforexaminingtheplasmadrivenchannelow. ............. 139 6-5Velocitymagnitudesmeasurementsattheexitoftheplasmadrivenchannel. 140 6-6Velocityandmassowmeasurementsatthechannelexit. ............ 141 6-7Pressuremeasurementsalongthecenterlineandsurfaceoftheplasmachannelforavaryingnumberofactuators. ......................... 142 6-8Pressuremeasurementsalongthecenterlineandsurfaceoftheplasmachannelwiththeadditionofanscreenimpedingthechannelow. ............ 143 6-9Maximumpressuredifferentialmeasuredwithintheplasmachannel. ...... 144 6-10Efciencyoftheplasmaactuation. ......................... 145 7-1Comparisonoftheoweffectsforplasmaactuatorsindifferentdomains. ... 149 12

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7-2Comparisonofplasmachannelgeometries. .................... 150 7-3Basicschematicoftheowdomainusedforsimulationsoftheplasmadrivenchannelow. ..................................... 151 7-4Velocityprolesfromthestreamwisegridresolutionstudy. ............ 152 7-5Velocityprolesfromthewallnormalgridresolutionstudy ............ 153 7-6Averagevelocityinthechannelasafunctionoftimeandthedistancebetweenthechannelinlet/outletandboundaries. ...................... 154 7-7Instantaneousviewofthetwo-dimensionalchannelowforDc=0.5inthesinglegeometry. ................................... 155 7-8Instantaneousviewofthetwo-dimensionalchannelowforDc=2.0inthesinglegeometry. ................................... 156 7-9Velocityprolesfromthesinglechannelgeometry. ................ 157 7-10Averagevelocityinthechannelwithasinglegeometryplasmaactuator. .... 158 7-11Boundsontheowseparationforlowerlevelsofplasmaactuation. ....... 158 7-12Instantaneousboundsontheowseparation. ................... 159 7-13Instantaneousviewofthetwo-dimensionalchannelowforDc=0.5inthedoublegeometry. ................................... 160 7-14Instantaneousviewofthetwo-dimensionalchannelowforDc=2.0inthedoublegeometry. ................................... 161 7-15Velocityprolesfromthedoublechannelgeometry. ................ 162 7-16Averagevelocityinthechannelwithadoublegeometryplasmaactuator. ... 163 7-17Averagevelocitiesintheshortenedchannelasafunctionoftime. ........ 164 7-18WalljetsformedbytheDBDplasmaactuatorunderquiescentconditions. ... 165 7-19Photographsoftheplasmachannel. ........................ 165 7-20TimemeanandRMSstreamwisevelocityforthesinglegeometryplasmaactuatorsat16kVpp. ...................................... 167 7-21TimemeanandRMSstreamwisevelocityforthesinglegeometryplasmaactuatorsat18kVpp. ...................................... 168 7-22Timemeanstreamwisevelocityforthesinglegeometryplasmaactuatorsat20kVpp. ........................................ 168 7-23Behavioroftheowneartheaplasmaactuatorat16kVpp. ........... 170 13

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7-24TimemeanandRMSstreamwisevelocityforthedoublegeometryplasmaactuatorsat16kVpp. ................................ 171 7-25TimemeanandRMSstreamwisevelocityforthedoublegeometryplasmaactuatorsat18kVpp. ................................ 171 7-26TimemeanandRMSstreamwisevelocityforthedoublegeometryplasmaactuatorsat20kVpp. ................................ 172 A-1Gridconvergenceofthenumericalmethodascomparedto Orszag ( 1971 )forPoisseuilleow. .................................... 180 A-2Calculationofthetransientgrowthinachannelow. ............... 182 B-1Distributionofmemoryforashiftandinvertstrategy(withashiftof). ..... 187 B-2DistributionofmemoryfortheArnoldialgorithm. ................. 187 B-3Optimaltransientgrowthinthechannelasafunctionofthegrid-resolution.Comparisonswithlocalstabilityanalysisandapastresultarealsoshown. .. 189 B-4Gridresolutionstudyforbi-globalstabilityanalysisofaductow. ........ 191 14

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyTHECONTROLOFBOUNDARYLAYERSANDCHANNELFLOWSUSINGPLASMAACTUATORSByPaulMarkeyRiherdIIDecember2013Chair:SubrataRoyMajor:MechanicalEngineeringDielectricbarrierdischargeactuatorscanbeusedtocontrolboundarylayerandchannelows.Theseactuatorsintroduceabodyforcetotheow,addingmomentumandvorticityinabenecialmannerforcontrolpurposes.Thekeycontributionsofthisstudyareanincreasedunderstandingofhowplasmaactuatorscanbeusedtomodifythestabilitypropertiesofalaminarboundarylayer,characterizationofhowserpentinegeometryplasmaactuatorscanbeusedtogenerateboundarylayerstreaks,andanexplorationthebulkowpropertiesandowstructureofplasmadrivenchannelows.Linearstabilityanalysesofaplasmamodiedboundarylayerhavebeenperformed,indicatingthatplasmaactuatorscanbeusedtostabilizeordestabilizealaminarboundarylayer,dependingonthemagnitudeandorientationoftheactuator.Thelocalandglobalstabilitypropertiesofplasmamodiedboundarylayershavebeencharacterized,andevidenceforadditionalinstabilitymodeshasbeendiscovered.Itisalsoshownthroughnumericalsimulationsthatwhentheelectrodegeometryoftheseactuatorsismodiedinasinuousmanner,boundarylayerstreakscanbegenerated,openingadditionalavenuesforcontrolusingthisclassofactuators.Theuseoftheseactuatorstodriveachannelowisdemonstratedexperimentally.Empiricalrelationshipsaredrawn,baseduponthedatacollected.Characterizationofthesechannelowsisalsoperformedusingnumericalsimulationsandexperimentaltechniques. 15

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CHAPTER1INTRODUCTIONFluidowispresentinmanyofthetechnologiesandnaturalphenomenathatareusedandexperiencedeverydaybyindividualsworldwide,aswellasinthecomponentsofthephysicalinfrastructurerequiredforamodernworld.Therangeofphenomenawhichcompriseuidowisincrediblydiverseandispresentonthelengthscalesrangingfromthemeanfreepathbetweencollidinggasmoleculesuptothesizeofgalaxies.Onamoretangiblescale,theowofgasesandliquidsareimportanttomanymoderntechnologies.Theseowscomprisethemovementofwaterandfuelinpipesandaroundcars,planes,boats,andtrainsastheymovepeopleandgoodsfromonelocationtothenext.Whilemanyofthesetechnologiesinvolvinguidowworkadequatelywithoutanyadjustments,theremaybesignicantgainstobemadebyusingpassiveandactivemethodsofowcontroltomanipulatetheseows.Themotivationforimplementingowcontrolwithinthesetechnologiesisformoreoptimaldevices,bymakingthemmoreenergyefcient,morereliable,expandingtheenvelopeofoperatingconditions,orgenerallypushingthetechnologiesunderconsiderationtowardsamoredesiredstate.Inparticular,thoughawidevarietyofpassiveandactiveowcontroldevicesexist( Cattafesta&Sheplak 2011 ),plasmabasedcontroldeviceshaveshownpromiseforowcontrolapplications.Inthisstudy,severalfacetsofowcontrolareexamined.OneaspectofowcontrolthatisexaminedisthecontrolofthelaminartoturbulenttransitionprocessofsimpleowsusingDielectricBarrierDischarge(DBD)plasmaactuation.Thenalgoaloftheappliedowcontrolwillbetoeitheraccelerateordelaytheonsetoftransitionandtheowbecomingturbulent.Eitherresultmaybedesired,dependingonthecircumstancesoftheowandthegoalsoftheappliedcontrol.Anapplicationforarelativelynewclassofplasmaactuator(i.e.theserpentinegeometryplasmaactuator) 16

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isalsoexamined.Thisparticulartypeofactuatorisintendedtobemoreversatilethanthestandardgeometryactuator,andthepotentialforthistypeofactuatorgeometrytogenerateboundarylayerstreaksforthecontroloflaminarboundarylayersisexplored.AnotheraspectofcontrolthatisexaminedistheapplicationofDBDplasmaactuatorstodrivelowspeedchannelows.Theintendedimpactofthisexaminationistodevelopsmalldeviceswhichcanpumpuidforheatandmasstransferapplicationsinsmallenvironments.DBDactuatorshavebeensuccessfulindemonstratingowcontrolinlaboratoryscaleexperimentsandnumericalsimulations.Tospeakgenerally,thisclassofactuatorspossessesaverybroadrangeofapplicability.InadditiontothemultitudeofapplicationsforDBDdevices,trendsandrelationshipsregardingthebehaviorofthebodyforce,owelds,powerconsumption,andperformancecharacteristicshavebeenexamined,whichcanbefoundinthemanyreviewsofthesedevices( Corkeetal. 2010 ; Moreau 2007 ; Rothetal. 1998 2000 ; Wangetal. 2013 ).Thereisextensiveliteratureexaminingtheeffectivenessoftheseactuatorsforseparationcontroloverhumps( Rizzetta&Visbal 2010 ; Schatzman&Thomas 2008 )andairfoils( Greenblattetal. 2008 ; Post&Corke 2004 ; Rizzetta&Visbal 2007 2011 ).Dragreductions,noisecontrol,andvortexsheddingcontrolhavealsobeenexaminedforbluntbodies( Jukes&Choi 2009 ; Rizzetta&Visbal 2009 ; Sungetal. 2006 ).Laminarandturbulentboundarylayercontrolhavebeenexamined,includingusesoftheseactuatorsforclosed( Grundmann&Tropea 2008a b 2009 )andopenloop( Duchmannetal. 2013a 2012 2010 2013b ; Gibsonetal. 2012 ; Grundmann&Tropea 2009 )hydrodynamicstabilitycontrol.Whiletheapplicationshavetypicallybeenlimitedtoowsoflessthan50m/s,therearesomestudieswhichindicatethattheseactuatorshavebeenusedforsupersonicowcontrol( Imetal. 2010 ; Schuele 2011 ),wherethefreestreamowvelocitieshavebeenupwardsof700m/s. 17

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SomeofthebenetsofDBDplasmaactuatorsarethattheyhaveextremelyquickresponsetimes(onelectronic,ratherthanmechanicaltimescales( Corkeetal. 2010 )),theycanbeconstructedinawaysothattheyaresurfacecompliant(andshouldhavenoimpactontheowwhentheyarenotactiveoractasatypeofsurfaceroughness( Corkeetal. 2010 )),theycanbepulsedanddutycycledatawidebandwidthoffrequencies(whichoffersanopportunitytousetheseactuatorstoinstigatecertaininstabilities,suchastheTollmien-Schlichtingwave( Riherd&Roy 2013c ; Rizzetta&Visbal 2011 ; Visbaletal. 2006 ; Visbal 2010 )),amongotherbenets.Thedownsidesofthistypeofdevicesarethattheyarenotveryenergyefcientandthattheyhaveonlylimitedcontrolauthority.Thebenetsoftheseactuatorssuggestthattheycanbeusedforsomeveryspecializedowcontrolapplications,butthedownsideslimitwhattheseapplicationscanbe,especiallyfromtheperspectiveofeconomicfeasibility.Goingforwardinthiseld,thereareanumberofobstaclesthatmuststillbeovercomeinthedevelopmentoftheseactuatorsandtheirapplications,includingimprovingtheefciencyoftheseactuatorsinthecontextofuiddynamics. Kriegseisetal. ( 2013b )predictsthatfortheretobeanoverallpowersavingsusingtheseactuators,theeffectontheowmustbeampliedmanytimesoverthepoweractuallydeliveredtotheowbytheactuators.Thisnecessaryamplicationmustovercometheextremelylowefciencyoftheactuatorsindeliveringpowertotheow,asitispredictedthatonlyabout0.1%ofthepowersuppliedtotheactuatorreachestheow.Therearesignicantlossesinpowerduetotheionizationprocessandweakmomentumtransferbetweenthechargedparticlesintheplasmaandtheunchargedparticlesinthegas.Thisexceedinglylowefciencydrivestheneedforinnovativeapplicationsthattakeadvantageoftheow'snaturalreactiontomodicationbyplasmaactuation,otherwisenetpowersavingswillnotbefeasible.Withouttheseimprovementsinperformance,thistypeofplasmaactuatormaynotbecomeeconomicallyfeasibleforwideruse. 18

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Theperformanceoftheseactuatorsatgroundtestingconditionsappearstoworkinmanyexperimentalsettings,buttherearephysicallimitationstothestabilityoftheplasmadischargeasthesedevicesareoperatedathigherandhighervoltages( Durscheretal. 2012 ).Benecially,thiseffectappearstobetemperatureandpressuredependent,andgroundtestingconditionsaresuggestedtobehighlynon-ideal.Rather,ightconditionsappeartobemoreappropriatefortheoperationoftheseplasmaactuatorsduetothedecreasedtemperatureandpressure,whichallowforagreaterdischarge( Soni&Roy 2013 ; Valerioti&Corke 2012 ),eventhoughthereisadropinactuatorperformancewhenthefreestreamvelocityisincreased( Kriegseisetal. 2013a ).Theobjectiveofthisdissertationistopresentresearchexploringhowplasmaactuationcanbeappliedtocontrolboundarylayerandchannelows.Theresultsofthisworkaddtotheexistingliteratureregardingtheuseoftheseactuatorsforowcontrolandindicatetheversatilityoftheseactuatorscontinuestoexpand.ThekeyndingsfromthisworkarethatDBDplasmaactuatorshavethepotentialtomodifythestabilityoflaminarboundarylayers,delayingoracceleratingtheonsetofturbulence,dependingonhowtheyareoriented.Inparticulartheeffectsofow-wiseactuationhavebeenexamined,anditisshownthetwoinstabilitiesimportanttolaminarboundarylayers,Tollmien-Schlichtingwavesandboundarylayerstreaks,arestabilized.Furthermore,theoweldsgeneratedbytheuseofserpentineplasmaactuatorsinlaminarboundarylayerhavealsobeencharacterized.Fromthischaracterization,itisfoundthattheseactuatorscangenerateboundarylayerstreaks.Whilethesestreaksarenormallyconsideredaninstability,whentheyareintroducedverycarefullytoaoweld,theycanbeusedtostabilizeordestabilizetheow,indicatingthatthistypeofactuatorcouldbeusedformultipleowcontrolapplications.Finally,theapplicationofusingDBDactuatorstodrivesmallscalechannelowshasbeenexaminedusingexperimentalmethodsand 19

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numericalsimulations.Thesystempropertiesandtheowstructureofthesechannelshavebeenexamined.Thisdissertationisstructuredinthefollowingmanner.InChapter 2 ,therelevantuidmechanicsarereviewed.Theeffectsofusingsteady,lineargeometryplasmaactuationforthecontrolofthehydrodynamicstabilityoflaminarboundarylayersareconsideredusinglocalstabilityanalysisinChapter 3 .Furthermore,thelocalstabilityproblemisexpandeduponusingamodelofaboundarylayermodiedbyplasmaactuation,allowingforamorerapidanalysisoftheboundarylayerstability.Afoundationforstudyingthisprobleminatwo-dimensionalenvironmentisalsoestablished.InChapter 4 ,theframeworkforstudyingtheeffectsofplasmaactuationontwo-dimensionalboundarylayersisexpandedupon.Amethodofbi-globalstabilityanalysisispresented,andtwodifferentinstabilitymechanismsareexamined.Withtheuseofowwiseorientedplasmaactuation,stabilizationofbothinstabilitymechanismsispredictednumerically.Plasmaactuatorswithcomplex,serpentinegeometriesareexaminedfortheirpotentialroleinlaminarboundarylayerowcontrolinChapter 5 .Theoweldsgeneratedbythistypeofplasmaactuationarecharacterized,andmotivationforexaminingtheirstabilitypropertiesfortransitioncontrolisalsopresented.TheconceptofaplasmachannelisintroducedinChapter 6 .Experimentalresultsarepresenteddemonstratingitseffectivenessandcharacterizingthebulkowpropertiesofanitelengthchannel.InChapter 7 ,theplasmachannelconceptisfurtherextended.Propertiesinsideofthechannelarecharacterizedanddiscussed,ratherthanthebulkowproperties.Numericalsimulationsandexperimentalvalidationareperformedinordertostudytheowinsideofthechannel.Finally,asummaryandconclusionsofthecompletedresearcharepresentedinChapter 8 20

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CHAPTER2OVERVIEWOFRELEVANTFLUIDMECHANICSTheowofliquidsandgasescanbecomeextremelycomplicated,dependingonthedensityandtemperatureoftheuid,aswellasthelengthandtimescalesinvolved.Foramajorityoftheowsrelevanttomoderntechnologiesandphenomenaexperiencedbyindividuals,acontinuummodelofuiddynamics(i.e.thecompressibleNavier-Stokesequations)canbeused( White 2006 ).ForthecompressibleNavier-Stokesequations,thecontinuity,momentum,andenergyequationsfortheuidare @ @t+ui@ @xi+@ui @xi=0 (2a)@ui @t+uj@ui @xj=)]TJ /F3 11.955 Tf 10.53 8.09 Td[(@p @xi+@ij @xj+fi (2b)@h @t+ui@h @xj=@p @t+uj@p @xj+@ @xik@T @xi+ij@ui @xj+uifi (2c)whereindicatesdimensionalvalues.xirepresentstheCartesiancoordinatesystem,uirepresentsthecomponentsofthevelocityvector,ijrepresentstheshearstresstensor,andp,h,Tandrepresentthepressure,enthalpy,temperature,anddensity.firepresentsthebodyforcevector.Intermsofthephysicalcoefcients,andrepresentthedynamicandbulkviscosities,whilekrepresentsthethermaldiffusivity.Theenthalpyoftheuidisdenedash=e+p (2)whereeistheinternalenergydensity.Theshearstressesontheuidaredenedasij=@ui @xj+@uj @xi+ij@uj @xj (2)whereijistheDiracdeltafunction.Toclosethissystem,anequationofstateisnecessary.Forgases,theidealgaslawisoftenused,whichisp=RT (2) 21

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whereRistheindividualgasconstant,whichdependsonthemolecularweightofthegas.Forthepresentstudy,theowsunderinvestigationareofrelativelylowvelocityandpossessonlysmallvariationsindensity.Assuch,itisassumedthattheycanbemodeledusinganincompressible,continuummodelofuidmechanics.Whilethisassumptionltersoutagreatdealofwhatispossibleinthebroadersenseofuidmechanics,itwillstillcapturetheessentialphysicsoftheproblemsunderinvestigation.TheseassumptionsallowfortheowstobeexaminedusingtheIncompressibleNavier-Stokes(INS)equations,whichonlyinvolvesthevelocityeldsandthepressureinthedomainandaredenedusingindexnotationas @ui @xi=0 (2a)@ui @t+uj@ui @xj=)]TJ /F4 11.955 Tf 12.74 8.09 Td[(1 @p @xi+@2ui @x2j+fi (2b)Theuid'sdensity()andkinematicviscosity()areassumedtobeconstant.Non-dimensionalizingtheoweldbyitscharacteristiclength(L)andvelocityscales(u1),suchthatu=u u1xi=xi Lp=p u21t=tL u1fi=f u21 (2)itcanbefoundthatformostincompressibleows,onlyasinglenon-dimensionalparameterisnecessary,theReynoldsnumber( Reynolds 1883 ),whichisdenedasRe=u1L (2)Usingthisnon-dimensionalization,theINSequationscanberecastas @ui @xi=0 (2a)@ui @t+uj@ui @xj=)]TJ /F3 11.955 Tf 11.54 8.09 Td[(@p @xi+1 Re@2ui @x2j+fi (2b) 22

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Thissimpliedmodeloftheuidmechanicsisrestrictedtolowspeedowswithnegligibledensityvariations,butthosearetheconditionsexpectedinthepresentwork.Evenundertheselimitedconditions,awidevarietyofphysicalphenomenahavebeendocumented.Fluidowscanlargelybeclassiedintothreedistinctcategories,laminar,transitional,andturbulent. Reynolds ( 1883 )wasthersttomakeascienticdistinctionofthistype.Hedidsobyusingdyetovisualizethemotionofuidinapipe,observingthattheowinthepipeappeareddirect(laminar)atlowvelocities,butsinuous(turbulent)athighervelocities.Thesimplestowsarelaminarows,ofwhichthereareanumberofexactsolutionstotheNavier-Stokesequations.Theselaminarowsarecharacterizedasconsistingofarelativelysimpleowstructure,thestreamlinesoftheseowsarelargelyparallel.Inalaminarow,ifanyvorticalstructuresexist,thentheyexistatdiscretetimeandlengthscales.Turbulentowsarequitetheopposite,beinghighlyunsteady,random(oftenchaotic),andpossessingarangeofvorticalstructuresthatcanexistonadiscreteandcontinuousrangeoftimeandlengthscales.Theseowshavetraditionallybeenstudiedusingstatisticalmethods.Inmorerecentdecades,withtheadventofhighqualityexperimentalmethods,suchasParticleImageVelocimetry(PIV),andimprovementsinthescaleandaccuracyofComputationalFluidDynamics(CFD)methods,theindividualstructuresinaturbulentowhavealsogarneredattention.Comparativelylessexploredtransitionalowsoccupyanintermediateplacebetweenlaminarandturbulentowsandaresubjecttoawiderangeofphenomena.TheseowswillbedicussedingreaterdetailinSection 2.3 2.1BoundaryLayerFlowsBoundarylayerowsarethosewhichimposehighlevelsofshearstressnearano-slipboundaryconditionalongasurface(i.e.ui=0).Thisisincontrasttofreeshearlayers,suchasjetsandplumeswhichexhibitconcentrationsofshearstress, 23

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butdonotinvolveano-slipcondition.Fluidmovementthroughpipes,channels,andaroundatsurfacesareexamplesofboundarylayerows.However,overtime,unlessotherwisespecied,thetermboundarylayerowhastakenonthemeaningofaspatiallydeveloping,semi-boundedowwithano-slipsurfaceononeside,afreestreamvelocityfarfromthissurface,andspanningsomedistanceinthedirectionsparallelandspan-wisenormaltotheow.AtlowReynoldsnumbers,incompressible,boundarylayerowsarewelldescribedbytheFalkner-Skan-Hartree(FSH)similaritysolution( Falkner&Skan 1931 ; Hartree 1937 ).AspecialcaseoftheFSHboundarylayersistheBlasiusboundarylayer,whichdescribestheowoveraatplatewithnopressuregradient(i.e.aplateisperfectlyalignedwithanoncomingow)andnocrossowcomponents.Forboundarylayers,therearethreeheightswhichareconsideredimportant(Figure 2-1 ).The99%height(99%)denestheouterrangeoftheboundarylayer.Thedisplacementheight()describesthevirtualchangetothesurfacegeometryduetotheboundarylayerdisplacinguid.Finally,themomentumheight()isalsoimportant,asit'sthestreamwiserateofchange(d dx)isdirectlytiedtotheskinfrictiononaatsurface.Theseboundarylayerheightsaredenedas 99%=yju(y)=0.99 (2a)=Z101)]TJ /F5 11.955 Tf 13.15 8.09 Td[(u(y) u1dy (2b)=Z10u(y) u11)]TJ /F5 11.955 Tf 13.15 8.08 Td[(u(y) u1dy (2c)AthigherReynoldsnumbers,aboundarylayerwilleventuallybecometurbulent(Figure 2-2 ).EventhoughthereisnoexactsolutiontotheNavier-Stokesequationsdescribingaturbulentboundarylayer,thereisstillahighdegreeofsimilaritybetweenturbulentboundarylayersatvaryingReynoldsnumbers.Examplesincludetheturbulentowsinpipesandchannels( Nagib&Chauhan 2008 ).Withturbulentboundarylayers,thereareadditionalvelocity,length,andtimescales,basedontheshearstressatthe 24

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Figure2-1. Blasiusboundarylayervelocityproleforthestreamwiseandwallnormalvelocities.CalculatedusingBlasiusboundarylayerequations( White 2006 ). wall.Thedimensional()andnon-dimensionalformsofthesescalesare u=s @u @ywu=s 1 Re@u @yw (2a)= u=1 Reu (2b)t= u2t=1 Reu2 (2c)Usingtheseviscousvelocity,length,andtimescales,theowcanbeputintermsofthewallunitssuchthat u+=u u=u u (2a)y+=y =y (2b)t+=t t=t t (2c) 25

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Whentheturbulentboundarylayerisscaledbytheviscouslengthscale,itcanbedecomposedintothreeparts,aviscouswalllayer,whereviscousdissipationdominates,anouterlayer,whichisdominatedbyinviscidconvectiveeffects,andabufferlayerwherebothoftheseeffectsarepresent.Foranattachedboundarylayerwithnopressuregradient( White 2006 ),theseareroughly viscouswalllayer:0y+<5u+=y+ (2a)bufferlayer:5
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2.2ChannelFlowsThetermchannelowcommonlyreferstoabounded,laminarorturbulent,pressureorgravitydrivenowbetweentwoinnitelywide,parallelplates.Here,theowpropertiesaresimpliedoncetheowissufcientlyfarfromanyinletsoroutlets,asthemeanpropertiesinthestream-wisedirectionbecomefullydeveloped(@() @x=0).Assuch,stream-wisevaryingeffectscanbeignoredintheseows,allowingforevenmorebasicstudyofuiddynamicphenomenathaninaboundarylayer.Thelaminar,pressure(orgravity)drivenchannelowhasanexactsolution.Ifaconstantpressuregradientof@p @x6=0,fulldevelopmentoftheowvelocities)]TJ /F12 7.97 Tf 6.67 -4.57 Td[(@ui @x,andnoslipconditionsatthechannelwallsareassumed,thenallofthestream-wisevaryingtermsintheNavier-Stokesequationscanbeneglected.Undertheseconditions,thetheNavier-Stokesequationssimplifydownto@u @t=1 Reh@2u @y2)]TJ /F3 11.955 Tf 13.19 8.09 Td[(@p @x (2)whereReh=u1h (hbeingthechannelhalfheight).Equation 2 canbetreatedasadiffusionequationwithanincludedsourceterm.Coupledwithno-slipboundaryconditionsatthechannelsurfaces(y=h=1),andassumingasteadyow,thevelocityproleofthechannelowissuchthatu(y)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(y2 (2)whereuisnormalizedbythecenterlinevelocity,pisnormalizedbythedynamichead,and@p=@x=2 Reh.Intheunsteadycase,asolutionisalsopossibleusingtheseparationofvariablestechnique( White 2006 ).Inadditiontothissimplestexampleofapressuredrivenchannelow,variationsinvolvingmovingwalls( Poiseuille 1840 ),expandingorcontractingchannelwalls( Hamel 1917 ; Jeffery 1915 ),sidewalls(commonlyreferredtoasductows,( Berker 1963 )),andporousblowingonbothwalls( Berman 1953 )havealsobeenexaminedandfoundtohaveexactsolutions,amongothers. 27

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SimilarlytohowthestabilityofboundarylayerowsisdependentontheReynoldsnumber,thehydrodynamicstabilityproperitesofarechannelowsarealsoReynoldsnumberdependent.Furthermore,asboundarylayerowswilleventuallybecomesaturbulentowastheReynoldsnumberincreases,sodochannelows.AtsufcientlyhighReynoldsnumbers,theturbulentboundarylayerproledescribedfortheboundarylayerowsalsoappliestochannelows,thoughthecoefcientsinvolvedaredifferent(=0.37,B=3.7forEquation 2 ,( Nagib&Chauhan 2008 )).Forthemostfundamentalstudiesofturbulentows,channelowsareoftenused,asthistypeofowisusuallywelldevelopedalongthelengthoftheow,whichsimpliesdatacollectioninexperimentsandtheboundaryconditionsthatmustbeappliedinnumericalsimulations. 2.3HydrodynamicStabilityandTurbulentFlowsTransitionalowsarethosewhicharecharacterizedasundergoingthechangefrombeinganunstablelaminarow,sensitivetosmallperturbations,toonewhichisfullyturbulent.Howthischangeoccursdependsontheparticularowandtheperturbationsitisexposedto.TheearliestmethodsofexaminingthestabilityoftheseowswerebasedondeterminingiftheeigenmodesofthelinearizedNavier-Stokesequations(basedaroundasteadybaseow)arestableornot( Rayleigh 1880 1887 ).Eigenvaluemethodsarestillemployedformodernresearch,thoughmoresophisticatedmethodshavealsobeendeveloped,allowingforagreaterunderstandingofmanydifferentinstabilitymechanisms.Foranincompressibleow,letui=[u,v,w]Trepresentthesteadybaseowvelocitiesand~ui=[~u,~v,~w]Tand~prepresenttheperturbationvelocitiesandpressure,respectivelysothatui=ui+~uiandp=p+~p.LinearizingtheINSequations(Equation 2 ),thelinearized,incompressibleNavier-Stokesequationsaregivenas @~ui @xi=0 (2a)@~ui @t+uj@~ui @xj+~uj@ui @xj=)]TJ /F3 11.955 Tf 11.53 8.09 Td[(@~p @xi+1 Re@2~ui @x2j (2b) 28

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Ifanoscillatorytemporalnatureisassumedtotheperturbations,suchthatu=266666664~u~v~w~p377777775=exp()]TJ /F5 11.955 Tf 9.3 0 Td[(i!t)266666664u0v0w0p0377777775=exp()]TJ /F5 11.955 Tf 9.29 0 Td[(i!t)u0 (2)Equations 2 2 canbeputintotheformAu=i!Bu (2)whereAandBrepresentappropriateoperatormatrices,respectively.Furthermore,iftheowissufcientlysimple(one-dimensional,slowlydeveloping,andparalleltothedirectionofow,thatisu=u(y),w=w(y),@() @x=,@() @z=v=0)andtheperturbationisassumedtobeperiodic,exceptforinthenon-homogeneousdirection(i.e.u=exp(i(x+z)]TJ /F3 11.955 Tf 11.96 0 Td[(!t))u0),thenthissystemofequationscanbesimplieddowntotheOrr-Sommerfeldequationforwallnormalvelocity,v0,(Equation 2a )andtheSquireequationforthewallnormalvorticity,!0y,(Equation 2b ). (i!+iu)@2 @y2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2)]TJ /F5 11.955 Tf 11.95 0 Td[(i@2u @y2)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2 @y2)]TJ /F5 11.955 Tf 11.95 0 Td[(k22!v0=0 (2a)()]TJ /F5 11.955 Tf 9.29 0 Td[(i!+iu))]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2 @y2)]TJ /F5 11.955 Tf 11.96 0 Td[(k2!0y=)]TJ /F5 11.955 Tf 9.3 0 Td[(i@u @yv0 (2b)Dependingonthedesiredanalysis,theseequationscanbeusedtoevaluatethespatial(complex=R+iI,real!=!R)ortemporal(complex!=!R+i!I,real=R)stabilitymodesasaneigenvalueproblemforor!.Forinviscidows(1 Re!0),theRayleighequationsareasimplicationoftheOrr-SommerfeldandSquireequations.Theseequationsare (i!+iu)@2 @y2)]TJ /F5 11.955 Tf 11.95 0 Td[(k2)]TJ /F5 11.955 Tf 11.95 0 Td[(i@2u @y2v0=0 (2a) 29

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()]TJ /F5 11.955 Tf 9.3 0 Td[(i!+iu)!0y=)]TJ /F5 11.955 Tf 9.3 0 Td[(i@u @yv0 (2b)FromtheRayleighequation,severalsimpliedtheoremsofowstabilityhavebeendeveloped. Rayleigh ( 1880 )theorizedthatforaninviscidowtobeunstable,theremustbeaninectionpointintheow'svelocityprole(i.e.@2 u @y2=0and@ u @y6=0). Fjrtoft ( 1950 )expandedonthistheorem,determiningamorerigorousinviscidinstabilityrequirement.Thisadditionalrequirementisthatatsomepointinthevelocityproled2u dy2(u)]TJ /F4 11.955 Tf 12.2 0 Td[(us)<0 (2)mustbesatised,whereusisthesteady,streamwisevelocityattheinectionpoint.Examiningthecomplexphasevelocitiesthatinstabilitiestravelatinavelocityprole, Howard ( 1961 )determinedthatallofthephasevelocitiesoftheperturbations(wherethephasevelocityofaperturbationisdenedasc=cR+icI=! )shouldexistwithinsomecircleofpotentialvalues,whicharedependentontheminimumandmaximumofthevelocityprole,suchthatcR)]TJ /F4 11.955 Tf 13.15 8.09 Td[(1 2(umax+umin)2+c2I1 2(umax)]TJ /F4 11.955 Tf 12.2 0 Td[(umin)2 (2)ExaminingtheresultsofeigenmodegrowthfromtheOrr-Sommerfeld-Squireequationsprovidesstabilityresultsofquestionableaccuracywhencomparedtoexperimentalresultsforsimpleproblems.Forsomeproblems,theseequationsaccuratelydescribesomeofthestabilitymechanismspresentinsomeproblems(suchasFSHboundarylayers),butdonotaddresstheentirespectrumofeffectsthatmayoccur.Forotherproblems,theresultsarelessaccurate.Forexample,thiseigenvalueanalysisincorrectlyaddressesthecriticalpointsofstabilityinsomeows(suchasPoiseuilleow),andpredictsabsolutestabilityforothers(pipeowandCouetteow)whenexperimentalevidenceshowsthattheyaremostdenitelyunstable,evenatlowReynoldsnumbers.Becauseofthis,itwaslongassumedthatperturbationgrowth,not 30

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accuratelycapturedbytheOrr-Sommerfeldequations,wasduetonon-lineareffectsinstead( Trefethenetal. 1993 ).Inthelate1980sandearly1990s,Farrellandhisco-workers( 1992 ; 1988 )developedamethodtostudytheoptimallytransientgrowthofperturbationsforviscousows.Previousattemptsatstudyingtransientmechanismhadpredictedthatshortterminstabilitygrowthcouldbeimportantinotherwisestableows( Ellingsen&Palm 1975 ; Gustavsson 1979 ; Salwen&Grosch 1981 ; Thomson 1887 ).ThismethodpredictedthatevenatlowReynoldsnumbers,lineargrowthmechanismswouldbeabletoamplifytheenergycontainedinsmallperturbationsbyseveralordersofmagnitude,eventhoughtheperturbationswouldeventuallydecaytotheleaststableormostunstableeigenmode. Trefethenetal. ( 1993 )introducedanadditionalmethodwhichshowedthatcertaintimeperiodicbodyforcedistributionswouldbeabletoproducesimilarlevelsofamplication.Thesemethodsprovidesufcientevidencethatlineargrowthmechanismsarehighlyrelevanttotheearlystagesofthelaminartoturbulenttransitionprocess,evenwhentraditionaleigenvaluemethodssuggestthattheowisstable.AnoverviewofthesemethodsisprovidedinTable 2-1 .Dependingonthemagnitudeoftheprimaryinstabilities,theseinitialinstabilitiescanalsobepronetosecondaryandhigherinstabilities( Herbert 1988 ),andcanleadtoevenfasterratesofinstabilitygrowth.Oncetheperturbationsrelevanttolineargrowtharesufcientlyamplied,theyenterthenon-linearregime,wheretheynormallybreakdowntoformaturbulentow.Non-linearstabilityanalysisalsopresentsitselfasadiverseproblem,andthetheoriesandmethodsusedvarywidely,dependingonthedesiredoutcome,andwhetherthetransitioneffectsincorporatedintothemethodqualiedasweaklyorfullynon-linear.Whiletheearlystagesoftransitionarefairlywellunderstood,lessisknownabouttheselaterstagesandhowadeterministicowwithdiscretefrequencyoscillationsbecomesamorerandomowwithacontinuumoffrequencyoscillations. 31

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Table2-1. Overviewofdifferentlinearstabilitymethods. Stabilityanalysis: Eigenvalueproblem OptimalInitialValueProblemForcingResponse Form:Initialconditionresponse:Forcingresponseu=exp(i!t)u0u=(A)]TJ /F3 11.955 Tf 11.96 0 Td[(!B))]TJ /F7 7.97 Tf 6.58 0 Td[(1fGrowth:Longtermdynamics(t!1)Shorttermdynamicsbasedontheinitialcondition,u0OptimalperiodicresponseCasespecicperiodicresponseOutput:Exponentialgrowthratesast!1MaximumgrowthofthekineticenergyoveraniteamountoftimeforanoptimuminitialconditionMaximumgrowthinthesystemforanoptimalforcingResponseofthesystemtoaspecicforcing(i.e.responsetocontrol)Stabilityisbasedon:GrowthofasingleeigenmodeNon-modalgrowthofsomeinitialconditionSVDoftheresolventApplicationoftheresolvent Thestudyofwhathappensinthenon-linearregimeispresentlyatopicofintenseresearchandbeyondthescopeofthepresentstudy.Differenttypesofstabilityanalysiswilloftenpredictsimilarphysicalbehavior,butthisisnotalwaysthecase.Sometimesseveraldifferentbehaviorsintheowcanbeshowntobeunstable.Thisindicatesthattheremaybemultiplepathsforaow'stransitiontoturbulence,andthatthetransitionprocesscanoccuratdifferentrates,dependingonwhichpathistaken.Thisisaveryimportantresulttonote,astheperturbationswhichaowwillexperienceinmanyapplicationsscenarios(suchasinaboundarylayeroveranaircraftwingortheowofchemicalsinpipe)donotalwaysmatchupwiththoseseeninexperimentsorinsimpliedtheories.Whentheseperturbationsdomatchup,itisoftenduetotheirintroductionintotheowbysomesortofactuator,notnecessarilythenaturalenvironmentinwhichtheowexists.Thatthere 32

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existsacertainrandomnesstowhichtransitionpathisselectedisimportantforowcontrolapplications,asitindicatesthattheremaybeawaytocontrolthetransitionpathselectedbytheow,providinganentrypointforowcontrol.Thisparticularsubsetofowcontrolisdenedastransitioncontrol.Oneimportantconsiderationfortheapplicationoftransitioncontrolistodeterminewhatpathaowwilltakewhiletransitioningfromlaminartoturbulent.Therelevantpathtotransitionisdeterminedbywhatperturbationsexistintheenvironment,andatwhatmagnitudetheyexistat.Howtheowrespondstoaspecictypeofperturbationisdenedastheow'sreceptivitytothattypeofperturbation( Morkovin 1969 ).Agivenowwillexperienceavarietyofperturbations,andmayalsobesubjecttocontrolledperturbationsthroughsomeowcontrolactuator.Dependingontheperturbation,theowmayexperiencesomecombinationofmodalandnon-modal,primary,secondary,transient,orby-passtransitionmechanisms.Foraboundarylayerow,thereceptivitychartfordifferenttypesofperturbationsisshowninFigure 2-3 .Forboundarylayerswithnocross-ow,thedifferenttransitionpathshavebeendescribedby Reshotko ( 2001 ).Inpath(A),thetransitionisinitiatedbythegrowthofsmallTSwaves,whichthenbecomeunstableontheirownduetosecondaryinstabilitiesbeforeeventuallybecomingturbulent.Path(B)indicatesasituationwheretransientgrowthinstigatesthemodalgrowthoftheTSwave.Path(C)designatesasituationwheretheexponentialgrowthofinstabilitiesisonlyofminorimportance,andthattherapid,algebraicgrowthofaperturbationeventuallycausesittobecomeunstabletoitsownsecondaryinstabilitiesandbreakdown.Paths(D)and(E)indicatethescenarioswherelargefreestreamperturbationseitherexciteandbreakdownalgebraicinstabilities,ordirectlytransitiontheowbygeneratingaturbulentspot.Theselatterpathsrequireperturbationsofsignicantmagnitude.TheTSmodeistraditionallytheonemostwidelystudiedforboundarylayerows.Thistransitionpathisusuallyinitiatedbysmallsurfaceperturbations,whichgrowand 33

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Figure2-3. Diagramofthetransitionpathswhichtheowisreceptivetobasedonthemagnitudeandnatureoftheperturbationforboundarylayers.From Morkovinetal. ( 1994 ). decayatexponentialratesalongthelengthoftheboundarylayerow.OncetheTSmodereachesasufcientamplitude,thelaterstagesoftransitionbegin(asindicatedinFigure 2-3 ).Two-dimensionalTSmodeshavebeenwidelyexaminedinthereportedliterature,asSquire'stheoremindicatesthattwo-dimensionalinstabilitiesarethemostampliedmodalinstabilities(forparallel,fullydevelopedows)( Squire 1933 ).Whileviscosityisnormallytakentobeastabilizingeffectinuidows,withregardtotheTSwave,theviscosityplaysanecessarypartofthedestabilizingeffect.ThistypeofperturbationonlygrowswhenthelocalboundarylayerReynoldsnumber(i.e.Re)hasalarge,butnitevalue.AsRe!1,theTSwavebecomesstableagain( Wazzanetal. 1968 ).AnexampleoftheTSwavecanbeseeninFigure 2-4 .TheTSwavecanbeclassiedasawallmode,duetoitslocationwithintheboundarylayer.Whilethewaveextendsfarawayfromtheboundarylayer,themagnitudeoftheperturbationdecays 34

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rapidlyawayfromthewall.Ingeneral,thewavelengthofthemostunstableTSmodeisapproximately20long( White 2006 ). A B C Figure2-4. TSwavecomponentsofA)u-velocityandB)v-velocityatanon-dimensionalfrequencyofF106=140.0,whereF=!R Re.TheinletReynoldsnumberisRe=500C)BoundarylayerprolesfromaspatialstabilityanalysisforaReynoldsnumberofRe=500. WhiletheTSmodeisanexampleofaninstabilitywherethetheoreticalresults( Schlichting 1933 ; Tollmien 1929 )overlapwiththeexperimentalresults( Rossetal. 1970 ; Schubauer&Skramstad 1947 ),theoverlapbetweentheresultshasnotbeenfreeofcontroversy.ExperimentalmethodstendtomeasurethecriticalReynoldsnumbertobelowerthanwhatthelocal,linearstabilityanalysespredictittobe( Rossetal. 1970 ; Schubauer&Skramstad 1947 ).Asthetheorybasedmethodsbecamemoresophisticatedintakingthetwo-dimensionalnatureoftheowintoaccountusingparabolicequations( Bertolottietal. 1992 ; Gaster 1974 ),andascomputationalmethodsbecamemorerobust( Bertolottietal. 1992 ; Fasel 1976 ),thedatacontinuedtofavorlinearstabilitytheory(thoughdiscrepanciesarenotedathigherfrequencies).However,outofthisdisagreement,severalfactshavebeenidentied. 1. Themetricusedtoquantifyperturbationgrowthmatters( Fasel&Konzelmann 1990 ; Gaster 1974 ).Theu-velocityandv-velocitycomponentsoftheTSmodegrowatsimilar,butnotidenticalrates.Consequently,theenergygrowthoftheTSwave,whetheratsomedistanceawayfromthewallorintegratedalongaline 35

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normaltothewallalsogrowsatitsownrate.Assuch,thespatialamplicationratemustbewelldenedinordertoperformmeaningfulcomparisonswithotherstudies.Theneutralstabilitycurvecanbesensitivetowhichmetricisused. 2. Thelocationoftheperturbationmeasurementalongtheheightoftheboundarylayeralsomatters( Fasel&Konzelmann 1990 ).Thegrowthoftheperturbationoccursatdifferentratesdependingonwheretheinstabilityismeasuredintheboundarylayer.Theneutralstabilitycurvecanbesensitivetowheredataiscollected. 3. Agreementbetweenthedifferentmethodsisbetteratlowernon-dimensionalfrequenciesthanathighernon-dimensionalfrequencies( Bertolottietal. 1992 ). 4. Theperturbationintroducedtotheowmatters( Bertolottietal. 1992 ).Dependingontheperturbation,transientornon-lineareffectsmaybeintroducedintotheboundarylayer.Eventually,itwasdeterminedthatthedifferencesbetweenthetheoretical,numerical,andtheexperimentalresultswasduetoaspatialtransientgrowthmechanism( Bertolottietal. 1992 ).Transientgrowthmechanismsusuallycanbeminimized,butnoteliminatedinexperimentalstudies.Onlyinnumericalsimulationscanaperturbationbedenedthatdoesnotprojectontoanytransientinstabilities.TwoempiricallyderivedconclusionshavearisenfromstudyingtheTSwaveforboundarylayers.Therstresultisknownastheuniversalcorrelationforboundarylayer( Wazzanetal. 1979 ).Basedonabroadsetofexperimentsandstabilitycalculationsofboundarylayers,itwasfoundthattheshapefactoroftheboundarylayer(H==)istheprimaryvariableofimportancewhendeterminingthecriticalReynoldsnumber(specically,Re).ThecriticalReynoldsnumberforalloftheowsexamined(includingsubsonic,supersonic,owswithadverse/favorablepressuregradients,etc.)appearstocollapsetoasinglelinewhencomparedbytheshapefactoroftheboundarylayer(Figure. 2-5 ).AnotherempiricallybasedresultisthatoncetheTSwavehasbeenampliedtoacertainamount,theowwilltransitionquickly.Whilenotentirelyaccurate,astheTSmodedoesnotinstantlytransitionfromorganizedarraysofspanwisevoriticestoafully 36

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Figure2-5. UniversalcorrelationofthecriticalReynoldsnumber.AftertheFalkner-Skandatain White ( 2006 ). turbulentow,thelaterstagesoftransitionoccurfairlyquickly,sothisapproximationisnottoounrealistic.ThemagnitudeofexponentialamplicationrequiredtoassumethattransitionwilloccurisreferredtoastheN-factor.Thetotalamplicationoftheperturbationcanbedenedthroughtemporalstabilityanalysis( vanIngen 1956 )asA A0=expZxtrans.xcrit.cidt=expZxtrans.xcrit.!idtexp(9) (2)orthroughspatialstabilityanalysis( Jaffeetal. 1970 )asA A0=exp)]TJ /F10 11.955 Tf 11.29 16.27 Td[(Zxtrans.xcrit.idxexp(10) (2)wherexcrit.isthepointwheretheinstabilitybeginstogrowatacertainfrequency(i.e.!R)andxtrans.referstothepointwheretheowcanbeconsideredfullyturbulentforacertainperturbationfrequency.A A0referstotherelativemagnitudeoftheperturbationtoitsinitialmagnitude(A0)atxcrit..TherighthandsidetermsofEquation 2 isexp(9)8000andEquation 2 isexp(10)22000refertothetotalamplicationofthemodalinstabilitywhichmustoccurfortransitiontooccur.Theamplicationrequiredfortheowtobecomefullyturbulentdependsonanumberoffactors(freestreamturbulencelevels,surfaceimperfections,vibrationsoftheboundarylayersurface,etc.),butcanbeassumedtobeoftheorder104.TheN-factorisdenedaslnA A0.While 37

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thespatialstabilitymethodofdeterminingtheNfactoristhemorephysicallyaccuratemethod,thetemporalstabilityismorecommonlyused,asitiseasiertocalculateusinglinearstabilitymethods.TheN-factoranalysisoftheTSwave,whileusefulinapplication,doesnotpresentaphysicallyaccuratedescriptionofthelaterstagesofthetransitionprocess.OncetheTSmodeofacertainfrequencyreachesacriticalmagnitudewithrespecttothefreestreamvelocity,itbecomessusceptibletosecondaryinstabilities( Herbert 1988 ).IntheTSwavepathtotransition,thesesecondaryinstabilitiespresentthemselvesasLambda()shapedvorticesintheboundarylayer,repeatinginthespanwisedirection.Interestingly,thissecondaryinstabilitymechanismcanbemanifestedintwodifferentways.Thelessunstableorientationofthevorticesoccurswhenthevorticesexistinastaggeredpattern(H-type,Figure 2-6 AandB),suchthattwostreamwiseadjacentvorticesarestaggered180degreesoutofphasewitheachother.Themoreunstableorientationofthevorticesoccurswhenthevorticesarealignedwitheachother(K-type,Figure 2-6 CandD).Athigherdisturbancelevels,thetransitionoftheowfromlaminartoturbulentislikelytobeinitiatedbyadifferentpathtotransition(shownasPath(C)inFigure 2-3 ).InsteadofthemodalgrowthoftheTSwaveintheboundarylayer,perturbationsfromthefree-streaminitiatenon-modalgrowthofvorticalstructuresintheboundarylayer.Transientinstabilityanalysis( Butler&Farrell 1992 )indicatesthatstreamwiseorientedvortexstreaksarethemostampliedmodeoveranitelengthoftime.Theselong(<<)structuresconsistofcounterrotatingstreamwiseorientedvortices,whichgenerateverylargevariationsinthestreamwisevelocity(ju0j>>jv0j,jw0j),astheytransportstreamwisemomentumintoandoutoftheboundarylayer(Figure 2-7 ).Theenergycontainedinthesenon-modalstructurescanbeampliedbyuptothreeordersofmagnitudebythesetransientamplicationmechanisms. 38

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A B C D Figure2-6. ExamplesofsecondaryinstabilitiesforTSwave.A,B)H-typeandC,D)K-typevorticesinacontrolledtransitionscenariofrom Sayadietal. ( 2011 ). JustastheTSwaveissubjecttosecondaryinstabilitygrowthandeventualbreakdownoftheorganizedowfeaturesintheboundarylayer,thesestreamwiseboundarylayerstreaksarealsosubjecttotheirownsecondaryinstabilityandbreakdownmechanisms.Anumberofdifferentfundamentalandsubharmoniceffectshavebeenidentiedby Anderssonetal. ( 2001 )usinganinviscidinstabilityanalysisandviscousDNS,whichdisplayreasonableagreementwiththeexistingexperimentalstudies.Athirdandequallyimportantpathtotransitionisthatduetobypasstransition(whichincludespaths(D)and(E)inFigure 2-3 ).Thispathtotransitionisusuallyinitiatedbylargerperturbationsintheow.Inbypasstransition,theselargeperturbationsusuallymanifestthemselvesasturbulentspots,withouthavingtorstbeampliedbyanyotherinstabilitymechanism(Figure 2-8 ).Whilethefulldetailsofbypasstransition 39

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A B Figure2-7. Examplesofstreamwisevortexstreaksinaboundarylayer.TheA)streamfunctionindicatingstreamwiseorientedvorticityandB)thestreamwisevelocitycomponentastheygrowinmagnitude.From Butler&Farrell ( 1992 ). arenotfullyunderstoodatthistime,itishypothesizedthatbypasstransitionisstronglyrelatedtothespatialoptimallygrowingperturbations( Tumin&Reshotko 2001 ). Figure2-8. Exampleofby-passtransitiongeneratingaturbulentspot.From Wu&Moin ( 2009 ) Withtheproperapplicationoftransitioncontrol,specicperturbationscanbeintroducedtotheow,andwiththeproperfeedbacksystem,existingperturbationsmaybecanceled.Passiveandactive,closedandopenloopowcontrolsystemshaveall 40

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beenimplementedusingavarietyofdifferenttypeofsensorsandactuators( Cattafesta&Sheplak 2011 ).Indoingthis,thelaminartoturbulenttransitionprocesscanbecontrolled,acceleratingordeceleratingtheprocessasdesired.Givenenoughtime,andiftheinstabilitiesintheowaresufcientlylarge,aowwilleventuallyreachaturbulentstate.Withtheadventofhighperformancecomputing,whichallowsforsimulationstoprovidetheentireoweldatspecicinstancesintime,specicphenomenaintheowcanbeidentied.However,thishasonlybeenthecaseinthelastfewdecades.Beforecomputationaluiddynamics(CFD)becameaviabletoolforexaminingturbulentows,statisticalmethodswerecommonlyusedtoexaminethepropertiesofturbulentows.Assuch,thestatisticalpropertiessuchasthemean,variance,co-variancesandfrequencyspectra(bothtemporalandspatial)oftheowhavebeenusedtocharacterizeandexploreturbulentows.Onestatisticalpropertyinparticular,the u0v0Reynoldsstress,hasbeensuccessfulindescribingtheunsteadymixingthatoccursinaturbulentow.Statisticaldescriptionsofaturbulentoweliminatetheoverwhelmingdetailsofexaminingthemultitudeofindividualvorticalstructuresthatexistintheow,andsimplifytheanalysis,allowingforthefocustobeplacedonmoreapplicationrelevantandmoreeasilyquantiedpropertiessuchasturbulentmixing,skinfriction,ortheproductionanddissipationofturbulentkineticenergy.Inadditiontothestatisticalmethodsthathavebeenemployed,advanceshavebeenmadeinunderstandingtheindividualcoherentowstructuresinaturbulentow.Themostprevalentstructuresinaturbulentboundarylayerorchannelowarehairpinvortices(Figure 2-9 ).Othercoherentstructuresalsoexist,suchasstreamwisevortices,whichoccasionallyburstandejectlowspeedturbulentowawayfromthesurfaceofaboundarylayer.Muchefforthasgoneintothestudyofthehairpinstructuresinparticular,andtheyhavebeenfoundtobegeneratedeitherdirectlybysomemeansofperturbationintheow( Acarlar&Smith 1987a b ),aswellasthroughvarioustransitionprocesses,includeprimarytosecondarywallmodetransition(Tollmein-Schlitching 41

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A B CD Figure2-9. Visualizationofturbulentstructuresinaatplateboundarylayer.A)HairpinstructuresinaturbulentzeropressuregradientatplateboundarylayerusingtheQ-criterion.From Wu&Moin ( 2009 ).B)Ejectionandsweepeventsinaturbulentboundarylayer.From Falco ( 1977 ).C)SchematicofahairpinvortexandD)sideviewofahairpinvortexalongthecenterofthestructure.From Adrianetal. ( 2000 ) waves+harmonicorsubharmonicsecondaryinstabilities( Sayadietal. 2011 )),andbypasstransition( Wu 2010 ; Wu&Moin 2009 ).Theorganizationofthesestructureshasalsobeenexamined( Adrian 2007 ),aswellasthemannerinwhichindividualstructuresreproducetoformpacketsofhairpinstructures( Zhouetal. 1999 ).Ithasbeendeterminedthatindividualhairpinvorticescangenerateextendedtrainsofhairpinvortices.Thisnaturalamplicationoftheturbulence 42

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structuresuggestsanentrypointforfutureowcontroleffortsforturbulentboundarylayers.Theseturbulentmotionscanbequantiedbywhatquadrant( Wallaceetal. 1972 )theybelongto,where,dependingonthesign(positiveornegative)ofthestreamwiseandwallnormalcomponents,theuidataspecicpointcanbeclassiedasejections(u()]TJ /F4 11.955 Tf 9.3 -.17 Td[()v(+)),asweep(u(+)v()]TJ /F4 11.955 Tf 9.3 -.17 Td[()),wallwardinteractions(u()]TJ /F4 11.955 Tf 9.3 -.17 Td[()v()]TJ /F4 11.955 Tf 9.3 -.17 Td[()),oroutwardinteractions(u(+)v(+)). Wallaceetal. ( 1972 )measuredtheseeffectsusinghot-wireanemometry,andconcludedthattheejectionsandsweepscontributemostgreatlytothegenerationoftheReynoldsstress,whilethewallwardandoutwardinteractionssubtract(toalesserdegree)fromtheReynoldsstress.Withtheavailabilityofhigh-qualityPIVdataithasbeenfoundthatthesesweepsandejectionslargelycorrespondtopacketsofhairpinvorticesintheboundarylayer( Adrian 2007 ). 2.4ImplementationofDBDBodyForceIntoNavier-StokesEquationsDBDplasmadevicesconsistoftwoelectrodeasymmetricallyplacedacrossasoliddielectric,withcomponentsranginginsizefromfrommmtom( Rothetal. 1998 2000 ; Zitoetal. 2012 ).Theelectrodesarethenpoweredusingahighvoltage,highfrequencysignal(ontheorderof0.1)]TJ /F4 11.955 Tf 12.26 0 Td[(10kVppand1)]TJ /F4 11.955 Tf 12.26 0 Td[(100kHz).Thissignalgeneratesanelectriceld,whichionizestheworkinggasandthoughtheTownsendionizationprocessandsecondaryemissionprocesses,sustainstheplasma.Thisalternatingelectriceldisthenabletoaddmomentumtothechargedparticlesviatheelectro-hydrodynamicforce(fi=qEi).Inturn,thechargedparticlesimpartmomentumtotheun-chargedparticles,eventuallyresultingintheformationofawalljetintheregionoftheDBDactuator( Opaitsetal. 2010 ).Variousmethodsofsimulatingtherstprinciplesplasmadischargeprocesshavebeendeveloped( Boeufetal. 2007 ; Roy 2005 ),thoughthistypeofcalculationdoesrequiresignicantcomputationalexpense.Whenitcomestocouplingtheplasmaactuationtotheuidow,theplasmaismostoftentreatedasanon-thermal 43

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bodyforce,appliedthroughthefitermintheNavier-Stokesmomentumequations(Equation 2 ).FirstprinciplesmodelshavebeenemployedtoserveasthebodyforceinCFDsimulations( Gaitondeetal. 2006 ),thoughsimplerbodyforcemodelsaremorecommonlyused.Thesimplerbodyforcemodelscanbegroupedintoanumberofdifferenttypes.Theearliestandsimplest,phenomenologicalmodelsofthebodyforcewerebasedonobservationofthattheplasmaactuatorsaddedmomentumtotheow( Shyyetal. 2002 ).Thesemodelsassumethattheplasmaregiongeneratesabodyforcedistributedoveracertainarea,andthatitisstrongestneartheelectrodes,andweakestattheedgeoftheplasma.Whilethesemodelsarenotthemostrobustmodelswithrespecttothephysicsthatarerepresented,theydogenerateawalljetdownstreamoftheplasmaactuator,providingaloworderapproximationofthedesiredeffect.Morerobustmodelsassumealoaddistributionofchargedparticles( Suzenetal. 2005 ),andcalculatetheelectriceldtogeneratethebodyforce.ExperimentalmeasurementsofthevelocityeldaroundtheplasmaactuatorallowforthebodyforcetobecalculatedbeforebeingemployedaspartofCFDsimulations( Kotsonisetal. 2011 ; Kriegseisetal. 2013c ).Whenitcomestobalancingtheeaseofuseandphysicalrobustnessofthedifferenttypesofmodels,approximatedbodyforcedistributionsmaybethebest.Approximatedbodyforcedistributionsgeneratethedistributedbodyforcebasedonnumericalsimulations( Singh&Roy 2008 )orexperimentalresults( Madenetal. 2012 )oftheDBDactuator.Thesedistributionsareusuallydenedascontinuouspolynomialorexponentialfunctionswithrespecttoxandy,allowingthemtobeeasilyimplementedintocomputationaltools,withouttheneedforinterpolatingexistingdataontoanewdomain.Allofthesedifferentmethodshavetheirprosandconswithrespecttoevaluatingthebodyforcedistributionandimplementingitforsomeuse.Arecentstudyby Madenetal. ( 2012 )comparedanumberofdifferentbodyforcemodels,andmeasuredthe 44

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differencesinthevelocityeldinandaroundtheplasmaregion.Theirstudyfoundthatthereareslightdifferencesintheoweldsgeneratedbythedifferentmodelsinthenearplasmaregion.However,thatstudyalsofoundthatwhenthemodelsareproperlycalibratedtheresultsarenearlyindependentofthebodyforcemodeloutsideoftheplasmabodyforceregion. 45

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CHAPTER3LOCALSTABILITYANALYSISOFAOFAPLASMAACTUATEDBOUNDARYLAYERInthepastfewyears,oneemergingapplicationofDBDplasmaactuatorshasbeentocontrolthelaminartoturbulenttransitionprocessinboundarylayerows.Thereareadvantagestousingtheseactuatorstocontrolthelaminartotransitionprocessoverothertypesofactuators.Thattheseactuatorscanbeushmountedcertainlyposesitselfasanadvantage,asthismeansthattheactuatorcannotaccidentallydisrupttheowwhenitisinactive,butonlymodiestheowwhenitisactivated.Furthermore,thistypeofactuatorcanberuncontinuouslyorinadutycycledmanner,whichallowsforittobeusedindifferentoperationalmodes,dependingontherelevantowconditions. Grundmann&Tropea ( 2008b 2009 )usedDBDactuatorsinaowwise(co-ow)orientationwithcontinuousactuation,constantlyaddingmomentumintotheboundarylayer.Operatingtheactuatorsinthismanner,theywereabletodelaythetransitionby200mmfora10m/sow. Duchmannetal. ( 2012 )madePIVmeasurementsoftheTSwaveintheregionaroundtheplasmaactuator,andshowedasignicantreductioninthewaveamplitudeandchangestothewavespeedrelativetotheowwithoutplasmaactuation.SometheoreticalworkwasalsoperformeddescribinghowmomentumadditionusingEHDdevicesmodiestheboundarylayeranditsstabilityproperties.Forexample,owstabilizationdownstreamoftheplasmaactuatorhasbeenpredictedforco-oworientedmomentumadditionusinglocalstabilityanalysisandemployingboundarylayerprolesfromexperiments( Duchmannetal. 2013a )andCFDsimulations( Riherd&Roy 2013b ; Riherdetal. 2012 2013 ).Morerobustworkusingbi-globalstabilitycalculationsforTSwavesandboundarylayerstreakshasalsobeenperformed,identifyingtheexactstabilizationmechanisms( Riherd&Roy 2013a b d ).ThedetailsofthatworkisdescribedinChapter 4 46

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Withregardtospecicapplicationsbeyondaatplate, Seraudieetal. ( 2011 )haveexaminedtheeffectsofusingDBDplasmaactuatorsfortheowoveranONERA-Dairfoilatanangleofattack(whichgeneratedanadversepressuregradientoveramajorityofthechord).TheirstudyemployedlowvelocitywindtunneltestingandLSTtopredictthetransitionpointsoftheow,usingamodelsimilartowhathasbeendevelopedinthisstudy.Recently,ighttestsusingcontinuousplasmaactuationhavebeenperformed( Duchmannetal. 2013b ).Theseighttestsdemonstratedatransitiondelayof3%ofthechordlengthforasmallaircraftforahighReynoldsnumberow(Rex1.15106).Activeowcontrolapproacheshavealsobeenexamined, Grundmann&Tropea ( 2008a b 2009 )usedthesedevicesasamethodoftransitiondelayinanadversepressuregradientboundarylayeraspartofaclosedloopcontrolsystemtocanceloncomingTollmien-Schlichting(TS)waves.ItwasfoundthatpulsedDBDactuatorscouldbeusedtoaccuratelyinjectmomentumintotheboundarylayer,cancelingtheTSwaves. Dadfaretal. ( 2013 )examinedthisproblemfromatheperspectiveofaimplementingamorerobustfeedbackcontrolproblemusinglinearquadraticcontrollers,andpredictedthatthedisturbancescouldbemitigatedby60-75%oftheirmagnitude. Hansonetal. ( 2010a b )haveperformedsomepreliminaryworkintotheuseofstreamwiseorientedDBDactuatorsforthecontroloftransientinstabilities,whichshouldideallyleadtothegenerationofclosedloopowcontrolapplications.Whileactiveowcontrolapproachessuchasthesemayultimatelyleadtoincreasedenergysavingsrelativetocontinuousforcingapproaches,thereisamuchhigherlevelofcomplexityinimplementingthistypeofapproach,whichmybeprohibitiveforapplicationsinthenearfuture.Inthischapter,theeffectsofaplasmaactuatorinalaminarboundarylayerareexamined.SimulationsatlowReynoldsnumbershavebeenperformed.Basedontheresultsofthesesimulations,differentspatialregionsintheboundarylayerareidentied 47

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fromtheirlocalowcharacteristics.Alocal,temporalstabilityanalysisisthenusedtoexaminethestabilityofthisregion,focusingontheonsetoftheunstableTSmode.BasedonthedifferenttypesofboundarylayervelocityprolesseenintheselowReynoldsnumbersimulations,alowordermodeloftheboundarylayervelocityproleisthenconstructed.Usingthesamelocal,temporalstabilityanalysis,thisgeneralmodelisusedtoperformawideparametricstudyoftheplasmaactuatedboundarylayer.Inthisparametricstudy,bothco-owandcounteroworientedplasmaactuationareexamined.Evidenceforaninviscidinstabilitymodeandanabsoluteinstabilityisdiscovered.Thedatafromthismodelisalsofoundtopredictdramaticchangesinthecriticalstabilitypointoftheboundarylayer,buttheresultsarestillinlinewiththeuniversalcorrelationof Wazzanetal. ( 1979 ). 3.1BaselineFlowModicationAsastartingpointforthestabilityanalysis,theeffectoftheDBDactuationontheZPGboundarylayerhavebeensimulatednumerically.ThisisdoneusingtheImplicitLargeEddySimulation/DirectNumericalSimulation(ILES/DNS)Navier-StokessolverFDL3DI( Rizzettaetal. 2008 ),detailsofwhichareprovidedinAppendix C .Atwodimensionalmesh(801151)isused,whichresolvesthenearwallboundarylayer,theeffectsofasharpleadingedge,andthesteadyadditionofmomentumthroughabodyforceterm.Atthelocationofplasmaactuation,thereare62pointsintheboundarylayer(99%)fortheReynoldsnumbertestedaspartofthisstudy.Thismeshisnerthanrequiredandisabletocapturetheowadequatelynearregionsofhighgradients,particularlyforthethinboundarylayerneartheplateleadingedge.Themeshisgeometricallystretchedneartheboundariesinordertopreventtheeffectsofreectionsthatcouldpotentiallybounceoffofthefareldboundariesandinterferewiththeow.AschematicofthedomainusedcanbeseeninFigure 3-1 A. 48

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Figure3-1. Schematicofthemeshusedforcomputations.A)Thetwo-dimensionaldomainandvelocityboundaryconditionsusedforthebaselineowmodications,andB)acloseupoftheco-oworientedbodyforceusedwithintheboundarylayer.Everyfourthgridpointisshown. 3.1.1BoundaryConditionsTheintentofthesesimulationsistomodeltheeffectsofaplasmamodiedboundarylayeraspartofanincompressibleow.However,thesimulationtoolused,FDL3DI,isacompressibleowsolver.WhilesettingthefreestreamMachnumbertoanappropriatelylowvalueof(M1=0.1)allowsforminimalcompressibilityeffectsinthedomain,therearesomeboundaryconditionconcernsthathavetobeaddressed.Thevelocityboundaryconditionsarethesimplesttounderstandandimplement,astheyarethesameregardlessofcompressibility.Onthesurfaceoftheplate,noslipconditionsareapplied(u=v=0).Upstreamoftheplate,asymmetryplaneisassumed(v=@u @y=0).Attheinlet,afreestreamvelocityconditionisapplied(u=1,v=0).Forthefareldnormaltotheplateanddownstreamoutboundaries,rstorderaccurateNeumannboundaryconditionsareappliedtouandv.Theseboundaryconditionstaketheformofun=un)]TJ /F7 7.97 Tf 6.58 0 Td[(1,wherenandn)]TJ /F4 11.955 Tf 12.48 0 Td[(1indicatethegridpointsattheboundaryandtherstpointinsideoftheboundary.Atthislevelofaccuracy,thisconditionisequivalenttoloworderapproximation(i.e.nearestneighbor).Whilethismayimplythatthereisadditionalerrortotheow,theseboundariesaresignicantlyfarawayfromtheareaofinterestandshouldnothaveasignicantimpact. 49

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Thepressureboundaryconditionsarenotaswelldenedforthecaseoftheincompressibleow,butnonetheless,thereisanintuitivewaytoapplythem.Upstreamoftheplate,thepressureissettoaconstantvaluep1.Astheboundarylayerisazeropressuregradient,itisexpectedthatthefreestreampressuregradientalongthelengthoftheplateiszero(i.e.@p @x=0).Whilethisassumptionwillnotbeadequatenearthesurfaceoftheplatewhenthereisaplasmaactuator,itshouldsufceforthefareld.Assuch,thepressureforthefareldboundaryawayfromthesurfaceisconstantaswell,resultinginp=p1.Thisboundaryconditionisalsoappliedattheoutletofthedomain,sothatthereisnopressuregradientdrivingtheow.Forthesurface,theassumptionusedtogeneratetheBlasiusboundarylayer,@p @y=0,isemployedalongthelengthoftheplate(usingathirdorderaccuratebiasednitedifferencestencil,@p @yjj1 y)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 7.97 Tf 10.49 4.71 Td[(11 6pj+3pj+1)]TJ /F7 7.97 Tf 13.15 4.71 Td[(3 2pj+2+1 3pj+3).Thisboundaryconditionalsosufcesasaboundaryconditionforthepressureupstreamoftheplate.Densityboundaryconditionsalsoneedtobedened.Itisassumedthatattheinletandinthefareld,theplatehasnoinuenceonthedensity,sodensityisassumedtobeconstant(=1=1).Upstreamoftheplate,asymmetryconditionisappliedusingathirdorderaccuratestencil(i.e.@ @y=0).Overthelengthoftheplateaconstantwalltemperatureconditionisapplied,assuchthedensityisproportionaltothepressureatthesurfaceoftheplate.Whilteisassumptionmayormaynotbeaccuratedueheatingatthewallbytheplasmaactuator,temperatureeffectsareintentionallyneglectedinordertosimplifytheproblem.Attheoutlet,therstorderNeumannboundaryconditionisapplied,whichdoublesasarstorderinterpolationofthedensityattheoutlet. 3.1.2CalibrationofthePlasmaModelTheplasmaactuationismodeledusinganapproximatebodyforcedistributionbasedonrstprinciplesimulationsoftheplasmadischarge( Singh&Roy 2008 )(Figure 3-1 B)inamannerconsistentwiththedescriptionin Rizzettaetal. ( 2008 ).Thesamesetupisusedwithnoslipconditions(u=v=0)fortheleft,right,and 50

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bottomboundariesandanoshearconditionontheupperboundary,leadingtoaquiescentconditionoveramajorityofthedomain.Theplasmaactuatorisrunatvariousmagnitudeswithquiescentinitialandamixofnoslip/noshearboundaryconditions.ThemagnitudeoftheplasmaactuationisproportionaltotheparameterDc,whichrelatesthemagnitudeofthebodyforcetothedynamicpressure(i.e.,Dc=jf0jL u21).Theeffectoftheactuationontheowisthencharacterizedbythemaximumvelocityseeninthewalljet(up,showninFigure 3-2 ).Alinearinterpolationisthenusedtocontrolthebodyforceforthesimulationundernon-quiescentconditions.Themagnitudeoftheimplementedforceischaracterizedbythenon-dimensionalparameter 0=upjx0 u1(3)Thisparameterisselectedinordertofocussolelyontheuiddynamiceffectoftheplasmaactuationanditsinuenceontheowstability,ignoringtheelectricalinputssuchasvoltage,frequencyandthewaveformdrivingthedevice.Thevaluesof0arecalibratedfortheReynoldsnumbertestedaspartofthisstudy. Figure3-2. Calibrationdatafortheplasmaactuation.A)ValuesofupusedtocalibrateDc.B)VelocityprolesatalocationdownstreamoftheplasmaactuationforvariousvaluesofDc. 51

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3.1.3SimulatedBaseowsInthesimulations,thebodyforceisplacedatapositioncorrespondingtoRex=100,000(Re=543)inaboundarylayerow.Thisactuatorlocationisinthetransitionalregime,whichisusefulforunderstandinghowthemomentumadditionmodiesthelaminartoturbulenttransitioninthecriticaldomain.Therstthingthatshouldbenoticedisthattheadditionofmomentumintotheboundarylayermodiestheboundarylayerows(Figure 3-3 A-C).Slightlyupstreamoftheactuator,theowispulleddownwardsintothewallasiftherewereboundarylayersuctionpresent(Figure 3-3 D).However,asthepressuredatainChapter 6 willshow,thereisalocalpressureriseabouttheactuator,contrarytothecommonbelief( Corkeetal. 2010 ).Immediatelydownstreamofthedevice,theboundarylayerproleshowsseveralinectionpoints,whichsuggeststhatinviscidinstabilitiesmaybecomeimportant(Figure 3-3 E),astheysatisfyFjrtoft'scriteria(Equation 2 ).Furtherdownstreamofthis,theprolesreturntosomethingresemblingtheBlasiusprole,thoughfullerthanthatoftheinitialow(Figure 3-3 F). Figure3-3. VelocityeldsaroundtheplasmaactuatorsforA)0=0.00,B)0=0.10,andC)0=0.20.BoundarylayerprolesatD)x=0.99,E)x=1.01,andF)x=1.10arealsoshown.ThedashedlinesinA-C)indicatelocationswheretheboundarylayerprolesareextractedfrom,andareshowninD-F). 52

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Thereisexpectedtobesomeimpactonthepressureeldsintheseowsaswell.Itisknownfromhydrodynamicstabilitytheorythatforlaminarboundarylayers( White 2006 ),owswithaapositivepressuregradient(i.e.adeceleratingow,suchasanexpandingnozzle)willbelessstable,whileowswithnegativepressuregradients(i.e.anacceleratingow,suchasonaforwardfacingramp)willbemorestable.Comparingthesurfacepressureswithandwithoutplasmaactuation(Figure 3-4 A),thereisaslowlydeveloping,reductioninthepressureupstreamofthebodyforce(duetoentrainmenteffectsacceleratingtheow).Rightattheactuator,thereisaveryrapidjumpinthesurfacepressures,whichthenslowlydecaysastheowdevelopsdownstream.Thisrapidriseinthepressureislikelyduetovariouseffectsresistingtheaccelerationoftheowbythebodyforce.Examiningasliceoftheowoutsideoftheboundarylayer(Figure 3-4 B),comparableeffectscanstillbeseen,buthesuddenriseislessrapid,andthemagnitudeoftheeffectsisreduced,indicatingthatthesechangestothepressureeldarelocalizedtotheboundarylayer. Figure3-4. Pressureonandnearthesurfacearoundtheplasmaactuator.A)SurfacepressureandB)pressurecollectedataheightof0.02,whichisoutsideoftheboundarylayerandbodyforceregion. Itisexpectedthatthesepressuregradientshavesomeimpactonthestabilityoftheow.Upstreamanddownstreamoftheactuator,thepressuregradientisnegative,whichcorrespondstothestabilization.Thisstabilizingeffectiscomplicatedbynon-parallel 53

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andowdevelopmenteffect,sothisimplicationontheow'sstabilityshouldbeinterpretedwithcautionuntilquantitativelyanalyzed(seeSection 3.2 ).Theevenmorecomplexregiontounderstandistheregionrightaroundtheplasmaactuator.Duetotwo-dimensionaloweldinthisregion,theimpactofthepressuregradientonitsownmayormaynotbeofimportancerelativetoothereffectsintheoweld.Astheboundarylayervelocityprolesaremodiedduetotheadditionofmomentumintotheboundarylayer,thedisplacementandmomentumdecitoftheboundarylayershouldbealteredaswell.ItcanbeseeninFigure 3-5 thatthereisamonotonicresponseoftheboundarylayerheightstothemagnitudeofthebodyforce.Therearesomelocalizedeffectsneartheboundarylayer,inwhichtheboundarylayerheightmayrise(99%,)duetomodicationsveryneartoorawayfromthewallorfall()duetoareductionintheboundarylayermomentumdecit.Overall,thereisadropintheboundarylayerheightsdownstreamwiththeadditionofmomentumintotheboundarylayer.Alloftheseeffectssuggestthatthereare3differentlocationsimportanttothestabilityofthesedevices(Figure 3-5 ).Thereistheregionupstreamofthedevices(I),whichmaynowbeslightlymorestableduetotheentrainmentofuidintotheboundarylayer.Thereistheregionoverandimmediatelydownstreamofthedevice(II),whichshowssignsofinviscidinstabilityduetoFjrtoftscriteria.Finally,thereistheregionfardownstreamofthedevice(III),whichshouldbemorestableduetoareducedboundarylayerheightcausedbytheinjectionofmomentumintotheboundarylayerbytheplasmaactuator.Asimportantaschangestotheboundarylayerheightsmaybe,thechangestotheboundarylayerprolesareevenmoreimportant.Thefullnessoftheboundarylayerprolecanbemeasuredbytheboundarylayershapefactor(H= ).Alowershapefactorindicatesafuller,morestableboundarylayerprole,andevensmallchangestotheshapeofthevelocityprolemayincurverylargechangestothecriticalReynolds 54

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Figure3-5. Boundarylayerheightsandshapefactorasafunctionofthevelocityratio,0,forvaluesrangingfrom0to0.25-A)99%,B),C),andD)theshapefactor,H==,alongwithacomparisontotheanalyticalsolutionforthecaseofReact=100,000.Inlaysshowazoomedinviewneartheactuatorlocation. numberoftheboundarylayer( Wazzanetal. ( 1979 ),Figure 2-5 ).ItcanbeseeninFigure 3-5 Dthattheadditionofmomentumintotheboundarylayercaninducelargechangestotheshapefactor.Basedonthesetworesults,itisexpectedthatthechanges 55

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totheshapeoftheboundarylayerprolesbythemomentumadditionwillinduceastabilizingeffectintheplasmamodiedboundarylayerows. 3.2LocalLinearStabilityTheory 3.2.1NumericalModeloftheEigenvalueProblemLinearstabilitytheorycanbeusedtopredicttheexistenceandgrowthratesofinstabilitiesthatmaymanifestthemselvesintheboundarylayer.Inordertoperformthistypeofanalysis,itmustbeassumedthattheowisslowlydevelopingandparalleltothesurface.Twometricshavebeendeveloped,EkandEd,whichquantifythevalidityoftheseassumptions.Thesemetricsaredenedas Ek=maxy2(0,1)tan)]TJ /F7 7.97 Tf 6.59 0 Td[(1v u(3)and Ed=maxy2(0,1)@u @x(3)andquantifytheparallelowandslowlydevelopingassumptions,respectively.TheseassumptionsarequantiedinFigure 3-6 .Thisow,whileitdoesexhibitsomerapidspatialchangesandnon-parallelbehaviorneartheactuator,canbeconsideredaslowlydeveloping,parallelowovertheremainderofthedomain.Assuch,onedimensional,linearstabilitytheorycanbeapplied,exceptforneartheactuator.Inordertounderstandtheeffectsneartheactuator,abi-globalstabilitymethodisemployed,theresultsofwhicharediscussedinChapter 4 .Forthepresentlocalstabilityanalysis,themethoddescribedinAppendixAisemployed.Thismodelfocusesononlyuidiceffects;thereisnofeedbackmechanismbetweentheuidandplasmapresentinthisanalysis.Inordertocheckfortheaccuracyandconvergenceofthesolution,agridresolutionstudyhasbeenperformed.Twoseparatecases,oneexaminingtheTSmode,theotherexaminingafastmode,whichwillbeexplainedinmoredetailinSection 3.3 ,havebeenexamined.TheconvergenceofthemostunstableeigenvaluecanbefoundinTable 3-1 andFigure 3-7 .Theorder 56

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Figure3-6. Evaluationofparallelowapproximation.A)EkandB)EdfortheoweldscalculatedinSection 3.1.3 forvelocityratiosof0=0to0=0.35. ofaccuracyofthestencilsemployedforthecalculationsweresecondandfourthorderaccurate,respectively.Theconvergenceoftheunstableeigenmodes,reectsthis,asitconvergesataratebetweenn)]TJ /F7 7.97 Tf 6.59 0 Td[(2yandn)]TJ /F7 7.97 Tf 6.59 0 Td[(4y.Examiningthisconvergencedata,thegridresolutionofny=201,y=0.0581Blasiusappearstobeintheasymptoticerrorrange.Thisgriddensityshouldbesufcientresolutionforthepresentmethod.Furthermore,thisgriddensityprovidesaresultinanappropriateamountoftimetoperformawideparametricstudy. Table3-1. Convergenceofthemostunstableeigenvaluefortwosamplecases.Forbothcases,Re,Blasius=1000,=0.3. nyBlasius,!TSModel(=1,=0.25),!Fast 510.11238325236)]TJ /F4 11.955 Tf 11.96 0 Td[(0.00090042619i0.23306979831+0.00220664208i710.11017358051+0.00059957882i0.23281253531+0.00242939870i1010.10924347588+0.00163256137i0.23273816623+0.00249039239i1420.10898287015+0.00213952218i0.23272109814+0.00250492030i2010.10892701325+0.00235576622i0.23271726363+0.00250853091i2830.10892058760+0.00243347102i0.23271652043+0.00250929338i4010.10892145449+0.00246038133i0.23271638394+0.00250945035i5660.10892233968+0.00246910010i0.23271636662+0.00250947873i8010.10892273162+0.00247197532i0.23271636809+0.00250948486i 57

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A B Figure3-7. Gridconvergenceofthecriticaleigenvaluefortwodifferentboundarylayerinstabilies.A)BlasiusboundarylayerproleandB)aboundarylayerprolegeneratedbythemodel(=1.0,=0.25).Forbothcases,Re,Blasius=1000,=0.3.Theverticallineindicatesthegridresolutionusedforthepresentstudy. 3.2.2Co-FlowActuationVelocityprolesfromSection 3.1 wereextractedfrom160pointsintheow(0
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movedfartherandfartherdownstream.ThisindicatesthattheboundarylayerstabilityissuccessfullybeingreinforcedbytheuseofDBDactuation. A B Figure3-8. ComputedeigenspectradownstreamofthepalsmaactuatoratRex=120,000,=0.3A).Aclose-upoftheTSwaveeigenvalueisalsoshownB). ItcanbeseenthatLSTpredictsincreasedowstabilizationintermsofRex(Figure 3-10 )aswellasReintheneutralstabilitycurves(Figure 3-9 ).Furthermore,thisowstabilizationextendsbeyondthenearactuatorregion,topointswheretheassumptionsrequiredfortheone-dimensionalstabilityanalysisarevalid.Thereasonsforthisowstabilizationarelikelytobeduetotwodifferenteffects,bothofwhicharecoupledtoeachotherandtheadditionofmomentumintotheboundarylayer.Theadditionofmomentumintotheboundarylayerisshowntoreducetheboundarylayerdisplacementheight(Figure 3-5 B).Inturn,thisreducesthelocalboundarylayerReynoldsnumber,whichdelaystheonsetofperturbationgrowthintheboundarylayer.Thesecondsourceofstabilizationresultsfromthemomentumadditionmodifyingtheboundarylayervelocityproles(Figure 3-3 ).Byaddingmomentumtotheow,theboundarylayerprolesaremadefuller,andtheshapefactor,H,ismodied.ThisparameterhasbeenshowntobeveryimportantinidentifyingtheonsetoftheperturbationgrowthinboundarylayersthroughauniversalcorrelationbetweenthecriticalReynoldsnumberandtheshapefactor( Wazzanetal. 1979 ),whereadecreasedshapefactorimplies 59

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Figure3-9. StabilitydiagramsofthemodiedboundarylayerowswhentheactuatorisplacedatRex=100,000forA)0=0.00,B)0=0.05,C)0=0.10,D)0=0.15,andE)0=0.20.InA-E),thelineofneutralstabilityismarked.F=Real(!)=Re.F)Neutralstabilitycurvesforvaluesof0rangingfrom0.00to0.25withaspacingof0.05 transitiondelayoftheTSwave.Fortunately,themannerinwhichmomentumisaddedintotheboundarylayerreducestheshapefactor,thusdelayingthecriticalonsetoftheTSwave'sgrowth. 3.3AModeloftheLocalBoundaryLayerProlesWhilethesesimulatedowsallowforanalysisunderthesespecicowscenarios,knowledgeofthestabilityofaboundarylayermodiedbyanarbitrarylevelofplasmaactuationisalsodesirable.Inordertoexaminetheeffectsofamoregeneralmodiedboundarylayer,alowordermodelapproximatingtheboundarylayervelocityprolescanbedeveloped.Itisknownthatunderquiescentconditions,plasmaactuationisabletocreatewalljets,whichmatchtheGlauertwalljetsimilaritysolutionsufcientlyfardownstream( Opaitsetal. 2010 ).Thecurrentsimulations(Figure 3-3 )aswellaspast 60

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Figure3-10. CriticalvaluesofRexfortheactuatedow. numericalandexperimentalresults( B.Jayaramanetal. 2007 ; Moreau 2007 )suggestthatthemomentumadditionintotheboundarylayercanformwalljet-likeeffectsifthelevelsofactuationarehighenough.Evenforlowerlevelsofactuation,themomentumadditionisstillseen.ThissuggeststhatasuperpositionofaboundarylayerandwalljetvelocityproleshouldsufcetoapproximatetheeffectsofplasmaactuationonaZPGboundarylayer.Thatis, ucomb.=uBoundarylayer+uWalljet(3)Inordertomanipulatethismodel,baseowsolutionsfortheboundarylayerandwalljetprolesarenecessary,aswellastwonon-dimensionalparametersinordertoscalethesizeandvelocitymagnitudeofthemomentuminjectionrelativetotheboundarylayer.TheBlasiusboundarylayer( Blasius 1908 )andGlauertwalljet( Glauert 1956 )similaritysolutionsarelogicalchoicesfortheZPGboundarylayer,thoughthereisnosuggestionthatthesuperpositionofthesesolutionswillresultinanexactsolutiontotheNavier-Stokesequations,onlyanapproximation.Concerningthenon-dimensionalparameters,whiletheglobalvalueof0canbeusedtocharacterizeanentiretwo-dimensionaloweld,butwhenexaminingindividual 61

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Figure3-11. Componentsoftheplasmamodiedboundarylayerowmodel. boundarylayerprolesitismoreusefultodenealocalizedvelocityratioparameter, =up(x) u1.(3)whichismorecloselytiedtothemomentumadditionthatexistsintheboundarylayerproledownstreamoftheplasmaactuator.Thislocallyvaryingvalueofisafunctionoftheglobalmagnitudeofthemomentuminjection(ascharacterizedthrough0)aswellastheconvectiveanddiffusivetransportofthemomentumintheboundarylayer,whichwillvaryasonemovesawayfromtheactuatorlocation.Themomentuminjectedbythewalljet p=Z10uWJ(y) updy(3)andmomentumdecitoftheboundarylayer(i.e.thedisplacementboundarylayerheight)canbeformulatedaslengthscales BL=Z101)]TJ /F4 11.955 Tf 13.4 8.09 Td[(uBL(y) u1dy(3)toformarelativelengthscalesuchthat =p BL(3) 62

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whichisalsoalocalparameter,duetothedevelopmentoftheboundarylayerandwalljetcomponentsintheow.Thesenon-dimensionalparameterscanthenbeappliedtotheinitialmodelfromEquation 3 suchthat u(y)=uBlasius(y)+uGlauerty (3)Theproductofthetwonon-dimensionalparametersdevelopedherecanbeusedtogenerateathirdphysicallyimportantparameter,,where =(3)thisparameteristheratioofthemomentuminjectedintotheboundarylayerbytheplasmabodyforceascomparedtothemomentumdecitintheboundarylayer.Whilethetotalamountofmomentumreplacedisdenitelyimportant,exactlyhowthismomentumisaddedintotheboundarylayeralsomatters.Momentumadditionoutsideoftheboundarylayerwillnotlikelybehelpful,norwillmomentumadditionthatoccursattooneartothesurface.Therefore,itisnotonlythevelocityratioortotalmomentumadditionthatwillaffecttheboundarylayerstability,butbothoftheseparameters,improperapplicationofwhichcouldbecounterproductive.Applyingthismodel,awiderangeofpotentialboundarylayerprolescanbecreated.Theseboundarylayerprolesareapproximatelymatchedtospecicparametersfromthesimulationsperformed.TheresultsofthisshowthatthereisareasonablygoodagreementbetweenthevelocityprolescreatedbythemodelandthosefromtheCFDsimulations(Figure 3-12 AandB).Agreementisgoodnearthewall,neartheregionofmaximumvelocity,andintothefareld.Howeverintheintermediateregionwherethewalljetdissipatesmomentumintotheboundarylayer,thereisanoticeablediscrepancy.Asthisdiscrepancyappearstobediffusiveinnature,itislikelytovarydependingonthelocalReynoldsnumberoftheboundarylayervelocityprole,whichisnotsomethingthatthecurrentmodeltakesintoeffect.Therealsoappearsto 63

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beadiscrepancyintheinectionpointsbetweenthemodelandtheextractedboundarylayerproles,asshownusingthesecondderivativeofthevelocityproleinFigure 3-12 C.Theinectionpointsinthemodelvelocityprolesdonotoccuratthesamelocationintheboundarylayerasthoseinthesimulatedvelocityproles,thoughtheyarestillwithin0.25BL.Thisdiscrepancyinthelocationofboundarylayerinectionpointshasimplicationswithrespecttotheexistenceandimportanceofinviscidinstabilitiestotheboundarylayer.Thisdiscrepancyseemstobesmallerfartherdownstreamoftheactuatorlocation,asthelocalvalueoftheboundarylayerratio,,hasincreasedandthevelocityratio,,hasdecreased.Asonemovesfartherandfartherdownstreamoftheactuator,thetrendsofincreasinganddecreasingcontinue,duetothedissipationofmomentumawayfromthewall. Figure3-12. ComparisonsbetweenthesimulatedandmodeledvelocityprolesareshownforthedownstreamprolesatA)x=0.01andB)x=1.1presentedinFigure 3-3 .Thevaluesof0listedindicatewhichCFDsimulationthevelocitieseldsarebeingextractedfromandmatchedto.Thevaluesofandvaryinordertottothemodel.C)Acomparisonofthesecondderivativeofuatx=1.1fortheboundarylayerprolesextractedfromtheCFDandgeneratedbythemodel. Agreementbetweentheboundarylayerprolesisbetterfartherdownstreamoftheactuators(atx=1.1,ascomparedtox=1.01,Figure 3-12 AandB),wherethe 64

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boundarylayerprolesdonotexhibitalargeovershootduetomomentumaddition.Thelocalparametersforthemodelare<0.20and0.64<<0.83atx=1.01(Figure 3-12 A)and<0.14and1.00<<1.15atx=1.1(Figure 3-12 B).Theco-owboundarylayerprolesseenfartherdownstreamoftheplasmaactuatoraremuchmoresimilartothoseinthecommonliteraturethanthoseseenveryclosetotheplasmaactuator,assuch,calculationsusingthelowordermodeloftheboundarylayerprolewillusecomparablevaluesofand. Figure3-13. Boundarylayerprolesusedinthecalculations,with=1.0andvaryingvaluesof. Itshouldbenotedthattherearenowtwodisplacementboundarylayerheightsrelevanttotheboundarylayerstability,bothofwhichareimportantfordifferentreasons.Fromaowcontrolperspective,thescalingbasedontheboundarylayercomponent(BL,i.e.theBlasiusboundarylayer)ofthecombinedvelocityproleismostrelevant.Holdingthisboundarylayerheightconstant,momentumcanbeaddedorsubtractedfromtheowandthechangesintheboundarylayer'sstabilitypropertiescanbeexamined.Whenitcomestounderstandingtherelevantphysics,thedisplacementboundarylayerheightbasedonthecombinedboundarylayervelocityprole()ismorerelevant,asitisthisboundarylayerheightdenedbythevelocityprole. 65

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3.3.1AdditionalInstabilityModesInadditiontotheviscid,convectiveinstabilityassociatedwithboundarylayers(theTSmode),additionalinstabilitiesmayalsobepresent.PreviousstudiesoftheeffectofDBDactuationonboundarylayerstabilityhavefocusedontheco-oworientationoftheplasmaactuationforowstabilization.However,thereareinstanceswhereowdestabilizationisagoalofowcontrol.Intheseinstances,operatingtheplasmaactuatorinacounterowmannermaybeofuse.Justasco-owactuationinjectsmomentumintotheboundarylayerandstabilizesit,counterowactuationremovesmomentumfromtheboundarylayerandshoulddestabilizetheboundarylayer.Inadditiontoremovingmomentum,forhighenoughlevelsofcounterowactuation(thatislarge,negativevaluesof),owseparationandreversalmayoccur.Thisowreversal,whilenotarequirementofanabsoluteinstability,suggeststhatonemaybepresent,addinganotherinstabilitymodetotheexistingconvectiveinstabilityassociatedwithZPGboundarylayers.Flowregimes(withrespecttoand)atwhichowreversalmayoccurareshowninFigure 3-14 ,whichareevaluatedusingthepresentmodeloftheplasmamodiedboundarylayerprole. Figure3-14. Regionwhereowreversaloccursintheboundarylayerprole. Examiningthedifferentboundarylayervelocityprolesgeneratedbythesimulationandthemodel,itcanbeseenthatinectionpointsmayoccurinthevelocityprole(Figure 3-12 C).Whileaninectionpointdoesnotnecessarilyindicatethataninviscid 66

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instabilityispresent,itdoesraisesuspicionsthatonemayexist.Fjrtoft'scriterionisastricterconditionforthepresenceofaninviscidinstability(Equation 2 ).Applyingthiscriteriontothemanyboundarylayerprolescomputedfordeterminingwhetherowreversalhadoccurred,itcanbeseenthatthereisonlyasmallregioninthe(,)planewhereFjrtoft'scriterionisnotmet(Figure 3-15 ).Thisresultsuggeststhattheregionforstabilizingowcontrolislimited,andthatonlyslightlevelsofco-owactuationcanbeusedforthispurpose.Iftoolargeofaco-owactuationisapplied,theninviscidinstabilitieswillbecomeasignicantproblem. Figure3-15. RegionwhereFjrtoft'scriterionismet. 3.3.2ComparisonBetweenComputedBoundaryLayerProlesandTheModelBeforeperformingalargenumberofcalculations,itwouldbebenecialtocomparethestabilitypropertiesofthecomputedboundarylayerproleswiththosegeneratedbythemodel.Thiscomparisonwouldbebenecialinestablishingthevalidityofthepresentmodelforthistypeofcalculation.Forthiscalculation,theboundarylayerprolesshowninFigure 3-12 Bhavebeenchosenforexamination.Atthispoint,Rex=110,000andRe=556.Whiletheseprolesmaynotbethemostoptimalduetotheirclosenesstotheplasmaactuator,theyexhibitenoughvariationintheboundarylayerproletodeterminewhetherornotthemodelandthecomputedboundarylayerprolespredictsimilarbehavior.TheexactparametersusedforthemodelaregiveninTable 3-2 67

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Table3-2. Parameterstocomparetheboundarylayervelocityproleowmodeltothesimulatedows. Case 0=0.050.01261.3330=0.100.03601.3180=0.150.06561.2980=0.200.09891.183 Incomparingtheresultofthetwomodels,itshouldbenotedthatboththeeigenmodesandtheeigenvaluesshouldbeconsidered,astheytogetherpredicttherelevantphysics.Comparingtheeigenvalues(Figure 3-16 ),itcanbeseenthattheerroroftherealandimaginarycomponentsislessthan20%ofthemagnitudeofthecriticaleigenvalue,andformostofthecomparisoncasesislessthan10%,especiallyathigherwavenumbers.Furthercomparingtheresults,itcanbeseenthatthegeneraltrendsofthefrequencyandgrowthratewithregardtotheincreaseinthemagnitudeoftheplasmaactuation(increasing0),aswellasthestreamwisewavenumber,arecomparablefortheboundarylayerprolesextractedfromtheCFDsimulationsandforthemodel.Examiningtheeigenmodes(Figure 3-17 ),itcanbeseenthatthegeneralshapeoftheeigenmodeiscomparablefortheboundarylayerprolesextractedfromtheCFDsimulationsandthosegeneratedbythemodel. 3.3.3LinearStabilityUsingthe1DFlowModelThehypothesesdevelopedinSection 3.3.1 suggestthatanumberofdifferenteffectscouldoccurtotheowstabilityasafunctionofthelocalvelocityratio,.Differentinstabilitiesmayoccur,andsignicantchangestothenatureoftheinstabilitiesmayoccurastheeffectofthemomentuminjectionisvaried.Forthesecalculations,themodeloftheboundarylayerowdevelopedinSection 3.3 isused,asitaffordsmoreeaseandexibilityingeneratingboundarylayerprolesthantheCFDbasedapproachusedinSection 3.2 68

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Figure3-16. Comparisonanderrorbetweenthecritialeigenvaluesofboundarylayerprolesfromthesimulationsandthemodel.A)RealandB)imaginarycomponentsofthecriticaleigenvaluesandpercentageerroroftheC)realandD)imaginarycomponentsoftheeigenvaluefortheboundarylayerprolesextractedfromtheCFDsimulationsandthemodeloftheboundarylayerprole. Inordertodeterminewhichinstabilitiesmayexist,aparametricstudyispresentedinordertodeterminetheeffectsofco-owandcounterowactuationonthedifferenteigenmodesoftheboundarylayer.Theeigenvaluesoftheowarecomputedforthecaseof=1.0and=0.3forvariousReynoldsnumbersandvaryingthelevelofactuationfrom=)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5to=0.5.Thevalueofforwhichtheeigenvaluesarecomputedhasbeenvariedslowly(=0.02)inordertoensurethatsmoothbehavior 69

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Figure3-17. Comparisonoftheeigenmodesfromthesimulationsandthemodel.A)0=0.05,B)0=0.10,C)0=0.15andD)0=0.20for=0.3fortheboundarylayerprolesshowninFigure 3-12 B. exists.ItcanbeseenthatatthelowestReynoldsnumberexamined(Re=150,Figure 3-18 A),thatthemostunstablemodeforagivenvalueofvariescontinuouslyasvariesfrom)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5to0.5.However,atahigherReynoldsnumber(Re=450and600,Figures 3-18 CandD),themostunstablemodechangesasthevalueofisincreasedfrom)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5to0.5.AtthehigherReynoldsnumber,theTSmodebecomesmorestableasco-owactuationisappliedandisdestabilizedascounterowactuationisapplied.EventhoughtheTSmodeisstabilizedasco-owactuationisapplied,adifferentmode,whichhadpreviouslybeenverystable,becomesunstableasthistypeofowcontrolisutilized.ItcouldbeassumedthattheseeigenmodesaremovinginthecomplexplaneasthedispersionrelationshipwhichcontrolsthemisvariedwithrespecttotheReynoldsnumber.However,itcanbeseenthatthesetwobranchesofthedispersionrelationshipintersect(Figure 3-18 B)andthentradeportionsoftheirbranchtoeachotherastheReynoldsnumberischanged.ThecriticalpointatwhichthisoccursisRe,Blasius=315and=0.0675atthepoint!=0.155)]TJ /F5 11.955 Tf 12.83 0 Td[(i0.495forthewavenumber=0.3(thisvalueofwasnotoptimizedtondtheabsolutelowestvalueofRewherethiseffect 70

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occurs).Theremaybeanumberofdifferentimplicationswiththisbranchswitching.TheimplicationmostrelevanttothisstudyisthatatverylowReynoldsnumbers,onlyasingleeigenmodeisrelevanttothestabilityofthesecombinedwalljetandboundarylayerows,butatmoderatetohighReynoldsnumbers,twoseparatemodesexistthatareconnectedtoeachother. A B C D Figure3-18. Eigenspectraofthemodelasafunctionof()]TJ /F4 11.955 Tf 9.29 0 Td[(0.5<<0.5)for=0.3andaboundarylayerheightratioof=1.0forA)Re,Blasius=150,B)Re,Blasius=300,C)Re,Blasius=450,andD)Re,Blasius=600.Greydotsindicateeigenvaluesforacertaincounterowvelocityprole.Blackdotsindicateeigenvaluesforacertainco-owvelocityprole. 3.3.4Co-FlowActuation(>0)Examiningtheeffectsofco-owactuation,itisexpectedthatthismannerofoperationwillstabilizetheboundarylayeruptoacertainpoint,abovewhich,inviscid 71

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instabilitieswillbecomerelevantandtheowwillbedestabilized.NeutralstabilitycurveshavebeencalculatedforthecasesofpositiveandareshowninFigure 3-19 .TwodifferentscalingsbasedonthedisplacementboundarylayerheightandthedisplacementheightofBlasiuscomponentofthecombinedvelocityproleareused.However,neitherofthesescalingsprovideasufcientcollapseofthedata.Theseneutralstabilitycurvesconrmthattherearetwoseparatemodeswhichcanbecomeunstable,withwidelyvaryingproperties.Forthecaseofco-owactuation,thesetwomodescanbeseparatedasbeinga(slow)TSwaveanda(fast)outermode,basedontherealphasespeed(cR=!R ).Examiningthevelocityprolesofthesewaves(Figure 3-20 ),itcanbeseenthatthestructureofthewavesisdifferent.TheshapeofTSmodeisnotindependentoftheplasmaactuation,butitdoesretainitsbasicshapeasthemagnitudeofthewalljetisvaried.Thefastermodeshowsastrongerdependenceontheplasmaactuation.Asthemomentuminjectioneffectsbecomemoreandmorepronounced,theshapeofthisoutereigenmodealsochanges,indicatingthatthisfastermodeisstronglycoupledtothemomentuminjection,andmoresensitivetothemagnitudeofthewalljet.Thissecondmodedoesnotbecomerelevantuntilthewalljeteffectsintheboundarylayerreachacertainlevel.FromthecomputationsofFjrtoft'stheoremthiseffectshouldoccurat=0.1530.003.Whentheowstabilityisexaminedintheinviscidlimit(Re!1)usingthecurrenteigenvaluemethod,thecriticalvalueisfoundtobe=.143750.006.Theviscidresults(asamplingofwhichareshowninFigure 3-19 )indicatethattheinviscidinstabilitiesbecomerelevantnear=0.1560.006.Inall,thereisreasonablygoodagreementatwhatpointthismodeshouldbecomeimportantbetweenthesedifferentmethodsandthatthismodeisinviscidlyunstable.Examiningthestructureoftheoutermodeasthewalljetcomponentofthemodelbecomesincreasinglylarger(i.e.increasing),theshapeoftheperturbationtakesonathreemaximastructure(Figure 3-20 B)comparabletothethreemaximastructureofwall 72

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A B C D Figure3-19. NeutralstabilitycurvesintermsoftheA,B)reducedfrequency,F=!R=Re,andtheC,D)phasevelocity,cR=!R=scaledbythedisplacementboundarylayerheightsoftheA,C)combinedvelocityproleandtheB,D)Blasiuscomponentofthevelocitycomponent.TheTSwavesareindicatedbythesolidlines.Theoutermodeisindicatedbythegreylines. jetinstabilities( Amitay&Cohen 1997 ; Meleetal. 1985 ). Meleetal. ( 1985 )and Amitay&Cohen ( 1997 )alsocalculatedneutralstabilitycurvesforthetwoinstabilitiespresentinwalljets(bothofwhichhaveathreemaximastructure),theresultsofwhichsuggestthatthegrowthmechanismsforthewalljetinstabilitiesareinherentlyinviscidinnature,justliketheoutermodeinstability.Thesetwosimilaritiessuggestthatthisinstabilityhasmoretodowithwalljetcomponentofthevelocityprolethanthecombinedboundarylayervelocityprole,thoughtherearemodicationsduetotheadditionoftheboundarylayerow. 73

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A B C Figure3-20. EigenmodesoftheA)TSandB)outerinstabilitiesfortheconditionsofRedelta,Blasius=4000and=0.5.C)Thestabilityofthemodesisalsoshownforreference. 3.3.5CounterFlowActuation(<0)Iftheplasmaactuatorisorientedintheoppositedirectionsuchthatmomentumisremovedfromtheow,itisexpectedthattheboundarylayerwillbedestabilizedrelativetotheBlasiusboundarylayerprole.Neutralstabilitycurveshavebeencalculatedforthecaseofcounterowoperation(Figure 3-21 ).Thesecalculationsshowthatthismethodofoperationishighlydestabilizing,reducingthecriticalReynoldsnumbermorethananorderofmagnitudefromRe520fortheBlasiusboundarylayer(=0)Re<30(=)]TJ /F4 11.955 Tf 9.29 0 Td[(1).ItwasnotedinSection 3.3.1 ,thatthevelocityprolesforcounterowoperatedplasmaactuationmaycauseowreversal.Withthisowreversal,thereistheimplicationthatsomeeigenmodesoftheowmaytravelupstreamorremainstationary.Withinthissubsetofmodeswhichremainstationary,thepossibilityofanabsoluteinstabilityexists.Furthermore,theneutralstabilitycurves(Figure 3-21 )indicatethatforsufcientlylarge,negativevaluesof,thereareunstablemodeswithzerorealphasevelocity(ascR=!R=).However,thisobservationdoesnotsatisfythemorerigorous 74

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A B Figure3-21. Neutralstabilitycurvesofthecounter-owmodiedboundarylayerswhere=1.0.A)scaledbyandB)Blasius. requirementsofanabsoluteinstabilityasdenedby Briggs ( 1964 )and Bers ( 1972 ).Theserequirementsforanabsoluteinstabilityarethat: 1. Thereexistsadispersionrelationshipconnecting!and,denedasD(!,)=0.Inthecomplexdomainsof!and,theremustbesaddlepointswhere@!=@=0,i.e.thegroupvelocityequalszero. 2. Thesaddlepointsmustbepinchpointsofanupstreamanddownstreamtravelingmode. 3. Thesaddlepointmustalsobeunstable.Thatis,!I>0atthesaddlepoint.Constructinganetmapallowsforthevisualizationofthedispersionrelationship(whichisafunctionofthevelocityproleandReynoldsnumber)intermsofthecomplexvaluesofand!(Figure 3-22 ).Asaddlepointisimmediatelyvisibleineachoftheseplots.ItcanbeseenthatastheReynoldsnumberisincreased,thevalueof!Iatthispointincreasesfromnegativetopositive,whichsatisesthethirdrequirementforanabsolutelyinstability.ItcanbeseenthataboveacriticalReynoldsnumber,Re,abs.crit,thepreviouslyconvectiveinstabilitybecomesanabsoluteinstability.Thisabsoluteinstabilitypresentsaconictingviewoftheviscousconvectioninstability.MeasurementsoftheeigenvaluesasshowninFigure 3-18 indicatethatinstabilitiesseenwithcounterowactuationaredirectmodicationsoftheTollmien-Schlichting 75

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A B C Figure3-22. NetmapsforlowReynoldsnumbercounter-owmodiedboundarylayers.A)Re,Blasius=26,whichisabsolutelystable,B)Re,Blasius=30,whichisconvectivelyunstable,andC)Re,Blasius=32,whichisabsolutelyunstablefor=)]TJ /F4 11.955 Tf 9.3 0 Td[(1.0and=1.0.Solidlinesindicatethecontoursof!R.Dashedlinesindicatethecontoursof!I.Thevaluesof!atthepinchpointsare!=0.0926214)]TJ /F5 11.955 Tf 11.95 0 Td[(i0.046581,!=0.0885969)]TJ /F5 11.955 Tf 11.96 0 Td[(i0.010166,and!=0.0867067+i0.00496014,whicharemarkedbytheblacksquares. waveasmomentumisinjectedinthedirectionopposingthefreestreamvelocity.However,Fjrtoft'scriterion(andthecalculationsoftheowinstabilityintheinviscidlimit)suggestthataninviscidinstabilityshouldbepresent,evenforsmall,negativevaluesof.Whenoneconsidersthegrowthratesofinstabilitiesoccurringfromcounterowoperation(showninFigure 3-18 ),theevidencepushesmoretowardsthisinstabilitybeinginviscidinitsbehavior.Largegrowthratesarenormallyassociatedwithinviscidinstability,andthecalculatedgrowthratesaresignicantlyhigherforcounterowactuationthantheyareforanormalBlasiusboundarylayerandotherknownviscidinstabilities. 3.3.6ComparisonoftheOnsetofDifferentStabilityModesCombiningallofthestabilityresultsthathavebeengarneredfromthismodeloftheow,itcanbeseenthatthereexistawidevarietyofphenomenathatoccurwithinthecombinedwalljet/boundarylayervelocityprole.ThedifferentcriticalReynoldsnumbershavebeenplottedasafunctionofinFigure 3-23 .ItcanbeseenthattheTSwaveremainspresentforallvaluesof,thoughforlargerlevelsofco-owactuation,the 76

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TSinstabilityisstabilizedrelativetoitsimportanceintheBlasiusboundarylayer,onlybeingpresentatsignicantlyhigherReynoldsnumbers.Theupperlimitonimprovingthestabilityoftheboundarylayerappearstooccuraround=0.075,withthecriticalReynoldsnumberbeingincreasedfromRe,Blasius=520forthenon-actuatedowtoRe,Blasius=5818.Forthecounterowactuation,theTSwaveisdestabilized,andbecomesunstableatReynoldsnumbersafullorderofmagnitudelowerthanthatoftheBlasiusboundarylayer.Furthermore,forsufcientlystrongcounterowactuation,anabsoluteinstabilitybecomessignicant. Figure3-23. AcomparisonofthecriticalReynoldsnumbersofthedifferentinstabilitiescomparedtofor=1. 3.3.7ComparisontotheUniversalCorrelationItcanbeseenthatbothco-owandcounterowoperationoftheplasmaactuatorshaveaprofoundeffectontheboundarylayerstability.However,otherboundarylayerprolemodicationscanhaveanequallystrongeffectontheowstability.Itisthoughtthatthereisacertainuniversalcorrelationbetweentheowstabilityandtheshapefactor( Wazzanetal. 1979 ).ItcanbeseeninFigure 3-24 thatfortheconvectiveviscidinstability,thecriticalReynoldsnumbersreachedaspartofthisstudyareinagreementwithotherboundarylayerswhencomparedviatheshapefactor,H. 77

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Forsmalllevelsofactuation,thesestabilitylimitsarecomparabletootherboundarylayerproles.However,aslargerlevelsofplasmaactuationareexamined,itcanbeseenthattheydeviatefromthebehaviorseeninotherboundarylayerproles.Thiseffectislikelyduetothewalljetcomponentofthevelocityprolebecomingsignicantrelativetotheboundarylayercomponent,andtransformingtheowinsuchawaythatitcannotbecomparedtootherboundarylayers.DataispresentedforthecriticalReynoldsnumberscaledbybothBlasiusandthecalculatedvalueofforthevelocityprole.ThechoiceofReynoldsnumberscalingdoesseemtomakeadifferenceinmatchingthecurrentresultstotheuniversalcorrelation.However,oncethevelocityproleallowsforsignicantwalljetlikeeffectstodevelop,thecriticalReynoldsnumberdivergesfromthatofmoretraditionalboundarylayers. Figure3-24. Comparisonofcurrentresultstootherboundarylayerproles.Falkner-Skandatatakenfrom Wazzanetal. ( 1979 ). 3.3.8UnsteadyEffectsInalloftheseanalysis,thebodyforcegeneratedbytheplasmaactuatorhasbeentreatedasasteadysourceofmomentum.However,thisisapointofinaccuracythatisnecessaryfortheexistingstabilityformulations,whichcannothandleunsteadyow. 78

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Inreality,themomentumadditionprovidedbyDBDactuationisinherentlyunsteady,asthedevicesarepoweredbyahighfrequencyACsignal.Normallythetimescalesassociatedwithplasmaactuationareontheorderof10'sofkHz,whilethoseassociatedwithalowspeedowareontheorderof100'sofHz.Thesetwotimescalesareseparatedbytwoordersofmagnitude,anditisoftenassumedthatonlythemeancomponentofthemomentumadditionisrelevanttotheowcontrol.Itshouldbenotedthatthisorderofmagnitudeapproximationisstillbasedaroundtheuseofasteadyowforthestabilityanalysis.Foramoreaccurateanalysis,theunsteadyeffectoftheplasmaactuationontheboundarylayerwouldneedtobetakenintoaccountthroughaFloquetstabilityanalysis.ThepresentanalysisofinstabilitiesrelevanttotheplasmaactuatedowindicatesthatpotentialinviscidandabsoluteinstabilitiesmayoccuratsignicantlyhigherfrequenciesthantheTSwave.Assuch,assumptionsbasedontheseparationoftimescalesmustberevisitedinordertoestablishwhentheymayormaynotbevalid.Thenon-dimensionalfrequencyusedtocharacterizethepresentinstabilitiescanbedenedusing(dimensional)parametersas F=2f u21(3)maybearrangedtoprovideanupperlimitonthespeedwhereplasmaactuationisabletobeusedwithoutself-excitingtheow(u1,NE).Performingthisrearrangement,theupperlimitis u1,NE
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approximatevaluesareprovided.Fortheothervariables,dimensionalparametersare=1.510)]TJ /F7 7.97 Tf 6.59 0 Td[(5m2=s,andfplasma=10kHz.Theresultsofthisorderofmagnitudeanalysisindicatethatevenforrelativelylowspeedows,theinviscidandcounterowactuatedTSinstabilitymechanismsmaybeselfexcitedbytheplasmaactuation.Atmoderatelyhighervelocities(upwardsof68m/s),theDBDactuationmayevenexcitetheTSmodewhenco-owactuationisemployed,atwhichpointthistypeofactuationisactivelycausingtheowtotransition,notstabilizingit. Table3-3. Approximatefrequenciesfordifferentinstabilitymodesandthefreestreamvelocitywheretheplasmaactuationwillbegintoexcitethesemodes. InstabilityModeF0106u1,NE(m=s) TSWave-Co-Flow20068.4TSWave-Counter-Flow200021.7InviscidInstability100030.7 WhenitcomestostabilizingaowusingDBDplasmaactuation,thisresultplacesanapproximateupperlimitonthefreestreamvelocitywheresuccessmaybefound;thatisu1u1,NE),itappearsthattheDBDactuationhasthepotentialtosimultaneouslydestabilizetheowandtogeneratetheunstableperturbations.Thisselfexcitationmaybeonereasonwhytheseotherinstabilitymodeshavenotyetbeenreportedintheliterature,anditmayalsoprovidesomeadditionalcontrolauthorityathighervelocitieswhenattemptingtodestabilizetheboundarylayerathighervelocities. 3.4ConclusionsInthischapter,localstabilityanalysishasbeenutilizedtostudythestabilitypropertiesofboundarylayerprolesdownstreamofaplasmaactuator.BoundarylayerprolesfromCFDsimulations,aswellasalowordermodelhavebeenexamined,indicatingthattheadditionorremovalofmomentumfromtheboundarylayercanhavewiderangingeffectsonthestabilitypropertiesoftheboundarylayer. 80

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Inordertostudyspecicboundarylayerproles,boundarylayershavebeensimulatedincorporatingtheeffectsofplasmaactuationthroughaphysicsbasedapproximationofaDBDplasmaactuator,operatedinaco-owmanner,continuouslyaddingmomentumintotheboundarylayer.Basedontheseows,stabilitycalculationssuggestthattheonsetofinstabilities(basedonthecriticalReynoldsnumberswithrespecttoxand)canbepushedmuchfartherdownstreamasadditionalmomentumisaddedintotheboundarylayer.IntermsofthecriticalstreamwiseReynoldsnumber,thegrowthofinstabilitiescanbepushedupwardsof50%fartherdownstreamascomparedtowheretheywouldnormallyoccur.BasedontheboundarylayerprolescollectedfromtheCFDsimulations,alowordermodeloftheplasmamodiedboundarylayerproleshasalsobeendeveloped.Thismodelallowsforboundarylayerprolestoberapidlygenerated,allowingformucheasierparametricstudies.Thismodelhasbeencharacterizedwithrespecttothephysicsofboundarylayermomentumadditionaswellashowaccuratelyitpredictsthestabilitypropertiesofindividualboundarylayerproles.Parametricstudieshavebeenperformedusingthismodelforco-owandcounteroworientedplasmaactuators.Basedontheparametricstudiesperformed,evidenceforaviscidabsoluteinstabilityandaninviscidinstabilityhasbeendiscovered.ConsideringtheTSwave,dramaticincreasesanddecreasesinthecriticalReynoldsnumberarepredictedtobepossiblebasedonthislowordermodeloftheboundarylayer,extendinganorderofmagnitudeupwardsanddownwardsofthecriticalstabilityoftheBlasiusboundarylayer,dependingonwhethermomentumisaddedorremovedfromtheow.Evenastheboundarylayerisstabilizedordestabilized,thecriticalReynoldsnumberofboundarylayerprolestillcorrelateswellwithrespecttotheuniversalcorrelationof( Wazzanetal. 1979 ),uptoacertainpoint.Inall,theselocalstabilityanalysessuggestthattheadditionofmomentumintotheboundarylayercanhaveverydramaticeffectsontheboundarylayerstability.The 81

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physicsthatcanbeexaminedusingthistypeofanalysisislimited,buttheseresultssuggestthatfurtherstudyusingmoresophisticatedmethodsshouldbeemployed,specicallythosethatareabletohandlethetwodimensionalnatureofthisow,whichisexaminedinChapter 4 .Theresultsanddiscussioninthepresentchapterhavebeenpresentedandpublishedinseveraldocuments( Riherd&Roy 2013b ; Riherdetal. 2012 2013 ). 82

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CHAPTER4BI-GLOBALSTABILITYANALYSISOFAOFAPLASMAACTUATEDBOUNDARYLAYERInChapter 3 ,itwasshownthatlocalstabilityanalysismethodspredictowstabilizationofaboundarylayerwhenplasmaactuatorsareusedtoaddmomentumintotheow.However,intheregionimmediatelyaroundtheplasmaactuation,theassumptionsrequiredforalocalstabilityanalysisfailtohold(Figure 3-6 ).Inthisregion,two-dimensionalstabilitymethodsmustbeemployedtostudythestabilityproperties.Inthelastdecade,matrixbasedbi-globalstabilitymethodshavenallyreachedalevelofmaturitysuchthattheycanbeappliedtonon-trivialresearchproblems.Inthepast,parabolizedstabilityequations(PSE)( Bertolottietal. 1992 ; Herbert&Bertolotti 1987 )orDNS( Bertolottietal. 1992 ; Fasel&Konzelmann 1990 )havebeenemployedtostudythestabilityofboundarylayerows.Morerecently,matrixbasedbi-globalstabilitymethodshavebeenemployedtostudytheseows( Akerviketal. 2008 ; Alizard&Robinet 2007 ; Brandtetal. 2011 ; Sipp&Marquet 2012 ).AscomparedtoDNSandPSEbasedmethods,matrixbasedstabilitymethodssimplifytheanalysisoftheresults,astherequiredlinearalgebraoperationsoftenproducethedesiredinformation(temporalgrowthrates,maximumtransientamplication,eigenmodes,singularmodes,etc.).Thetwo-dimensionalimplementationprovidesadditionaccuracyoverlocalmethods,asnon-parallelandowdevelopmenteffectscanbetakenintoaccount.ItisknownthatthetwoprimaryinstabilitymechanismsinaboundarylayeraretheTSwaveandboundarylayerstreaks,whichareexponentialandalgebraicinstabilities,respectively.Thepresentworkaimstodescribetheresponseofaboundarylayerowmodiedbyplasmaactuationtothesetypesofinstabilities,aswellastoquantifythegrowthanddecayratesoftheseperturbationsoverthelengthoftheboundarylayerusingabi-globalstabilityanalysis.Parametricstudiesareperformedwithrespecttothemagnitudeoftheplasmaactuation,aswellasfrequencyandspanwisewavenumber. 83

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4.1Bi-GlobalStabilityAnalysisoftheTollmien-SchlichtingWaveInordertostudythegrowthoftheTSwavewithgreateraccuracyandinthenearactuatorregion,abi-globalstabilitymethodhasbeenemployed.Thismodel,whilemorerobustthanalocalstabilityanalysis,focusesononlyuidiceffects;thereisnofeedbackmechanismbetweentheuidandplasmapresentinthisanalysis. 4.1.1NumericalMethodforBi-GlobalStabilityAnalysisInordertodeterminetheresponsetheplasmamodiedboundarylayertooncomingTSwaves,abi-globalstabilityapproachisemployed.Thevelocityandpressureeldscanbedecomposedintothesteadyequilibriumandperturbationcomponents,whereui=ui+u0iandp=p+p0.Usingthisdecomposition,theincompressibleNavier-Stokesequationscanbelinearizedaroundthissteadypointsuchthat @u0i @xi=0 (4a)@u0i @t+uj@u0i @xj+u0j@ui @xj+@p0 @xi)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2u0i @x2j=0 (4b)Assumingthattheseperturbationsareperiodicintime,suchthatu0i=~uiexp()]TJ /F5 11.955 Tf 9.3 0 Td[(i!t),theproblemcanbeputintomatrixoperatorformsuchthati!Bu=Au+f (4)whereu=266666664~u~v~w~p377777775,A=266666664@() @x@() @yi0C)-222(D+@u @x@u @y0@() @x@v @xC)-222(D+@v @y0@() @y00C)-223(Di377777775,B=2666666640000I0000I0000I0377777775 (4) 84

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TheconvectionandviscousdiffusionoperatorsCandDaredenedasC=u@() @x+v@() @yandD=1 Re@2() @x2+@2() @y2)]TJ /F3 11.955 Tf 11.95 0 Td[(2.findicatesacomplexperiodicforcingtothesystem,anduistheresponseofthesystemtothatforcing.Theresponseofthesystemcanbemeasuredbysolvingthesystemofequationsshownabovesuchthatu=[i!B)]TJ /F15 11.955 Tf 11.95 0 Td[(A])]TJ /F7 7.97 Tf 6.59 0 Td[(1f (4)Forthepresentstudytheperiodicforcingtermfrepresentsanon-homogeneousboundaryconditionattheinletofthedomain. 4.1.1.1NumericaldiscretizationandboundaryconditionsInordertoperformthesecalculations,thematrixoperationsdescribedinEquations 4a 4 arediscretizedonasemi-staggeredmesh.ThevelocityeldsandtheirgradientsproducedinSection 3.1.3 wereinterpolatedontothismesh.Themeshesusedtoperformthesestabilitycalculationsrepresentionaportionofthedomainusedtocalculatethebaseows.Relativetotheoriginaldomain,thissubdomainisentirelydownstreamoftheleadingedge(0.52
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perturbationsareshowninFigure 4-1 .Fortheboundarylayerwallandthefree-stream, Figure4-1. Perturbationsenforcedattheinletofthedomainatvaryingfrequencies.A)uandB)vvelocitycomponents. noslipconditionsareemployed.Inthelatterportionofthedomain,aspongeregionisimplementedinordertodampthevelocityperturbationsastheyapproachtheoutlet.Thisspongeregioneffectivelypreventstheperturbationsfromreectingoffoftheoutletandaffectingtheupstreamow. 4.1.1.2GridresolutionstudyAgridresolutionstudyhasbeenperformedinordertoensurethattheproblemisadequatelyresolved.ThedetailsofthedomainandmeshparametersareindicatedinTable 4-1 .Thisstudyhasbeenperformedontheunforcedboundarylayer,butadditiongridresolutionstudiesforboundarylayerswheremomentumadditionwasappliedwerealsoperformed,indicatingsimilarconvergence.InordertoquantifythegrowthanddecayoftheTSwaveasitpropagatesdownstream,themagnitudeoftheperturbationsalongalineintheowaremeasured,asampleoftheresultscanbeseeninFigure 4-2 A.Examiningtheconvergenceoftheresultsasafunctionofthestreamwisegriddensity,itcanbeseeninFigure 4-2 Cthattheresultsareapproachingconvergenceasthemeshdensityincreases.Comparingthespatialexponentialgrowthratesoftheseperturbations(Figure 4-2 B),itcanbeseenthatthereisgoodagreementbetweenthedifferentmeshesusing 86

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Table4-1. DetailsofthegridresolutionstudyperformedforthelinearTSwavecalculations.TheVeryFine-L,Fine-H,andTallcasesareidentical. StudyCasenxnyLx=0Ly=0x=0ywall=0 StreamwiseresolutionCoarse-L513651061.7402.07360.02409Medium-L725651061.7401.46640.02409Fine-L1025651061.7401.03680.02409VeryFine-L1449651061.7400.73320.02409 WallnormalresolutionCoarse-H1449331061.7400.73320.09631Medium-H1449491061.7400.73320.04282Fine-H1449651061.7400.73320.02409 HeightShort1449331061.7200.73320.04815Medium1449491061.7300.73320.03212Tall1449651061.7400.73320.02409 thisbi-globalmethod.Comparisonscanalsobemadewithlocalstabilitymethods,whichpredictsimilarbehavior,thoughtheexactvaluesdifferbyasmallamount.Inordertomakeagoodcomparisonwiththebi-globalstabilityanalysis,thelocalgrowthrateshavebeennormalizedbythelocalboundarylayerdisplacementheightandmultipliedbyafactorof2toaccountfortheenergy(ratherthanvelocity)perturbationgrowth.Convergencestudiesforthewallnormalmeshdensity(Figure 4-2 D)anddomainheight(Figure 4-2 E)havealsobeenexamined,andshowgoodconvergence.Basedonthisgridresolutionstudy,allcomputationswillbeperformedusingtheVeryFine-Lmesh,andthemagnitudeoftheboundarylayerperturbationsisassumedtobeaccuratetowithinafewpercent. 4.1.2ResultsComparingtheperturbationoweldsinFigure 4-3 ,itcanbeseenthatasthemagnitudeoftheplasmaactuationisincreased,theperturbationsareincreasinglydampeddownstreamoftheplasmaactuator.ThemetricA0(x)=R10ju0j2+jv0j2dyx R10ju0j2+jv0j2dyx=inlet (4) 87

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Figure4-2. ConvergenceofamplitudesandspatialgrowthratesforthegridconvergencestudyofthelinearTSwave.A)amplitudesandB)spatialgrowthratesoftheincomingTSwavesfordifferentmeshdensitiesfor5differentfrequencies.TheconvergenceoftheF106=150casecanalsobeseenfortheC)amplitudesD)spatialfrequencies. isusedtoquantifythemagnitudeoftheboundarylayerperturbation.Thismetricquantiestheamountofkineticenergyintheowatagivenpointinxrelativetotheinlet.Applyingthismetric,themagnitudesoftheperturbationshavebeencalculated,samplesofwhichcanbeseeninFigure 4-4 forseveraldifferentforcingfrequencies.Inexaminingthemagnitudesoftheseperturbations,theamountofdampingcausedbytheadditionofowwisemomentumintotheboundarylayerisfoundtobesignicant.Asthevelocityratiooftheplasmaactuationisincreased,themagnitudeoftheenergyperturbationsaredecreasedbyuptotwoordersofmagnitude(onlyasingleorderof 88

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Figure4-3. Realcomponentoftheuperturbationvelocityforthenon-dimensionalfrequencyofF106=100andvaryinglevelsofplasmaactuation.A)0=0.00(Noplasmaactuation),B)0=0.10,andC)0=0.20.Thedashedlineindicatesthelocationoftheplasmaactuator.Thecontourvaluesinthisplotdosaturatefor)]TJ /F4 11.955 Tf 9.3 0 Td[(0.1urinordertoshowgreaterdetailforsmallervaluesoftheperturbationvelocity. magnitudeforthevelocityperturbations).IntermsofN-factor(seeEquations 2 & 2 andtherelateddiscussion),thedampingoftheseperturbationsisequivalenttoadropintheN-factorofapproximately2.3relativetothenon-actuatedcase.ConsideringthattransitionisnormallyassumedtooccurinaboundarylayerforanN-factorof9to10( vanIngen 1956 ; Jaffeetal. 1970 ),thisdampingoftheperturbationcoulddelaytransitionbyasignicantamount.Basedonthisdata,thegrowthratesoftheseperturbationscanbecalculated.TheexponentialgrowthrateisdenedasI=)]TJ /F4 11.955 Tf 14.02 8.09 Td[(1 AudAu dx (4) 89

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Figure4-4. MagnitudeoftheTSperturbationsatvaryingfrequenciesandasafunctionofthemagnitudeoftheplasmaactuation.A)F106=80,B)F106=100,C)F106=120,andD)F106=140.Theverticaldashedlineindicatesthelocationoftheactuator,andtheinsetguresareacloseupofthenearactuatorregion. ThegrowthratesforanumberofcasesareshowninFigure 4-5 forseveraldifferentforcingfrequencies.Forthesecalculations,anegativevalueofIindicatesinstability(perturbationgrowth),whileapositivevalueofIindicatesstability(perturbationdecay).Examiningtheowelds,perturbationmagnitudesandexponentialgrowthrates,itcanbeseenthatthebehavioroftheTSwaveisnothomogeneousalongthelengthoftheboundarylayerwhenplasmaactuationisintroducedintotheboundarylayer.Rather,thereappeartobeanumberofdistinctregionsintheowwherethisbehaviorchanges.Theseregionscanbedescribedastheregionupstreamoftheactuator,theregionsimmediatelyaroundanddownstreamoftheactuatorwherethemomentumaddition 90

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Figure4-5. SpatialgrowthratesoftheTSperturbationsatvaryingfrequenciesandasafunctionofthemagnitudeoftheplasmaactuation.A)F106=80,B)F106=100,C)F106=120,andD)F106=140. bythebodyforcestronglymodiestheoweld,andnallytheregionsufcientlyfardownstreamsuchthatthemodicationstotheboundarylayerhaverelaxedenoughtobecomenegligible.Therstregionintheowistheregionupstreamoftheactuator(x=0<240),whichexhibitsonlyweakchangesinitsresponsetoperturbationsatthedomaininlet.Inthisregion,thebehavioroftheowisslightlystabilizedwiththeadditionofplasmaactuationrelativetothenon-actuatedcase,buttherearenosignicantchangestothestructureoftheTSwave.Asecondregionintheowcanbedenedastheregiondirectlyaroundtheplasmaactuatoranditsbodyforce(240
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beginstodecayinmagnitude(Figure 4-4 ,inset)toalowervaluethanitpossessedupstreamofthedevice.Inordertoevaluatethedampingeffectsaroundtheplasmaactuator,aRANSbasedkineticenergybalancefortheseperturbationscanbeused.D(TKE0) Dt+@Ti @xi=P)]TJ /F3 11.955 Tf 11.95 0 Td[( (4)where TKE0=1 2)]TJ ET q .478 w 249.76 -157.14 m 273.61 -157.14 l S Q BT /F5 11.955 Tf 249.76 -167.36 Td[(u0iu0i (4a)P=)]TJ ET q .478 w 228.52 -186.06 m 254.99 -186.06 l S Q BT /F5 11.955 Tf 228.52 -196.28 Td[(ui0u0j@ui @xj (4b)=2 Re)]TJ ET q .478 w 239.82 -218.1 m 267.61 -218.1 l S Q BT /F5 11.955 Tf 239.82 -228.32 Td[(sij0s0ij (4c)and@Ti @xirepresentaccelerativetransporttermsduetotheperturbation'sownpressureandstrainuctuations.TherighthandsidetermsinEquation 4 ,Pand,representtheproductionanddissipationofkineticenergyintheperturbation,respectively.Asthedisturbancemovesoverthebodyforceregion,productionofthewave'skineticenergy(heredenedusingtheturbulentkineticenergyproduction)isreduced,andevenbecomesnegativeforlargeenoughamountsofplasmaactuation(Figure 4-6 B,E,andH).Kineticenergyproductionisnormallyoffsetbydissipation(Figure 4-6 C,F,andI).Inthepresentscenario,theproductionoftheTSwave'skineticenergyisreversedbytheadditionofthebodyforce,whichimmediatelybeginstoattenuatetheperturbationmagnitude,withdissipativeeffectscompoundingthisattenuation.Thisnearimmediatedropinperturbation'sgrowtharoundtheplasmaactuatorindicatesthatthelocalizedchangesinthevelocityeldaroundthebodyforceregionarestabilizing.Again,thereisnosignicantchangeinthestructureoftheTSwaveatthispoint(Figure 4-8 A),eventhoughitsmagnitudedoesbegintodropatthispoint.Whentheindividualproductiontermsareexamined,severalthingsbecomeapparent(Figure 4-7 ).The v0u0@ v @xand v0v0@ v @ytermsarefoundtobenegligiblewhencomparedtothe u0v0@ u @yterm.Fortheunforcedow,thestreamwiseowdevelopment 92

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Figure4-6. ComparisonofuR,kineticenergyproduction,andkineticenergydissipationofTSwavesinthenearplasmaregion.A,D,G)Realcomponentoftheuperturbationvelocityforthenon-dimensionalfrequencyofF106=140,(B,E,H)perturbationkineticenergyproduction,andC,F,I)perturbationkineticenergydissipationforplasmaactuationlevelsofA-C)0=0.00(Noplasmaactuation),D-F)0=0.10,andG-I)0=0.20.Theblacklineindicatesadomainwithinwhichthebodyforceisatleast10%ofitsmaximum.Thecontourvaluesinthisplotsaturateinordertoshowgreaterdetailforsmallervaluesoftheperturbationvelocity,perturbationkineticenergyproduction,andperturbationkineticenergydissipation. term)]TJ ET q .478 w 33.01 -460.22 m 52.13 -460.22 l S Q BT /F5 11.955 Tf 33.01 -469.78 Td[(u0u0@ u @xalsodoesverylittletoamplifyorattenuatetheperturbations.However,whenplasmaactuationisappliedtotheow,therearesomehighlylocalizedstabilizationeffectsthatoccur.Inthebodyforceregion,therapidaccelerationoftheowleadstosignicantowacceleration,whichimpliesnegativeproductionofkineticenergy.Astheowdeceleratesmovingdownstream,thereisagaininproductionoffsettingthis,butitdoesnotaccountforthesignicantlossofperturbationkineticenergythatoccursinthenearplasmaregion.Overalongerdistance,itwouldseemthattheproductiontermassociatedwiththewallnormalgradientofthevelocityprolehasagreaterrolein 93

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stabilizingtheow.Whenplasmaactuationisapplied,thefullervelocityprolesthataregeneratedleadtochangesinthekineticenergyproduction.Overall,thistermplaysagreaterroleinstabilizingtheowthanthestreamwisedevelopmentterm. Figure4-7. ComparisonofdifferentkineticenergyproductiontermsfortheTSwavesinthenearplasmaregion.A,C)streamwisekineticenergyandB,D)wallnormalkineticenergyproductiontermsA,B)withoutandC,D)withplasmaactuation(0=0.20)forthenon-dimensionalfrequencyofF106=140. Examiningtheoweldsdownstreamoftheactuator(260
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preserved.However,asoneexaminesthevelocityprolesfartherdownstreamwheretheperturbationhasmovedawayfromthewall,theshapeoftheperturbationchangesgreatlyforthecasesinvolvinghigherlevelsofplasmaactuation,andmoreoftheperturbation'senergyistransferredawayfromthewall(i.e.for0=0.20and0=0.25). Figure4-8. u-velocityprolesoftheboundarylayerdisturbancedownstreamoftheplasmaactuatorforvaryinglevelsof0,normalizedbyeachprole'smaximumvalueforF106=100.A)x=0=250,B)x=0=300,C)x=0=350,D)x=0=400,E)x=0=400,andF)x=0=450. Asthemodicationstotheboundarylayerbythebodyforcerelaxbacktozeroasonemovesdownstream,theTSwaveeventuallyreturntoitsoriginalstate.Itcanbeseenintheperturbationgrowthrates(Figure 4-5 ),thatthegrowthoftheperturbationsmatchesthebaselinenon-actuatedcasedownstreamoftheplasmaactuator,indicatingthatthisnalregionbeginsatx=0600.Beyondthispoint,itwouldseemthattheimpactofthemomentumadditionisessentiallyzero. 4.1.3StabilizationoftheTSWaveUnderstandingthelocalizedbehavioroftheowstabilizationprovidesinsightintotheunderlyingphysicalmechanismsatplay,butatthesametime,itisalsobenecialtoreducetheentireTSwavestabilizationprocessdowntoasinglevalue,suchthattotaleffectofthestabilizationcanbeunderstood.InordertounderstandthistotaleffectoftheadditionofmomentumintotheboundarylayerusingDBDplasmaactuation,theratiooftheperturbationmagnitudeswithandwithoutplasmaactuationcanbeused. 95

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InFigure 4-9 A,itcanbeseenthatsufcientlyfardownstreamoftheplasmaactuation(i.e.theregionwherethemodicationstothestabilitypropertieshaverelaxedaway),theratiooftheperturbationmagnitudesisapproximatelyconstant.Anaverageoftheratiooftheperturbationmagnitudescanbetakenoveralengthoftheboundarylayerdownstreamoftheactuation(Bu)inordertoquantifythisratioinamoregeneralmanner.DeningthismetricasBu=1 xe)]TJ /F5 11.955 Tf 11.95 0 Td[(xsZxexsAu(x,0) Au(x,0=0)dx (4)wherexs=0=700andxe=0=1000.Theaveragedratioofthemagnitudesisweaklydependentonthefrequencyoftheperturbationsbeingexamined(Figure 4-9 B).Itappearsthattherelativemagnitudeoftheseperturbationsismuchmoredependentonthemomentumadditionintotheboundarylayerthroughplasmaactuation. Figure4-9. ComparisonofrelativeTSwavemagnitudeasafunctionofvelocityratio.A)Comparisonoftheratiooftheperturbationmagnitudesforvaryingvaluesofthevelocityratio0forthenon-dimensionalfrequencyF106=100.B)Averagevaluesofthevelocityratioalongalengthoftheboundarylayerdownstreamoftheplasmaactuationasafunctionofthenon-dimensionalfrequency. ItcanbeseeninFigure 4-10 Athatforsmallvaluesofthevelocityratio,thelevelofdampingisproportionaltothevelocityratio(thatis,1)]TJ /F5 11.955 Tf 12.53 0 Td[(Bu/0).However,asthemagnitudeofthevelocityratioincreases,thislinearapproximationofthedamping 96

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breaksdown.Forlargerlevelsofplasmaactuation,itappearsthattherelativedampingoftheTSwaveismorecloselyrelatedtotheamountofbodyforceappliedtotheboundarylayerbytheplasmabodyforce(Figure 4-10 B)andthattherelativemagnitudeoftheperturbationsdecreasesexponentiallywithrespecttoDc(thatis,Bu/exp()]TJ /F5 11.955 Tf 9.3 0 Td[(kDc),wherekisaconstant) Figure4-10. DampingratioforTSwavesdownstreamoftheplasmaactuatorasafunctionoftheA)velocityratio,andB)proportionaltotheamountofmomentumaddition.ThedashedlinesindicatetheA)linear(1)]TJ /F5 11.955 Tf 11.96 0 Td[(Bu/0)decreaseintherelativemagnitudesoftheperturbations. 4.2StabilizationofBoundaryLayerStreaksStabilizingtheTSpathtoturbulencedoesnotimplythatalloftheremainingpathstoturbulencearealsostabilized.Thetransientgrowthofboundarylayerstreaksandtransitionduetofreestreamturbulenceareareaswheretheuseofplasmaactuationforstabilizationisstilllargelyunknown.Thesetwogrowthmechanismsarethoughttoberelatedtoeachother( Reshotko 2001 ; Tumin&Reshotko 2001 ).Furthermore,thistypeofdisturbanceisthoughttobetheoneresponsibleforturbulentowintheuncontrolledenvironmentsthatairandgroundvehiclesexperience.Assuch,thisisthemorerelevantforpracticalowapplications.WhiletheTSpathtoturbulencemaystillplayapartinboundarylayersbecomingturblent,iftheboundarylayerstreakpathto 97

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turbulencecannotalsostabilizedwiththeuseofplasmaactuators,thentheirusefulnessforowstabilizationwillbeseverelylimited. 4.2.1GenerationofBoundaryLayerStreaksBoundarylayerstreaksarealgebraicallygrowingdisturbances,andhavebeenidentiedasthemostampliedperturbationovernitelengthsoftimeanddistance.Typically,themostampliedperturbationsaregeneratedusingasuperpositionoftheeigenmodesofthesystem( Butler&Farrell 1992 ; Farrell 1988 ),thecomputationofwhichisdescribedintheprevioussubsection,suchthat u0=neigXiciu0i (4a)v0=neigXiciv0i (4b)w0=neigXiciw0i (4c)Inordertostudythegrowthofthesedisturbances,aquantifyingmetricmustbeemployed.ThemetricG(x)=K(x) K0 (4)isselected,asitdescribestheamplicationoftheperturbation'skineticenergyasitdevelopsinthestreamwisedirection,where K0=Z101 2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(u0u0+v0v0+w0w0dyx=inlet (4a)K(x)=Z101 2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(u0u0+v0v0+w0w0dyx (4b)Furthermore,themostampliedperturbationsaredenedasthosewhichmaximizethegrowthinkineticenergyatsomepointalongthestreamwisedirection,i.e.Gmax(x)=maxK06=0K(x) K0 (4) 98

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Farrell ( 1988 )describesndingthemostampliedmodesbyformingasecondeigenvalueproblem,K)]TJ /F7 7.97 Tf 6.58 0 Td[(10Kxc=c (4)whereK0andKxaremassmatricesrepresentingthemodalproductsofthedifferenteigenmodesattheinletandatdownstreamlocationssuchthat K0,ij=Z101 2)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(ui0u0j+vi0v0j+wi0w0jdyx=inlet (4a)Kx,ij=Z101 2)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(ui0u0j+vi0v0j+wi0w0jdyx (4b)ThelargesteigenvalueoftheproblemdescribedinEquation 4 representstheamplicationofthekineticenergyatadownstreampoint(Gmax(x)).TheinitialconditionsattheinletcanbeextractedbythelinearsuperpositionofeigenvalueperturbationsdescribedinEquation 4 .ThiseigenvalueproblemissolvedusingtheLAPACKsubroutineZGEEV(thedetailsofwhicharedescribedin Andersonetal. ( 1987 )). 4.2.2GridResolutionStudiesAgridresolutionstudyhasbeenperformedinordertoensurethattheproblemisadequatelyresolved.ThedetailsofthedomainandmeshparametersareindicatedinTable 4-2 .Thisstudyhasbeenperformedprimarilyontheunforcedboundarylayer,butadditiongridresolutionstudiesforboundarylayerswheremomentumadditionwasappliedwerealsoperformed,indicatingsimilarconvergence.Forthegridresolutionstudy,perturbationswithspanwisewavenumbersof=0.45,0.60&0.75andatemporalfrequencyof!=0areexamined.Themaximumamplicationofperturbationsatstreamwiselocationofx=0=600isconsidered.ItcanbeseeninFigure 4-11 AandB,thatthestreamwiseresolutionschosenconvergeverywellintheabsenceofmomentuminjectionintheboundarylayer.Furthermore,thewallnormalgridresolutionappearstobesufcient(Figure 4-11 C).Astudyofthedomainheightalsoshowsconvergenceoftheperturbation'smagnitude(Figure 4-11 D). 99

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Table4-2. Detailsofthegridresolutionstudyperformedforthelinearboundarylayerstreakcalculations.TheFine-L,Fine-H,andTallcasesareidentical. StudyCaseNxNyLx=0Ly=0x=0ywall=0neig StreamwiseresolutionCoarse-L513651061.7402.07360.0240964Medium-L725651061.7401.46640.0240964Fine-L1025651061.7401.03680.0240964 WallnormalresolutionCoarse-H1025331061.7401.03680.0963132Medium-H1025491061.7401.03680.0428248Fine-H1025651061.7401.03680.0240964 HeightShort1025331061.7201.03680.0481532Medium1025491061.7301.03680.0321248Tall1025651061.7401.03680.0240964 Figure4-11. Convergenceofboundarylayerstreakmagnitudesforthegridresolutionstudiesinanunforcedboundarylayer.A)Kineticenergyamplicationforperturbationsofspanwisewavenumber=0.45,0.60,0.75thataremostampliedatthelocationx=0=600asafunctionofthestreamwiseresolution,alongwithB)datacollectedatindividualpointsalongthestreamwisedirectionforthe=0.60case.ConvergencedatafortheC)wallnormalresolutionandD)domainheightarealsoshown. Acasewithabodyforceappliedtotheboundarylayerisalsoconsidered.Thestreamwiseresolutioninthenearactuatorregionmaypresentitselfasdifculttoresolveduetothesteepgradientsinthestreamwisedirection.However,basedonthegrid 100

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resolutionstudyperformed(Figure 4-12 ),theFine-Lmeshshouldsufcientlyresolvealloftheimportanteffectsintheow. Figure4-12. Convergenceofboundarylayerstreakmagnitudesforthegridresolutionstudiesinaplasmamodiedboundarylayer.A)Kineticenergyamplicationforperturbationsofspanwisewavenumber=0.45,0.60,0.75thataremostampliedatthelocationx=0=600casewherebodyforceisinjected(0=0.20)intheboundarylayerasafunctionofthestreamwiseresolution,alongwithB)datacollectedatindividualpointsalongthestreamwisedirectionforthe=0.60. 4.3Results 4.3.1BaselineCaseBeginningwiththeunforcedboundarylayer,severalthingsshouldbenotedaboutthestreaksthataregeneratedusingthismethod.First,themostampliedboundarylayerstreaksdisplaysimilartrendsingrowthandstructureregardlessofwheretheoptimizationisperformedinthestreamwisedirection,buttherearequantitativevariationsinthegrowthdependingonwherealongthestreamwisedirectiontheoptimizationtakesplace(Figure 4-13 A).Thegrowthofthesestreaksisdrivenbystreamwiseorientedvorticesintheboundarylayer(Figure 4-13 B).Thesestreaksstartoffashavingasmall,ifnotnegligiblestreamwisevelocitycomponent.However,thestreamwiseorientedvorticestransferlowmomentumuidupwardsandawayfromthesurfaceoftheboundarylayerononesideofthevortex,whilesimultaneously 101

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transportinghighmomentumuidfromthefreestreamdownwardsclosertothesurfaceontheother.Thisresultsinlocalizedhighandlowspeedstreaksinthesurface(Figure 4-13 C).Asthisprocessoccurs,themagnitudeofthestreakgrowsasthevorticitycontinuestotransportstreamwisemomentumintoandoutoftheboundarylayer,butafteracertainlengthofboundarylayer,viscouseffectsdissipatethestreamwisevorticity,whichlimitthemaximumgrowth,aswellastheboundarylayerstreaks,whichleadstotheireventualdecayintheabsenceofnon-lineareffects.Thisprocessisverywelldocumentedintheliteratureforbothone-dimensional( Butler&Farrell 1992 ; Ellingsen&Palm 1975 ; Landahl 1980 )andtwo-dimensionalows( Luchini 2000 ; White 2002 ).Themaximumlevelofgrowthforthistypeofperturbationisverystronglytiedtothespanwisewavenumberoftheperturbation.Widerperturbationsgrowmoreslowlyatrst,butultimatelyreachlargermagnitudes,butatpointsmuchfartherdownstreamofnarrowerperturbations. Figure4-13. GrowthandstructureofaboundarylayerstreakforA)=0.6(thinlines)asafunctionofwherethemostampliedperturbationisdetermined(circles).Thethicklinerepresentstheenvelopeofmaximumgrowthasafunctionofthestreamwiselocationfor=0.6.B)SpanwiseslicesoftheoweldattheinitialconditionandC)owatx=0=600areshownforthemostampliedperturbationatx=0=600for=0.6.Thecontourlinesindicatethestreamwisevorticity,whiletheshadingindicatesthestreamwisevelocity. 102

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4.3.2WithPlasmaActuationWhenaplasmabasedbodyforceisappliedtotheow,addingmomentumintotheboundarylayer,itappearsthattheboundarylayerstreaksarereducedinamplitude.Figure 4-14 showsthatdampingofthestreaksdownstreamofthebodyforcecanoccur,regardlessofwheretheyareconsideredtobethemostampliedstreak.Furthermore,dependingonthemagnitudeofthebodyforceandwherethestreakamplitudeisconsidered,thisdampingappearsbeupto20%oftheoverallstreakmagnitude. Figure4-14. Growthofkineticenergycontainedinboundarylayerstreaksfor=0.6withandwithoutplasmaactuation.ThestreaksshownarethemostampliedperturbationsatA)x=0=200andB)x=0=600.Thedashedverticallineindicatesthelocationoftheplasmaactuator,andthecircleindicateswherethisperturbationisthemostampliedperturbation,aswellasGmaxfortheunforcedcase. Havingobservedthatthekineticenergycontainedinthesestreaksisreduceddownstreamoftheplasmaactuation,thenextitemofconcernistodeterminewhythisdampingoccurs.Inordertoidentifythesourceofthisdamping,aRANSbasedkineticenergybalancefortheseperturbationsisused.Evaluatingtheperturbationkineticenergy,production,anddissipationterms(Figure 4-15 ),itbecomesapparentthattheadditionofabodyforceregionintheboundarylayerinterfereswiththeproductionoftheperturbation'skineticenergy.Insideofthebodyforceregion,itappearsthatnotonlyistheproductionofkineticenergyattenuated,butiftheamountofbodyforceishigh 103

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enough,itcanevenbecomenegative.Examiningtheeffectsofviscousdissipation,itappearsthatthissinkfortheperturbation'senergyislargelyunaffectedbytheadditionofplasmaactuation,thoughtherearesomevariationsduetothedecreasedmagnitudeoftheperturbation. Figure4-15. Comparisonofstreaktubulentkineticenergy,kineticenergyproduction,andviscouskineticenergydissipationaroundtheplasmaactuator.A,D,G)Turbulentkineticenergy,B,E,H)kineticenergyproduction,andC,F,I)kineticenergydissipationofboundarylayerstreaksfortheA-C)unforced,D-F)0=0.10andG-I)0=0.20cases,forthemostampliedperturbationatx=0=200. Decomposingtheproductionofkineticenergyintoitsindividualterms,thelocalizedeffectsbecomeevenmoreapparent(Figure 4-16 ).Theeffectsofthe v0u0@v @xand v0u0@v @xontheenergyproductionarenegligiblewithorwithoutthebodyforceaddedtotheow,butthechangestothe u0u0@u @xand u0v0@u @ytermsandthechangestothemaresignicant.Itappearsthatthemajorityofthenegativeproductioninthebodyforceregionisduetothe u0u0@u @xterm.Thisterm,whichisthelessdominantproductiontermintheunforcedboundarylayer,isreliantonthestreamwisedevelopmentoftheow.Astheadditionofaverylocalizedbodyforcegeneratessignicanthighgradientsinthestreamwise 104

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direction,itlogicallyfollowsthatthisproductiontermwouldbeaffected.Astheowisacceleratedbytheco-owactuation,thestreamwisevelocitygradient@u @xincreasesinmagnitude,leadingtoareductionintheperturbation'skineticenergyproductionbythe u0u0@u @xterm.Theothertermofsignicanceisthe u0v0@u @yterm,whichisresponsibleforamajorityoftheenergyproduction.Thistermalsoshowsaverylocalizedattenuationinproduction,aswellasaslightdecreaseinproductiondownstreamoftheactuator.However,thiseffectdoesnotseemtobeasdominanttoreducingthestreakmagnitudeasthe u0u0@u @xterm. Figure4-16. Comparisonofboundarylayerstreakkineticenergyproductiontermsaroundtheplasmaactuator.A,C)streamwisekineticenergyandB,D)wallnormalkineticenergyproductiontermsA,B)withoutandC,D)withplasmaactuation(0=0.20). ThechangeinperturbationkineticenergyproductionhasalsobeennotedinthestabilizationoftheTSwave.Howeverinthatscenario,thestabilizationisprimarilyduetomodicationofthe u0v0@u @yterm,thoughlocalstabilizationduetothe u0u0@u @xtermalsoplaysasmallpart.FortheTSwave,thestreamwiseandwallnormalvelocitycomponentsareofcomparablemagnitude.However,forboundarylayerstreaks,thestreamwisevelocitycomponentisfargreaterinmagnitude,andalargedifferenceinmagnitudeofthe u0u0and u0v0termsreectsthis.Assuch,forboundarylayerstreaks, 105

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changesto@u @xwillhaveagreaterimpactontheproductiontermthan@u @y.Therefore,thelocalizedaccelerationoftheowbytheco-oworientedbodyforceisstabilizing,butonlyintheregionimmediatelyaroundtheplasmaactuator.Downstreamoftheactuator,theseeffectsreverseastheowrelaxesbacktoazeropressuregradientboundarylayer,andthenormalizedproductionduetothe u0u0@u @xtermwillbeincreasedrelativetothebaselineow.Thisoffsetsmostofthestabilizationduetothefullerdownstreamboundarylayerprole.Evenso,thelocalizedstabilizationeffectsaresufcientenoughtoensureareducedmagnitudeofthestreakinthedownstreamregion.TheextentofthesestabilizingeffectscanbeseenmoreclearlyinFigure 4-17 Figure4-17. Totalandnormalizedkineticenergyproductionintheboundarylayerstreaks.A)Totalproductionofkineticenergyintheboundarylayer(Ptot(x)=R10P(x,y)dy)forvaryinglevelsofplasmaactuation.B)Totalproductionofkineticenergyintheboundarylayernormalizedbythetotalkineticenergycontainedintheperturbation(K(x)). 4.3.3ScalingoftheDampingTheobservationthattheadditionofaplasmabasedbodyforcecanreducetheamplitudeofboundarylayerstreaksisinterestingandsuggestswiderpotentialandapplicabilityforowstabilization,anditwouldbeverybenecialtocharacterizethebehaviorofthesedevicesoverabroadrangeofparameters,aswellastoreducethescalingeffectsdowntoasinglevalueforcomparison,similarlytowhathasbeendonefortheTSwaveearlierinthischapter. 106

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ParametricstudieshavebeenperformeddeterminingthevalueofGmaxasafunctionofthespanwisewavenumber,aswellasthevelocityratioalongthestreamwisedirection(Figure 4-18 ).Thisapproach,whileitdoesnotfocusonthegrowthanddecayofindividualstreaks,doesprovideinformationaboutwhatthemaximumpossiblegrowthwillbe.Assuch,thisapproachdescribesthegrowthofthemostampliedperturbationwithorwithouttheeffectsoftheplasmaactuator.InFigure 4-18 A,thevaluesofGmaxareshownacrossaspectrumofspanwisewavenumbersfortheunforcedboundarylayer,indicatingthatwiderperturbationsgrowmoreslowlyinitially,buteventuallygrowtolargermagnitudes.Withtheadditionofplasmacontrol(0=0.20,Figure 4-18 B),thistrendcontinuestohold,butthereisanoticeabledipintheperturbation'senergylocatedaroundtheplasmaactuatorforallofthewavenumbersexamined.Furtherexaminingthisdata,themaximumofGmaxforallofthewavenumbersexaminedisshownin 4-18 C.Itcanbeseeninthatgurethattheadditionofthebodyforcedampsthetheentireenvelopeofmostampliedpotentialperturbations.Inordertoquantifythedampingoftheseperturbations,themetricHmax=Gmax(x,,0) Gmax(x,,0=0) (4)isused,whichcomparestheratioofthemaximumdisturbanceamplicationwithandwithoutplasmaactuation.Evaluatingthismetricatapointdownstreamoftheplasmaactuator(x=0=300),itcanbeseenthatthereisdampingoftheperturbations,andthatthisdampingcanbesignicant,upto25%oftheoverallmagnitudeoftheperturbation,dependingonthemagnitudeoftheplasmaactuationisexamined.Whiletheamplicationoftheseperturbationsisobviouslywavenumberdependent(Figure 4-19 A),thedampingeffectsarenotassensitivetothespanwisewavenumber(Figure 4-19 B).Itwouldappearthatwhilethereisaslightspanwisewavenumber 107

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Figure4-18. Effectofplasmaactuationonindividualstreakamplitudeandtheenvelopeofstreakenergywithandwithoutplasmaactuation.Amplitudeofthemostampliedperturbations(Gmax)A)withoutandB)with(0=0.2)aplasmaactuatorasafunctionofthespanwisewavenumberandlocation.C)ThemaximumofGmaxtakenoverallofthespanwisewavenumbersexaminedforvaryinglevelsofplasmaactuation. dependence,thedampingismuchmorestronglydependentonthemagnitudeoftheplasmaactuation.Comparingthedampingofthestreaksacrossathevelocityratioandtotalamountofbodyforceaddedtotheow(whichisproportionaltotheparameterDc),severaltrendsbecomeapparent.Themagnitudeofthedampingincreasesmonotonicallywithrespecttobothparameters.Whilethedampingisverylow(5%)fortheweakerlevelsofplasmaactuation(0<0.1),oncetheplasmaactuationreachesahighenoughlevel,alineartrendinthedampingappearswithrespecttothetotalamountofbodyforce 108

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Figure4-19. Overallcomparisonofstreakmagnitudewithandwithplasmaactuationappliedtotheow.A)GmaxandB)Hmaxatthelocationx=0=300asafunctionofthespanwisewavenumberandthemagnitudeoftheplasmaactuation. addedtotheow.Assuch,itfollowsthatthedampingfollowsapowerlawtrendforthevelocityratioasupisproportionaltoDc. Figure4-20. [Dampingoftheboundarylayerstreaksbytheplasmaactuatorsasafunctionof0andDc.HmaxintermsoftheA)walljetvelocityratioandB)totalamountofbodyforceaddedtotheow.Hmaxisshownforspanwisewavenumbers,,between0.3and0.9 4.4ConclusionsThepresentbi-globalstabilityanalysishasexaminedhowcontinuousmomentumadditionintoazeropressuregradientboundarylayerthroughaplasmabasedbodyforcecanbeusedtostabilizetheTSandboundarylayerstreakpathstotransition. 109

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ParametricstudieshavebeenperformedvaryingthefrequencyoftheoncomingTSwaveandspanwisewavenumberoftheboundarylayerstreak,aswellasthemagnitudeofthebodyforceinjectedintotheow.Asmultiplepathstotransitionhavebeenexamined,itisapparentthattheeffectsofco-oworientedplasmaactuationhasabroadeffectonstabilizingtheowagainstgenericperturbations.Thisisverypromising,asitcannotalwaysbepredictedwhattypeofperturbationsaowwillexperience,especiallyifthoseconditionsdependonexternalfactors,suchaslocalatmosphericconditions.Theresultscontainedinthischapterhavebeendistributedthroughseveraldifferentconferenceproceedings( Riherd&Roy 2013b )andjournalpublications( Riherd&Roy 2013a d ). 4.4.1ConclusionsontheStabilizationoftheTSWaveExaminingthelocalowphysicsaroundanddownstreamofthemomentuminjectionpoint,anumberofdifferenteffectsareseentooccur.Thereisasuddenstabilizationoftheowaroundtheactuator.Immediatelydownstreamoftheactuator,wherethedisplacementboundarylayerheightsandshapefactorarereduced,theTSperturbationisfurtherstabilized,leadingtoasignicantdecreaseinitsmagnitude.Sufcientlyfardownstreamofthepointofmomentumaddition,theeffectsofowcontrolbecomeminimal,andtheowreturnstoitsuncontrolledstate.Basedontheparametricstudies,therelativemagnitudeoftheoncomingTSwavescanbemonotonicallydecreasedasthebodyforceinjectingmomentumintotheowisincreased.Themagnitudeoftheperturbationcanbedecreasedbyuptotwoordersofmagnitudeacrossawiderangeoffrequencies,thoughmoremodestreductionsinperturbationmagnitudeareseenforsmallerlevelsofplasmaactuation.Forsmalleramountsofplasmaactuation,thestabilizationeffectsareproportionaltothevelocityratiobetweenthefreestreamvelocityandtheinducedvelocityofthebodyforceunderquiescentconditions.Forlargermagnitudesofplasmaactuations,thestabilizationofthe 110

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TSwaveappearstobeexponentialwithrespecttotheamountofmomentumaddedtotheboundarylayer. 4.4.2ConclusionsontheStabilizationofBoundaryLayerStreaksUsingabi-globalstabilityanalysis,ithasbeenshownthatDBDplasmaactuatorsmaybeusedtostabilizeboundarylayerstreaks,llingagapintheunderstandingofhowtheseactuatorscanbeusedtodelaytheonsetofturbulentowinboundarylayerswithrespecttoalgebraicandby-passtransitionscenarios.ThephysicalmechanismsresponsibleforowstabilizationbeenexaminedusingaRANSapproachtotheperturbationgrowth.Parametricstudiesexaminingthesestreaksoverabroadrangeofwavenumbersandlevelsofplasmaactuationhavealsobeenperformed.Thedampingofthesestreaksispredictedtobeontheorderof5)]TJ /F4 11.955 Tf 12.06 0 Td[(25%ofthetotalstreakmagnitude,andoccursoverabroadrangeofspanwisewavenumbers.Thestabilizationofthesestreakscanbeattributedtohighlylocalizedeffectsaroundtheplasmainducedbodyforce.Theadditionofthebodyforcelocallydeformstheoweld.Inturn,thelocalizedvariationsintheoweldattenuatethelinearizedproductionofthestreak'skineticenergy,andforsufcientlyhighlevelsofactuation,theseoweldvariationscaninducenegativekineticenergyproduction,similartothelocalizedstabilizationoftheTSwave,butlackingthecontinuedstabilizationoftheowdownstreamduetothelowervaluemagnitudeofv0.Basedontheparametricstudiesperformed,theoveralleffectofstabilizationscaleslinearlywiththemagnitudeofthebodyforceappliedtotheowbytheplasmaactuators.Thereissomewavenumberdependencetothestabilization,buttheattenuationofthestreakystructuresintheowismorestronglytiedtothemagnitudeoftheplasmaactuation. 4.4.3FutureWorkFutureworkonthistopicshouldfocusonextendingtheuseoftheseactuatorsoverabroaderrangeofconditionsandundermorerealisticowandapplicationconditions 111

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anddisturbanceenvironments,particularlythosethatareexperiencedregularlybyairandgroundvehicles.Experimentalvalidationwillalsobenecessaryinthefuturebeforetheseactuatorscanbewidelyadoptedforstabilization,anareathathasnotbeenignored,buthasnotreceivedadequateattentionatthispoint.Verylittleattentionhasbeengiventotheeffectsofusingarraysoftheseactuators,ratherthanasingleactuator(asidefromthestudyof Duchmannetal. ( 2013a ),whousedaveryloosedescriptionoftheplasmamodiedoweld,butsuggestedthatenormousgainsinstabilityarepossible).Theuseofmultipleactuatorscouldbeextremelybenecial,asitwouldallowforperturbationstobedampedbeyondthelimitedeffectsofasingleactuator.ItwouldalsoallowforstabilizationeffectstoberealizedatveryhighReynoldsnumberswherethevelocityratiooftheplasmaactuationisontheorderof0.01.Theremayalsobeedgeeffectswhentheseactuatorsareapplied. Visbaletal. ( 1998 )predictedthattheedgeeffectsforwalljetowshaveasignicantimpactonthedownstreamowandonthestabilityoftheow.Inthatstudy,itwasfoundthatthereisacontractioninthewidthofthewalljetasitdevelopeddownstream,whichledtothegenerationofcross-oweffectsandaresultinginstabilityatthespanwiseedgesofthewalljet.AsthemomentumadditiontoaboundarylayerbyDBDplasmaactuatorscanbecomparedtoaddingawalljetcomponenttotheow,theseeffectswouldalsobelikelytooccurifnitewidthplasmaactuatorswereemployedtostabilizethearealow.However,theywouldlikelyoccurataslowerratealongthelengthoftheowduetotheaddedconvectiveeffectsofthefreestream(ratherthanquiescent)ow.Goingforward,theimpactoftheseedgeeffectwillneedtobequantiedandunderstood,astheymaybecomealimitingfactorintheeffectivenessofthecontrol.Shouldtheeffectivenessoftheseactuatorsstillholdasthey'reimplementedinamorerigorousenvironment,itwouldindicatethatthesedevicescanbeusedtoreduce 112

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turbulentskinfrictionandpotentiallyincurreducedfuelusageforarangeofeverydaytransportationapplications. 113

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CHAPTER5FLOWSTRUCTUREINABOUNDARYLAYERWITHSERPENTINEGEOMETRYPLASMAACTUATIONInadditiontothesimplegeometriesrstexploredbyRothandhisco-workers( Rothetal. 1998 2000 ),othergeometriccongurationsforDBDplasmaactuatorshavealsobeenexploredintheresearchliterature.Theconceptofaplasmasyntheticjet,whereactuatorsareplacedfacingeachother,andgenerateaplaneorcircularjetowingnormaltothesurfacehasbeenexplored( Santhanakrishnan&Jacob 2007 ).Othergeometries,suchastheserpentinegeometry( Durscher&Roy 2012 ; Roy&Wang 2009 ; Wangetal. 2011 )areabletoproducethree-dimensionaleffectsduetotheirmorecomplex,periodicallyvaryingelectrodegeometries.Manyofthedifferentactuatorgeometriescanbegeneralizedintowhatisreferredtoasaperiodicserpentinegeometry,examplesofwhichareshowninFigure 5-1 ,andcanbecharacterizedbytheirperiodicelectrodeshape,aswellastheperiodiclength()andthelengthoftheelectrodegeometryfromit'smedianposition(A). A B C D Figure5-1. Schematicofvariousplasmaactuatorgeometries.A)linearB)arc,C)rectangle,andD)combgeometriesareshown. 114

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Theprimaryaimofusingthistypeofactuatorgeometriesisthattheyshouldallowforplasmaactuatorstotapintoawidersetofowphysicsthanthestandardgeometryactuatorwouldbeabletodo.Themodicationtotheelectrodegeometryprovidesadditionaldegreesoffreedomfortheseactuatorstobeusedforowcontrol.Inparticular,thespanwisewavelengthisofthemostimportance,asthisallowsforperturbationsofaspecicspanwiselengthtobeaddedtotheow.Withthisadditionalspanwiselengthscale,itispossiblethatsteady(boundarylayerstreaks)andunsteady(obliquemodalandsecondary)instabilitiescouldbeexcitedbytheactuator,somethingthatisnotpossiblewithasinglestandardgeometryplasmaactuator.Previousapplicationofthistypeofactuatorgeometryhavefocusedonseparationcontroloverairfoils( Riherd&Roy 2012a ; Rizzetta&Visbal 2011 )andcontrolofswirlincombustionchambers( Wang&Roy 2011 ).Thecurrentworkaimstousethisclassofactuatorstoexciteboundarylayerstreaksinalaminarboundarylayerfortransitioncontrol.Atlowmagnitudes,boundarylayerstreakshavebeenshowntobeusefulfordampingTSwavesandrandombackgroundnoiseaffectingaboundarylayer. Franssonetal. ( 2006 )foundthatwiththeadditionofstreaksintotheboundarylayer,thevelocityprolecouldbesufcientlymodiedsuchthatthestabilitypropertiesoftheboundarylayerarealsomodied.Itwasfoundthatforstreaksofsufcientmagnitude,transitionoftheowcanbedelayed,ifnotavoidedentirely.Whentheseboundarylayerstreaksaregeneratedatorampliedtoasufcientlyhighmagnitude,theypossesstheirowninstabilityproperties( Anderssonetal. 2001 ),ofwhichfoursimilarbutdistinctinstabilitymodeshavebeenidentied(Figure 5-3 ).Thesecondaryinstabilityresultsof Anderssonetal. ( 2001 )indicatethatthesecondaryinstabilityissensitivetoslightchangesinthemeaneld(whichmaypresentitselfasachallengewhenexaminingtheowgeneratedbyarealisticplasmadevice).Dependingonthemannerofthesecondaryinstability,theseinstabilitiesmaynotbecomecritical 115

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Figure5-2. Comparisonofunsteadyowmeasurementsinaboundarylayerwithandwithoutboundarylayerstreaksapplied.A)MeasurementsoftheRMSstreamwisevelocitymagnitudeofaTSwaveina5m/sboundarylayerwith(bluesquares)andwithout(reddots)boundarylayerstreaksgeneratedbyupstreamroughnesselements.Temporalmeasurementsmadeusinghot-wireanemometryshowthattheowislaminarforbothcasesatB)x=400mm,buthasbecometurbulentintheabsenceofcontroldownstreamatC)x=1650mm.TheresponsetobackgroundnoiseintermsoftheD)RMSstreamwisevelocityandE)intermittancyfactoralsoindicateowstabilization.From Franssonetal. ( 2006 ). untilthestreakamplitudeisupwardsof25%ofthefreestreamvelocity.Furthermore,theperturbationstendtobelargestintheregionofthehighestshearstressinthestreak.Researchershavealsoexaminedthetransientbreakdownoftheboundarylayerstreaks,identifyingthatnon-modalgrowthofperturbationsonthestreakscanoccurevenwhenthestreakyboundarylayerowisstablewithrespecttoexponentialinstabilities( Hpffneretal. 2005 ),whichsuggeststhatthesestreaksmaybeusedtorapidlytransitionalaminarboundarylayerevenformodestlevelsofowcontrol.Astheseboundarylayerstreakspossessbothstabilizinganddestabilizingeffectsdependingontheirmagnitude,theymaybeextremelyusefulandversatilefortransitioncontrol.AsDBDactuatorscanbeusedasactiveowcontrolactuators,oneactuatorcouldbeusedtogeneratelowmagnitudestreaksinordertostabilizetheoweldwhenoperatedatonevoltageandtodestabilizetheoweldusinghighermagnitudestreaks 116

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Figure5-3. Differentboundarylayerstreakbreakdownmodesasseenfromabovetheboundarylayer.Solidlinesindicatethelowspeedstreakregionsanddashedlinesindicatehighspeedstreakregions.From Anderssonetal. ( 2001 ). whenoperatedatahighervoltage.Asmanytypesoflandandairvehiclesundergoavarietyofoperatingconditionsoverasingletrip,thisversatilitymaybehighlybenecial,asitwouldallowforoneactuatortooperateforseveraldifferentpurposes.Inthischapter,numericalsimulationsofaserpentinegeometryplasmaactuatorunderquiescentarepresented,whichshowgoodqualitativeandquantitativeagreementwithcomparableexperiments( Durscher&Roy 2012 ).Numericalsimulationsofserpentinegeometryplasmaactuatorsinalaminarboundarylayerarethenpresented,theresultsofwhicharecharacterized.Theresultsofthesesimulationsindicatethatboundarylayerstreaksareformedwhenserpentinegeometryplasmaactuatorsareappliedinaco-owmannertoalaminarboundarylayer. 5.1SerpentineGeometryActuatorUnderQuiescentConditionsUnderquiescentconditions,theseplasmaactuatorshavebeenshowntoinduceaverycomplexow.Inparticular,thisclassofactuatorshasbeenshowntogeneratestreamwisevorticalstructuresdownstreamoftheactuator( Durscher&Roy 2012 ; Roy 117

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&Wang 2009 ; Wangetal. 2011 ),aneffectthatisnotseenforthestandardgeometryofplasmaactuators.SlicesofdatafromastereoscopicPIVstudyareshowninFigure 5-4 ,illustratingthiseffect. Figure5-4. SpanwiseslicesofthestreamwisevoriticitygeneratedbyaserpentinegeometryplasmaactuatorunderquiescentconditionsatA)7.5mm,B)12.5mm,andC)17.5mmdownstreamofatheleadingedgeofacurvedgeometryplasmaactuatorwithspanwisewavelengthof20mmandanamplitudeof5mm.Thesliceshownin(A)istakeninaregionwheretheplasmabodyforceisnon-zero.InD)iso-surfacesofthestreamwisevorticityarealsoshown.From Durscher&Roy ( 2012 ). Theseactuatorshavealsobeenshowntoejectuidawayfromthesurfaceatthepinchpointsoftheactuator( Durscher&Roy 2012 ; Roy&Wang 2009 ; Wangetal. 2011 ).FlowvisualizationofthiseffectisshowninFigure 5-5 .Betweenthesetwoeffects,itisexpectedthattheserpentinegeometryactuatorsareabletogenerateasufcientperturbationtotheowinordertoexcitethestreamwisevorticalstructuresintendedtobeusedforowcontrol. 118

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Figure5-5. [FlowvisualizationofserpentineandstandardgeometryplasmaactuatorsatA)thepinchpointofaserpentinegeometryactuatorandB)byastandardgeometryactuatorunderquiescentconditions.From Durscher&Roy ( 2012 ). Beforemovingontotheeffectsoftheseactuatorsinaboundarylayer,simulationsoftheseactuatorsinaquiescentowareperformedinordertocomparewiththeseexperimentalresults. 5.1.1NumericalDetailsInordertosimulatetheseows,theImplicitLargeEddySimulation(ILES)Navier-Stokessolver,FDL3DI( Rizzettaetal. 2008 )isemployed.Thiscodesolvesthecompressible,three-dimensionalNavier-Stokesequations,andisdescribedinmoredetailinAppendix C .Forthesesimulations,theowwassimulatedinathreedimensionalboxofsize[0.3,8.0][0,1.5][0.0,0.1]inthexyzdirections.Thegridusedtoresolvethemeshcontains7.5milliongridpoints.Themeshishighlyrenedattheinletandaroundtheactuator1
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Theserpentinegeometryplasmaactuatorislocatedatthepointx=1.025.ThestreamwiseamplitudeoftheserpentinegeometryisA=0.025,sotheleadingedgeofthebodyforceislocatedatthepointx=1.Thetrailingedgeoftheactuatorgeometryiscenteredatx=1.05.Thespanwisewavelengthoftheactuatorgeometryis=0.1.ThisgeometryisequivalenttothatshowninFigure 5-6 C.Tomodeltheeffectsoftheplasma,anapproximatebodyforcedistributionisemployed( Singh&Roy 2008 ),asliceofwhichisshowninFigure 5-6 B.Thisbodyforcedistributionisappliedalongthelengthofserpentinegeometryelectrodes,andthebodyforceisrotatedsothatitisalwaysorientednormaltotheelectrodeedge. Figure5-6. Computationalmeshusedtoperformserpentineplasmaactuatorowsimulations.A)Mesh,B)two-dimensionalsliceofthebodyforce,andC)geometryusedtosimulatetheserpentinegeometryplasmaactuation.Everyotherpointisshown. Asintheprevioussimulations,themagnitudeoftheplasmaactuatorischaracterizedbythevelocityratio(Equation 3 ).Detailsofthedimensionalandnon-dimensionalparametersusedareinTable 5-1 5.1.2FlowCharacterizationExaminingtheoweldsaroundtheserpentinegeometryactuatorsoperatedunderquiescentconditions,itcanbeseenthatseveralinterestingowfeaturesarepresent(Figure 5-7 ).TheseresultsareshownalongsidestereographicPIVresultsofacomparableexperimentperformedby Durscher&Roy ( 2012 )foracomparable 120

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Table5-1. Dimensionalandnon-dimensionalvaluesusedtocomputetheowwithaserpentinegeometryplasmaactuator. ReferenceParameterValue Dimensionalvalues u15.0m=sL0.30m11.20kg=m31.510)]TJ /F7 7.97 Tf 6.59 0 Td[(5m2=s Non-dimensionalvalues Re100,000Pr0.72Ma0.1Dcvaries,seeFigure 3-2 a experiment(Figures8AandBof Durscher&Roy ( 2012 )),andshowgoodqualitativeagreement.Atthepinchpoint,avectoredjetisproduced(Figure 5-7 A).Thiscontrastswiththemuchsimplerbehavioratthespreadingpointoftheactuatorgeometry,whereasimplewalljethasformed(Figure 5-7 B).AtwistedvectoredjetisvisualizedusingstreamlinesinFigure 5-8 A,whichcomparedfavorablytoexperimentalresultsinFigure 5-8 b.Inthesestreamlines,theeffectsofastreamwiseorientedvortexcanbeseendownstreamoftheactuator(identiedbythetwistingbraidofstreamlines).ThestreamwiseorientedvorticityintheoweldcanbemoreclearlyseeninFigure 5-9 .Astheseowshavebeendiscussedby Wangetal. ( 2011 )and Durscher&Roy ( 2012 ),andareinqualitativeagreementwiththosestudies,theywillnotbediscussedinfurtherlengthhere. 5.2BoundaryLayerModicationUsingSerpentineGeometryPlasmaActuationHavingconrmedthattheeffectsofserpentinegeometryplasmaactuationarecomparableinsimulationsandexperimentsunderquiescentconditions,anumberofsimulationshavebeenperformedinordertoexaminetheeffectsofaserpentinegeometryactuatorinalaminarboundarylayer.ThesesimulationsarebasedaroundowconditionsatrelativelylowReynoldsnumbers,wheretheactuatorisplacedatastreamwiseReynoldsnumberofRex=100,000.Thesesimulationsarenotperturbedin 121

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Figure5-7. Comparisonofsimulatedandexperimentalvelocityeldsinducedbyaserpentinegeometryplasmaactuatorunderquiescentconditions..VelocityeldsattheA,C)pinchpoint(z=0),B,D)spreadingpoint(z=0.05)oftheserpentinegeometryactuatoroperatedunderquiescentconditionsforA,B)simulationswithaprescribedinducedvelocityofup=u1=0.1andC,D)experimentsperformedby Durscher&Roy ( 2012 ).Theexperimentalresultshavebeennon-dimensionalizedsothattherelativesizesoftheactuatorsmatch. Figure5-8. Comparisonofsimulatedandexperimentalvelocitystreamlinesinducedbyaserpentinegeometryplasmaactuatorunderquiescentconditions..StreamlinesintheoweldsfromtheA)simulationofaup=u1=0.1serpentineactuatorandB)experimentsforacurvedserpentineactuator.Ablacklineisusedtoindicatethelocationoftheactuator.Experimentaldatafrom Durscher&Roy ( 2012 ). 122

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Figure5-9. Streamwisevorticityatx=1.025oftheserpentinegeometryactuatoroperatedunderquiescentconditionsforsimulationwithaprescribedinducedvelocityofup=u1=0.1 anyactivemanner,otherthanthesteadyadditionofmomentumtotheowthroughtheplasmabodyforce. 5.2.1FlowCharacterizationTherearetwoprimaryowfeaturesintheowthathasbeenmodiedwiththeuseofaserpentinegeometryplasmaactuator.Therstoftheseistheforcingofuidawayfromthesurfaceintheregionimmediatelydownstreamoftheactuatorinavectoredjet,comparabletotheeffectsofaplasmasyntheticjet( Santhanakrishnan&Jacob 2007 ).EvidenceofthiseffectcanbefoundinFigure 5-10 AandB.Thesecondofthese,isthecounterrotatingstreamwiseorientedvorticesthatpropagatedownstream( Durscher&Roy 2012 ; Roy&Wang 2009 ),evidenceofwhicharegiveninFigure 5-10 EandF,aswellFigures 5-15 through 5-17 .Itisalsoexpectedthatthestreamwisevorticalstructuresgenerateboundarylayerstreaks,whicharetheprimaryaimofimplementingthisdeviceinaboundarylayer.Examiningtheoweldatthepinchpointoftheactuator(Figure 5-10 AandB),itcanbeseenthatthevectoredjetproducedinthepresenceofameanowisofmuchshalloweranglethanthatproducedunderquiescentconditions.Theangleofthejetisstronglydependentonthemagnitudeoftheplasmaactuation,wherealargervelocityratioproducesasteeperjet.Atthespreadingpoint(Figure 5-10 CandD),uidispulleddown,towardsthesurface,aneffectthathasbeenseenforstandardgeometryplasmaactuatorsinboundarylayerspreviously.Overall,basedonthespanwiseaveraged 123

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velocityeld,theeffectofthevectoredjetisminimal,indicatingthatitisalocalizedeffect. Figure5-10. Slicesoftheboundarylayeroweldstakenatthepinchingandspreadingpointsandstreamwisevorticitywhenaserpentineplasmaactuatorisapplied.A,C,E)0=0.05andB,D,F)0=0.10.ThestreamwisevelocitiesareshownfortheA,B)pinchingandC,D)spreadingpoints.E,F)Streamwisevorticityatthecenteroftheseprentineactuator(x=1.025). ThepresenceofthestreamwisevorticescanbeseeninFigures 5-10 C,Dand 5-12 B,DandF.Thestrengthofthesevorticesincreasesasthemagnitudeoftheplasmaactuationisincreased.However,itcanalsobeseenthatasthestrengthoftheactuationisincreasedbeyond0=0.05,thevorticesareejectedfromtheboundarylayer.Thismaybedetrimentalforapplicationsofowcontrol,asthesevorticeswilllikelyhavelessofaneffectontheboundarylayerastheymovefartherandfartheraway.Examiningthevelocityprolesoftheow(Figure 5-13 ,itcanbeseenthatthereareseveraleffectsoccurringsimultaneously.Atx=1,itcanbeseeninthemeanboundarylayerprolesthattheowproleisbeingmadefullerbytheentrainmentofadditional 124

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Figure5-11. Angleofthevectoredjetasthevelocityratioisvaried.Thisanglewasmeasuredasthemaximumowangleattheheightofy=0=0.0079alongtheimpingementplane. momentumintotheboundarylayerbytheplasmaactuator.ExaminingthestreamwisevelocityvariationsinFigure 5-13 D,H,L,itcanbeseenthattherearesignicantspanwisevariationsinthestreamwisevelocity,andforthelowerlevelsofplasmaactuation,theshapeofthevariationsappearstobeverysimilar.However,asthemagnitudeoftheactuationisincreased,non-lineareffectsbecomeimportant,andtheeffectofthebodyforceontheboundarylayernolongerresemblestheintendedboundarylayerstreaks.Deningthestreakvelocitymagnitudeashalfthedifferencebetweenthevelocityatthecenterofthehigh(i.e.thespreadingpoint,z=0.05)andlowspeedstreak(i.e.thepinchpoint,z=0.00)regions,Au(x)=maxy2[0,1)jupinch(x,y))]TJ /F5 11.955 Tf 11.95 0 Td[(uspread(x,y)j 2u1 (5)andthestreamwisevortexmagnitudeasthetotalamountofvorticityalongthelengthoftheboundarylayer,A!x=Zz=0Z1y=0j!xjdydz (5)themagnitudeofthestreakscanalsobequantied(Figure 5-14 ).Inapplyingthesemetrics,itcanbeseenthatmagnitudeofthevortexstreakscalesnicelywithrespect 125

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A B C D E F Figure5-12. Visualizationoftheowstructurearoundanddownstreamoftheserpentineplasmaactuatorinalaminarboundarylayer.A,C,E)Q-criteria(coloredbyvelocitymagnitude)andB,D,F)streamtracesforthecasesofA,B)0=0.05,C,D)0=0.10,andE,F)0=0.20.InthebackgroundofB,D,F),asliceofthestreamwisevelocitycomponenttakeninthe(y,z)planeatx=2isshown.Thedatasetisrepeatedtwiceinthez-direction,onlyasinglewavelengthwassimulated. 126

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tothevelocityratio0untilnon-lineareffectsbecomeimportant.Itappearsthatstreakvelocitymagnitudecanbeampliedanorderofmagnitudelargertheinducedvelocityoftheplasmaactuation.ThisamplicationshouldalsobeafunctionofthespanwisewavenumberandtheReynoldsnumberoftheow,effectsnotexaminedinthepresentstudy.Themagnitudeofthestreamwiseorientedvorticityseemstobemorerobustagainstnon-lineareffects,andappearstoscalenicelywithrespectto1=20evenafternon-lineareffectssetin.Examiningtheeffectsforlargerlevelsofplasmaactuation,itappearsthattheapplicationofowcontrolmaybetoolargeinmagnitudetoexcitethedesiredboundarylayerstreaks.Thatthevariationsintheowelddonotdisplaysimilarprolesindicatesthatthemagnitudeoftheperturbationsisnotbehavinginalinearmanner.Forthelinearbehavioroftheboundarylayerstreakstooccur,themagnitudeoftheactuationmustbereduced.Astheowcontrolisapplied,theboundarylayerstreaksseparatefromthesurface.Whilethismaystillleadtothedesiredeffectofacceleratingthetransitionoftheow(asitappearsthatthestreaksdisplaymultipleinectionpointsabovetheboundarylayer,suggestinganinviscidinstability),itwillnotnecessarilybeduetothesamephysicsastheboundarylayerstreaksthatarehopedtobeexcited.Evenforthemoremoderatemagnitudesoftheplasmaactuation,itcanbeseenthattheboundarylayerstreaksarebeingtransportedoutoftheboundarylayer,whichmayreducetheeffectivenessoftheseactuatorsforowcontrol. 5.3ConclusionsSimulationsofaserpentinegeometryplasmaactuatorhavebeenperformedunderquiescentconditions,aswellasinaboundarylayer.Underquiescentconditions,comparisonsweremadetoexistingresultsintheliterature( Durscher&Roy 2012 ),indicatingsimilarowfeatures.Whenappliedtoaboundarylayer,itwasconrmedthattheseactuatorsareabletogenerateboundarylayerstreaks. 127

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Theeffectsofusingserpentineplasmaactuatorsinaboundarylayerwerecharacterized.Whenplasmaactuationwasappliedatasufcientlylowmagnitude,thestreaksgeneratedweresimilartotheoptimallygeneratedstreaks.Thesestreakscouldbeusedtocontrolthelaminartoturbulenttransitionmechanismsascurrentlyunderstood,whethertoaccelerate( Anderssonetal. 2001 ; Hpffneretal. 2005 )ordelay( Franssonetal. 2006 )theonsetofaturbulentow.Whenlargerlevelsofplasmaactuationwereapplied,thestreaksdeviatedfromtheexpectedbehavior,duetonon-lineareffects.Inparticular,itshouldbenotedthattheserpentinegeometryplasmaactuatorsgeneratedvorticalstructuresintheboundarylayerthatmaybetoolargetobeofuse,astheyliftedthemselvedoutoftheboundarylayer.However,ifthefreestreamvelocityweretobeincreased,thiseffectwouldbecomealesslikelyscenario,duetothelimitationsofthisclassofactuators.Thepresentsimulationsindicatethattheuseofserpentinegeometryplasmaactuatorsmaybeverybenecialforhigherspeedapplications.Thenaturalamplicationoftheboundarylayerstreakscanbeusedtoincreasethemagnitudeoftheperturbationsgeneratedbytheplasmaactuator. 128

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Figure5-13. Variousvelocityprolestakendownstreamofaserpentineplasmamodiedboundarylayerasafunctionofthevelocityratio.A,E,I)Meanboundarylayerproles,B,F,J)boundarylayerprolesatthespreadingpoint,C,G,K)boundarylayerprolesatthepinchpoint,andD,H,L)spanwisespandarddeviationintheboundarylayerprolesatvariouspointsalongthelengthoftheboundarylayerforA-D)0=0.01,E-H)0=0.025,andI-L)0=0.05. 129

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Figure5-14. NormalizedboundarylayerstreakA)velocitymagnitudeandB)streamwisevortexmagnitude. Figure5-15. Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.01).A,C,E)StreamwisevariationsinthevelocitymagnitudesandB,D,F)streamwisevorticityandvelocityvectorsatA,B)x=1.2,C,D)x=1.5,andE,F)x=2.0.The99%boundarylayerheight(99%)ismarkedbythethicksolidline. 130

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Figure5-16. Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.025).A,C,E)StreamwisevariationsinthevelocitymagnitudesandB,D,F)streamwisevorticityandvelocityvectorsatA,B)x=1.2,C,D)x=1.5,andE,F)x=2.0.The99%boundarylayerheight(99%)ismarkedbythethicksolidline. 131

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Figure5-17. Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.05).A,C,E)StreamwisevariationsinthevelocitymagnitudesandB,D,F)streamwisevorticityandvelocityvectorsatA,B)x=1.2,C,D)x=1.5,andE,F)x=2.0.The99%boundarylayerheight(99%)ismarkedbythethicksolidline. 132

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Figure5-18. Boundarylayerstreaksinaserpentineplasmaactuatormodiedboundarylayer(0=0.10).A,C,E)StreamwisevariationsinthevelocitymagnitudesandB,D,F)streamwisevorticityandvelocityvectorsatA,B)x=1.2,C,D)x=1.5,andE,F)x=2.0.The99%boundarylayerheight(99%)ismarkedbythethicksolidline. 133

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CHAPTER6USINGPLASMATODRIVEACHANNELFLOWThereexistmanytypesofchannelows,whichcanbepoweredbyanynumberofdifferentmechanisms.Oncefullydevelopedalongthelengthofthechannel,thesesimpleowscanbeconsideredtodrivenbyauniformbodyforce,wherethetimemeanpressuregradientorgravitymimicstheeffectsofabodyforce(i.e.fx=@p @xorfx=g).Thatthesechannelowsareessentiallydrivenbyabodyforceleadstotheconceptofdrivingachannelowusingaplasmabasedbodyforce.Thereareobviouslysomedistinctdifferencesbetweentheapproximationofpressuregradientsandgravityasauniformbodyforceandaplasmabasedbodyforce.Themostobviousoftheseisthattheplasmabodyforceisanunsteady,localizedforce,whilethepressure/gravityforceisusuallysteadyandlocalized.However,comparisonsontheplasmadrivenchannelasawholeshouldbemadewithothertypesofpumps,astheplasmachannelisactuallydrivingtheow,whereasthepressuredropassociatedwithchannelowsisactuallyanimpedimenttowhateverpumpisdrivingtheow.Beforegoingintoextensivedetailexploringtheinternalowstructureasitisdrivenbyaplasmachannel(whichispresentedinChapter 7 ),aproofofconceptandanexplorationofgeneralsystemcharacterizationwouldbeverybenecial.Knowledgeofthesystemasawholewouldhelptoguideanynumericalsimulations,asitwouldprovideageneralrangeofowconditionsthatcouldbeexpectedundernormaloperation.Theseparameterswouldalsobebenecialinunderstandingwhereapumpofthistypecouldbeusefulinapplications,wheresystemparameterssuchasthemaximumpressuredifferentialorvolumeowratecanbeofgreaterimportancethanmeasurementsoftheReynoldsstressorturbulentkineticenergy.StudiesoftheowphysicswillbeperformedinChapter 7 .AschematicofanitelengthplasmadrivenchannelisshowninFigure 6-1 A.Thelengthscalesshownarethechannelheight(h),thedistancebetweentheleadingedges 134

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2A B Figure6-1. Schematicoftheplasmachannelandofaplasmaactuator.A)theplasmachannelandB)theplasmaactuator. oftwoconsecutiveactuators(s),thetotallengthofthedevice(L1)andthelengthfortheowtodevelopdownstreamofthedevices(L2).Foreachplasmaactuator,therearealsoseveralimportantlengthscales(Figure 6-1 B).Therearetheupperandlowerelectrodewidths(lUandlL),aswellastheelectrodegap(g)andthedielectricthickness(t). 6.1ExperimentalMeasurementsofaFiniteLengthChannelFlowAsarststepintounderstandingtheseows,experimentshavebeenperformedinwhichthesystemcharacteristicsaremeasured.Intheexperimentsperformedaspartofthisstudy,eachplasmaactuatorispoweredbythesamehighvoltage,highfrequencysignal,andconnectedtothesameground.Alloftheplasmaactuatorshavealsobeenbuiltwithuniformelectrodegeometries,sothattheyallproduceasimilarbodyforce.ThissignaloriginatesinaTektronicsAFG3022Bfunctiongeneratoras14kHzsinusoidalsignal.ThesignalisthenampliedtoahighercurrentusingaQSCRMX2450audioamplier.ACoronaMagneticshighvoltagetransformerisusedtoconvertthelowvoltage,highcurrentsignaltoahighvoltage,lowcurrentsignal,whichisthenconnectedtothepowered(upper)electrode.Ahighvoltageprobeandoscilloscopeareusedtomeasurethesignalandensurethatanaccuratevoltageisappliedtotheplasmaactuators.Whenalargernumberofplasmaactuatorsareoperatedsimultaneously(i.e.whenthecurrentnecessarytopowerthedevicesexceedstheratingofthetransformers),thissystemisdoubled,suchthattwoidentical 135

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Figure6-2. Circuitdiagramoftheplasmachannel. signalsoriginatingfromthefunctiongenerator,whicharethenamplied,transformed,andconnectedtotheplasmaactuators.AdiagramofthissystemisshowninFigure 6-2 .TheplasmachannelwasconstructedoutofPMMA(plexiglass),whichalsoservedasthedielectricfortheplasmaactuators.Fourpairsofactuatorswereinstalledinthechannel,foratotalof8actuatorsinsideofthechannel.ThegroundedelectrodesweresandwichedbetweentwolayersofPMMA,whichwereepoxiedtogether.Thistwolayerapproachwaseffectiveatcontrollingthermaldeformation,whichhadbeensignicantinearlierexperimentswhereonlyelectricaltapehadbeenusedtoencapsulatethegroundedelectrodes.WhiletheDBDactuatorcreatesanon-thermalplasma,thedevicesstillbecamehotenoughtocreatesignicantdeformationswhenthechannelwasnotstiffened.FourDBDactuatorswereinstalledontheuppersurfaceofthechannelandfourmoreontheloweractuatorswereinstalledonthelowersurfaceofthechannel,foratotalofeightDBDactuatorsinthechannel.Supergluewasusedtoattachsidewallstothechannel,allowingforachannelwidthof10cm.Theelectrodesfortheplasmaactuatorsextendedallthefullwidthofthechannel.Whilethisnitechannelwidthimpliesthattheowinsidethechannelisnottwodimensional,butisthreedimensionalinstead,withanaspectratioof5,thischannelshouldprovidearelativelytwodimensionalowatthecenterofthechannel'sspan.Furthermore, 136

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measurementswereprimarilytakenatthechannelcenterline,wherethemeanowshouldbesymmetric.Forthemeasurementsandsimulationsperformed,achannelofthefollowingdimensionswasexamined. Table6-1. Dimensionsoftheplasmachannelusedforvelocitymeasurements. ParameterValue ChannelLI2.1cmLtot30.0cmLdev20.0cms2.5cm2h2.0cmw10.0cmN1-4 ActuatorsLU0.4cmLL0.4cmg0cmt0.25cm Beforeanydatawascollected,theplasmawasoperatedfor30secondstoremoveanytransienteffects.Aftereachsetofsampleswascollected,thedevicewasallowedtocooldownfor180seconds.Measurementswereperformedforavaryingnumberofactuatorsoperatedatasingletime,andvaryingtheappliedvoltageforeachactuator.Aspartofthisstudy,thedenotationofnactuatorsimpliesthatnactuatorswereoperatedontheuppersurfaceoftheplasmachannelandnmoreactuatorswereoperatedonthelowersurfaceinordertopromoteasymmetricchannelow.Theseexperimentswereperformedina0.6m1.2m1.2mquiescentchamber.Nomeanowwaspresent,exceptfortheowinducedinandaroundtheplasmachannelbytheDBDactuators.Whilevoltageandcurrentdatawasonlyobservedandnotrecordedduringthecollectionofvelocityandpressuredata,electricaldatahasbeencollectedinordertoquantifytheamountofpowerconsumptionofthesedevicesduringoperation.Asingleactuator,ofthesamedimensionsusedintheexperimentswasoperatedunderthesameoperatingconditionsasintheexperiments.Thepowerdatacollectedshowsasimilar 137

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powerlawrelationshipasdescribeinrecentreviews( Corkeetal. 2010 ; Moreau 2007 ),wherePower/V3.5. Figure6-3. PowerconsumptionbyasingleDBDactuatorwiththesamedimensionsanddielectricasusedinthechannel. 6.1.1VelocityMeasurementsandSimulationsoftheFiniteLengthChannelFlowALaVisionPIVsystemwasusedtomakemeasurementsoftheowexitingthechannel.A532nmNd:YAGlaserwasusedtoilluminatetheOndinaseededuid.250samplesweretakenatarateof15samplespersecondusingaPhantom7.3highspeedcamera,whichhasa600800pixelresolutionfocusedona4cm5cmregionneartheexitoftheplasmachannel.Measurementsinsideofthechannelwereattempted,butanopaqueresiduefromtheuseofsuperglueonthechannelsidewallsandreectionsofthelaserbouncingoffoftheupperandlowerchannelwallsandelectrodespreventedhighqualityfrombeingcaptured.ThesesampleswerethenanalyzedusingLaVision'sDaVissoftware.Thevelocityeldwascalculatedusinga1616pixelintegrationwindowwith50%overlap.A3232pixelintegrationwindowwasalsousedwith50%overlapinordertoverifythattheresultswereinsensitivetothePIVprocessing. 138

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Figure6-4. PIVSetupforexaminingtheplasmadrivenchannelow. ItcanbeseeninFigure 6-5 A,thatuponit'sexit,theowproducedbytheplasmachannelproducesunsteadyjet-likeeffects.However,whentimeaveragedovertheavailablesamples,theowcanbeseentobenearlysymmetricinnature(Figure 6-5 B).Theselargescaleunsteadyeffectscanlargelybeattributedtoinstabilitiesinthejet( Bickley 1937 ),ratherthanlargeowperturbationsinthechannelow.Basedontheexitvelocity,theReynoldsnumberofthechannelowsrangesfromapproximately1500to4000(forasinglesetofactuatorsoperatedat16kVppto4setsofactuatorsoperatedat19kVpp)intheseexperiments,whichisarangewhereunsteadyeffectsandhydrodynamicinstabilitiescanbecomeimportant.WhilethismayleadtoarelativelylowReynoldsnumberturbulentchannelow,itdoesnotprecludethechannelowfromeventualturbulentoweffectsinlongerchannelswithhigherReynoldsnumbersduetoincreasedowvelocityorchannelheight.Exploringtheeffectsofvoltageandoperatingdifferentnumbersofplasmaactuatorssimultaneously,itcanbeseenthatthemaximumvelocityandthetotalmassowperunitwidth(Q=w)containedinthechannelowasitexitsincreaseswithrespecttoboththeappliedvoltageandthenumberofactuatorsrunforaspeciccase(Figure 6-6 ).Thisisalogicalconclusiontoarriveat,asthetotalamountofbodyforcepushingtheuiddownstreamincreaseswithrespecttobothofthesevariables.Asthetotalamountof 139

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Figure6-5. Velocitymagnitudesmeasurementsattheexitoftheplasmadrivenchannel.A,B)InstantaneousandC,D)timeaveragedvelocitymagnitudesmeasurementsattheexitofthechannelwithanappliedvoltageof18kVppforA,C)1andB,D)3actuatorspoweredonthetopandbottomofthechannel. bodyforceinsideofthechannelincreases,themomentumadditionsimilarlyincreases,whichwillinturnleadtohighervelocityowinthechannel.Assumingthatapowerlawrelationshipcanalsobeappliedtothemaximumvelocityexitingthechannel,umaxnactV (6)where,,andaretheempiricallyderivedconstants=3.9810)]TJ /F7 7.97 Tf 6.58 0 Td[(3,=0.307and=2.02fromapowerlawcurvet.Inthisrelationship,thevoltageVmustbeinkVpp,themaximumvelocityinm=s.Performingthesamepowerlawcurvetforthemassowperunitwidth,theconstantsare=1.8710)]TJ /F7 7.97 Tf 6.58 0 Td[(5,=0.352and=2.29. 140

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Figure6-6. Velocityandmassowmeasurementsatthechannelexit.A)MaximumvelocityandB)totalmassowperunitwidth(Q=w)oftheow0.5cmdownstreamofthechannelexit. 6.1.2PressureMeasurementsAlongerchannelwasalsoexamined(L1=45cm)inordertobettergagehowthepressurevarieswithinthechannelaroundanddownstreamoftheactuators.Pressuretapswereinstalledatnumerouslocationsinthechannel.Neartheactuators,tapswereinstalledonthesideofthechannelatthecenterlineinordertomeasurethepressure.Thecenterlinelocationwasselectedinordertoavoidanyeffectsofthewallnormalvelocityimpingingdownonthesurfaceofthechannel,whichcouldcreateleadtogreateruncertaintyinthepressuremeasurements.Downstreamoftheactuators,wherethesurfacenormalvelocityinducedbytheactuatorsisnotofsignicantconcern,pressuretapswereinstalledonthesurfaceofthechannel.AFurnessControlsModel332Differentialpressuretransmitterwasusedtotakemeasurements.512samplesweretakenateachlocationshownatarateof20samplespersecond.Again,a30secondwarmupperiodwasallowedbetweenthestartofplasmaactuationandthestartofdatacollection,anda180secondcooldownperiodaftereachpressuresampling.Thedifferentialpressurewasmeasuredagainstthepressureinthequiescentchamberlocatedapproximatelyonemeterawayfromtheplasmachannel.TheThompson-Tauoutlierremovalmethodwasappliedtoremovespuriousresults. 141

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Samplesareshownfor1,2,and3plasmaactuatorsrunningsimultaneously.Theseplasmaactuatorsarelocatedbetweentherstandsecond,secondandthird,andthirdandfourthpressuretaps.ItcanbeseeninFigure 6-7 thatthereisaninitialriseinpressureattheinletofthechannel,correspondingtoboththerstplasmaactuatorandanypotentialinleteffects.Withtheadditionofasecond(Figure 6-7 B)andthird(Figure 6-7 C)plasmaactuatordownstreamoftheinitialactuator,furtherincreasesinpressureeffectscanbeseentooccur.Asonemovesdownstreamoftheplasmaactuators,thepressurecanbeseentocontinuetoincreaseastheowdevelopsinthechannel.Fartherdownstream,thepressureisnallyseentoslowlydrop,indicatingthattheviscouseffectsofskinfrictionwiththechannelwallscreateanegativepressuregradient. Figure6-7. [Pressuremeasurementsalongthecenterlineandsurfaceoftheplasmachannelforavaryingnumberofactuators.A)1,B)2,andC)3actuators. Experimentswerealsoperformedwithascreenimpedingtheowofairthroughthechanneldownstreamoftheactuators,withthescreenlocatedatx=15cm.Thescreenwashexagonalinnatureandapproximately2.5cmthick,with3mmholesarrangesinahexagonalpattern.Thisscreenwasintendedtosteadytheow,andremoveanyvorticesintheow,buttheresultwasanearblockageofowthroughthechannel.However,pressuremeasurementswerestillrecordedwiththesamesamplingrateastheunimpededplasmachannel.Withthisimpedance,thepressurebuildsupatthe 142

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mesh,andthepressuredropprimarilyoccursacrossthemesh,withnosignicantchangesoccurringdownstreamofthemesh(Figure 6-8 ). Figure6-8. Pressuremeasurementsalongthecenterlineandsurfaceoftheplasmachannelwiththeadditionofanscreenimpedingthechannelow.A)1,B)2,andC)3actuators. Whilethesetwochannelowexperimentsshowsignicantdifferencesinthedifferentialpressurealongthelengthofthechannel,whenthemaximumpressuredifferentialwithintheinteriorofthechannel(thatismax(p))]TJ /F5 11.955 Tf 12.47 0 Td[(min(p))isexamined,astrikingresultisfound(Figure 6-9 ).Themaximumpressuredifferentialforthenumberandappliedvoltageoftheplasmaactuatorsisseentomatchupfairlywell.Thisindicatesthattheplasmaactuatorsaregeneratingthesamepressuredifferentialacrosseachactuator,independentlyofwhethertheowismovingdownstreamornotundertheselowvelocityconditions.Consideringthewellbehavednatureoftheserelationships,anempiricalmodeloftheexpectedpressurerisecanbedeveloped.Aproportionalincreaseinthepressurewithrespecttothenumberofplasmaactuatorsemployedisexpected.Powerlawrelationshipshavebeendrawnbetweenthethrustproduction,powerconsumptionandoperatingvoltageoftheplasmaactuator.Assuch,apowerlawrelationshipisexpectedwiththevoltage.ThisleadstotheapproximatecorrelationpmaxnactV (6) 143

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Figure6-9. Maximumpressuredifferentialmeasuredwithintheplasmachannel.Linesdenotepowerlawcurvetstothedata. whereandaretheempiricallyderivedconstants=1.2110)]TJ /F7 7.97 Tf 6.59 0 Td[(7and=5.57.Inthisrelationship,thevoltageVmustbeinkVpp,theresultingpressuredifferentialwillbeinPascals. 6.1.3DeviceEfciencyImplementingtheseplasmaactuatorsinaclosedenvironmentprovidesanopportunitytoevaluatetheefciencyofplasmaactuatorsinimpartingmomentumtouidinthechannel.Deningthisefciencyastheratioofhydrodynamicpowertotheinputelectricalpower=Pow Pin=Rh0(up)dA Pin (6)whichcanbeapproximatedintwodifferentways.Iftheaverageowisassumedtobeplug-like,uumax,thentheapproximationuumaxhpmax Pin=L (6)arises,whichcanalsobeconsideredanupperboundonthedeviceefciency.Foramoreaccurateapproximationusingthepowerandmassowrateperunitwidth,the 144

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efciencycanbeapproximatedasQQ=Lpmax Pin=L (6)Computingtheseefcienciesfromtheexperimentaldata(andusingapproximateddatabasedonthepowerlawcurvetswhennecessary),theefciencyoftheplasmachannelcanbeseentobelessthat0.1%(Figure 6-10 ).Thechannelefciencyappearstobehighlydependentontheoperatingvoltageandincreasesanorderofmagnitudeastheoperatingvoltageisincreasedfromalowtomoderatevalue. Figure6-10. Efciencyoftheplasmaactuation. Usingthedifferentempiricalpowerlawrelationshipsdevelopedforumax,pmaxandPin,theefciencyofthechannelcanalsobeputintermsofapowerlawrelationshipwithrespecttotheinputvoltage,suchthatQ)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(QnQactVQ)]TJ /F3 11.955 Tf 12.95 -9.69 Td[(pnactVp enactVe (6)u)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(unuactVu)]TJ /F3 11.955 Tf 12.96 -9.68 Td[(pnactVp enactVe (6)wherethevariouscoefcientsaredenotedbyQforthemassowrateperunitwidth,uforthemaximumvelocity,pforthemaximumpressureandefortheelectricalinput.Ignoringtheconstants,thereisanimportantpropotionalitytotheseefciency 145

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calculations.Q,u/VQ,u+p)]TJ /F12 7.97 Tf 6.58 0 Td[(e (6)Computingthisexponentbasedontheempiricalcurvets,Q+p)]TJ /F3 11.955 Tf 12.55 0 Td[(e=4.35andu+p)]TJ /F3 11.955 Tf 12.5 0 Td[(e=4.09,indicatingthattheseplasmadevicesbecomeincreasinglymoreefcientatimpartingmomentumtotheowastheoperatingvoltageisincreased.However,theupperrangeofoperatingvoltagesislimitedbytheeffectsofplasmastreamerformationanddielectricbreakdown( Thomasetal. 2009 ). 6.2ConclusionsTheresultsoftheseexperimentsandsimulationsshowthatwiththeapplicationofdielectricbarrierdischargeactuators,channelowsofuptoseveralmeterspersecondcanbegenerated.InusingDBDactuationtodrivethisow,theactuationiseffectivelyworkingasalowspeedpumpforsmallows.Adistinctdifferenceexists,asthistypeofpumpisinherentlytwo-dimensionalinnatureandcanbeextendedtoaverylargewidth,somethingwhichtraditionalcentrifugalandaxialpumpscannotdo.ResultsofparametricstudiesvaryingthenumberofDBDactuatorsoperatedinachannel,aswellastheoperatingvoltageappliedtotheseactuators,havebeendiscussed.Theseexperimentsindicateseveralconclusions.Theexitvelocities,maximumpressure,massux,anddifferentialacrossanarrayofactuatorstwelltoapowerlawrelationshipwithrespecttotheoperatingvoltage.Theserelationshipsindicatethatthereisanincreasingmarginofreturnforthemaximumvelocities,massux,andpressureriseduetotheactuatorswithrespecttovoltage.However,onlythepressuredifferentialcontinuestorisewithrespecttoinputelectricalpowerasthevoltageisincreased.Themaximumvelocitiesandmassuxperunitpowerinputdecreaseasthevoltageisincreased.Eachplasmaactuatorgeneratesthesamepressureincreaseacrossitssurface,independentlyofthebulkowinthechannel.Whiletheremaybesomeupperboundon 146

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thetotalpressuredifferentialthancanbecreatedinthechannel,themaximumpressureincreaseappearstohavealinearrelationshipwiththetotalnumberofactuatorsinthechannel.TheefciencyoftheDBDplasmachannelcaneasilybeinferredfromthemassowandmaximumpressuredifferentialinthechannel.Usingthepreviouslydevelopedpowerlawrelationshipsforthemassow,maximumpressuredifferential,andelectricalpowerinput,thechannelefciencyalsoappearstodisplayapowerlawrelationshipwithrespecttotheoperatingvoltage,withanexponentontheorderof4.09to4.35.Whiletheefciencyofthedevicesislow,thisexponentsuggeststhatlargeincreasesinefciencymaybepossible.Theseresultshavebeenpublishedintheresearchliterature( Riherd&Roy 2012b ). 147

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CHAPTER7FLOWSTRUCTUREOFPLASMADRIVENCHANNELFLOWSWhilethegeneralcharacteristicsofaplasmadrivenchannelareusefulforcertainapplicationsandinordertodenearangeofoperatingparametersforadeviceofthistype,thereareanumberofoutstandingquestionsregardingthecharacterizationandphysicalunderstandingofthisowitself,howmanipulatingtheowphysicscanbeusedtoimprovetheefciencyofthechannel,andhowitcomparestomorecanonicalchannelows.Withthelocalizationofthebodyforce,theassumptionsofaone-dimensionalowarenolongervalid,andissuesthatcannormallybeneglectedbecomeimportant.Severalstudieshavebeenperformedexaminingtheinternalstructureoftheseowsinthepast( Debiasi&Jiun-Ming 2011 ; Morgan&Visbal 2013 ; Riherd&Roy 2012b ),butthesehaveprimarilybeencomputationalinnature,andtherehasbeenlittleinthewayofexperimentalvalidationoftheowstructureinsideofthechannel.Theonlyexperimentalstudythatdidlookinsidethechannelisofquestionableaccuracyanddoesnotprovidemuchinthewayofdataorexplanationoftheow'seffectsaroundtheactuator( Debiasi&Jiun-Ming 2011 ).Moreindepthdescriptionsoftheoweldarepresentedin Morgan&Visbal ( 2013 ).AsdescribedinChapter 6 ,underquiescentconditionsandinasemi-boundeddomain,DBDactuatorsareknowntoproducewalljets.ThesewalljetshavebeenshowntomimictheGlauertwalljetsimilaritysolutionsufcientlyfardownstreamoftheplasmaactuator( Opaitsetal. 2010 ).Thebodyforcecreatedbytheplasmaactuatorentrainsuidfromupstreamofandabovetheactuatorrelativetothewalljet(Figure 7-1 A).Thisuidisthenconvectedbythewalljet,whichgraduallydecreasesinmagnitudewhileincreasinginitsheightnormaltothesurface.However,whenthedomainoftheplasmaactuationisbounded,suchasinachannelow,theregionsfromwhichuidisentrainedchanges(Figure 7-1 B).Intheboundeddomain,theuidentrainedbythewalljetislargelydrawnfromupstreamof 148

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A B Figure7-1. Comparisonoftheoweffectsforplasmaactuatorsindifferentdomains.A)Plasmaactuatorinansemi-unboundeddomain.B)Pairofoppositelyplacedplasmaactuatorsinachannel. thewalljet,thoughsomeisstillpulleddownwardstowardthesurface.Forapairofplasmaactuatorsplacedonoppositesidesofachannel,awalljetwillformnearbothoftheactuators.However,asthewalljetsexpandanddecelerate,theywillcoalescewitheachother,eventuallyformingachannelowsufcientlyfardownstreamoftheplasmaactuators.Asowispushedthroughthechannel,theplasmaactuatorsareeffectivelyactingasapumporafan,withthebodyforcereplacingtheeffectsofanymovingparts. 7.1ProblemDescriptionTherearetwogeometriesofplasmadrivenchannelsthatarecurrentlyunderinvestigation.Inoneconguration,plasmaactuatorsarelocatedonthebottomofthechannelataperiodicdistance,Lx.Thisgeometrywillbereferredtoasasingleconguration.Inthesecondconguration,therearesymmetricallyplacedpairsofplasmaactuatorslocatedperiodicallyalongthelengthofthechannelatadistanceofLx.Thisgeometrywillbereferredtoasadoubleconguration.ThesetwocongurationsareshowninFigure 7-2 .InChapter 6 ,itwasshownthatacrosseachplasmaactuator,thereisanincreaseinthepressureacrosseachplasmaactuator,aresultalsoindicatedinthesimulationsof Morgan&Visbal ( 2013 ).Thispressureincreaseisduetotheowresistingtheappliedbodyforcegeneratedbyeachofthesetsofplasmaactuators.Assuch,itwouldappearthatasthenumberofactuatorsincreasesandtheowreachesaperiodicmeanstate, 149

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A B Figure7-2. Comparisonofplasmachannelgeometries.Streamwisebodyforce(fx)forA)singleandB)double'geometryplasmadrivenchannelowsforLx=2. thepressureintheowdoesnotreachaperiodicstate,ratherthepressuregradientbecomesperiodic.Becauseofthis,anysimulationsofaplasmadrivenchannelowwithperiodicboundaryconditionswouldnecessarilyrequireameanpressuregradienttobeappliedinordertoaccountforthisinducedpressuregradient.Whilethismeanpressuregradientcouldbeestimated,basedonpreviousexperimentsorsimulations,itcannotbedeterminedexactlyapriori.Duetothisconstraint,anysimulationwouldbespeculativewithrespecttotheactualpressuregradientappliedtothechannelandtheinducedpressuregradient.Basedontheneedtohaveaclearlydenedpressuregradientappliedtothechannelow,thepresentworkfocusesonnitelengthchannelowswithinletsandoutlets,thoughitishopedthattheowreachesanearlyperiodicstateinsideofthechannelasitdevelops. 7.22DLaminarFlowCharacterizationAtsufcientlylowReynoldsnumbers,itisexpectedthattheseowsremainlaminar.Inordertoexaminethebehavioroftheseowsunderlaminarconditions,two-dimensionalsimulationsoftheseowshavebeenperformedusingthetwo-dimensionalvariantofthecodeFDL3DI(Appendix C ). 150

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7.2.1NumericalDetailsInsimulatingtheseplasmadrivenchannelows,channelsofxedlengthandheighthavebeendiscretizedonabodyttedmesh.Attheinletandoutlet,thesurfaceofthechannelisroundedinordertoproducesmoothmesh.Outsideoftheinletandoutletofthechannelow,themeshisextendedforafurtherdistanceinordertoallowfortheappropriatelevelofinowandoutowfromthechannel.AschematicofthedomainusedcanbeseeninFigure 7-3 Figure7-3. Basicschematicoftheowdomainusedforsimulationsoftheplasmadrivenchannelow. AReynoldsnumberof3300wasusedforthesesimulations,whichcorrespondstoacharacteristicvelocityof5m/sandachannelhalfheightof1cmusingairastheuid(=1.2kg/m3,=1.510)]TJ /F4 11.955 Tf 7.08 -4.34 Td[(5).Basedontheplasmamodelused,theintegratedbodyforceperunitlengthfromeachplasmaactuatorisequaltoDc39.3mN/m.Foramajorityofthesimulations,atimestepoft=0.005wasused(thoughtemporalresolutionstudieswereperformedinordertoestablishthatthiswasasufcientlylowtimestep).Severaldifferentmeshdensitiesweretestedaspartofa 151

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gridresolutionstudy,thedetailsofwhichareinTable 7-1 .Forthestreamwisemeshresolutionstudy,thetimestepwasalteredsothattheCFLnumberremainsthesame. Table7-1. Meshesusedforgridresolutionstudiesofthechannelgeometry. StudyMeshLxLyNx,totNx,channelNyxchannelywall StreamwiseresolutionFine-L562691063511010.010.00505Baseline562345531751010.020.00505Coarse-L562186717271010.040.00505 WallnormalresolutionFine-H562345531752010.020.00252Baseline562345531751010.020.00505Coarse-H56234553175510.020.01012 Basedonthesimulationsofasinglegeometrychannel,velocityprolesafter750unitsoftimeareshowninFigures 7-4 and 7-5 .Itshouldbestatedthatatthistime,theowisunsteady,butthatthereisstillrelativelygoodagreementbetweenthedifferentmeshresolutionsexamined.Assuch,itappearsthattheBaselinegridresolutionissufcientforthesesimulations. Figure7-4. VelocityprolesfromthestreamwisegridresolutionstudyatA)x=h=19.0andB)x=h=44.2unitsdownstreamfromthechannelinlet. Inordertoproperlycapturethefareldbehavioroftheplasmadrivenchannelow,adaptivespongeregionshavebeenappliedatadistancefromthechannelinletandoutlets.Thesespongeregionshavebeenimplementedinordertodampoutanyacousticwavesthataregeneratedbythechannelandpreventthemfromreectingoff 152

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Figure7-5. VelocityprolesfromthewallnormalgridresolutionstudyatA)x=h=19.0andB)x=h=44.2unitsdownstreamfromthechannelinlet. theboundaryconditions,whichcouldinterferewiththeowinsideofthechannelinanon-physicalmanner.However,astheupstreamanddownstreamconditionsontheowareunknown,thebasestateforthespongeregionisdeterminedusingatemporallter,asdescribedin Akerviketal. ( 2006 ).Themeshwasextendedradiallyfromtheinletandoutlet,alongwiththestartandendofthespongeregionhasalsobeenvaried.Basedontheaveragevelocityinthechannel,itappearsthatthedistancebetweentheinlet/outletandtheboundaryconditionsdoesplayaroleinthevelocityoftheowinsideofthechannel(Figure 7-6 ),butsolongasthedomainisextendedmorethan30unitsoflengthfromthechannelinlet/outlet,itwouldseemthattheow'sdependenceontheboundaryconditionsislimited. 7.2.2DescriptionoftheResultingFlowFieldsInordertoexaminetheseows,anumberofdifferentparametricstudieshavebeenperformed,varyingthemagnitudeoftheplasmabodyforceandtheheightofthechannel.Simulationsperformedoftheseowsshowaninteresting,butpredictableoweld.Neartheplasmaactuators,uidisentrainedtowardstheactuator.Thehighestvelocitiesarefoundneartheactuator.Betweenthetwogeometries,therearesomedistinctdifferences,affectingthestructureoftheowelds. 153

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Figure7-6. Averagevelocityinthechannelasafunctionoftimeandthedistancebetweenthechannelinlet/outletandboundaries.ThesinglechannelgeometryandDc=1.0areused. First,thesinglegeometryoweldisexamined.InstantaneousoweldscanbeseeninFigures 7-7 and 7-8 .Forthisow,itcanbeseenthatattheleadingedge,owseparationcanoccur,intheformofalaminarseparationbubble.Forsufcientlylargelevelsofplasmaactuation,thevelocityinthechannelbecomeshighenoughthattheselaminarseparationbubblescanbecomeunstable,periodicallyreleasingvorticesintothechannel,whichensuresthattheowbecomesandremainsunsteady.Betweentheleadingedgeofthechannelandtheplasmaactuators,theowdevelopsinapredictablemanner,withtheboundarylayersthickeningasitdevelops.Astheowapproachestherstplasmaactuator,uidisentrainedfromthecenterofthechannelandamajorityofthemomentumbecomesconcentratedonthesideofthechannelwheretheplasmaactuatorsarelocated,formingwhatisessentiallyawalljet.Forlowerlevelsofplasmaactuation,theowontheunactuatedsideofthechannelcanseparate,formingamassiveowseparationinthechannel.Asoneexaminestheowfartherandfartherdownstreaminthechannel,theamountofmomentumcontainedinthewalljetsincreasesasmoreandmoreuidisentrainedintothewalljetwitheachadditionalplasmaactuator,andthenegativevelocitiesintheowseparationregionincreaseinmagnitude.Thisiscausedbytheconuenceofaninducedpressuregradientbythe 154

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plasmaactuator,resistingthemovementofuidtravellinginthedownstreamdirection,withtheconcentrationofthemomentumonasinglesideofthechannel.Inthehighmomentumregion,neartheplasmaactuators,theowisabletoovercomethepressuregradient,butawayfromtheplasmaactuators,wheretheowvelocityisreduced,thepressuregradientistoostrongfortheowtocontinuetomovedownstream.Assuch,theowisforcedtomoveupstream.Forhigherlevelsofactuation,thiseffectismuchlesssignicant,andwhiletherearesmallowseparations,theydonotgenerateanegativemeanowvelocity.Anotherimportantfeatureoftheseoweldsathighervelocitiesistheonsetofinstabilitiesinthewalljets.Theseinstabilitiesgrowastheyareconvecteddownstream.Theyalsoreachasufcientlyhighmagnitudesuchthatsecondaryinstabilitiesmaybecomeimportant,whichcannotbecapturedinthesetwo-dimensionalsimulations. Figure7-7. Instantaneousviewofthetwo-dimensionalchannelowforDc=0.5inthesinglegeometry. 155

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Figure7-8. Instantaneousviewofthetwo-dimensionalchannelowforDc=2.0inthesinglegeometry. Inexaminingthechannelvelocityproles,theunsteadynatureoftheowbecomesmoreapparent(Figure 7-9 ).Forthehighestandthelowestlevelsofplasmaactuation,thewideextentoftheseparationbubblebecomesmoreapparent.Whilethisbubblecantakeupalargespanofthechannelheight,thevelocityoftheowinthereversalisonlyafewpercentofthevelocityinthewalljet.Thesevelocityprolesalsoindicatethatforlowerlevelsofactuation,thebubbleshouldbesteadierthanathigherlevels,thoughthiswillonlybeabletobeconrmedwiththeuseofthree-dimensionalsimulationsandexperimentalvalidation.Examiningtheseowsoveranextendedlengthoftime,theaveragevelocityinthechannelsuggeststhatthereisaverylowfrequency,quasi-periodiceffectoccuringinthesechannelswhenthevelocitybecomeshighenough,andthatthishasanenormousimpactonthemassuxthroughthechannel(Figure 7-10 ). 156

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Figure7-9. VelocityprolesfromthesinglechannelgeometryforA-E)Dc=0.25andF-J)Dc=2.0.Thesevelocityprolesaretakentwounitsoflength(2h=2cm)downstreamofeachplasmaactuator,i.e.A,F)x/h=19.0,B,G)x/h=25.3,C,H)x/h=31.6,D,I)x/h=37.9,andE,J)x/h=44.2. Examiningthesizeandshapeoftheowreversalmorecarefully,itcomestoseemthatthesizeandshapeoftheow'sreversalapproachasteadystateforallbutthehighestlevelofactuationthatwassimulated(andperhapsthesecondhighestlevelonamustlongertimescale).Forthehighestlevelofactuation,theowseparationstartsoffasseveralsmallowseparations,whichextendinlengthastheyareslowlyconvecteddownstream(Figure 7-12 ).Theratethattheseseparationsaretransporteddownstreamislessthantheratethattheygrowat,allowingfortheseparationbubblestojoineachother,forminglargerowseparations.Theseeventuallyformoneenourmousseparationbubble,chokingtherateofowinthechannel.However,oncetheseparationbubble 157

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Figure7-10. AveragevelocityinthechannelwithasinglegeometryplasmaactuatorA)asafunctionoftimeandB)AveragevelocityinthechannelnormalizedbyDc.TheverticaldashedlinesindicatethetimessampledforFigure 7-12 reachesacertainsize,itburstsandisrapidlytransporteddownstreamandpassedthroughtheoutlet,allowingforanincreasedrateofowinthechannel.Itlogicallyfollowsthattheformationandburstingofthemassiveseparationbubbleinthechannelistherootofthequasi-periodicpulsinginthechannelow. Figure7-11. BoundsontheowseparationforlowerlevelsofplasmaactuationwiththesinglegeometryandDc=0.25,0.50,1.00. Forthedoublegeometrychannels,theowappearstobemuchsimpler.InstantaneousoweldcanbeseeninFigures 7-13 and 7-14 .Themeanvelocitiesinthechannelareroughlydoublewhattheyareforthesinglegeometrychannels,andtheeffectsoftheleadingedgeseparationbubblearemoresignicantbecauseofthat.ThevelocityoftheowinthewalljetsisalsoofcomparablemagnitudeforaxedvalueofDcwiththesinglegeometrychannelowcases,eventhoughthemeanvelocityinthechannel 158

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Figure7-12. InstantaneousboundsontheowseparationforthechannelwiththesinglegeometryandDc=2.0. isroughlydoublewhenactuatorsareplacedonbothsidesofthechannel.Asuidapproachestheregionwhereplasmaactuatorsareplacedonbothsidesofthechannel,insteadofbeingentrainedtoasinglesideofthechannel,uidisnowentrainedtobothsidesofthechannel.Thus,oneachwall,thereexistsahighmomentumregion,whiletheowatthecenterofthechannelisoflesservelocity.However,whilethevelocityatthecenterlineofthechannelisreduced,itdoesnotbecomenegativeandformaowseparationinthecenterofthechannel.Theseeffectsofthewalljetdevelopmentandentrainmentofmomentumcanbemoreclearlyseeninthechannelvelocityproles(Figure 7-15 ).Theseprolesshowthatasonefollowstheowpastadditionalplasmaactuators,moreandmoreuidandmomentumisentrainedinthewalljets,whicharegrowinginheightandvelocitywitheachadditionalplasmaactuator.Theendresultofthisisthatverylitttlemomentumexistsatthecenterofthechannelaroundtheactuators.Examiningtheaveragevelocityinthechannelowwithactuatorsonbothsidesoftheow(Figure 7-16 ),itcanbeseenthatoncetheowreachesitsmaximum,therearenosignicantoscillationsintheaveragevelocityinthechannel.Whilesmallscaleoscillationspersist,thereisnogenerationandejectionofamassiveowseparationasthereisforthesinglegeometrychannel(Figures 7-10 and 7-12 ). 159

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Figure7-13. Instantaneousviewofthetwo-dimensionalchannelowforDc=0.5inthedoublegeometry. 7.2.2.1EffectsofheightinthechannelChannelswithaheightof1.0cmhavealsobeenexamined.Theemphasishereisonexaminingwhathappensastheseplasmaactuatorsarepushedmoretowardssmallerscalepumpingapplications.Oneresultofreducingthechannelheightisthatthevelocitiesandunsteadinessinthechanneldecreases(Figure 7-17 AandC).Thiscanlargelybeattributedtotheowbecomingrelativelymoreviscouswiththereductioninthechannelheight.Asthechannelheightisdecreased,viscouseffectsbecomeincreasinglymoreimportantandacttorestrictthemovementofuidinthechannelanddampoutanyperturbations.Withthereducedchannelvelocities,theeffectiveReynoldsnumberislessthat1500atmost,andinthehundredsinforsomeofthecases.Thiswouldsuggestaverylowspeed,laminarow,whichwouldn'tnecessarilyrequirefurtherthree-dimensionalmodelingto 160

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Figure7-14. Instantaneousviewofthetwo-dimensionalchannelowforDc=2.0inthedoublegeometry. accuratelydescribethebehaviorinsideofthechannel.ThereducedmarginalgainsinvelocitycanalsobeseeninFigure 7-17 BandD,indicatingthatjustaswiththetallerchannel,asmorebodyforceisappliedtothechannel,thevelocitiesdonotincreaseproportionally.Thegeneralbehavioroftheowinsideofthechannellargelyremainscomparable.Largescaleowreversaloccursforgeometrieswhereplasmaactuatorsareonlylocatedononesideofthechannelbutdoesn'toccurwhenplasmaactuatorsarelocatedonbothsidesofthechannel.Onesignicantdifferenceisthatthegeometrieswithreducedchannelheightsdonoseetheformationofalaminarseparationbubbleatthechannelinlet.Thisowfeatureisasignicantcontributortohighfrequencyunsteadinessinthechannel.Anotherbigdifferencebetweentheshorterandtallersinglechannelgeometriesisthesizeandstabilityoftheseparationbubblesaroundtheactuators.For 161

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Figure7-15. VelocityprolesfromthedoublechannelgeometryforA-E)Dc=0.25andF-J)Dc=2.0.Thesevelocityprolesaretakentwounitsoflength(2h=2cm)downstreamofeachplasmaactuator,i.e.A,F)x/h=19.0,B,G)x/h=25.3,C,H)x/h=31.6,D,I)x/h=37.9,andE,J)x/h=44.2. theshorterchannelgeometries,thedevelopmentlengthbetweenanddownstreamoftheactuatorsisdoublethatoftallerchannels.Assuch,theviscouseffectsintheowaregivenadditionalspacetotakeeffect,andinsteadofrelyingonunsteadinessandinstabilitiestoreattachtheow,viscouseffectscanaccomplishthattaskinstead.Theendresultisthattheseparationbubblesdon'textendasfardownthelengthofthechannelwhenthechannelheightisreduced. 7.3ExperimentalValidationInordertoensurethataccurateowphysicsisbeingcapturedbythesimulationsdescribedinthischapter,experimentalvalidationisalsonecessary.AsinChapter 6 ,PIV 162

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Figure7-16. AveragevelocityinthechannelwithadoublegeometryplasmaactuatorA)asafunctionoftimeandB)AveragevelocityinthechannelnormalizedbyDc. wasusedtoexaminetheoweldsgeneratedbytheplasmaactuators.Thepresentemphasisisplacedontheowinsideofthechannels,though. 7.3.1CharacterizationoftheDBDPlasmaActuatorsBeforemovingontotheentirechannelow,theperformanceoftheplasmaactuatoritselfshouldbecharacterized.Aplasmaactuatorconstructedoutof3mmthickPMMAandtwo10mmwideelectrodeswasmade.Zerogapwasusedbetweentheelectrodesinthestreamwisedirection.Ahighvoltage,highfrequency(14kHz),ACsignalwasappliedtotheactuator,andtheresultingoweldswerecollectedusingPIV(Figure 7-18 ).Intheexperiments,voltagesbetween14and20kVpparetested.Thesamevoltagerangewastestedforthecharacterizationoftheplasmaactuators.Theresultingowsproducewalljetswithvelocitiesbetween1and5m/s. 7.3.2ConstructionoftheChannelExperimentForthepresentexperiments,anexperimentalsetuphasbeenconstructed(Figure 7-19 ),whichisdifferentfromtheoneforusedinChapter 6 .Allofthepartsoftheexperimentalsetuparemadeof6.35mmthick(1/4)PMMAandaredesignedtoberemovableandinterchangeable.Thesidewallsconnectallofthepieces,andsidewalls 163

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Figure7-17. Averagevelocitiesintheshortenedchannelasafunctionoftime.uavgintheA)singleandC)doublegeometrieschannels.uavg=DcintheB)singleandD)doublegeometrychannels. ofvariousheightshavebeenmanufacturedinordertovarythechannelheight.Theinletandoutletsegmentsareseparatefromtheportionofthechannelwheretheactuatorsarelocated.Inordertodrivetheow,theactuatorsareinstalledona30cmplate,whichliesbetweentheinletandoutletsegments.Allofthesepiecesareattachedtothesidewallsusinghotglue,whichdoesnotgenerateanopticalimpedancealongtheentirethesidewalls,onlyinlocalregions,allowingforPIVdatatobetakeninsideofthechannel.TheexperimentsareperformedinthesamechamberasdescribedinChapter 6 164

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Figure7-18. WalljetsformedbytheDBDplasmaactuatorunderquiescentconditionsatvaryingvoltages.A)14kVpp,B)16kVpp,C)18kVpp,andD)20kVpp. A B Figure7-19. Photographsoftheplasmachannel.A)Assembledplasmachannelexperiment.B)Close-upofthesegmentoftheplasmachannelwheretheactuatorsarelocated. 7.3.3DetailsoftheExperimentalMethodTheseexperimentswereconductedmultipletimesduetounforeseendifculties,butwitheachadditionaliteration,improvementsweremadeinthedesignandconstruction 165

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ofthechannelsandtheexecutionoftheexperiments.Amongthedifcultiesencounteredinconstructingandtestingthesechannelswere 1. Overthecourseofrepeatedexperiments,thesidewallsofthechannelscanbedegradedduetointeractionswiththeplasmaactuators.Thisdeformationpreventsthecollectionofqualitydatainsidearoundtheplasmaregion,whichisofprimaryinterest.Inordertocorrectthis,theplasmaactuatorsweremovedawayfromthewallssothatthereare2to3mmofspacebetweentheendoftheplasmaandthesidewalls,whichlessenedthedegreeofopticaldegradation. 2. Whilemovingtheplasmaactuatorsslightlyawayfromthesidewallspreventedthepreventedsomephysicalandopticaldegradation,theplasmaactuatorswerestillabletodegradetheopticalqualityofthePMMAsidewallsbycausingaresiduetoformaroundtheactuator.Itwasfoundthroughexperimentationthatwipingdownthesidewallswithadampragbetweentestscouldremovemuch,butnotalloftheresideandallowforimproved,butnotperfectdatacollection. 3. Overextendedperiodsoftesting,theOndinaseedmaterialusedforthePIVdatacollectionwouldbuilduponthesurfaceoftheplasmachannel,particularlyonthebottomsurfaceofthechannel.Thebuildupoftheseedmaterialcanleadtoincreasedreectionsoffofthechannelsurfacesandreducedactuatorperformance.Inordertominimizetheseeffects,thechannelsurfaceswerewipeddownwithadampragbetweentests.Inordertodothis,thetopofthechannelwasmaderemovableandnotattachedwithanytypeofsemi-permanentadhesive.Instead,theendsofthechannelweretapedon,andaheavyobjectusedtoholditinplace. 4. Thesedevicesarehandmadeandthereissomeerrorassociatedwiththeconstructionintheplasmaactuators.Furthermore,theappliedvoltagedoesvaryfromsampletosampleandevenwithineachsample.Theerrorassociatedwiththesevoltagevariationsisapproximately0.2kVpp,whichisapproximately1-2%oftheappliedvoltage.Thesetwofactorsleadtoacertaindegreeofinconsistencyinoperatingtheactuators.Inperformingtheseexperiments,theplasmaactuatorswererunfor30secondsinordertowarmupandestablishastatisticallysteadyowinthechannel.500imagepairswerecollectedoveratimeof33secondsatarateof15Hz.Theplasmaactuatorswerethenturnedoffforaminimumof90secondsbeforemovingontothenexttest.Often,theplasmaactuatorswereallowedtocooloffforagreateramountoftimeinorderfortheamplierpoweringthecircuittocooldownaswell. 166

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Diagramsofthecircuitpoweringthesystem(Figure 6-2 )andthesetupofthePIVsystem(Figure 6-4 )canbefoundinChapter 6 .Datawasonlycollectedalongapartiallengthofthechannel,intheregionoftheplasmaactuators(13.5cmx41.5cm),butnotaroundtheinletoroutlet.Inordertoaccuratelycapturedataalongthislength,thechannelwasplacedonamovablehorizontalrail,whilethelaserandcamerawereputinstationarypositions,allowingforthechanneltomovewithouthavingtorecalibrateandre-focusthelaser/camerasetupforPIV. 7.3.4PIVResults 7.3.4.1SinglegeometrychannelsFromthenumericalsimulationsdiscussedearlierinthischapter,thetwomostimportantowfeaturesinthesinglegeometrychannelarethegenerationofawalljetthatincreasesinheightandvelocityasitdevelopsdownstreamwithadditionalplasmaactuatorsandtheexistenceofaowseparationontheoppositesideofthechannel.InexaminingthetimemeanoweldscapturedusingPIV(Figures 7-24 7-25 ,and 7-26 ),itcanbeseenthatthereissomequalitativeagreementwiththesimulationsthathavebeenperformed. Figure7-20. Timemeanstreamwisevelocityforthesinglegeometryplasmaactuatorsat16kVpp.ThewhitelineinA)andB)representswheretheowseparationoccurs. 167

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Figure7-21. Timemeanstreamwisevelocityforthesinglegeometryplasmaactuatorsat18kVpp. Figure7-22. Timemeanstreamwisevelocityforthesinglegeometryplasmaactuatorsat20kVpp. Itcanbeseenthatthewalljetsforminthevicinityoftheplasmaactuators,andthatthesizeandmagnitudeofthesewalljetsincreasesastheydevelopovereachadditionalactuator.Theeffectsofuidentrainmentarealsovisibleduetothisincreasinglysignicantwalljet,whichleadstolowervelocitiesontheoppositesideofthechannel,consistentwiththeCFDsimulations.Onesignicantdifferencebetweenthesimulationsistheexistenceofthemassiveseparationbubbles.InFigure 7-20 ,itcanbeseenthatthereisasmallseparation 168

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bubbleinthetimemeanow.However,thisisincontrastwiththesimulations,whichpredictmuchmorerobustseparationbubbles,andbubblesthatexistathigherandlowervelocities.TheDc=1.0generateswalljetvelocitiescomparabletoexperimentswithanappliedvoltageof16kVpp.Inthosesimulations,thereisarelativelysteady,verylarge,separationbubble.Intheexperiment,thereismuchmoreunsteadiness,andtheremaybeseveraldifferentreasonsforthis.ItcanbeseeninthetimemeanPIVdata(Figure 7-20 A)fromaroundanactuator,thattheowisonaveragenotreversed.WhentheinstantaneousPIVdataisexamined(Figure 7-20 C-F),thereisaveryunsteadyowreversalnearthesurfaceawayfromtheplasmaactuators.However,becauseoftheseoscillations,theseparationdoesn'tseemtoremainconstant,andonceaveraged,leadtoaforwardow,ratherthanareversedone. 7.3.4.2DoublegeometrychannelsThenumericalsimulationspresentedearlierinthischaptersuggestthathedoublegeometrychannelcongurationshouldbecomprisedoftwosymmetricwalljetsthatgrowinheightandvelocityastheydevelopdownstreamwithadditionalplasmaactuators.InexaminingthetimemeanoweldscapturedusingPIV(Figures 7-24 7-25 ,and 7-26 ),itcanbeseenthatthereisreasonablequalitativeagreementwiththesimulationsthathavebeenperformed.Forallofthesesimulations,walljetsofcomparablemagnitudeformoneachsideofthechannel.Thesizeandmagnitudeofthewalljetsalsoincreasesastheydevelopdownstream.Itcanalsobeseenthatuidisentrainedonbothsidesfromthecenterofthechannelintothewalljets,bythereductionofvelocitiesinthecenterofthechannel. 7.4ConclusionsTheinternalstructureofplasmadrivenchannelowswithtwoactuatorgeometrieshasbeeninvestigatedusingnumericallyandexperimentallyusingPIV.Thedifferent 169

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Figure7-23. Behavioroftheowneartheaplasmaactuatorat16kVpp.A)Timemeanstreamwisevelocity.B)RMSvaluesofthestreamwisevelocity.C-F)Instantaneousvelocitiestakenatdifferentinstancesintime. approachesyieldvaryingresults,butthereisadegreeofcommonalitybetweentheresultingowelds.Two-dimensionalsimulationspredictthattheowsgeneratedbythechannelsshouldberelativelysteady,exceptforathigherlevelsofplasmaforcing.Whenactuatorsareplacedononesideofthechannel,theowonthatsideformsasingle,longwalljet.Thiswalljetgrowsinheightandvelocityasitdevelopsalongeachadditionalplasmaactuator,suchthatamajorityofthemomentuminthechannelendsupbeinginthewall 170

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Figure7-24. Timemeanstreamwisevelocityforthedoublegeometryplasmaactuatorsat16kVpp. Figure7-25. Timemeanstreamwisevelocityforthedoublegeometryplasmaactuatorsat18kVpp. jet.Inthesesimulations,theowontheoppositesideofthechannelusuallyseparates,sothataverylong,vorticalseparationbubbleformsinthechannel,whichmayormaynotbesteadyorstable.Thisowseparationisduetothepressuregradientinducedbytheplasmaactuators.Astheplasmabodyforcepushesowdownstream,thedownstreampressureisincreasedrelativetotheupstreampressure.Whilethiseffecthasbeennotedforplasmaactuatorsinsemi-boundeddomainsbefore,itsimpactinboundarydomainshasonlybeenexaminedrecently( Morgan&Visbal 2013 ; Riherd 171

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Figure7-26. Timemeanstreamwisevelocityforthedoublegeometryplasmaactuatorsat20kVpp. &Roy 2012b ).Thisinducedpressuregradientpushestheowbackwards,andinthecongurationwhereplasmaactuatorsareonlyplacedononesideofthechannel,thisbackwardspushmaybeabletocauseowseparation.Furtherexperimentaleffortstovalidatetheseeffectshavesuggestthattheowseparationislessimportantwhentheoperatingenvironmentismoreunsteady.Theunsteadinessintheseadditionalexaminationsappearstolessentheeffectsoftheowseparation,andwhilethereappearstobeinstantaneousamountsofowseparation,thelargescaleowseparationseeninthetwo-dimensionalsimulationsdoesnotappeartobevalid.Whenplasmaactuatorsareplacedonbothsidesofthechannel,amuchmorefavorableowstructureresults.Walljetsformonbothsidesofthechannel,whichentrainamajorityofthemomentumcontainedintheow.Forthisconguration,noinstantaneousormeanowseparationsareencounteredintheregionoftheplasmaactuators.Withthisincreasedpredictabilityintheowstructure,thiscongurationwilllikelybepreferredforanyfuturestudiesorapplications.Continuedworkonthistopicshouldfocusonincreasinglysmallchannelsandinthedevelopmentofadditionalapplicationsforthesechannels.Theboundarylayercontrolapplicationdevelopedby Morgan&Visbal ( 2013 )seemspromising,butfurther 172

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developmentofthisapplicationwillbenecessaryinlowspeedandlaboratorysettingsbeforemorerealisticapplicationscanbeapproached.However,itislikelythatthistypeofapplicationcanbeextendedtomimicthebehaviorofother,moretraditionalowcontrolactuatorsaswell.Asthesedevicesaredevelopedatsmallerandsmallerlengthscales,theeffectsofunsteadinessshouldbecomelessenedduetoareducedReynoldsnumberforthechannel. 173

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CHAPTER8SUMMARYANDRECOMMENDATIONSFORFUTUREWORKSeveralaspectsofusingDBDplasmaactuatorstocontrolboundarylayerandchannelowshavebeeninvestigated.Theintendedimpactofthesestudieshasbeentoexpandtheversatilityofthesedevicesacrossarangeofpotentialapplicationsaswellastoquantifytheimpactoftheplasmaactuatorsfortheseapplications. 8.1ControlofBoundaryLayerStabilityLocalandbi-globalhydrodynamicstabilitycodeshavebeenwrittentoquantifythestabilityofincompressibleows.Forthebi-globalstabilitycode,aparallelimplementationofthecodehasbeenperformed,allowingformuchlargerdomainstobeexaminedthancouldbeonasingleprocessor.Thesecodeshavebeenbenchmarkedandfoundtobeaccuraterelativetootherhydrodynamicstabilitycodes.Parametricstudiesofthelocalstabilityanalysisofaplasmamodiedboundarylayerhavebeenperformed.Fromsimulationsoftheboundarylayerwithplasmaactuatorsoperatinginaow-wise(co-ow)manner,ithasbeendeterminedthattheadditionofmomentumintotheboundarylayerleadstochangesintheboundarylayerproles.Thelocalstabilityanalysisperformedindicatesthatthechangestotheshapeoftheboundarylayerprolesaretheprimarystabilizingeffect,thoughtheremaybesomesmallimpactduetoboundarylayerheightreduction.Furthermore,alowordermodeloftheboundarylayerproleshasbeenconstructedandvalidatedincomparisontothecomputationalresults.Thismodelallowsfortheexaminationofowswheretheplasmaactuatorisoperatedinaco-oworacounterowmanner.Theevidencesuggeststhatthereareadditionalinstabilitymechanismspresent,includingpreviouslyundocumentedinviscidandabsoluteinstabilityprocesses.Bi-globalstabilityanalyseshavebeenperformedexaminingtheeffectsofusingcontinuouslyoperatedco-oworientedplasmaactuatorstostabilizeTSwavesandboundarylayerstreaks.Flowstabilizationisindicatedusingthismannerofstability 174

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analysisoverabroadrangeofthefrequencyspectrafortheTSwavesandoverasignicantrangeofthespanwisewavenumberspectrafortheboundarylayerstreaks.ARANS-typeanalysisoftheperturbation'skineticenergyindicatesthattheowisstabilizedinthenearplasmaregion.Interestingly,asthemagnitudeoftheplasmaactuationisincreased,theapplicationofaplasmabodyforceleadstonegativekineticenergyproductionterm.LinearstabilityanalysissuggestthatTSwavescanbedampedbyupwardsof90%oftheirkineticenergy,butboundarylayerstreakscanlikelyonlybedampedbyupto25%oftheirkineticenergy.WhilethestabilizationoftheTSwavesismuchmorepronouncedthatoftheboundarylayerstreaks,thestabilizationforbothoftheseinstabilitiesissignicant.FutureworkonthistopicshouldplaceemphasisonhowthisproblemscalesupathigherReynoldsnumbers.Aconcertedfocusonexperimentalvalidationwouldalsobeverybenecial,astheexistingliteratureislimitedwithrespecttostudiesfocusingontheindividualinstabilities.Iftheseactuatorscanbedemonstratedtobeeffectiveatcontrollingallofthedifferentpathstoturbulenceinboundarylayers(alongwiththeadditionalrequirementsofbeingrobustagainstenvironmentalconcerns,reliability,powerconsumption,etc.),thentheycouldhaveasignicantimpactforreducingturbulentskinfrictionforthoseapplications.Thereshouldalsobeapushintotheimplementationofthesedevicesforrealapplications,andtheunderstandingsofthelimitsinimplementation.Thepresentanalyseshavefocusedontheeffectsofasingle,innitelylongactuator,butinpractice,arraysofnitewidthactuatorsaremorelikelytobeused.Thelimitationsofsuchimplementationwillneedtobeunderstoodforpracticalapplications.Anotherpotentialavenueforapplicationsisinthemodicationofunsteadyowfeaturesinturbulentboundarylayers.Asplasmaactuatorshavebeenshowntomodifywhatisessentiallyaturbulenceproductiontermfordifferentlinearizedperturbations, 175

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itspossiblethatDBDactuatorscoulddothesameinahighlynon-linear,turbulentboundarylayer.Areductionintheunsteadyowfeatureswouldimpactthe u0v0Reynoldsstressinturbulentows,whichcoulddirectlyimpacttheboundarylayerprolesandtheviscousskinfriction.Inorderforthistypeoftechnologytomovefromthelaboratorysettingtomoreimpactfulapplications,moreworkinexperimentaltestingwillbenecessary,inordertoaddresstheconcernsofreliability,robustness,andpowerconsumptionthatfacethesedevices. 8.2SerpentineGeometryPlasmaActuatorsSimulationsofserpentinegeometryplasmaactuatorsinalaminarboundarylayerhavealsobeenperformed.Characterizationoftheseowsindicatesthattheyareabletogenerateboundarylayerstreaksandcouldbeusedfortransitioncontrol.Dependingonthemagnitudeoftheactuation,theseactuatorsmaybeusedtoeitherdelayoracceleratethetransitionoftheow.Whiletheapplicationofusingboundarylayerstreaksforowcontrolapplicationsisbynomeansanewcontribution,thatthiscouldbedoneusingDBDplasmaactuatorsallowsforgreatereaseandinconstructingowcontrolsystems.Normallythesestreaksaregeneratedbyxedstructuresonanairfoilortheroofofacar.Assuch,thesexeddevicescanbeoptimizedforasinglecondition,offeringreducedperformanceoverthewiderrangeofconditionswherecontrolis(orisnot)desired.Plasmaactuatorsallowforgreaterexibilityintheiroperation,astheycanbeturnedoffwhennotneededforcontrol,andtheirperformancecanbemodulatedwhentheyaredesired.Thisexibilityindicatesthattheseactuatorshavegreaterversatilitythanmoretraditionalmeansofgeneratingboundarylayerstreaks.Thepresentworkhasonlyexaminedtheimpactofusingtheseactuatorstogenerateboundarylayerstreaksinalaminarboundarylayer,butboundarylayerstreaksarealsoverycommoninturbulentboundarylayers,particularlyintheviscoussublayer. 176

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Theintroductionofstreaksbyaserpentinegeometryactuatorintotheviscoussublayercouldbeusedtomodifythestructureoftheowinthenearwallregion.Furthermore,iftheseactuatorswereoperatedinapulsedmanner,itmaybepossibletogenerateorganizedarraysofhairpinvorticesforthepurposesofowcontrolandmodulatingthemixingpropertiesinaboundarylayer.Bothoftheseapproachescouldbemodiedtotakeadvantageoflinearandnon-linearamplicationoftheseperturbationsbytheow,allowingforincreasedcontrolauthorityforeachunitofpowerdeliveredtotheactuators.Goingforward,thisactuatorgeometryshouldbeusedtoextendthecontrolauthorityDBDactuatorstoincreasinglymoredifcultowcontrolproblems,butforthemostdifcultgoalstobemet,theseactuatorsmustbeusedindeliberateandintentionalwaystooptimizetheeffectsthattheyintroducetotheow.Sloppyimplementationofthesedevicesmaybemoredetrimentalthanbenecialtocontrolefforts.Theimpactofthree-dimensionalperturbationshasnotyetbeenfullyrealized,andserpentinegeometryplasmaactuatorsareaexibleandversatiletoolforfutureapplications. 8.3PlasmaDrivenChannelFlowsExperimentsusingplasmaactuatorstodriveachannelowhaveshownthatitispossibletodrivealowspeedchannelowuptoseveralmeterspersecondusingplasmaactuators.Functionalrelationshipsbetweentheoperatingvoltage,owvelocity,andpressuredifferentialinthechannelhaveallbeendeveloped.Thesefunctionalrelationshipshavebeenusedtodetermineasimilarrelationshipfortheefciencyoftheplasmaactuatortodrivetheowasafunctionofthevoltage.Basedontheseresults,theplasmaactuatorsbecomemoreefcientastheyareoperatedatahighervoltage.Numericalsimulationsandexperimentshavealsobeenperformedexaminingtheinternalstructureofthesechannelows.Itisfoundthattheplasmaactuatorscauseamajorityofthemomentuminthechanneltobeentrainedintowall-jetfeatureswherevertheplasmaactuatorsarelocatedatintheoweld.Ifplasmaactuatorsareonlylocated 177

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ononesideofthechannelow,itispossiblethattheowseparatesinamassivewayontheoppositesideofthechannel.Futureworkonthistopicshouldaimtoimplementthesedevicesforowcontrolapplications.Someresearchtothiseffecthasbeenperformedby Morgan&Visbal ( 2013 ).Therearealsoongoingeffortstousethistypeofplasmadrivenchannelowtoreduceheattransfereffectsincoldaircurtains(suchasthoseinopenairrefrigerators).Theremayalsobeapplicationsatincreasinglysmallerscales(mtocm)foruidandheattransferpurposes,wheretraditionalfansandblowerswouldnotbeaseffective. 178

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APPENDIXAMODELOFTHELOCALTEMPORALSTABILITYANALYSISLocallinearstabilitytheorycanbeusedtopredicttheexistenceandgrowthratesofinstabilitiesthatmaymanifestinatheboundarylayerorchannelow,supposingthatcertainassumptionsregardinghowparalleltheowisandhowslowlytheowcontinuestodevelop.Thoughtemporalinstabilitiesareexaminedhere,usingtheGastertransformationspatialinstabilitiescouldbesimilarlyapproximated( Gaster 1962 ),orbyreducingthespatialorderofthegoverningequationsusingauxillaryequationsasdescribedin Schmid&Henningson ( 2000 ). A.1EigenvalueProblemFormulationThelinearizedNavier-Stokesequations(Equation 2 )canbesimpliedusingtheassumptionsofafullydeveloped,parallelow,withwavelikeperturbations,suchthatitcanbeformulatedasageneralizedeigenvalueproblem iu0+@v0 @y+iw0=0 (Aa)iuu0+v0@u @y+ip0)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re)]TJ /F3 11.955 Tf 9.3 0 Td[(2u0+@2u0 @y2)]TJ /F3 11.955 Tf 11.96 0 Td[(2u0=i!u0 (Ab)iuv0+@p0 @y)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re)]TJ /F3 11.955 Tf 9.3 0 Td[(2v0+@2v0 @y2)]TJ /F3 11.955 Tf 11.95 0 Td[(2v=i!v0 (Ac)iuw0+ip0)]TJ /F4 11.955 Tf 17.21 8.09 Td[(1 Re)]TJ /F3 11.955 Tf 9.3 0 Td[(2w0+@2w0 @y2)]TJ /F3 11.955 Tf 11.95 0 Td[(2w=i!w0 (Ad)Thisformulationonlyaccommodatesone-dimensionalowvelocityproles,whichcanbedescribedatu=u(y).Thephysicsofmorecomplexowsisentirelyneglectedbythisformulation.FurthersimplicationoftheseequationswouldresultintheOrr-Sommerfeld-Squireequations(Equation 2 ).However,thismoreextendedformulationisused,asitissimplertoimplementduetothelowerorderderivativesandeasierimplementationofboundaryconditions.Forthepresentcomputations,thissetofequations(Equations A )wasthendiscretizedontoauniformstaggeredmesh( Patankar 1980 ).A4thorderaccurate, 179

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FigureA-1. Gridconvergenceofthenumericalmethodascomparedto Orszag ( 1971 )forPoisseuilleow.Re=10,000,=1,=0,!=0.23752649+i0.00373967. centered,nitedifferencestencilwasusedforthedifferencingoveramajorityofthedomain.A2ndorderaccurate,centered,nitedifferenceschemewasusedneartheboundaries.ThecodewasbenchmarkedagainstthatofthePoisseuilleowasdiscussedby Orszag ( 1971 ).TheReynoldsnumberofthiscaseisRe=10,000.Thewavenumbersare=1and=0.Thesingleunstableeigenvalueis!=0.23752649+i0.00373967,andisconsideredaccuratetowithin8signicantdigits.Thecalculatedvalueofthiseigenvaluewasusedtodeterminetheorderofaccuracyofthepresentcode.Theorderofaccuracyofthecodehasbeenfoundtobesomewherebetween2and4(Figure A-1 ).Theasymptoticregionisreachedrelativelyquicklyforthisproblem. A.2TransientStabilityAnalysisWhilethestraightforwardeigenvalueanalysisallowsforaclearunderstandingofthebehaviorofgrowthofinnitesimallysmallperturbationsintheverylongtimescales,itisalsobenecialtounderstandhowperturbationsgrowonmuchshortertimescales.Fortheseconditions,atransientanalysis,examiningthealgebraicgrowthofperturbationsmustbeperformed.Theoriginsofatransientanalysisliewith Orr ( 1907 ),butthepresentformulationforaviscousowislargelyadaptedfrom Farrell ( 1988 ). 180

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Inthetransientanalysis,wearemostinterestedinthegrowthofthekineticenergyoftheperturbations,thatisu0u0+v0v0+w0w0.Wecandescribetheseperturbationsasasumoftheeigenmodessuchthat 266664u0v0w0377775=Xjcj266664u0v0w0377775exp()]TJ /F5 11.955 Tf 9.3 0 Td[(i!t) (Aa)=Ecexp()]TJ /F5 11.955 Tf 9.3 0 Td[(i!t) (Ab)wherethematrixEcontainsalloftheeigenvectors,andthematrixccontainsthevaluesofthecoefcients.ThematrixA0isdenedasA0=EEAttimet=0,thekineticenergyhassomevaluesuchthatK(t=0)=(Ec)(Ec)=cEEc=cA0c (A)andattimet,thekineticenergyhassomevaluesuchthatK(t)=cAtc (A)whereAtrepresentsthemultiplicationoftheeigenmodesasthey'vebeenprogressedintime.Moreexplicitly,At=exp()]TJ /F5 11.955 Tf 9.29 0 Td[(i!t)EEexp()]TJ /F5 11.955 Tf 9.3 0 Td[(i!t).Fromthis,itcanbeshownusingvariationalmethodsthatAtc+A0c=0 (A)whererepresentsthefactorofenergygrowthofaperturbation,denedasthesumofthemodeswiththecoefcientsinthevectorc.Thisproblemisessentiallyaneigenvalueproblem,wherethelargestvalueofrepresentstheoptimalenergygrowth,andthesuccessivelysmallervaluesrepresentorthonormalmodesofenergygrowth.Thiseigenvalueproblemcanbehandledusingstandardtechniques,asthesizeofthismatrixisusuallyonly(O)(102)(O)(102). 181

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Performingthiscalculationforachannelow,itcanbeseenthat,thetransientgrowthmechanismproducesalargerperturbationthanthemodalgrowthoftheunstableeigenvalue,andthattheresultsarecomparabletothoseof Farrell ( 1988 ). FigureA-2. Calculationofthetransientgrowthforaperturbationinachannelowof=1,=0,Reh=10,000. 182

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APPENDIXBMODELOFTHEBI-GLOBALSTABILITYANALYSISForowsthatvaryintwo-dimensions,theassumptionsnecessarytoforone-dimensionalstabilityanalysisareviolated.Undersomecircumstances,theseassumptionsmaybeonlyweaklyviolated,andaone-dimesionalanalysiscanstillholdacertainlevelofaccuracy(suchasinslowlydevelopingboundarylayers).However,whentheseassumptionsaremorestronglyviolated,morecomputationallyexpensivemethods,suchasbi-globalstabilityanalysisbecomenecessarytools.Onlyinthepastdecadehavematrixbasedbi-globalstabilitymethodsreachedalevelofmaturitysuchthattheycanbeusedforphysicallyimportantproblems.Beforethistime,bi-globalstabilityanalysisfocusedongeometricallysimpleliddrivencavityandpressuredrivenductows( Tatsumi&Yoshinmura 1990 ; Theolisetal. 2004 ).Morerecently,bi-globalstabilitymethodshaveextendedthemselvesfromcanonicalowproblemssuchassimpleductandcavityowstoseparatedowsoverplates( Merleetal. 2010 ; Theolis&Rodriguez 2009 )andcavities( Bres&Colonius 2008 )andtwo-dimensionalchannelows( Chedevergneetal. 2006 ).Morecomplexgeometriesoverairfoils( Theolis&Rodriguez 2009 )andturbineblades( Abdessemedetal. 2009 ; Sharmaetal. 2011 )havealsobeenexamined,indicatingthatthepresentmethodscannallybeappliedtoindustriallyimportantowelds.Inadditiontothemodalgrowthproblems,bi-globalstabilitymethodshavebeenextendedtotransientgrowthproblems( Abdessemedetal. 2009 ; Akerviketal. 2008 ; Alizard&Robinet 2007 ; Sharmaetal. 2011 ),periodicFloquetstabilityanalysis( Abdessemedetal. 2009 ; Sharmaetal. 2011 ),andtheoptimalforcingproblem( Brandtetal. 2011 ; Sipp&Marquet 2012 ).Tri-globalstabilitymethodshavebeenimplemented,butthecomputationcostofthesemethodsremainsexceedinglyhigh,andonlyspecializedmethodshaveshownanysuccess( Bagherietal. 2009 ; Natarajan&Acrivos 1993 ). 183

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UsingthelinearizedincompressibleNavier-Stokesequations(Equation 2 )andonlyrequiringthattheowbetwo-dimension,simplicationofthegoverningequationsleadstothesystemofequations @u0 @x+@v0 @y+iw0=0 (Ba)u@u0 @x+v@u0 @y+wiu0+u0@u @x+v0@u @y+@p0 @x)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2u0 @x2+@2u0 @y2)]TJ /F3 11.955 Tf 11.96 0 Td[(2u0=i!u0 (Bb)u@v0 @x+v@v0 @y+wiv0+u0@v @x+v0@v @y+@p0 @y)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2v0 @x2+@2v0 @y2)]TJ /F3 11.955 Tf 11.96 0 Td[(2v0=i!v0 (Bc)u@w0 @x+v@w0 @y+wiw0+u0@w @x+v0@w @y+ip0)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@2w0 @x2+@2w0 @y2)]TJ /F3 11.955 Tf 11.96 0 Td[(2w0=i!w0 (Bd)Thissystemofequationscanbeputintoageneralizedeigenvalueproblemform. B.1NumericalConcernsandImplementationDuetothesizeofthenecessarymatricesforthebi-globalhydrodynamicstabilityproblem,thenumericaldifferentiationbecomesaverytrickyproblemtoaddress.Thematricesusedmustbestoreableontheavailablememoryresources,otherwisetheproblemcannotbesolved.WiththepresentuseoftheArnoldiAlgorithmandLAPACKsubroutinesfortheeigenvaluecomputation,someofthememoryrequirementscanbedistributed,asinthecaseofthematrix~A=A)]TJ /F7 7.97 Tf 6.59 0 Td[(1B,however,otherscannot.Inordertoreducethesizeofthematrices,therearetwoavailableoptions 1. Reducethenecessarynumberofdegreesoffreedom. 2. Reducethenumberofelementsstoredviasparseandbandedmatrixmethods.Therstoftheseoptions,reducingthenumberofdegreesoffreedomimmediatelysuggeststhathigherorderandspectralmethodsareappropriateforthebi-globalstabilityproblem,whichhavehigherresolvabilitythanlowerorderschemes.However,ashighordermethodsareemployed,thedensityofthematricesusedincreases. 184

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Therefore,thereisatradeoffpresentbetweentheaccuracyandresolvabilityofaschemeandthedensityoftheresultingdifferentiationmatrices.Sparsematrixmethodswereconsidered,buttheyarenotoptimalforusewiththepresentLAPACKsubroutines.Sparsematrixmethodsallowfortheleastmemorystorage,buttheypresentdifcultiesinimplementation,andtheavailablelibrariesdonothaveextensivesupportforcomplexnumberarithmetic.Bandedmatrixmethodsarewellsupportedintheexistingnumericallinearalgebralibraries,includingLAPACK( Andersonetal. 1987 ).Usingbandedmatrices,thetotalnumberofelementsstoredcanbegreatlyreduced,whilestillretainingnumericalaccuracyandresolvability.Usingabandedmatrixmethodinconjunctionwithsparsenitedifferenceorniteelementdifferentiationisstillsub-optimalhowever,astherearealargenumberofemptyelementsinthematrices.Inordertotakeadvantageoftheavailablememorystorageforabandedmatrixmethod,aChebyshevcollocationdifferencingschemecanbeappliedinonedirection,andanitedifferentschemeisappliedintheother.Thiscongurationprovidesthefewestpossibleemptyelementsinthematrix,hopefullyreducingthenecessarynumberofelementsinthematricestoanearminimumwhilestillretainingnumericalaccuracy.Evenifthesetypesofmethodsareemployed,thistypeofproblemcontinuestobeconstrainedinthatitisageneraleigenvaluemethod,thatisi!Au=Bu (B)ThenormalmethodofapproachingthisproblemistoemploytheshiftandinvertorCayleyspectraltransforms,whichrequiretheinversionofmatrix( Theolis 2003 2011 ).Evenso,theinversionofthismatrix(evenifitissparse),leadstoadensematrix,whichwillbereferredtoas~A.Therelativesizeofthenecessarymatrices,isdescribed 185

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inTable B-1 .Itcanbeseenthatthelargestconstraintonmemoryisthestorageofthematrix~A. TableB-1. Memoryrequirementsforthesizeofcertainmatricesinvolvedinbi-globalstabilityanalysis MatrixDensematrixstorageBandedmatrixstorageSparsematrixstorage AO(n2xn2yn2gov)O(nxn2yn2gov(2ndi+1))O(nxnyn2gov(4n2di+1))BO(n2xn2yn2gov)O(nxn2yngov)O(nxnyngov)~AO(n2xn2yn2gov)O(n2xn2yn2gov)O(n2xn2yn2gov) B.1.1DifferencingSchemesUsedImplementingthisbi-globalstabilityanalysis,a4thorderaccuratenitedifferenceschemeisappliedinthex-direction.AChebyshevcollocationdifferencingschemecanbeappliedinthey-direction.Acompact10thorderdifferentiationschemehasalsobeenimplementedintothecodefordifferencinginthey-direction,basedontheworkof Lele ( 1992 ).Thisschemeusesacombinationofstaggeredandnon-staggereddifferentiationstencilsof10thorderaccuracyinthecenterofthedomain,butoflesseraccuracyneartheboundaries.Thecontinuityandmomentumequationsaresolvedonasemi-staggeredmesh.Themomentumequationsaresolvedandthevelocitydatastoredonpointsco-incidentwiththeboundaries.Thecontinuityequationissolvedandthepressuredataisstoredonintermediatepoints. B.1.2MemoryDistributionAsthememorycanbedistributedforthematrix~A,thematrixBisalsodistributed.However,inordertousetheLAPACKsubroutinesZGBSVandDGBSVtoinvertA,thatmatrixcannotbepresentlydistributedacrossmultipleprocessors.Thememorydistributionof~AandBisdoneinsuchawaythatanumberofconsecutivecolumnsfromthematrixarestoredoneachprocessorindependentlyfromeachother.ThistypeofmemorydistributionisalsoveryeasytoimplementwithrespecttotheArnoldialgorithm.TheindividualArnoldibasisvectors(qn)canalsobedistributedovertheavailableprocessors.Indoingthis,thematrix-vectormultiplication~Aqnis 186

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~A=~A1 ~A2 ~AnprocB=B1 B2 Bnproc~Ai=(A+B))]TJ /F7 7.97 Tf 6.59 0 Td[(1Bi FigureB-1. Distributionofmemoryforashiftandinvertstrategy(withashiftof). reducedtoonlythecomponentsstoredwitheachprocessor.Indoingthis,theneedforcommunicationbetweentheprocessorsisreduced,exceptforthesummationrequiredtoformtheupperHessenbergmatrix,H,whichapproximatestheeigenvaluesof~A(~AQHQ) Q=[q1,q2,,qn]=26664Q1 Q2 ... Qnproc37775 FigureB-2. DistributionofmemoryfortheArnoldialgorithm. B.1.3ArnoldiAlgorithmInadditiontothememoryconstraints,thereisalsotheconsiderationofperformingthistypeoflargescalecomputationwithinareasonableamountoftime.EigenvalueandSVDproblemsaresolvedusingsimilaralgorithms,astheSVDproblemsareusuallymodiedeigenvaluemethods,thecomputationalcostofwhichareoutlinedinTable B-2 TableB-2. OperationcountsforthesizeofcertainalgorithmsinvolvedintheEVandSVDproblemsforadensematrix. ArnoldiAlgorithmQRAlgorithm O(n2xn2yn2govn2arno)O(n3xn3yn3gov) TheArnoldialgorithmhasbeenimplementedasthepresenteigenvaluesolver.Thisalgorithmiseasilyadjustableforthememorydistributionusedatpresent.TheArnoldialgorithmisdescribedindetailin Trefethen&Bau ( 1997 ).InimplementingaparallelArnoldialgorithm,novectorsneedtobepassedusingaparallelimplementation.Instead,thenecessarymatrixvectorproductscanbe 187

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performedlocallyoneachprocessor.Thevector-vectorproductsandvectornormscanalsobeperformedlocallyoneachprocessor,onlythepartialsumsneedtobecombinedandpassedattheendofthemultiplication.TheArnoldialgorithmproducesamatrixHandanincompletebasissetofvectorsQn,whichcanbeusedtoapproximatetheeigenvaluesandeigenvectorsof~A.TheeigenvalueproblemofHissmallenoughthatitcanbecomputedonasingleprocessorinareasonableamountoftime(ontheorderof10-60seconds). B.2VericationInordertovalidatethiscase,twoexampleshavebeenexamined,thecanonicalcaseofchannelowasexaminedby Orszag ( 1971 ),andaductowcase. B.2.1ChannelFlowThepurposeofthisexampleistoshowthattheresultsofthebi-globalstabilitycodeagreewiththesingledimensionresults.Theuseofabi-globalstabilitycodeforaowsuchasthisisavastlyinefcientuseofcomputationalresources,butwillprovideassurancethattheproblemisbeingaddressedcorrectly.Inordertoapplyperiodicboundaryconditionswhilemaintainingahighorderofaccuracy,thechannelowisrotated90degrees,suchthatv=v(x)=1)]TJ /F5 11.955 Tf 12.74 0 Td[(x2andu=w=0.Threegridresolutionsweretestedforthiscase.800Arnoldiiterationsareusedtoevaluatetheeigenvalueproblem.TheresultsareshowninTable. B-3 .Withthecoursegridresolution,theeigenvalueisaccuratetoapproximately1%,whichiscomparabletotheerrorfromalocalstabilityanalysiswiththesamegridresolutionandsimilar4thorderaccurate,semi-staggered,nitedifferencescheme(describedinAppendix 8.3 ).Astheresolutionisincreased,theaccuracyoftheresultalsoincreases.Basedontheseeigenvaluecalculations,thischannelowcanbeaccuratelyanalyzedusingthepresentbi-globalstabilitymethod.Consideringthetransientinstabilityanalysis,asimilargridresolutionstudyhasbeenperformed.Comparingthenon-modalgrowthofperturbationsoveraniteamount 188

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TableB-3. Gridresolutionsandmostunstableeigenmodesforthechannelow. Casenxnyxywall! Bi-Global-Coarse65320.031250.19634950.240163-i0.000284Bi-Global-Medium129320.0156250.19634950.237686+i0.003064Bi-Global-Fine257320.00781250.19634950.237532+i0.003660Local-Coarse65n/a0.03125n/a0.238947+i0.000203Local-Medium129n/a0.015625n/a0.237435+i0.003245Local-Fine257n/a0.0078125n/a0.237479+i0.003700 Orszag ( 1971 )Exactn/an/an/a0.23752649+i0.00373967 oftime(Figure B-3 ,thelocalandbi-globalstabilityanalysisproducecomparableresults.Thehighergridresolutionsappeartoconvergeforallofthetimesexamined.Thelowergridresolutionsproduceagoodresultforsmalleramplicationtimes(10-40),buteventuallydivergefromthemoreresolvedtimesduetoexponentialamplicationoferrorinthecalculatedeigenvalues. FigureB-3. Optimaltransientgrowthinthechannelasafunctionofthegrid-resolution.Comparisonswithlocalstabilityanalysisandapastresultarealsoshown. B.2.2DuctFlowThepurposeofthisexampleistovalidatethebi-globalstabilitycodeforaowthathasgradientsintwodirections.Whereasthechannelowcaseisatwo-dimensionalextensionoflocalstabilitytheory,thiscaseissomethingthatcanonlybeexaminedusingbi-globalstabilityanalysis.TheowhererepresentsalowReynoldsnumber 189

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pressuredrivenowinaniteaspectratioduct.Thebaseowcanbefoundbysolvingtheequation1 Re@2w @x2+@2w @y2=@p @z (B)whichcanbederivedbyassumingasteady,two-dimensionalowfullydevelopedowinthez-direction,andassumingthattherearenostandingstreamwisevortices.Thesolutiontothisequationcanbefoundusingseparationofvariablesandisw=1Xn=04()]TJ /F4 11.955 Tf 9.3 0 Td[(1)n (2n+1)sinh((2n+1)(y+A)=2) sinh((2n+1)A)cos((2n1)x=2) (B)Forthisow,threegridresolutionswereexamined,whichareindicatedinTable B-4 .TheReynoldsnumberischosentobe100andthestreamwisewavenumbertobe=1.Basedonthemostunstableeigenmode,allofthesegridresolutionsappeartoresolvethemostunstabledynamics.TheeigenspectracomputedforeachoftheseconditionsisshowninFigure B-4 A. TableB-4. Gridresolutionsandmostunstableeigenmodesfortheductowofaspectratio1. Gridnxnyxywall! Coarse17170.1250.01921470.325631-i0.115880Medium33330.06250.00481530.325713-i0.115654Fine65650.031250.00120450.325729-i0.115606 190

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A B C D FigureB-4. A)EigenvaluesoftheductowforRe=100,A=1,and=1forthethreegridresolutionstested.Theabsolutevalueofthestreamwisevelocityperturbation(jw0j)oftheleaststableeigenmodefortheB)Coarse,C)Medium,andD)Finegridsisalsoshown. 191

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APPENDIXCDESCRIPTIONOFFDL3DIInordertosimulatetheseows,theImplicitLargeEddySimulation(ILES)Navier-Stokessolver,FDL3DI( Rizzettaetal. 2008 )isemployed.Thiscodesolvesthecompressibletwo-dimensionalNavier-Stokesequations(Equation 2 )inabodytted,diagonalized,conservationform. @ @t1 JQ+@ @F)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReFv+@ @G)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReGv+@ @H)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReHv=DcS(C)Thedependentvariablesaredenedas Q=,u,v,w,ET(C)withtheinviscidandvisciduxesaredenedas F=1 J266666666664UuU+xpvU+ypwU+zpEU+xiuip377777777775,G=1 J266666666664VuV+xpvV+ypwV+zpEV+xiuip377777777775,H=1 J266666666664WuWV+xpvW+ypwW+zpEW+xiuip377777777775(C) Fv=1 J2666666666640xii1xii2xii3xi(ujij)]TJ /F5 11.955 Tf 11.96 0 Td[(Qi)377777777775,Gv=1 J2666666666640xii1xii2xii3xi(ujij)]TJ /F5 11.955 Tf 11.95 0 Td[(Qi)377777777775,Hv=1 J2666666666640xii1xii2xii3xi(ujij)]TJ /F5 11.955 Tf 11.95 0 Td[(Qi)377777777775,(C)andtherighthandsidesourcetermaredenedas S=1 J0,fx,fy,fz,uifxiT(C) 192

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where E=T ()]TJ /F4 11.955 Tf 11.95 0 Td[(1)M21+1 2(u2+v2+w2)(C) Qi=)]TJ /F10 11.955 Tf 11.29 16.86 Td[(1 ()]TJ /F4 11.955 Tf 11.96 0 Td[(1)M21 Pr@j @xi@T @j(C) ij=@k @xj@ui @k+@k @xi@ui @k)]TJ /F4 11.955 Tf 13.16 8.09 Td[(2 3ij@l @xk@uk @l(C) U=@ @xiui,V=@ @xiui,W=@ @xiui(C)Inthissystemofequations,representstheuiddensity,uandvaretheowvelocities,pispressure,andEisthespecicenergy.Allofthesevariablesarenon-dimensionalizedbytheirreferencevalues.Pressureistheexceptiontothis,asithasbeennon-dimensionalizedbythedynamichead(1u21).ijrepresentsthestresstensorandQiistheheatuxvector.fx,fyandfzdescribethenormalized,spatiallyvaryingbodyforce.ThemagnitudeofthisbodyforceismodulatedbythevalueofDc.,,andrepresentthebodyttedcoordinatesystem,JisthegridJacobian,U,VandWarethebodyttedvelocities.Thenon-dimensionalvariablesRe,Pr,andMa1representthefreestreamReynolds(Re=u1L ),Prandtl(Pr=cp k),andMach(M1=u1 p RT1)numbers,respectively.Here,representstheratioofspecicheatsandisequalto1.4.Theidealgaslaw(Equation 2 )isalsousedinordertoclosethesystemofequations.WhilethisisthecompressibleformoftheNavier-Stokesequations,andincompressibleowcanbesolvedbysettingtheMachnumber,M1,toanappropriatelylowvalue.IthasbeendeterminedthatavalueofM1=0.1providesareasonablebalanceofincompressibilityandnumericalstabilityforthisparticularcode.Inordertoimplementtheplasmabodyforceintheuidsystem,thebodyforcetermSisemployed.Thistermcanbemadetobesteadyorunsteady,dependingonwhatmannerofbodyforceisdesired.Forthenitedifferencingscheme,compact,highorderschemeshavebeenimplementedintoFDL3DI( Rizzettaetal. 2008 ).Thesecompactschemesallowfor 193

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greaterresolvabilityoftheoweldusingadjacentmeshpoints( Lele 1992 )thanmoretraditionalexplicitdifferencingschemes.Thesestencilsgenerallytaketheformof)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(0i+2+0i)]TJ /F7 7.97 Tf 6.58 0 Td[(2+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(0i+1+0i)]TJ /F7 7.97 Tf 6.59 0 Td[(1+0= (C)c 6h(i+3)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F7 7.97 Tf 6.58 0 Td[(3)+b 4h(i+2)]TJ /F3 11.955 Tf 11.96 0 Td[(i)]TJ /F7 7.97 Tf 6.59 0 Td[(2)+a 2h(i+1)]TJ /F3 11.955 Tf 11.96 0 Td[(i)]TJ /F7 7.97 Tf 6.58 0 Td[(1)whereisthevariablebeingdifferenced,thesubscriptiindicatesthemeshindex,andhisthemeshspacing,thoughthewidthofthestencilcanbevariedtobewiderornarrower.Thecoefcientsof,,a,b,andccanbevariedinordertocontroltheaccuracyandresolvabilityofthedifferencing,subjecttocertainconstraints.Nearboundaries,thistypeofcompactdifferencingschemecanbeadaptedtonon-centeredstencils( Visbal&Gaitonde 1999 ).ForFDL3DI,asixthorderaccurate,tri-diagonaldifferencingschemeareemployedoveramajorityofthedomain.Non-centered,fourthandfthorderaccurate,compactdifferencingschemesareemployedneartheboundariesoftheow( Rizzettaetal. 2008 ).ThetemporalintegrationemployedinFDL3DIistheBeam-Warmingapproximatefactorizationmethod( Beam&Warming 1978 ),whichdiagonalizestheNavier-Stokesequationsasthey'rebeingsolved.Forthesimulationsperformed,athree-point,backwardsimplicitschemehasbeenused,whichhassecondordertemporalaccuracy( Rizzettaetal. 2008 ).Inordertoaddressnon-linearterms,threeNewtonlikesub-iterationshavebeenperformedforeachtimestep.Thefactorizationofthissystemissolvedindeltaform(wherexnindicatesthespatialdifferencinginthexdirectionwith 194

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nthorderaccuracy)as1 J+2t 32@Fp @Q)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@Fpv @QJ1 J+2t 32@Gp @Q)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 Re@Gpv @QJ1 J+2t 32@Hp @Q)]TJ /F4 11.955 Tf 17.2 8.08 Td[(1 Re@Hpv @QQ=)]TJ /F10 11.955 Tf 11.29 16.85 Td[(2t 31 2t3Qp)]TJ /F4 11.955 Tf 11.95 0 Td[(4Qn+Qn)]TJ /F7 7.97 Tf 6.58 0 Td[(1 J+6Fp)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReFpv+6Gp)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReGpv+6Hp)]TJ /F4 11.955 Tf 17.2 8.09 Td[(1 ReHpv)]TJ /F5 11.955 Tf 11.95 0 Td[(DcSp (C)wherethesuperscriptnindicatesthetimestep,andthesuperscriptpindicatesthesub-iterationbeingsolvedforattimestepn+1.Q=Qp+1)]TJ /F15 11.955 Tf 13.21 0 Td[(Qp.Fortherstsub-iterationofeachtimestep,therstsub-iterationisequaltothesolutionattheprevioustimestep(i.e.Qp=Qn).Fortheimplicit,spatialdiscretizationtermsinthisscheme,FDL3DIusesanexplicitsecondorderaccuratescheme.Fortheexplicit,spatialdifferencingterms,thehighorder,compactschemesareused.Thismethodofsolvingtheequationsdoesnotemployedasubgridstress(SGS)model,commoninmostLESmethods( Lesieuretal. 2005 ).Instead,thediffusionofsmallereddiesattheunresolvedscalesaremodeledbytheLESlter( Grinsteinetal. 2007 ).Solongasenoughofthelengthscalesareresolved,theILESmethodisabletoresolveturbulentowfeaturesandtheirstatistics( Visbal&Gaitonde 1999 ; Wachtoretal. 2013 ).Assuch,itisimportanttonotethatamongthesimulationsperformed,theminimumaccuracyofthelterswas8thorderaccurate.Theltersusedwerealsocompact,suchthat^i)]TJ /F7 7.97 Tf 6.59 0 Td[(1+^i+^i+1=a0i+a1 2(i+1+i)]TJ /F7 7.97 Tf 6.59 0 Td[(1)+a2 2(i+2+i)]TJ /F7 7.97 Tf 6.59 0 Td[(2)+... (C)where^representsthelteredvalueof.Similarlytohowtotheasymmetricstencilsforthecompactdifferencingschemecanbeusednearthedomainboundaries,asymmetricstencilscanalsobeemployedforthelter( Visbal&Gaitonde 1999 ),whichhasbeen 195

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implementedintoFDL3DI( Rizzettaetal. 2008 ).Theparameterusedtomodulatethelowpassnatureofthelterwas=0.4)whichroughlycorrelatestoneedingfourgridpointstoresolveafullwavelengthofsomeowfeature. 196

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BIOGRAPHICALSKETCH MarkRiherdisanativeofLakeButler,Florida.HeperformedhisundergraduateworkatDukeUniversityinDurham,NC,receivingaBachelorofScienceinmechanicalengineeringandaminorinmathematics,graduatinginMay2009.HeworkedforseveralsummersduringgraduateschoolwiththeAirVehiclesDirectorateoftheAirForceResearchLaboratoryatWright-PattersonAirForceBaseoutsideofDayton,OhioandonesummeratNASA'sFlowPhysicsandControlBranchattheLangleyResearchCenterinHampton,Virginia. 208