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The Mechanisms for Passive Suppression of Fluctuating Surface Pressure in a Supersonic Cavity Flow

Permanent Link: http://ufdc.ufl.edu/UFE0042351/00001

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Title: The Mechanisms for Passive Suppression of Fluctuating Surface Pressure in a Supersonic Cavity Flow
Physical Description: 1 online resource (225 p.)
Language: english
Creator: Dudley, Jonathan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: cavity, compressible, control, cylinder, des, flow, fluctuating, passive, piv, rossiter, supersonic, suppressed, tones, turbulence
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The control of supersonic cavity flows using a rod in cross-flow is a unique and challenging fluid mechanics problem that has applications ranging from the automotive to aerospace industry. This research combines experimental and numerical data acquisition of a baseline and controlled cavity flow at M=1.4 over a open rectangular cavity. The measurements included unsteady surface pressure measurements and spatially resolved particle image velocimetry. Analysis of the fluctuating pressures on the cavity surfaces included investigations of the root-mean-square fluctuating pressure, spectral analysis, correlation and coherence analysis and joint time-frequency spectrograms. The shear layer flowfield and turbulence was studied using ensemble averaged mean turbulent statistics including two-point turbulent velocity spatial correlations and Proper Orthogonal Decomposition.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan Dudley.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ukeiley, Lawrence S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042351:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042351/00001

Material Information

Title: The Mechanisms for Passive Suppression of Fluctuating Surface Pressure in a Supersonic Cavity Flow
Physical Description: 1 online resource (225 p.)
Language: english
Creator: Dudley, Jonathan
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: cavity, compressible, control, cylinder, des, flow, fluctuating, passive, piv, rossiter, supersonic, suppressed, tones, turbulence
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The control of supersonic cavity flows using a rod in cross-flow is a unique and challenging fluid mechanics problem that has applications ranging from the automotive to aerospace industry. This research combines experimental and numerical data acquisition of a baseline and controlled cavity flow at M=1.4 over a open rectangular cavity. The measurements included unsteady surface pressure measurements and spatially resolved particle image velocimetry. Analysis of the fluctuating pressures on the cavity surfaces included investigations of the root-mean-square fluctuating pressure, spectral analysis, correlation and coherence analysis and joint time-frequency spectrograms. The shear layer flowfield and turbulence was studied using ensemble averaged mean turbulent statistics including two-point turbulent velocity spatial correlations and Proper Orthogonal Decomposition.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan Dudley.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Ukeiley, Lawrence S.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0042351:00001


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THEMECHANISMSFORPASSIVESUPPRESSIONOFFLUCTUATINGSURF ACE PRESSUREINASUPERSONICCAVITYFLOW By JONATHANG.DUDLEY ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010 1

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c r 2010JonathanG.Dudley 2

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ToMeganandMalia 3

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ACKNOWLEDGMENTS Iwouldrstandforemostliketothankmywifewhoseconstant patienceandsupport haveguidedmethroughtheupsanddownsofthepastfewyears. Iwouldliketoextend mygratitudetomyfamilywhoseuninterruptedandunwaverin gsupport,sacriceandlove throughoutthisarduousjourneywasnotunrecognizedormis guided.Aspecialthanks isgiventoMr.RobertBallforhishelpindesignandfabricat ionoftheSWTandtest modelsandanythingelseIneededyesterday.Iamalsoindebt edtoallmycollegeswho havecomeandgoneattheResearchandEngineeringEducation Facilityfortheirtimeand helpwitheveryaspectoftheresearchincluding(butfarfro mlimitedto)experimental setup,facilitymaintenance/operationanddiscussions. IwouldliketoacknowledgeDr.LawrenceUkeileyforprovidi ngguidancethroughout myresearchattheUniversityofFlorida.Iwouldliketothan ktheothermembersofmy committee,Dr.AndreasHaselbacher,Dr.LouisCattafestaI II,Dr.JasonButlerand Dr.SrinivasanArunajatesanforprovidinginsight,review ingmyresearchandproviding valuablefeedback. Additionally,thanksgoouttothemembersoftheAirForceSE EKEAGLEOce andAirForceResearchLabnotonlyforfundingbutforprovid ingresourceandtrue center(s)ofexpertise.Specically,Iwouldliketoextend mygratitudetoDr.John Martel,Dr.MagdiRizk,Mr.SergeyKernazhitskiyandMr.Jas onTorresforthemany hoursofdiscussionandtroubleshootingthenumerics.Ithi nkInallyhavetimetojoin allofyouFridayafternoonontheboat. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................8 LISTOFFIGURES ....................................9 ABSTRACT ........................................14 CHAPTER 1INTRODUCTION ..................................16 1.1Background ...................................16 1.1.1DissertationOutline ...........................16 1.1.2Objectives ................................17 1.2FundamentalsofCavityFlow .........................18 1.3CavityFlowwithControl ...........................23 2STATISTICALANALYSISTECHNIQUES ....................32 2.1UnsteadyPressureMeasurements .......................32 2.1.1SpectralCalculations ..........................33 2.1.2JointTime-FrequencyAnalysis .....................34 2.1.3CorrelationAnalysis ...........................35 2.2TurbulentStatisticsandFloweldMeasurements ..............36 2.3ProperOrthogonalDecomposition .......................39 2.4Non-Dimensionalization ............................42 3THEFACILITYANDEXPERIMENTALTECHNIQUES ............45 3.1FacilityOverview ................................45 3.2CavityDesign ..................................48 3.3SWTCharacterization .............................50 3.4Instrumentation .................................53 3.4.1WindTunnelControl ..........................53 3.4.2UnsteadyPressureMeasurements ...................55 3.5FlowVisualization ...............................56 3.5.1Schlieren .................................56 3.5.2ParticleImageVelocimetry .......................59 3.6ExperimentalSetup ...............................62 3.7MeasurementUncertainty ...........................63 3.7.1UnsteadyPressureMeasurement ....................64 3.7.2PIVMeasurement ............................64 5

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4NUMERICALSIMULATIONAPPROACH ....................68 4.1Turbulence ...................................68 4.1.1CompressibleNavier-StokesEquations .................74 4.1.2Spalart-AllmarasDES .........................76 4.2FlowSolver ...................................78 5CYLINDERIMMERSEDINASUPERSONICBOUNDARYLAYER ......80 5.1ComputationalGrids ..............................83 5.2Results ......................................87 5.2.1MeanFloweld .............................88 5.2.2InstantaneousVorticityFloweld ...................96 5.2.3TurbulentStatistics ...........................103 5.3Summary ....................................106 6UNSTEADYPRESSUREMEASUREMENTS ...................108 6.1FullandFiniteSpanBaselineCavityComparison ..............108 6.2FiniteSpanCavitywithSuppression .....................113 6.2.1EectsofRodMounts .........................114 6.2.2EectsofRodDiameterandGapHeight ...............116 6.3FullSpanCavitywithSuppression ......................123 6.4Summary ....................................128 7FLOWFIELDMEASUREMENTS .........................131 7.1FullSpanCavity ................................131 7.1.1MeanFloweld .............................131 7.1.2TurbulentFloweld ...........................137 7.1.3EvolutionofTwo-PointStatistics ...................144 7.1.4ProperOrthogonalDecomposition ...................148 7.2FiniteSpanCavity ...............................157 7.2.1FullandFiniteSpanFloweldComparison ..............157 7.2.2MeanFloweld .............................159 7.2.3TurbulentFloweld ...........................160 7.3Summary ....................................162 8SIMULATIONOFBASELINEANDCONTROLLEDCAVITY .........166 8.1ValidationofCavitySimulations ........................168 8.2InstantaneousFloweldandPressureMeasurement .............173 8.3MeanFloweld .................................177 8.4MeanTurbulentFloweld ...........................183 8.4.1EvolutionofTwo-PointVelocityCorrelations .............184 8.4.2POD ...................................188 8.5Summary ....................................192 6

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9CONCLUSIONSANDFUTUREWORK ......................196 9.1FiniteandFullSpanCavities .........................197 9.2DemonstrationofDES .............................197 9.3CavityControl .................................198 9.4FutureWork ...................................201 9.4.1ComputationalFluidDynamics ....................201 9.4.2Experimental ..............................202 9.4.3FutureConsideration ..........................203 APPENDIX ATURBULENCEMODELFORMULATION ....................204 A.1CompressibleNavier-StokesEquations ....................204 A.2CompressibleReynoldsAveragedNavier-StokesEquatio ns .........206 A.3Spalart-AllmarasTurbulenceModel ......................208 BVIBRATIONANDDEFLECTIONOFSELECTEDRODS ...........212 B.1RodFundamentalFrequency ..........................212 B.2DerectionofRod ................................213 REFERENCES .......................................215 BIOGRAPHICALSKETCH ................................225 7

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LISTOFTABLES Table page 3-1Standardoperatingconditions. ...........................62 3-2Cavityroorpressuresensorlocations. ........................63 4-1Summaryofnumericalstrategiesandcost. .....................71 5-1Summaryofcomputationaldetails .........................88 6-1Rossiterresonanttonepredictions. .........................109 6-2Finitespancavitycongurations. ..........................114 8-1Listofvariabledimensions ..............................167 8-2Computationalgridsizesforthebaselinecavity. ..................168 8-3Cavitywavespeed,timelagsanddistances. ....................175 A-1SA-DESturbulencemodelcoecients ........................211 B-1Rodgeometry .....................................212 B-2Transversefundamentalvibrationofrod ......................213 8

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LISTOFFIGURES Figure page 1-1Schematicandcommonnomenclatureoftypicalcavitycon gurations. ......18 1-2Opencavityrowillustratingacousticfeedbackloop. ................19 1-3Adaptedrowcontrolclassication. .........................23 1-4Passiverodspoilersuppressionmechanism .....................27 3-1Thesupersonicwindtunnel. .............................45 3-2RANSsolutionofthesupersonicwindtunnelnozzle. ...............46 3-3Test-sectionwithanalyticalwavererection. ....................47 3-4Test-sectionwithratplate. .............................47 3-5Finitespancavitywithcloseupofrodmountedonleading edgeblock ......48 3-6Simulationofrowaroundrodmountingstructure. ................49 3-7Rodmountinglocationrelativetocavityleadingedge ...............50 3-8Experimentalandnumericalboundarylayerprolecompa rison. .........51 3-9FlowvisualizationusingSchlierenfortheSWTtest-sec tion. ...........53 3-11Bendingoflightduetodensitygradientoftwomediums. .............57 3-12Z-typeSchlierenarrangementforqualitativerowasse ssment. ...........58 3-13Schlierenimagingofabullet .............................58 3-14ParticleImageVelocimetry(PIV). .........................59 3-15PIVandpressuresynchronization. .........................61 3-16Pressuresensorlocations ...............................63 3-17PIVuncertaintyforturbulentintensitiesthatareles sthan30%. .........65 3-18DEHSparticlesizefrequencydistribution .....................67 4-1Transitionofaturbulentboundarylayeroveraratplate .............69 4-2TurbulenceenergyspectrumdepictingRANSthroughDNSs imulationscales ..72 4-3NumericalSchlierenofsupersoniccavityrow ....................74 5-1Sketchofcylinderimmersedinasupersonicboundarylay er. ...........84 9

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5-2Gridtopology. .....................................85 5-3Gridandtime-stepsensitivityonspectralresultsfor d =0 : 40. ..........86 5-4Meanstreamwisevelocitycontours. .........................89 5-5Meanstreamwisevelocityprolesatgivenaxiallocatio ns. ............91 5-6Timeaveragedpressuredistributiononcylindersurfac e. .............93 5-7CylinderPSDasafunctionoffrequencyandStrouhaltake nat x d ; y =(1 ; 0). .94 5-8Strouhalnumberbasedon U 1 and U c .......................95 5-9Smalldiameterrodmeanroweldcontoursfor G =0 : 11. ............97 5-10Instantaneousz-vorticityfor G =0 : 11. .......................98 5-11Instantaneousz-vorticityfor G =0 : 21. .......................99 5-12Instantaneousz-vorticityfortopofrodnearthetopof boundarylayer. .....100 5-13Instantaneousz-vorticityforrodoutsideboundaryla yer. .............101 5-14 d =0 : 20Instantaneousiso-vorticitycoloredbyvelocitymagnit ude. .......103 5-15 d =0 : 40Instantaneousiso-vorticitycoloredbyvelocitymagnit ude. .......104 5-16Meanresolved RSS = u 0 v 0 U 1 2 onz=0plane. ......................105 5-17MeanresolvedTKEprolesatgiven x d locations. .................106 6-1FullandnitespanbaselinePSDcomparison ...................110 6-2Fullandnitespanbaseline p rms onthecavityroor. ...............111 6-3Baselinecavitysurfacepressurecorrelation. ....................112 6-4Fullandnitespanspectrogramsmeasuredontheaftwall ............113 6-5Finitespancavityrodmountingeects .......................115 6-6MeanSchlierengradientsforisolatedrodmountsassupp ressiondevices. ....116 6-7Fluctuatingsurfacepressuresbasedon d ......................117 6-7Fluctuatingsurfacepressuresbasedon d and G .................120 6-8Cross-correlationcoecientfortopofrodneartopofbo undarylayer. ......121 6-9Cross-coherencecoecientfortopofrodneartopofboun darylayer .......122 6-10Finitespanspectrogramfor d =0 : 43. .......................124 10

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6-11Fullspanbaselineandcontrolledructuatingsurfacep ressure ...........125 6-12Fullspancavitycross-correlation ..........................126 6-13Fullspancavitycross-coherence ...........................127 6-14Fullspancavityaftwallspectrogramforgiven d and G values. .........129 6-15Percentreductionof p rms ...............................130 7-1Fullspanmeanstreamwisevelocitycontourswithvector s. ............132 7-2Zoomedcavitystreamwisemeanvelocitycontourswithve ctors. .........134 7-3Fullspancavitystreamlines. .............................135 7-4Fullspancavitymeanstreamwisevelocityproles. ................136 7-5Fullspancavitymeanstreamwiseturbulentvelocitycon tours. ..........137 7-6Fullspancavitymeanstreamwiseturbulentvelocitypro les. ...........138 7-7Fullspancavitymeannormalturbulentvelocitycontour s. ............139 7-8Zoomedcavitynormalturbulentvelocitycontours. ................140 7-9MeanRSSrowcontoursforthefullspancavity. ..................141 7-10FullspanmeanRSSproles. .............................142 7-11Fullspancavityvorticitythickness. .........................145 7-12Evolutionoftwo-pointspatialaxialvelocitycorrela tionforvarying y D and x L ..146 7-13Evolutionoftwo-pointspatialnormalvelocitycorrel ationforvarying y D and x L .147 7-14Evolutionoftwo-pointspatialaxialvelocitycorrela tionforxed y D =0. .....149 7-15Evolutionoftwo-pointspatialnormalvelocitycorrel ationforxed y D =0. ...150 7-16ConvergenceofPODModes .............................152 7-17ConvergenceofPODEnergy .............................153 7-18FullspancavityPODtemporalconvergenceandenergydi stribution. ......153 7-19StreamwisePODmodesforthefullspancavity ..................155 7-20NormalPODmodesforthefullspancavity ....................156 7-21FullspanPODspatialcorrelationmapfortherst20mod es. ..........157 7-22Exampleofseededroweldillustratingedgeofcavitys idewall. ..........158 11

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7-23Comparisonoffullandnitespancavitystreamwisevel ocitycontours. .....158 7-24Finitespancavitymeanstreamwisevelocitycontoursa ndproles. ........160 7-25Fullspancavitymeanstreamlines. .........................161 7-26Finitespanmeanstreamwiseturbulentvelocitycontou rs. .............162 7-27Finitespanmeannormalturbulentvelocitycontours. ...............163 7-28FinitespanmeanstreamwiseRSScontours. ....................164 7-29RandominstantaneousSchlierenimages ......................165 8-1Geometryandboundaryconditionsforthenumericalsimu lations. ........166 8-2Structuredoversetgrid. ...............................167 8-3Eectofgridrenementon z vorticitycontours. .................169 8-4Eectofgridrenementoncavityructuatingsurfacepre ssures. .........170 8-5Instantaneoussnapshotoftracerparticleincylindern eareld ..........170 8-6CFDandexperimentmeanaxialvelocitycomparison. ...............171 8-7CFDandexperimentmeanstreamwisevelocitycomparison ...........172 8-8Aftwallexperimentalandnumericalspectra. ...................173 8-9Baselineandsuppressedcavityructuatingsurfacepres sures. ...........174 8-10Baselineandcontrolledcavityautoandcrosscorrelat ions. ............175 8-11Instantaneousnon-dimensionalvorticitycontours. .................176 8-12Baselineandcontrolledmeanstreamlines. .....................177 8-13Meanvelocityroweldcontours. ..........................179 8-14Meanstreamwiseandnormalvelocityprolesatvarious axiallocations. .....180 8-15Baselinecavity yz planerecirculation ........................181 8-16Suppressedcavity yz planerecirculation ......................182 8-17Computationalvorticitythickness ..........................183 8-18Meanvorticitycontours. ...............................183 8-19Meanturbulentroweldcontours. .........................185 8-20Meanroweldandmeanturbulentroweldproles. ...............186 12

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8-21Evolutionofstreamwisespatialvelocitycorrelation ................187 8-22Evolutionofnormalspatialvelocitycorrelation. ..................189 8-23Fluctuatingsurfacepressurecorrelation .......................190 8-24EnergydistributionofPODmodes. .........................190 8-25PODmodesbasedonthenormalructuatingvelocity. ...............193 B-1Suppressiondeviceproperties. ............................213 B-2Estimatedrodderectionbasedondiameterand L foragiven U 1 ........214 13

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy THEMECHANISMSFORPASSIVESUPPRESSIONOFFLUCTUATINGSURF ACE PRESSUREINASUPERSONICCAVITYFLOW By JonathanG.Dudley December2010 Chair:LawrenceUkeileyMajor:MechanicalEngineering Thestudyofthepassivesuppressionofsupersoniccavityro wusingarodimmersed intheupstreamboundarylayerisauniqueandchallengingru idmechanicsproblem. Theroweldincludesacompressibleshearlayerinteractin gwithacomplexpatternof compressionandexpansionwaves.Theturbulentructuation sinsidetheshearlayermay beampliedthroughafeedback-receptivitycycleresultin ginincreasedpressureloading onthesurfacesofthecavity.Studyingthemechanismsdicta tingthesuppressionofthese ampliedturbulentructuationswhencontrolispresentmak esforanenlighteningand challengingproblem.Acombinedexperimentalandtimeaccu ratenumericalstudyusing detached-eddysimulationwasconductedtostudythesuppre ssionofpressureructuations duetosupersoniccavityrowat M 1 =1 : 4overanopenrectangularcavitywitha length-to-depthratioofsix.Inthisstudy,thefocusiscon nedtosuppressionduetoarod spoiler.Theexperimentalmeasurementsincludedtemporal lyresolvedructuatingsurface pressuremeasurementscoupledwithspatiallyresolvedpar ticleimagevelocimetry. Analysisoftheructuatingpressuresonthecavitysurfaces includedinvestigations oftheroot-mean-squareructuatingpressure,spectralana lysis,correlationandcoherence analysisandjointtime-frequencyspectrograms.Theshear layerroweldandturbulence wasstudiedusingensembleaveragedturbulentstatisticsi ncludingtwo-pointspatial turbulentvelocitycorrelationsandProperOrthogonalDec omposition.Resultsindicate thatthemosteectivesuppressionoftheructuatingpressu reswasachievedwhenarod 14

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sizedroughly40%oftheboundarylayerwasplacedsuchthatt hetopoftherodwasnear thetopedgeoftheboundarylayer.Itwasshownthattherodle adstoathickershear layerthatinitiallyspreadsmorerapidly.Theturbulentst ructuresinthewakeoftherod interactwiththecavityshearlayerwithatimeperiodicexc itationwhichliftstheshear layernearthecavityleadingedge.Thestructuresaresmall erandlessorganizedwhich isbelievedtoleadtoashearlayerthatislessreceptivetot hedisturbancespropagating upstreaminsidethecavity.Thecontrolledcavityexhibits analteredaftwallimpingement pointwhichisduetotheliftingandalteredrappingnatureo ftheshearlayer.The upstreampropagatingdisturbanceemanatingfromtheaftwa llisthusweakeneddue inparttothelowerspeedrowimpingingontheaftwall.These coupledeventsleadto drasticallyreducedtonalcomponents(whichareloweredto nearbroadbandlevels)and notableloweringofthebroadbandlevelsoftheructuatingp ressuremeasuredonthe cavitysurfaces. 15

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CHAPTER1 INTRODUCTION 1.1Background Thegrazingrowpastacavity,asdescribedin Rockwell&Naudascher ( 1979 ),can generateanintenseaeroacousticresonantphenomenadueto theinteractionoftheshear layerwiththeaftwall.Thistypeoffundamentalruidrowpro blemhasmanypractical applicationsrangingfromautomotiverowsoversunroofsan dwindowstoaerospace rowsofopenwheelwellsandweaponsbays.Studiesoftheaero acousticenvironment encounteredinthesebayshavebeenongoingsincetheearly1 950'sinitiatedbytheworks of Krishnamurty ( 1955 )and Roshko ( 1955 ). Modernmilitaryaircrafthaveresortedtointernalweapons carriagesandjettisonsat supersonicfreestreamconditionswhichpresentmanychall engesduetothepresenceof complexunsteadyrowelds.Technologicaldevelopmentsar eimprovingtheaccuracyand reducingthesizeandmassofmunitions.Thisdictatesthene edtounderstandandcontrol theintensepressureructuationsobservedincavityrowsce narios.Further,thedelicate electronicsthatmightexistintheenvironmentofrealairc raftalsopredicatethenecessity toreducetheseructuatingpressures.Toachievemaximuman deectivesuppressionof theseloadsitisvitaltounderstandtheinherentfundament almechanismsdrivingtherow andhowtocoupleitwiththedesiredcontrol.1.1.1DissertationOutline Thisdissertationisorganizedintoninechapters.Chapter 1presentsthemotivation fortheresearchandliteraturereview.Thestatisticalana lysistechniquesandplotting conventionsforthisworkwillbedescribedinChapter2.Cha pter3willlaythefoundations fortheexperimentalinvestigationoftheresearchincludi ngthedesignandfabrication oftheSupersonicWindTunnel(SWT)attheUniversityofFlor idaResearchand EngineeringEducationFacility(REEF).Chapter4introduc esthenumericalmodeling techniquestobeusedthroughouttheresearchincludingade tailedlookattheselected 16

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methodsandclosuremodels.Chapter5presentstheinitialc omputationalresultsofa rodimmersedinasupersonicboundarylayerwhichlaysthegr oundworkforthepassive cavitycontrolwhichwillbeusedinthisstudy.Chapter6foc usesondiscussingtheresults oftheexperimentallyobtainedructuatingsurfacepressur emeasurements.Chapter7 presentstheroweldmeasurementsandvisualizationusing Schlierenimagingandparticle imagevelocimetry.Chapter8willdetailthenumericalsimu lationresultsforthebaseline andsuppressedcavitycongurations.Chapter9concludesw ithareviewoftheresults andfuturework.Lastly,appendicesareincludedthatdiscu sstheintricatedetailsof theturbulencemodelandthephysicalandmaterialproperti esoftherodsusedinthe experimentaleort.1.1.2Objectives Theprimaryobjectiveofthisresearchistogainabetterund erstandingofthe physicalmechanismsdictatingthesuppressionofalowfree streamsupersonicMach numberonopencavityrowwhenarodisimmersedintheupstrea mboundarylayer priortothecavityleadingedge.Intuitively,therowpheno menainthewakeoftherod willinteractwiththeshearlayerabovethecavitycausingc hangesinitsproperties.To date,thereislittlespecicunderstandingoftheeectsof thisonthemeanorturbulent velocitiesforsupersonicfreestreamconditions.Inthise ortdetailsofthecylinder diameterandplacementontheructuatingsurfacepressurew illrstbeevaluatedto determinethemosteectiveconguration.Detailedmeasur ementsandComputational FluidDynamic(CFD)simulationswillbeconductedinanatte mpttoquantifytheeects ofthepassiverowcontroldeviceontherow.Specically,qu antitiesassociatedwiththe growthofthecavityshearlayeralongwiththeamplitudeand scaleoftheturbulencewill beexamined.Throughadetailedcomparisonbetweenthebase lineandcontrolledcasesa quantitativeassessmentofthecontrolontherowcanbeesta blishedandhypothesesfor themechanismswillbeputforth.Asecondaryobjectiveisto demonstratethatnumerical 17

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simulationsofthishighlyunsteadyrowcanbereasonablyac curatelycapturedusing DetachedEddySimulation(DES)techniques. 1.2FundamentalsofCavityFlow Thebasicroweldstructureofcavitiesaretypicallyclass iedaseitheropenor closedbasedonthegeometricallength(L)todepth(D)ratio L D .Aschematicoftypical openandclosedcavityrowareillustratedinFigure 1-1 .Cavitieswithan L D 10are designatedasopen, L D 13areconsideredclosed(shallow)andcavitieswherethe L D 2arecommonlyreferredtoasdeepcavities.Inclosedcavity rowstheleadingedge Figure1-1.Schematicandcommonnomenclatureoftypicalca vitycongurations. boundarylayerseparatesandformsashearlayerthattendst oreattachitselfonthecavity roor.Therowseparatesagainupstreamofthecavitytrailin gedgewhichleadstotwo recirculationzonesinsidethecavity.Closedcavitiesten dtobeassociatedwithhigher dragandheattransferpropertiesasreportedby Dusing etal. ( 1994 )whencomparedto opencavitiesandarethuslessdesirableformostpractical engineeringpurposes.Sincethe studyhereisaimedatopencavitiesthefollowingdiscussio nwillfocusmorecloselyonthis classofcavityrows. Theprimaryparametersdictatingthenatureofcavityrowsa slistedby CattafestaIII etal. ( 2003 )arethelengthtodepthratio L D ,lengthtowidthratio L W ,Reynoldsnumber (Re),boundarylayerthickness( ),displacementthickness( ),momentumthickness ( )andthefreestreamdynamicpressure( Q 1 ).Discretetonesaregeneratedbyboth laminarandturbulentboundarylayersthoughlaminarbound arylayerstendtobethicker resultinginathickerinitialshearlayerandhighertones Tam&Block ( 1978 ). 18

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Figure1-2.Opencavityrowillustratingacousticfeedback loop. Usingtheframeworkforanacousticfeedbackloopsimilarto thatillustratedinFigure 1-2 foropencavityrow. Powell ( 1961 )explainedthemechanismofselfsustainedacoustic resonanceforedgetoneswhichisanalogoustoaresonantcav ity.Hearguedthetimefor largescalestructurestoreachthetrailingedgeofthecavi tyandthetimerequiredfor acousticwavestopropagatebacktothereceptivitypoint,s houldbeanintegralmultiple oftheresonanttimescale.Theadditionofatermtoaccountf orthephaseshiftbetween thesevelocitieswouldsatisfyanytimelagnoticedbetween thehydrodynamicandacoustic waves. Rossiter ( 1964 )wasthersttodevelopasemi-empiricalmodelbasedonsubs onic andtransonicexperimentaldatausingthisfeedbackloopco ncept.Hebelievedthe pressurewashighernearthetrailingedgeofthecavityduet otheaccelerationofthe shearlayercausedbytherowenteringthecavity.Thispress uregradientallowedforthe formationoflargescalestructureswithintheexpandingsh earlayer.Healsobelievedthat vortexsheddingfromtheseparatingregionoftheshearlaye rcontributedtotheacoustic feedback.Hepostulatedthatwhenthefrequencyoftheseeve ntsapproachedthenatural frequencyofthecavitytheconditionsweremetwhichallowr esonancetooccur.The frequencywiththehighestpeakmagnitudeisknownasthedom inantorfundamental 19

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cavitytone.Thefrequenciescouldthereforeberepresente dempiricallyby f = U 1 L ( m ) 1 k + M 1 (1{1) wheretheratiooftheconvectionspeedtothefreestreamvel ocityisgivenbytheconstant k andthefreestreamvelocityis U 1 .Theintegermodenumberisgivenby m and isthe phasedelayconstant. Rossiter'sformulaisbasedontheexistenceofafeedback-r eceptivityloopand anintegernumberofphaseshiftsaspreviouslydiscussedwh eredisturbancesare propagatedintheformofanacousticwaveinsidethecavity. CattafestaIII etal. ( 1999 ) haspreviouslynotedthatthisphaseshiftisrequiredforse lf-sustainingoscillations. Rossiter'sequationislimitedinthatitdoesnotaccountfo rthreedimensionalityofthe roweld.Specically,itisonlycapableofpredictingthel ongitudinalcavitymodesand theempiricalconstantsareonlyvaliduptomoderateMachnu mberstypically M 1 2. Manyresearchers,suchas Unalmis etal. ( 2004 ),haverettheconstantstoextendthe regionofvalidity. Rossiter'soriginalequationwasslightlymodiedby Heller&Bliss ( 1975 )who assumedthetemperatureinthecavityisthesameasthefrees treamstagnationtemperature thuscorrectingtheratioofthespeedsofsound a e a c .Equation 1{2 iscommonlyreferredto asthemodiedRossiterequation St = fL U 1 = m M 1 p 1+ r 1 2 M 2 1 + 1 k (1{2) wheretheintegermodenumber m and k areasdenedabove.Theconstants k and havedependenceonthefreestreamconditionsandboundaryl ayershapefactor.The constant k hasbeenshowntorangefrom0 : 5 0 : 75forfreestreamMachnumbersranging from0 : 4 2 : 5asreportedby Unalmis etal. ( 2004 ).Forthinboundarylayerswith M 1 2 : 0theacceptedvaluesare =0 : 25and k =0 : 57asreportedby Dix&Bauer ( 2004 )and Unalmis etal. ( 2004 ).Undertheseconditions,themodiedRossiterequation 20

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hasdemonstratedconsiderablesuccessinpredictingthepo ssiblefrequenciesormodesthat arelikelytoappearinsupersonicrows.Althoughthesesemi -empiricalmodelshaveproven quiteeectiveatpredictingthefrequenciesofthetonesth ereisnomeansbywhichto predicttheiramplitudesorwhichtoneswillexisthencethe needforcontinuedstudy. Cavityrowhasbeenpreviouslymodeledbyassuminganinvisc idrowandsplitting theruiddomainintoaninteriorandexteriorregionseparat edbyathinmixinglayer asproposedby Bilanin&Covert ( 1973 ).Thethinmixinglayerwasapproximatedbya vortexsheetwiththedominantpressureoscillationsoccur ringatthetrailingedge.These pressureoscillationsweremodeledbyaperiodicacousticm onopole.Theyassumedthe drivingdisturbanceoftherowemanatedfromtheleadingedg eandwaslaterveriedby controlsensitivityexperimentsconductedby Kegerise etal. ( 2004 ).Thoughthismodelis animprovementtoRossiter'sequationasiteliminatesthen eedforempiricalconstants,it faltersifthevortexsheetbecomesstablewhichtypicallyo ccursnear M 1 2 : 5. Tam&Block ( 1978 )performedexperimentalinvestigationsofsubsoniccavit yrowfor M 1 0 : 4.Fromtheseinvestigationstheypostulatedthatthecavit yformedrowinduced oscillationsbymeansoftheshearlayerrollingupandgener atingacousticwavesanalogous toFigure 1-2 .Twoassumptionswererequiredforthisproposalwhichwere thecavity rowwasdominantlytwodimensionalandthemeanrowinterior tothecavityiszero.It hasbeenpostulatedthatthreedimensionaleectsofcavity row,orthecouplingofthe longitudinalandverticalmodes,arelikelyofsecondaryim portanceaslongasthewidthis smallsothathigherorderspanwisemodesarenotgenerateda snotedby Bilanin&Covert ( 1973 )and Zhuang ( 2007 ). Rossiter ( 1964 )notedthewidthofthecavitydoesaect theamplitudeoftheoscillationsand Cain etal. ( 1999 )againnoteditisofsecondary importance.TamandBlockarguedthattheshearlayerrapsin toandoutofthecavity nearthetrailingedge.Theyfurtherpostulatedthattheupw ardmotionoftheshearlayer mustbeuncorrelatedwiththegenerationofacousticwaves. Thiswasjustiedbasedon thenotionthatfortheacousticwavestobeemitted,impinge mentontheaftwallofthe 21

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cavitymustoccur.Iftheshearlayeronlyoscillatedabovet hecavity,inessencecovering thetrailingedgeofthecavity,therewouldbenomeansforac ousticwavegeneration astheshearlayerwouldrowfreelyabovethecavity.Theythe narguedthatonlythe downwardmotionoftheshearlayercouldgeneratesignican tdisturbancesandradiation ofacousticwavesthroughouttheinteriorofthecavity. FollowingthemodelofTamandBlockacavityexposedtounifo rmfreestream supersonicrowtypicallyexhibitsafullydevelopedturbul entboundarylayerupstream ofthecavityleadingedge.Theboundarylayerseparatesot heleadingedgeandforms ashearlayer.TheKelvin-Helmholtzinstabilityisdrivenb ythemeanshearinthe velocityprole.Thismeanshearresultsinshearlayergrow thandtheorganizationof theunderlyingvorticalstructures.Thesesmallscalestru cturesconvectdownstreamand evolveintolargescalecoherentstructures.Consequently ,theshearlayerrolls-upand impingesontheaftwallofthecavity.Thisimpingementgene ratesresonantconditions whichinruencetheshearlayerthroughcomplexinteraction swiththeupstreamadvancing acousticwavesandtheshearlayer.Asubsonicincompressib lerowtheoreticallysensesthis feedbackinstantaneouslywhereasforacompressiblerowth ereexistsatime-lagbeforethe rerectingwavesconstructively(areinphaseandthussynch ronizedwiththeshearlayer) interferewiththeshearlayerdisturbancesasdiscussedin Boydston etal. ( 2008 ). Rowley etal. ( 2002 a ); Rowley&Williams ( 2006 ); Rowley etal. ( 2006 )suggested thatthedominanttonesincavityrowareduetotheinstabili tiesintheshearlayerand itscoupledinteractionwiththeacousticeldandrow.Thes trengthoftheseoscillations weredeterminedbynonlinearsaturationanddrivenbyexter naldisturbances.They usedasecondordersystemwithaonedimensionalstandingwa vemodeltoestimate theacousticaleectsandtheyderivedtheirownversionofR ossiter'sequationwhose derivationisnicelylaidoutintheappendicesof Song ( 2008 ). 22

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1.3CavityFlowwithControl Flowcontrolattemptstoalterthenaturalstateofaparticu larrowintoamore desiredstate.Flowcontrolstrategiesareclassicallycat egorizedasactiveorpassiveyet theydonothaveclearlyacceptedrigorousdenitions.Fort hepurposesofthisresearch theclassicationchosenisoutlinedinFigure 1-3 .Thisclassicationisbasedonwhether externalenergyisusedtocontrolanactuator,whetherther eexistparametersthatcan bemodied(suchassystemgainoroscillationfrequencies) afterthecontrolsystemis designedorwhetherthecontrolissteadyorunsteady.Activ econtrolprovidesenergyto therowbutrequiresexternalenergytooperatewhereaspass ivecontrolapproachesdo notrequireexternalenergybutstillmayprovideenergytot herow.Passiverowcontrol approacheshavebeenimplementedwithvaryingdegreesofsu ccess.Examplesofpassive rowcontroltechniquesarevortexgenerators,geometrical modicationssuchasslantedaft wallsordevicesplacedatthecavityleadingsuchasramps,s poilersorarodincrossrow. Figure1-3.Adaptedrowcontrolclassication.Reprintedw ithpermissionfrom CattafestaIII etal. ( 2008 ). Activerowcontrolmaybefurtherdividedintoopenandclose d-loopapproaches. Aclosed-loopsystemhasawelldenedfeedbackloopwhereas ensorisusedtochange theoutputofanactuatorwhichinturneectsthesensorclos ingtheloop.Inthecase 23

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ofacavity,thepressuresaretypicallymonitoredonthecav itywallsandprocessingof thedataoccursbeforethesignalisreceivedbytheactuator .Closed-loopsystemscan furtherbecategorizedonthebasisoftimescales.Ifthecon trolparametersareadjusted slowlycomparedtothedynamicsoftherowitisreferredtoas quasi-steadyrowcontrol. Ifthecontrolparametersareadjustedonatimescaleontheo rderofthatwiththerow dynamicsthecontroliscommonlyreferredtoasdynamiccont rol.Ifnosuchclosed-loop existsthesystemisconsideredopen-loopwhererowcontrol strategiesaretypically predeterminedwithnosensormeasurementandaretypically feed-forwardsystems. Activerowcontrolisattractiveduetotheabilitytoadaptt ovaryingfreestream conditions.Activecontroltechniqueshavebeensuccessfu latreducingnoiselevels inawiderangeofoperatingconditionssomeofwhichwillbeb rierydiscussednext. Detailedreviewsofstudiesonthecontrolofcavityrowscan foundinthereviewsby CattafestaIII etal. ( 2008 ), Williams etal. ( 2007 )and Colonius ( 2001 ).Thediscussion belowwilldiscusssomehighlightstosetupthestudiesofth isdissertation. CattafestaIII etal. ( 1997 )demonstrateddynamicclosed-loopcontrolinsubsonic cavityrow.Theywereabletoobtainatimetransferfunction tomodelthesystemwhich isrealizableforreal-timecontrolhardware.Cattafestae tal.wereabletoreducekeytones ontheorderof20dBwithupto20timeslesspowerthanrequire dforopen-loopcontrol. CattafestaIII etal. ( 2001 )usedadaptivepiezoelectricrapactuatorsforMach0.74 subsonicrowsuccessfullysuppressingtherstmodebymore than10dB.Improvements totheseoriginalstudieswerereportedin Kegerise etal. ( 2007 a b )wheretheydevelopeda real-timeadaptivecontrollerusingageneralizedpredict ivecontrol(GPC)algorithmwhich wassuccessfulforfreestreamMachnumbersrangingfrom0.2 75to0.38. Rowley etal. ( 2000 a b )appliedPODbasedGalerkinmethodsthatattempttoobtain low-ordermodelsoftherow.PODbasedmethodsallrequireth ebasisfunctionstochange oncethecontrolleristurnedonandthechallengeisderivin gamethodologytoaccount forthis.Inanattempttogainmoreinsightintothebehavior ofthecontrolledcavity, 24

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Rowley&Williams ( 2006 ); Rowley etal. ( 2002 b )havealsoconsideredalinearmodelfor cavityoscillationswhichincorporatestheeectofextern aldisturbances.Theyfoundthe systemtobeunstableundersomeconditionswhereperturbat ionsgrowuntilnonlinearities becomeimportantandthelinearmodelisnolongervalid.For otherconditions,the systemisfoundtobestablebutlightlydampedandactsasano iseamplier.They notethenonlinearitiesmaybeimportantforcavityoscilla tionsinotherparameter ranges.Theyhopethecontrolledsystemwillalwaysbestabl eimplyingthelinearmodels presentedintheirworkshouldalwaysbeusefulforundersta ndingthecontrolledsystem. Open-loopcontrol,ontheotherhand,doesnotusetheabovem entionedfeedback mechanism.Someofthetechniquesusedincludeupstreamste adymassinjection asstudiedby Alvi etal. ( 2003 )and Vakili&Gauthier ( 2004 ).Theeectivenessof piezoelectricactuatorswereinvestigatedby Kegerise etal. ( 2003 )and CattafestaIII etal. ( 2001 ).Suppressioncharacteristicsofharmonicblowingwerest udiedby Shaw&Northcraft ( 1999 )and Smith etal. ( 2000 ).Oscillatingrapsweredetailedin Smith&Shaw ( 1975 ) whilepoweredresonancetubeswereinvestigatedby Kastner&Samimy ( 2003 )and Stanek ( 2005 ). Stanek etal. ( 2000 )studiedfourhighfrequencyactuatorsinsubsonicand supersonicrow:rodincrossrow,resonancetube,spoiler(s aw-tooth)andpiezo-ceramic wedges.Theyreportedthemosteectivedeviceforreducing tonesatpeakfrequencies wastheresonanttube. Ukeiley etal. ( 2003 )investigatedpoweredwhistlestoimpose steadymass(nitrogenandhelium)blowingattheleadingedg eofasupersoniccavityand reporteddominantpeakreductionsofnearly85%withappare ntbroadbandsuppression. Zhuang ( 2007 )and Zhuang etal. ( 2003 2006 )havereportedupto10dBreductionin thebroadbandspectrumand20dBreductionsattonalfrequen ciesusingmassinjection throughleadingedgemicrojetswiththecavityrowatsupers onicspeeds. Williams etal. ( 2007 )conductedexperimentsandstudiedtheeectsofopen-loop forcingontheacoustic tonesgeneratedinasupersoniccavityrowandtheabilityof aclosed-loopcontroller tosuppressthetones.Theyfoundtheamplitudeofthedistur bancesscaledwiththe 25

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freestreamdynamicpressureandtheacoustictoneswerelin earlyproportionaltothe actuatorinputandwereabletoeectivelyreducetonesbyro ughly10dB. Commonlimitationsrealizedforopen-loopactivecontrola retherequirement forlargemass-ruxinjectionintothebaserow,thephysical structure(piezoelectric) tendtobepronetofatiguefailuresorcomewiththetrade-o ofaddeddragtothe row.Theinterestedreaderisdirectedtothemoreexhaustiv elistofreferencesfrom CattafestaIII etal. ( 2008 )and Colonius ( 2001 )formoredetailsandexamplesregarding openandclosedloopactivecontrol. Incontrast,passiverowcontroldoesnotdirectlyprovidee nergyinputtotherow andistypicallyimplementedbymeansofgeometricmodicat ions.Whilethesemethods reducethecomplexityofthecontrolsystemstheirbiggestd rawbacksaretheytendto beapointdesignandaresometimesdetrimentalatodesignc onditions.However,they arestillthesourceofagreatwealthofstudiesandhaveprov enusefulinmanypractical situations. Oneexampleofpassivestudiesisthenumericalstudyofslan tedleadingandtrailing edgewallswereperformedby Nayyar etal. ( 2005 ). Vikramaditya&Kurian ( 2009 ) conductedanexperimentalinvestigationofsupersonicrow overwallmountedcavities withaftwallanglesof15to90degrees.Thestrengthoftheup streamtravelingacoustic wavewasweakenedwithdecreasingaftwallangles.Furtheri nvestigationofpassive techniquescanbefoundintheWeaponsInternalCarriageand Separation(WICS)report by Dix&Bauer ( 2004 )whichprovidesanimmensedatabaseofexperimentalresult s forructuatingpressuresuppressionatvariousfreestream Machnumbersusingdierent leadingedgespoilersincludingfencesandsawtoothspoile rs. Arodspoiler,asillustratedinFigure 1-4 ,immersedinaleadingedgeboundary layerhasbeenstudiedindetailforthepastdecadewithvari ousdegreesofsuccess.In thistechniquetosuppresstheructuatingsurfacepressure withinthecavityacylinderis immersedsomedistance(G)othewallbutstillatleastpart iallyimmersedinsidethe 26

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Figure1-4.Supersonicdeepcavitywithapassiverodspoile rsuppressionmechanism. boundarylayeraheadofthecavityleadingedge.Thischange stherowfeedingintothe cavityshearlayerresultinginanalteredinteractionofth eshearlayerwiththecavity trailingedge. Shaw ( 1998 )appearstobetherstdocumentedstudytousetherodin cross-rowasatonesuppressiondeviceincavityrows. Stanek ( 2005 )and Stanek etal. ( 2002 a 2003 )continuedandexpandedShawetal.'sexperimentsinvestig atingrowsin theMachnumberrangeof M 1 =0 : 5 1 : 2.Theyelaboratedonthesignicanceof rodendeectsinsupersonicrowtoachieveadequatesheddin gcharacteristics.Some supersonicexperimentsfailedandStaneketal.haveattemp tedtolinkthistomissing orinappropriateendconditionsontherodwheretheeectiv enessoftherodwasgreatly reducedforsupersonicfreestreamconditions.Thatworkfu rthershowedthatthelackof endcapsleadtosignicantalterationsinthevortexsheddi ngcharacteristicsoftherods. Withthisclaim,itwasalsosuggesteddesigncriteriaforro dsusedfortonesuppression insupersonicrows.Hesuggestedagoodstartingpointtoach ievereliableoptimum suppressionwastouselargediameterrodsatleast50%ofthe boundarylayerthickness immersedroughly 2 3 intotheboundarylayer. Stanek etal. ( 2002 a )hypothesizedthatVon-Karmantypevortexsheddingby thecylinderathighfrequenciessuppliedthenecessarydis turbancestotheshearlayer consistentwithhighfrequencyforcing. Stanek etal. ( 2002 b )alsolaterpostulated thattheinteractionoftherodsheddingandshearlayerenha ncedtheshearlayer 27

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stabilitycharacteristics.Thiswasarguedtobeaccomplis hedbyalteringthemean shearlayergrowthrate.Staneketal.werejustifyingtheir assumptionsbasedonthe workof Wiltse&Glezer ( 1998 )whosuggestedthathighfrequencyforcing(introducing disturbancesatfrequenciesthatcorrespondtothespectra 'sinertialrange)wasthedriving mechanismbehindjetshearlayernoisesuppression.Theyno tedadrasticincreasein theturbulentdissipationratewhentheshearlayerwasexci tedbyfrequenciesthreeto fourtimeshigherthanit'smostunstablefrequency. Cain etal. ( 2001 2003 )performed numericalsimulationswhichagreedwithWiltseandGlezerb utalsoshowedareductionin turbulenceproductionrates. Ukeiley etal. ( 2002 )demonstratedthatarodfullyimmersedintheapproaching boundary,whichisupstreamandspansthecavitywidth,isam oreeectivesuppression devicethanatypicalfence.Theyshowedreductioninthebro adbandlevelsaswellas theresonanttonesforbothsupersonicandsubsonicfreestr eamconditionsandreported reductionsontheorderof20dB.Consistentndingsweredis coverednumericallywith investigationspreparedby Arunajatesan etal. ( 2002 2003 ).ThroughtheuseofCFD, theyhaveshownthatcontrolatsubsonicconditionsisparti allyachievedbyliftingand spreadingoftheshearlayerandthustheturbulentkinetice nergyintheshearlayer. Theypostulatethisminimizestheimpingementontheaftwal landthusthefeedback receptivitymechanismfortherow. Panickar&Raman ( 2008 a b )studiedsurfacepressuresuppressionbyhighfrequency excitationoftheshearlayerbyusingthecylinderincrossr owataMachnumberof M 1 =0 : 60.Thestabilitycharacteristicsoftheshearlayerwakepr oleswerestudiedwith anewmodeltoexplainthesuppressioncharacteristics.The irclaimisthatsuppression isachievedthroughtheintroductionofthehighfrequencye xcitationintheshearlayer whichdecaysrapidlyasitisconvecteddownstream.Thesein troducedfrequenciesalterthe meanvelocityproleneartheleadingedgeofthecavityandw ereshowntobeeectiveat reducingthecavityresonance. 28

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Sarpotdar etal. ( 2009 )and Sarpotdar&Raman ( 2009 )studiedthecylinderin crossrowcongurationforsubsonicrowwith M 1 =0 : 50 0 : 80inordertounderstandthe mechanismofresonancesuppressionusinglinearstability characteristicsofthemodied meanvelocityprole.Theyassumedtheshearlayerwasnearl yparallel(locally),inviscid andcompressibleandincorporatedthemodiedmeanvelocit yprole.Theyexaminedthe instabilitygrowthratesofexperimentallymeasuredmeanv elocityprolesinanattempt tocorrelatethemtotheirresultsforthecontrolledcases. Theyfoundthesuppression resultsvarygreatlyasthelocationofthecylinderischang eddespitenearlyconstantshear layergrowthrates.Itshouldbestressed,thecylinderused inthisstudywassizedsuch that d 1whichismuchlargerthanhasbeenrecommendedinpreviousr esearchby Stanek ( 2005 )and Smith etal. ( 2002 ).Theysuggestthatthelinkbetweenthecavitytone suppressionandtheinruenceofthecylinderontheshearlay erstabilityisweak.They stateothermechanismsforcavitysuchashighfrequencyfor cingandshearlayerlift-o needtobeexaminedmorecloselybeforeaconclusivestateme ntcanbemade. Both Bastrzyk etal. ( 2009 )and Sarpotdar etal. ( 2009 )investigatedhowthe streamwiselocationoftherodeectedtheructuatingsurfa cesuppression.They reportedthemosteectiveplacementoftherodwasnearthel eadingedgeofthecavity. Bastryzkstudiedtwodierentroddiameterswiththerodalw ayslyingontheroor leavingquestionsregardingtheabilitytoenhancesuppres sionatvariousgapheightsfor xedstreamwisepositions. Sarpotdar etal. ( 2009 )reportedsimilarresultsbutdidnot considerdierentroddiameters.Interestingly,theoptim alsuppressionreportedinthese studieswaswithmostoftherodlyingoutsidetheboundaryla yer.Theroddiameterwas alsoslightlylargerthanthemeasuredboundarylayerheigh twhichisinconsistentwith investigationsreportedpreviously.However,thesearefo rdierentfreestreamconditions thanthosebeingexaminedinthisresearch. Previousresearchers,someofwhomarediscussednext,have performedCFD computationsofsubsonicandtransonicrowoveracavityusi ngacylindricalrodas 29

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apassivesuppressionmechanism. Levasseura etal. ( 2008 )reportedsimilarresults as Ukeiley etal. ( 2007 )whencomparingarodandspoilerassuppressiondevices. Comte etal. ( 2008 )reportedastrongsensitivityoftherodsperformancebase donthe turbulencelevelsoftheupstreamboundarylayer.Theyquan tiedthisbyconsideringtwo casesfortheReynoldsAveragedNavier-Stokes(RANS)-Larg eEddySimulation(LES) coupling.Therstsimulationutilizedaclassiccouplingo ftheRANS-LESequations wherenoupstreamturbulenceisaddedtothesimulation.The secondsimulationadded turbulencegeneratedbytherecyclingmethodproposedby Lund etal. ( 1998 ).The upstreamforcingislimitedtothesizeoftheintegralscale oftheturbulencefoundin thecylinderwake.Theynotedsuppressioncaseswiththeinj ectedturbulentructuations agreedmorefavorablywiththeexperimentaldatawhiletheb aselinecavitysimulations remainedunsensitivetotheaddedructuations.Theyconclu dedthattheKelvin-Helmoltz instabilityimpingingontheaftwallofthecavityisreduce dbutthemeanrowproles betweenthebaselineandsuppressioncasedonotshowsigni cantdierences.Theyalso postulatedthatthemeanupwardderectionoftheshearlayer (lofting)doesnotseem toplayasignicantroleinthesuppression.Theyfurtherco njecturethatsmallscale vorticalstructuresinjectedintotherowfromthewakeofth ecylinderseemtoenrichthe frequencycontentalteringtheshearlayerprolewhichisc onsistentwithargumentsby Stanek etal. ( 2003 ). Smith etal. ( 2002 )summarizedexperimentsconductedbyLockheedMartin Aeronauticsintheirtransonicin-draftwindtunnelinFort WorthTexas.Theexperiments includedtheplacementoftheroddownstreamandupstreamof thecavityleadingedge, lowdragmodelrodsincorporatinga20 includedangleandarat-toprodwitha10 includedangle.Theconclusionswereconsistedwiththatre portedpreviouslybyStanek andaresummarizedasfollows: Largerrodsweregenerallymoreeectiverowcontroldevice s. 30

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Eectivesuppressionrequiredtherodtobeimmersedroughl y30%oftheboundary layerthickness. Rodsappeartomaintaintheirsuppressioncharacteristics forsupersonicrows. Todate Williams etal. ( 2007 )and Stanek etal. ( 2003 )haveprovidedthemost completesummaryofthemechanismsbelievedtobeatworkinr educinganddisturbing theacousticfeedbackandreceptivityofcavityrows: Thehighfrequencyrodsheddingconvectsdownstreamandbre aksdownthe coherentstructuresintheshearlayer.Theacceleratedene rgycascadesinthe inertialrangedestructivelyinterferewithandstarvethe lowerfrequencymodes. Theloftingoftheshearlayeralterstheimpingementpointo ntheaftwalland consequentlyaltersthemeanshearprole. Thethickeningoftheshearlayeralterstheinviscidinstab ilitycharacteristicsofthe shearlayer. Thecancelationoftheacousticfeedbackwavewithinthecav ityduetoany combinationoftheabove. Theultimategoalofthisresearchistogainanunderstandin gofhowtherowis alteredwhenarodisplacedinsidetheboundarylayeratthel eadingedgeofsupersonic cavityrow.Thiswillbeaccomplishedbydetailedanalyseso ftheacquiredquantitative roweldvelocityandsurfacepressuremeasurementsforbot hbaselineandmultiple suppressedcongurations.Thebroadbandnoiselevels,key spectrapeaksandwall pressureructuationswillbeinvestigatedtodeterminethe eectivenessoftherodasan actuatorandtheoptimalsizeandpositioningbasedonthesu personicfacilityatthe REEF.Meanroweldmeasurementsusingacomplimentofexper imentalandnumerical simulationsthroughthecavityshearlayerwillbeconsider edtodeterminethelikely suppressionmechanismusingtherodincrossrow. 31

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CHAPTER2 STATISTICALANALYSISTECHNIQUES Throughoutthisdissertationatimeseries,denedasanord eredsequenceof observations,willbeusedinthediscreteformwhereaseque nceofobservationsare takenatevenlyspacedpointsintime.Theacquiredtimeseri esisassumedtobeweakly stationaryimplyingthemeanvaluesandautocorrelationsa retimeinvariant.Thetime seriesisfurtherassumedtobeergodicimplyingallofthest atisticscomputedfromthe singletimeseriesareequaltothosecalculatedfromanense mbleaverage. 2.1UnsteadyPressureMeasurements Severalstatisticalconceptswillbeusedfortheanalysiso funsteadypressure measurementsinthisstudyandaresubsequentlyintroduced .Theructuatingcomponent ofthepressurewillbetheprimaryvariableofinterestforp ressuremeasurementsand arefoundfromaclassicalReynoldsdecomposition.Equatio n 2{1 givestheructuating pressure p 0 ( t )= p ( t ) p ( t )(2{1) wheretheprimerepresentstheructuatingcomponentofthev ariable,theoverbar representsthemeanandavariablewithnodemarcationrepre sentstheinstantaneous value.ThemeantakesthetraditionaldenitiongiveninEqu ation 2{2 p ( t )= 1 N N X 1 p ( t )(2{2) wherethenumberofindependentobservationsisgivenby N whichissucientlylargeto ensurethemeanvaluehasconverged.Thevariationofthemea suredunsteadypressure p rms = vuut 1 N 1 N X 1 p 0 2 ( t )=std( p ( t ))(2{3) isstudiedusingtheroot-mean-square(rms)oftheructuati ngvariableandiscommonly referredtoasthestandarddeviationinstatisticallitera ture.Computedpercentreductions 32

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follow %di= j baseline suppressed j baseline 100%(2{4) andarefoundbyconsideringthesuppressedandbaselineval uesofthevariableofinterest. 2.1.1SpectralCalculations TheFourierTransform(FT)convertsasignalfromthetime( t )domaintothe frequency( f )domainwhichisthecyclicalfrequency.TheFouriertransf ormisthebasis formuchofthepressuresignalanalysisinthisdissertatio n.ADiscreteFourierTransform (DFT)isappliedtothetimeseries f n ( t )andsubsequentlytransformedyielding A k ( n )= N 1 X n =0 f n ( t ) e i 2 N kn ;k =0 ; 1 ; 2 ;:::;N 1(2{5) wherethediscretesignalisassumedtorepeatinaperiodicf ashion.TheFastFourier Transform(FFT)issimplyaclassofspecialalgorithmswhic himplementtheDFT withconsiderablesavingsincomputationaltimeandisthus usedextensivelyinthis dissertation. Powerspectraldensityfunctions(PSD)showthestrengthof powervariations asafunctionoffrequencynormalizedtoa1Hzbin.Simplysta ted,itshowsatwhich frequenciesvariationsarestrongandatwhichfrequencies variationsareweak.Throughout thisworktheuseoftheperiodogrammethodwillbeimplement edforestimatingpower spectra.Followingtheworkof Welch ( 1967 )letasamplefromastationarystochastic sequencebegivenby X ( j )where j =0 ; 1 ; 2 :::N 1.Thesequencemaybesplitinto segmentsoflength L ,thatmayuseoverlapping,withthestartingpointsseparat edby distance D .Therearethen K segmentsof X k ( j )suchthattheycovertheentirerecord ( N )and( K 1) D + L = N .Amodiedperiodogramiscomputedforeachsegment oflength L whereawindowisselected W ( j )for j =0 ; 1 ; 2 :::L 1.Thesequences X k ( j ) W ( j )areformedandtheniteFouriertransformsarecomputedon eachsegment A k ( n )= 1 L L 1 X j =0 X k ( j ) W ( j ) e 2 kijn L (2{6) 33

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where i = p ( 1).The K periodogramsarethencomputed I k ( f n )= L U j A k ( n ) j 2 (2{7) where k =1 ; 2 ;:::K f n = n L for n =0 ; 1 ;:: L 2 and U = 1 L L =1 P j =0 W 2 ( j ).Thespectralestimate, givenby ^ P ( f n ),isthemeanoftheseperiodograms ^ P ( f n )= 1 K K X k =1 I k ( f n )(2{8) witharesultingspectralwindowwhoseareaisunityandwhos ewidthisoforder 1 L .The frequencyresolutionofthePSDestimateisgivenby f r = f s nt where f s isthesampling frequencyandntisthelengthofanindividualblockoftheP SDestimate. 2.1.2JointTime-FrequencyAnalysis Onelimitationoftimeaveragedfrequencyspectraisthatit isnotcapableof distinguishingwhethertheresonantmodescoexistorexper iencemodeswitching.Mode switchingoccurswhenthepeakwiththedominantamplitudes witchesmodesasa functionoftime.Jointtime-frequencyanalysisprovidesi nformationcontainingfrequency, timeandamplitudeallowingonetovisualizethetime-evolu tionofthefrequencycontent. ThespectrogramsarecomputedusingShort-TimeFourierTra nsform(STFT) STFT ( t;f )= Z 1 1 z ( t ) w ( t ) e i 2 f d (2{9) where z ( t )isawindowfunctionwithashorttimeduration.Thetimedep endentinput signalispartitionedintoseveraldisjointedoroverlappe dblocksbymultiplyingthesignal withaHanningwindow.TheFastFourierTransformisthenapp liedtoeachblock.The blocksoccupydierenttimeperiodssotheresultingSTFTco mputesthespectralcontent oftheinputsignalateachperiod.TheSTFTthereforerepres entsthetime-dependent powerspectrumoftheinputsignal. AwellknownpropertyoftheFouriertransformpair s ( t )and S ( )istheuncertainty principledescribedin Qian&Chen ( 1996 ).Theprinciplestatesthetimeduration t of 34

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s ( t )andthefrequencybandwidth arerelatedby t 1 2 .Thisimpliesalongertime durationoftheinputsignalwillresultinasmallerfrequen cybandwidthandconversely thelargerthefrequencybandwidthofthetransformedinput signal,theshorterthetime durationoftheinputsignal.Thisillustratesthetradeob etweenthefrequencyresolution andtimeresolutionwhenusingthismethod.Theprecisionof thetransformisdetermined bythesizeoftheselectedwindowandisxedforallfrequenc ies.Itisrecognizedthat thiscouldbealimitingcaseintheanalysisofturbulentrow elds.Jointtimefrequency analysisusingSTFThasbeenusedsuccessfullyforhighlyre sonantcavityroweldsas evidentintheworks Kegerise etal. ( 2004 )amongothers. 2.1.3CorrelationAnalysis Cross-correlationandcross-coherenceanalysiswasusedt ohelpdeterminehow stronglytwosignalsarerelated.Ifthesignalsareidentic althecorrelationcoecient isunityandiftheyareunrelatedthecorrelationcoecient iszero.Ifthesignalsare identicalexceptthatthephaseisshiftedbyexactly180deg reesthenthecorrelation coecientis-1.Whentwoindependentsignalsarecomparedi nthetimedomainthe procedureisknownascross-correlation.Autocorrelation isaspecialcaseofcross-correlation wherethesamesignaliscomparedtoitself.Theautocorrela tioncoecientofasignal x ( t )isgivenas ^ R xx ( r t )= 1 N r N r X n =1 x 0n x 0n + r r =0 ; 1 ; 2 ;:::;m (2{10) wherethetimelagisgivenby = r t r isreferredtoasthelagnumberandthe maximumnumberoflagsisgivenby m .Thecross-correlationoftwosignals x ( t )and y ( t ) isthen ^ R xy ( r t )= 1 N r N r X n =1 x 0n y 0 n + r ^ C xy ( r t )= ^ R xy ( r t ) q ^ R xx ( =0) q ^ R yy ( =0) r =0 ; 1 ; 2 ;:::;m (2{11) where ^ R xy and ^ C xy areisthecross-correlationandnormalizedcross-correla tioncoecient estimatesrespectively. 35

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Thecross-correlationperformedinthetimedomainisacomp utationallyinecient approach,asdescribedin Papoulis ( 1962 ),whichmaybecomputedfasterbytheuseof theFFT.Theenhancedcorrelationiswrittenbyconvolvingt herstfunction, g ( t ),with thetime-reverseofthesecondfunctiongivenby h ( t ) g ( t ) h ( t )= g ( t ) ?h ( t )(2{12) wherethe ? denotesaconvolution, isthecomplexconjugateand indicatesthe cross-correlation.Theconvolutionisdenedas g?h = Z 1 1 g ( ) h ( t ) d: (2{13) Theresultingcross-correlationasdemonstratedby Papoulis ( 1962 )istherefore g h ( t )= Z 1 1 g ( ) h ( t ) d: (2{14) Theordinarycoherencefunctionrevealshowwellonedatase tcouldbeexplainedby alinearttedversionofanotherdataset.Theordinarycohe rencefunction,givenin Equation 2{15 ,isameasureoftheaveragecorrelationbetweentwotimeser iesateach frequency ^ r 2 xy ( f )= j ^ S xy ( f ) j 2 ^ S xx ( f ) ^ S yy ( f ) (2{15) where ^ S xy ( f )isthecrossspectrabetweenthetwosensorsand ^ S xx ( f )and ^ S yy ( f )are theauto-spectraofthosetwosensors,detailedin Bendat&Piersol ( 1986 ),whichareall estimateddiscretely.Normalizationvaluesofonereprese ntperfectcorrelationbetweenthe twosensorsatthatgivenfrequency. 2.2TurbulentStatisticsandFloweldMeasurements Particleimagevelocimetrywasusedextensivelyinthisres earchthereforeabrief descriptionofthecomputedvariablesarepresentednext.I magepairswereacquired atanintervalof7Hzwhichforthisroweldmakesthemappear asrandomlysampled independentmembersofanensemble.Thestreamwiseandnorm alcomponentsofthe 36

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velocityvectorsarefunctionsofspaceandsnapshotandthe ( x;y;N )nomenclaturewill bedroppedforconvenienceinthedenitionsbelow.Velocit ycomponents( u;v;w )willbe takenasthestreamwise,normalandspanwiserowdirections whichcorrespondstoxed coordinatedirections( x;y;z )respectively.Tensornotationforthevelocityeldmaybe usedforconciseness( u i )where i =1 ; 2 ; 3correspondto u;v;w velocitiesrespectively. Theensembleaverageofthevelocityis h u i i = 1 N N X n =1 u i n (2{16) where N isthetotalnumberofvalidvectorsatagivenpointinthevec torroweld.The ructuatingvelocityisfoundfromclassicalReynoldsdecom position u i 0 = u i h u i i : (2{17) wherelowercasevariablesrepresenttheinstantaneousvel ocities, hi representtheensemble averagedquantitiesandprimesmarktheructuatingcompone ntofagivenvariable.The root-mean-squareofthevelocitiesare h u i; rms i = vuut 1 N 1 N X n =1 ( u i 0 ) n 2 (2{18) wheretheunbiasedestimatorofthepopulationvarianceisu sed. Quantitativedensitymeasurementscouldnotbemeasuredfo rtheexperimentsso turbulentstresscalculationsneglectdensityaswasdonef orconsistencywithnumerical calculations.TheReynoldsstresstensorsimpliesaccord ingly R ij = h u i 0 u j 0 i (2{19) wheretheshearstresscomponentsarefoundwhen i 6 = j andnormalstressesarecomputed when i = j .ThesimpliedequationfortheTurbulentKineticEnergy(t ke)is tke= 1 2 u i 0 u i 0 (2{20) 37

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whichissimplythetraceoftheReynoldsstressmatrix.Itsh ouldbenotedsinceall velocitieswereavailableforthenumericalsimulationsth eturbulentkineticenergywas computedusingallthreecomponentsofthevelocityeld. Thelengthscalevariationsintheshearlayerarestudiedby employingatwopoint spatialcross-correlationonthecomponentsoftheructuat ingvelocitywithanorigin denedatsomexedpoint( ~x )withaknownseparation( ~r ).Thetwopointspatialtensor isthen ij ( ~x;~r )= h u i 0 ( ~x ) u j 0 ( ~x + ~r ) i q n u i 0 2 ( ~x ) q n u j 0 2 ( ~x + ~r ) (2{21) whereatzeroseparationthediagonalelementsrepresentth enormalturbulentstresses andtheodiagonalsrepresenttheturbulentshearstresses .Withthespatialseparation representedinthecorrelation,onemaylookathowtheveloc ityatthedenedoriginis relatedtothevelocityatanotherpointwhichinherentlyde nesalength( ~r )ofhowfar apartthevelocitiesarerelated.Similarly,pressurecorr elationswereperformedonthe cavitysurfacesusingtheexpression P ( ~x;~r )= h p 0 ( ~x ) p 0 ( ~x + ~r ) i q n p 0 2 ( ~x ) q n p 0 2 ( ~x + ~r ) : (2{22) Thez-vorticity( z )eldcalculationwasbasedonacirculation( ij )estimate z i;j = i;j 8 X Y i;j = X ( u i 1 ;j 1 +2 u i;j 1 + u i +1 ;j 1 )+ Y ( v i +1 ;j 1 +2 v i +1 ;j + v i +1 ;j +1 ) X ( u i +1 ;j +1 +2 u i;j +1 + u i 1 ;j +1 ) Y ( v i 1 ;j +1 +2 v i 1 ;j + v i 1 ;j 1 ) (2{23) withdierencingincludingtheneighboringeightpointsas detailedin Rael etal. ( 1998 ).Theapplicationofthisdierentiationschemeiscompara bletoapplyingacentral dierencescheme,whichhasatruncationerrorestimateof0 : 7 U X ,toasmoothedvelocity 38

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eldwherethevelocityuncertaintyisgivenby U .Applicationofthisdierencingtendsto resultinalowertruncationerrorwhichmaybeestimatedby0 : 61 U X 2.3ProperOrthogonalDecomposition TheProperOrthogonalDecomposition(POD)asproposedby Lumley ( 1967 )isa commonlyusedtoolforextractingabasissetfromasetofexp erimentalornumerical datawhichexhibitnon-homogeneousconditions.Thedatais representedasafunctionof spaceandtimeandthePODdeterminesabasissetoforthogona lfunctionsofspacewhich spanthedata.Giventhat u i ( ~x;t )isafunctionofspaceandtimethePODdetermines orthogonalfunctions i ( ~x ) ;j =1 ; 2 ;:::; wheretheprojection u ( ~x;t )= n X i =1 a i ( t ) i ( ~x )(2{24) isoptimizedsuchthattheprojectionontotherst n functionsresultsinthesmallesterror E ( jj u i ^ u i jj 2 )whereErepresentsthetimeaverageand( jj u i ^ u i jj 2 )isthe L 2 Euclidian normonspatialfunctions. ThePODcoecientsgivenby a i ( t )areuncorrelated.Themodesareoptimal,in convergingonameansquaresense,forreconstructingasign alimplyingthatamongall otherlineardecompositionsEquation 2{24 isthemostcompact.Foragivennumberof PODmodestheprojectionwillcontain,onaverage,themostt urbulentkineticenergy. Theempiricaleigenfunctions,commonlyreferredtoasPODm odes,arerepresented by i assumingthetwopointspatialvelocitytensorisusedasthe kernel.ThePOD modesarecomputedbysolvingthefollowingFredholmintegr alequationasdescribedin Berkooz etal. ( 1993 ) Z K ( ~x;~x + ~r ) ( ~x + ~r ) dy = ( ~x )(2{25) whosekernel K ( ~x;~x + ~r ).Thekernelisalsoreferredtoasthetwo-pointturbulentv elocity correlationtensorandisgiveninEquation 2{26 K ( ~x;~x + ~r )= h u i ( ~x ) u i ( ~x + ~r ) i (2{26) 39

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wherethe notationdenotesthecomplexconjugateand x and ~x + ~r arevectorsthat representanytwopointsofacartesiancoordinatesystem.E quation 2{25 maybemore compactlywrittenasaneigenvalueproblem K = : (2{27) AccordingtotheHilbert-Schmidttheoremexplainedin Riesz&Nag ( 1955 )thereexistsa diagonaldecompositionofthekernelgivenby K ( ~x;~x + ~r )= 1 X i =1 i i ( ~x ) i ( ~x + ~r )(2{28) where i and i aretheeigenvectorandeigenvaluepairofthekernel K and i 0 since K isapositivedenitematrix.Theensemblemaybereproduced byamodal decompositionusingtheeigenvaluesandeigenvectorsasgi venbyEquation 2{24 Equation 2{27 iscommonlysolvedusingtheMethodofSnapshotsdevelopedb y Sirovich ( 1987 ).Thismethodwasintroducedasanecientapproachwhenthe resolution ofthespatialdomain(N)exceedsthenumberofindependento bservations(M).The discreteformofthekernel(K)givenby K = 1 N YY T (2{29) where Y isavectorgivenby Y =[ u ( x 1 ;t ) ;u ( x 2 ;t ) :::u ( x N ;t i )] T Y T isthetransposeof Y and x j isthe j th gridpointof Y .Thedatasetcanbeobtainedfromeitheranumerical simulationorexperimentalobservationsoracombinationo fthetwo.Anumerical solutionforthePODModesfromEquation 2{27 istypicallysolvedusingasingularvalue decomposition svd K = T YY T = (2{30) whereisan m m orthonormalmatrixformingleftsingularvectorswiththec olumn vectorswhoformanorthogonalbasisandarethePODmodes.= N isan m N 40

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semi-positive-denitematrix. T isthetransposeofandformstherightsingular vectors. ThemethodofsnapshotsisappliedassumingthePODmodesare alinearcombination ofthesnapshots = YA (2{31) where Y isan m N matrix, m isthenumberofspatialpointsand N isthenumberof snapshots.Thematrix A isdeterminedbysubstitutingEquation 2{31 intoEquation 2{30 whichyieldstheeigenvalueproblem Y T YA = A CA = A (2{32) where A representstheeigenvectorsoftheempiricalcorrelationm atrix C = Y T Y .The PODmodesmaythenbedeterminedbynormalizingthecolumnve ctor j ofbyit's L 2 norm j = j jj j jj ;j =1 ; 2 ;:::;n: (2{33) Statisticallyspeaking, j representsthevarianceofthedatasetonthedirectionof thecorrespondingPODmode.Inthecaseofthisdissertation thisimpliesthatsinceuor vwillrepresentcomponentsofaructuatingvelocityeld,t hen j measurestheamountof turbulentkineticenergycapturedbythecorrespondingPOD mode.Thetotalturbulent kineticenergycapturedinaPODmaythenberepresentedby E = n X i =1 n : (2{34) Therelativeenergycapturedatthe n th modemaythenbeexpressed E n = n n P i =1 i : (2{35) 41

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2.4Non-Dimensionalization Forsimplicity,thisresearchattemptstomaintainconsist entnon-dimensionalplotting wherestreamwise(+ x )lengthswilltypicallybedividedbythecavitylength( L )and normalheights(+ y )bythecavitydepth( D )whichistypicalconventionforcavityrows. TheexceptionfortheseguidelinesfallsinChapter5wheret hesimulationconsistedof tworodsofdierentdiameter( d )immersedinaratplatesupersonicboundarylayerwith aheight( )atvariousgapheightswithnocavitypresent.Theappropri ateturbulent lengthscalesforthistypeofboundarylayerrowmaybetaken astheboundarylayer heightinthenormaldirectionandtheroddiameterinthestr eamwisedirectionand thereforethenormalandstreamwisecoordinatedirections arenon-dimensionalizedby and d respectively.Toattempttoreduceambiguity,thecapitall etter D willrepresenta givencavitydepthwhilethelowercaseletter d willrepresentagivencylinderdiameter throughouttheremainderofthisresearch. Thefreestreamdynamicpressure( Q 1 )canbeshowntobeafunctionofthe freestreamtotalpressure( P 0 ).Thefreestreamdensity( 1 )maybefoundfromthe idealgaslawgiveninEquation 2{36 1 = p 1 RT 1 (2{36) where p 1 isthefreestreamstaticpressure, T 1 isthefreestreamstatictemperatureand R isthespecicgasconstant.Thecompressiblerowrelationf orthefreestreamvelocity ( U 1 )is U 1 = M 1 a 1 = M 1 p rRT 1 (2{37) where a 1 and M 1 arethefreestreamspeedofsoundandMachnumberrespective ly. SubstitutionofEquation 2{36 andEquation 2{37 intheequationfordynamicpressure yields Q 1 = 1 2 1 U 1 2 = r 2 p 1 M 1 2 : (2{38) 42

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Thevariable r representstheratioofspecicheatsandiscalculatedfrom r = C p C v = 1 : 4.Theisentropiccompressiblerowrelationfortheratioof thestaticandstagnation pressuresderivedin John&Keith ( 2006 )is p 1 P 0 = 1+ r 1 2 M 1 2 r r 1 : (2{39) Substitutionofthestaticpressureasafunctionofthestag nationpressureintoEquation 2{38 yieldsanexpressionforthefreestreamdynamicpressureas afunctionofthe freestreamstagnationpressureyielding Q 1 = r 2 P 0 M 1 2 1+ r 1 2 M 1 2 r r 1 : (2{40) Powerspectraldensityplotsandroot-mean-squarepressur eswillbenon-dimensionalized bythemeanfreestreamdynamicpressure PSD = p PSD f s Q 1 (2{41) p rms = p rms Q 1 (2{42) followingEquations 2{41 andEquation 2{42 respectively. Whenacavityispresentthestreamwiseandnormaldirection sarenon-dimensionalized bythecavitylengthanddepth x = x L ;y = y D : (2{43) Whennocavityispresent,asisthecaseforChapter5,theapp ropriateturbulentlength scalesaretheroddiameterandtheboundarylayerheight.Th estreamwiseandnormal directionsarethereforenon-dimensionalizedaccordingt o x = x d ;y = y : (2{44) Therodkeynon-dimensionalvariablesaregivenbelow d = d ;G = G (2{45) 43

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wherethevariable G isthegapheightbetweenthebottomofthecylinderandthewa ll, istakenastheboundarylayerheightand d istheroddiameter. Allvelocities(mean,ructuatingorroot-mean-square)are non-dimensionalizedbythe meanfreestreamvelocitygivenbyEquation 2{37 whichgives u i = u i U 1 ;u i rms = u i rms U 1 : (2{46) ThecorrespondingReynoldsstressesandturbulentkinetic energyarenon-dimensionalized bythefreestreamvelocitysquaredgiving R ij = R ij U 1 2 ;tke = tke U 1 2 : (2{47) Thez-componentofthevorticityeldisnon-dimensionaliz edbytheratioofthecavity lengthandfreestreamvelocityyielding z = z L U 1 : (2{48) 44

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CHAPTER3 THEFACILITYANDEXPERIMENTALTECHNIQUES Thischapterwillbeginwithadiscussionregardingthedesi gnofthesupersonicwind tunnelwhichwasdesignedandinstalledattheREEF.Thisdis cussionisfollowedby explanationsoftheexperimentaldataacquisitiontechniq uesandthespeciccavity geometriesstudied.Thechapterisconcludedwithasummary oftheanticipated uncertaintyoftheexperimentalobservations. 3.1FacilityOverview AblowdownSWTasillustratedinFigure 3-1 anddetailedin Dudley etal. ( 2008 ) wasrecentlydesignedandinstalledattheREEF.Thetunnele mphasizedamodular designandincludesopticalaccessthroughthetest-sectio ntoallowfortheuseofPIVand Schlierenimagingtechniques.Thefacilityusesa44m 3 storagetankwithhighpressure airsuppliedfromarotaryscrewQuincyCompressor(modelQS I1000A/C)capableof producing26.4m 3 = minat1448kPathrough50.8mmdiameterpiping.Fromtheexit ofthe compressortheairrowsthroughadesiccantdryerwhichlowe rsthedewpointoftheairto 232.2Kbeforeenteringthestoragetank. Figure3-1.Thesupersonicwindtunnel. Thetunnelgenerallyoperatesatastagnationpressureof17 2kPaallowingfor approximately180sofrun-time.Subsonicairisbroughtint otheplenumandltered throughahoneycombsubstrateusedtobreakdownanyremaini ngorganizedlarge scaleturbulentmotionsintherowconditioningsection.Th eairthenpassesthrough therstcontractiondesignedtosmoothlytransitionthecr oss-sectionfromcircularto 45

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rectangularwhichmatchestheinletfaceofthenozzle.Then ozzleisatwodimensional converging-divergingassemblywithanexitcrosssectiono f76.2mmwideby101.6mmtall. Thenozzlewasdesignedusingclassicalmethodofcharacter istics,foundin John&Keith ( 2006 ),andfurtherevaluatedwithRANSnumericalsimulationsre sultingintheMach contoursillustratedinFigure 3-2 .Theairnextpassesthroughaconstantarea Figure3-2.RANSsolutionofthesupersonicwindtunnelnozz le. test-sectionwithauniformrectangularcross-section.Ai rthenentersthediuser withdiverginganglestotaling5 : 5 tokeeptherowfromseparating.Theairisnally exhaustedthrough304.8mmpipingasshowninFigure 3-1 above.Thecross-sectionof thetest-sectionhasaheightof101.6mmandawidthof76.2mm .Itshouldbenoted thattheaspectratioofthistest-sectionissomewhatuncon ventionalinsupersonicwind tunneldesignwiththeheightbeinggreaterthanthewidth.T headditionalverticalheight allowedforamaximumlengthcavitytobeinsertedbygivingt heanticipatedoblique shockwaveattheleadingedgeofthecavityalongerstreamwi sedistancetotraverse beforebeingrerectedothetopwallasshowninFigure 3-3 .Computationalruid dynamicsandanalogyfromsimilarexperimentalstudieswer eusedinestimatingthe obliqueshockangleillustratedby .Assumingaperfectrerectionothetopwall,the test-sectionheightwaschosensothererectedshockwavewo uldfallaminimumof2D downstreamoftheaftwallofacavitywithalengthof228mm. 46

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Figure3-3.Test-sectionwithanalyticalwavererection. Thetest-section,asillustratedinFigure 3-4 ,wassymmetricallydesignedsoeither atoporbottommountedcavitycouldbeinstalled.Thedesign consistsofaonepiece 6061-aluminumframewithfourrectangularcutoutsforopti calaccess.Theprimary opticalinsertswerefabricatedwith19mmthick N BK 7Schottopticalgradeglass machinedandpolishedto0 : 5 1wavesratnessper25.4mm60 = 40scratch/diganda maximumallowablechipof0.25mm.Plexiglasandaluminumin sertsareavailablewhen visualaccesstothetunnelisnotrequired.Theassemblyiss ealedwiththeuseof1.6mm diameterO-Ringinserts. Figure3-4.Test-sectionwithratplate. 47

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3.2CavityDesign Twocavitieshavebeendesignedandbuiltforthisresearch. Therstcavityisanite spancavitywhichhasbeendesignedwitha L D =6, L W =3andalengthof76.2mm.The fullspan,ortwodimensionalcavity,isincludedintherese archwithanidentical L D ratio andlengthbutspanstheentiretest-section,ie W =76 : 2mm.Thecavityisafourpiece constructionasshowninFigure 3-5 .A12.2mmlongleadingedgeblockismountedto thealuminumframefollowedbyarectangularroorsection.T hetrailingedgeblockis constructedof6061-aluminumwheretwoplugs(piecesthats itrushwithtrailingedge blockfrontsurfacetomakethefull-spancavity)orsidewal lblocks(thatareillustrated inorangeinFigure 3-5 )areattached.Dependingonthedesiredapplicationthecav ity sidewallsarefabricatedwitheitherBK7glassorplexiglas swhenopticalaccessisdesired oraluminumwhenitisnot.Thisdesignallowsforeasytransi tionbetweenthefulland nitespancavitysharingaroorwherethebulkofthepressur etransducersweremounted. Figure3-5.Finitespancavitywithcloseupofrodmountedon leadingedgeblock Theleadingedgeblockofthecavitywasttedwithaxtureth atsupportstherod spoilerwhosesuppressioncharacteristicswerestudiedwi thspecicdetailsreportedin Chapter6.Anobliquetriangularcrosssectionmountwithan angleof20 wasdesigned forthecylinderendmountsforshockmanagement.Therodspa nsthefullcavitywidth forthenitespancavityanddetailsofthemountingstructu reforthetwodimensional 48

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cavitywillfollow.Itisunderstoodtheeectoftherodspan lengthmaybefurther parameterizedwhichisbeyondthescopeofthecurrentresea rch.Figure 3-6 illustratesthe meanpressurecontourstakenfromaRANSCFDsimulation.The numericalapproach forthesimulationispresentedindepthinChapter4.Thisso lutionwasacquiredat aconstanty-planeintersectingthesupportwhenpositione dataheightforthemost eectiverodplacementcorrespondingto d =0 : 43 ;G =0 : 57.Thepressuredisturbances interferingwiththecavityrowduetotheexpansionwaveso theinsidecornerare evidentandunavoidablewiththecurrentmountlocation.Th eleadingedgeattached obliqueshockrerectsothetest-sectionsidewallandsubs equentlybecomesveryweak uponreachingthecavitysidewalls.Suppressionstudiesus ingthefullspancavityhad Figure3-6.Simulationofrowaroundrodmountingstructure toensuretherewasminimalderectionorbendingoftherodbe causeitisnotfeasibleto stretchtherodthewholespanofthetest-sectionduetothes mallscalesoftheresearch (therodwouldcertainlybend).Withthatinmind,therodlen gthandwallthickness wasselectedtominimizerodderection.Therodsselected(f orboththreedimensional andfullspancavities)werehollowcross-section304stain lesssteelwithaminimumwall thicknessof0.2mm.Thelengthchosenwas38mm(25.4mmforth enitespancavity) resultinginderectionsoflessthan0.90mmforthesmallest diameterrodbasedonthe freestreamrowandsimplebeamcalculations.Itwasfoundth atneithertheanticipated 49

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longitudinalortransversenaturalfrequencyoftherodwer eofmajorconcernasthe fundamentalfrequencyexceededthecavityresonanttonesa ndtheanticipatedshedding frequenciesoftherods.Detailsofthederectionandnatura lfrequencyanalysiscanbe foundinAppendixB. Figure 3-7 illustratesacloseupoftherodandvariablesusedtodenei tslocation relativetothewallandcavityleadingedge.Thegapdistanc emeasuredfromthewallto thebottommostpartofthecylinderisdenotedby G .Thedistancetothecavityleading edgefromthemostforwardpartoftherodisdenotedby X 2 whichisxedat3.2mm.The extradistanceallowsforapressuretransducertobemounte dusedtomeasurethevortex sheddingfrequencyoftherod(s).Themountingstructuresa repocketedandlledwith asmallamountofepoxytoholdthecylinderinplace.Thelowe rcase d denestherod diameterwhile G cl and X cl givetheverticalandstreamwisepositionoftherodcenterl ine. Figure3-7.Rodmountinglocationrelativetocavityleadin gedge 3.3SWTCharacterization Qualitativeandquantitativemeasurementswereobtainedt oascertainthequality oftherowthroughthetest-section.Therewaslessthan2%sp anwisepressurevariation 50

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measuredatvariousstreamwisepositionsonthebottomwall .Thestreamwisepressure distributionmeasuredonthecenterlinevariedbylessthan 3%overthelengthofthe cavityconsideredinthisstudy.Themeasuredvariationisl ikelyattributabletothegrowth oftheboundarylayerwithaconstantcross-sectionofarea. Forafurtherdiscussionofthe sidewallboundarylayerresultsandpressurevariationsth einterestedreaderisreferred to Dudley etal. ( 2008 ).AUnitedSensorstainlesssteelboundarylayerprobewith a sensingheaddiameterof0.64mmratteneddownto0.20mmwasu sedtomeasurethe boundarylayerprolesonthebottomwall.Measurementswer etakenontheroor41.3mm downstreamofthenozzleexit. Figure 3-8 overlaystheboundarylayerasmeasuredintheSWTandthecom putational boundarylayerfoundattheexitofthenozzle.Thereisgooda greementbetweenthe numericalandexperimentalprolesdowntotheminimummeas urementdistanceinthe experimentof0.30mmothewall.Themaximumpercentdiere ncewasontheorderof CFD Exp =3 : 8 mm =1 : 1 mm =0 : 9 mm H =1 : 2y ( mm )M 0 0 : 250 : 500 : 751 : 00 1 : 25 1 : 50 0 1 2 3 4 5 6 Figure3-8.Experimentalandnumericalboundarylayerpro lecomparison. 1.7%andthemeandierenceacrossallmeasurementlocation swas0.7%.Theboundary layerthickness( )inFigure 3-8 at99%offreestreamvelocityis3.8mm.Theresulting momentum( )anddisplacementthickness( )arethen0.9mmand1.1mmrespectively computedasshowninEquation 3{1 andEquation 3{2 .Thesecalculationswereperformed assuminganincompressiblerow o =1 .Thisassumptionistypicallyvalidwhere 51

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densityructuationsareassumedtobenegligiblewhichisco mmonlythecaseforboundary layerrowswhere M 1 2. = Z 0 1 u ( y ) U 1 dy (3{1) = Z 0 u ( y ) U 1 1 u ( y ) U 1 dy (3{2) H = (3{3) Theboundarylayercalculationsreportedareingeneralagr eementwithpreviousresearch conductedby Zhuang ( 2007 )wheretheapproachingboundarylayermomentumthickness wasfoundtobe0.80mmfor M 1 =2freestreamrow.ThecriticalReynoldsnumberfor transitionfromalaminartoturbulentboundarylayerforro woveraratplateisgenerally believedtooccurat Re = U 1 L > 10 5 whichisanorderofmagnitudelowerthanthe operatingReynoldsnumberfortheseexperiments.Theshape factorcalculatedaccording toEquation 3{3 wasfoundtoequal1.2.Asreportedin White ( 2006 )forrowoverarat plateaboundarylayeristypicallyconsideredlaminarif H 2 : 6andturbulentwhen H 1 : 3.Basedoncomparisonoftheboundarylayershapefactorand Reynoldsnumber calculationsitwasassumedtheboundarylayerwasfullytur bulentupstreamofthecavity. Itshouldbenotedthatalargeportionoftheboundarylayeri ssupersonicsothe introductionofthePitottubewillinduceabowshockinfron toftheprobe.Duetothis, acorrectionwasmadeforthedatapresentedinFigure 3-8 byrelatingthemeasured stagnationpressuretotheconstantstaticpressureatthem easuredcrosssectionaftera checktoverifythemeasuredpressureratioresultedinsupe rsonicrow.Thecorrection appliedisthewellknownRayleigh-Pitotformulafoundin John&Keith ( 2006 )wherean iterativeprocedureisusedtondtheMachnumberbasedonth emeasuredpressureratio p 02 p 1 = r +1 2 M 2 1 ( r +1) 2 M 2 1 4 rM 2 1 2( r 1) # 1 ( r 1) (3{4) where p 1 isthefreestreamstaticpressureand p 02 isthestagnationpressuremeasuredon theprobesideoftheshock.Aqualitativeassessmentofther owqualityintheSWTwas 52

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madeusingtheSchlierenimagingtechniqueshowninFigure 3-9 .Theguredisplaysan instantaneousSchlierenimagethroughthetunneltest-sec tionmidplanewiththerowfrom lefttorightandthelocationofthecavityoutlined.Wavesw ithanangleofroughly48 areevidentinFigure 3-9 thatformatthejointofthenozzleandtest-sectionframeat theupperandlowerjoints.Theobliqueshockatthelowerjoi ntlandsfardownstream (outoftheimage)afterrerectingothetopwall.Theuppert est-sectionandnozzlejoint disturbancehitsnearthecavitycenterandrerectso.Thej ointofthetop-panelinsert andtest-sectionframeresultinanoblique-shockwavethat fallsdownstreamotheaft wallofthecavity.Thestrongerwavesrerectdownstreamoft hecavitylocationofinterest andshouldnotinruencetherowofthecavity. Figure3-9.FlowvisualizationusingSchlierenfortheSWTt est-section. 3.4Instrumentation 3.4.1WindTunnelControl Thewindtunnelisinstrumentedtomonitorstaticpressurea tthenozzleexit,the stagnationtemperatureintheplenumandthestagnationpre ssureintherowconditioning section.ADruck TM PMP4015pressuretransducerisusedtomonitorthestaticpr essure upstreamofthenozzleatthesurfaceoftherowconditioning sectionwheretherowis assumedtobeslowenoughthatstaticandstagnationpressur esarenearlyequivalent. Thestagnationandstaticpressureweremeasuredintherowc onditioningsectionfor 53

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varyingdynamicpressuresinsomeinitialexperimentsandw ereallfoundtobewellwithin themeasurementuncertainty.Thetransducerhasapressure rangeof0-345kPawithan accuracyof0.08%fullscale.Asecondpressuretransducerm easurestheabsolutestatic pressureattheentranceofthetest-sectionanditsspecic ationsareidenticalexceptthe pressurerangeisfrom0-103.4kPa.Afourleadwire100nResi stanceTemperature Detector(RTD)ismountedintheplenumendrangetomonitort hestagnation temperature.TheRTDsensoris50.8mmlongwithanexposedpl atinumelementto increasetheresponsetimeandsuppliesmeasurementswith+ /-0.06%accuracyofthe measuredtemperature.Additionalmonitoringofthesystem stagnationconditionsis achievedthroughaOmegaprobemountedinthestoragetankwh ichmonitorsstagnation temperature,relativehumidityanddew-point.ADruck2068 kPapressuretransducer monitorsthetankpressureallowingforafeed-forwardcont rolifdesired. Thetunneliscontrolledbya152mmFisherControlvalveoper atedbyanPC-based LabVIEWalgorithmimplementedonaField-ProgrammableGat eArray(FPGA)card. Thecontrolvalveisoperatedbyaproportionalgaincontrol algorithmwhichsamples dataat1kHzandaveragesat4Hztoaccountforsystemrespons esanddelays.ANational InstrumentsPXI-1033chassisandPXI-7831Rcardusesmeasu rementsfromDruck 103.4kPaand345kPapressuretransducerstomeasurethenoz zleexitstaticpressureand thetunnelstagnationpressureintherowconditioningsect ion.The7831Rhas8analog inputsand8analogoutputs,eachwitha16Bitresolutionand adirectcurrentinput voltage-10to10V.Italsohasadigitalinput/outputinterf acewith96channelswitha 0to5.5Vrange.TheSWTiscontrolledusingaLabVIEWvirtual instrument(vi).With theassumptionofanidealgasandisentropicrow,theMachnu mberiscontrolledby measuringthestaticandstagnationpressureratioaccordi ngtotheisentropiccompressible rowequationgivenbelowwhere r =1 : 4. p 0 p = 1+ r 1 2 M 2 r r 1 (3{5) 54

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3.4.2UnsteadyPressureMeasurements Theructuatingsurfacepressuresonthecavitywallsaremea suredwithKulite TM pressuresensors.Twomodelnumberswereusedfortheexperi mentseachoftheXCQ series:XCQ-062-5DandXCQ-062-30A.Thenomenclatureisas follows: 1.062-1.5mmnominalouterdiameter2.5D-Dierentialpressuremeasurementof0-35kPa3.30A-Absolutepressuremeasurementwitharangeof0-207k Pa Thetransducersaremountedrushwiththecavitysurfacesan dheldinplacewith siliconeduringmeasurements.Thenaturalresonancefrequ encyspeciedbythevendoris 150kHzforXCQ-062-5Dand300kHzforXCQ-062-30Amodels.Fl atfrequencyresponses areexpectedupto20%oftheirnaturalfrequencieswithmaxi mumcombinednon-linearity andhysteresisof 0.5%offullscalepermanufacturerspecication.Thediame terofeach sensormodelsensorwasnominally1.67mmwhiletheXCQ-0625Dtransducersinclude a0.41mminchouterdiameterreferencetubeforsettingthep ressureonthebackofthe diaphragm.Thesensormeasuresthedierencebetweenthepr essureonthefrontsurface ofthesensordiaphragmandthepressureinthereferencetub eformeasuringthepressure dierential.Thebackpressuretubeswerehookedtotapswhi chweretypicallylocatedata spanwisedistanceof6.35mmoncenterlinesfromthedesired measurementlocation. Figure3-10.Dierentialpressuretransducers.Reprinted from Kulite ( 2010 ). Theoverpressureisratedastwicethelistedoperatingpres surewhiletheburst pressureisthreetimestheoperatingpressure.Iftheoverp ressuredierenceisexceeded 55

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thediaphragminthesensorwillexperienceaplasticdeform ationandcalibrationcurve wouldbepermanentlyaltered.Theuseofsuchtransducersto measuretimedependent dynamicwallpressureructuationshasbeenwelldocumented by Ukeiley etal. ( 2004 a ) forsubsonicandby Dix&Bauer ( 2004 )and Ukeiley etal. ( 2004 a )forhighspeedcavity rows. 3.5FlowVisualization Thefollowingsectionwilldiscusstheopticalcapabilitie sfortheSWTmeasurement techniquesthatwillbeusedinthisstudy.Thesectionwill rstdiscusstheinstrumentation andsetupforthequalitativeSchlierenmeasurementsinthe tunnel.Thiswillbefollowed byadiscussionoftheParticleImageVelocimetrysystemand itssetupforuseinthe SWT.3.5.1Schlieren Schlierenimagingcanbemadeusinganyoneofseveraldiere ntsetups.Thecommon drivingprincipleofeachmethodisthatthespeedoflightwi llchangeinmediumsof dierentdensity.Thisspeedchangeisreferredtoastheref ractiveindexofthemedium andisanon-dimensionalnumbergiveninEquation 3{6 .Therefractiveindexforairhas alinearrelationbasedonthespeedoflightinavacuum( c )andthephasevelocity( v p )in themediumthelightispassingthrough. n = c v p = p (3{6) Thevalue referstothematerial'srelativepermittivityand istherelativepermeability. Thisimplieslightwilltravelslowerwithincreasingmediu mdensity.Whenthisisapplied toadensityvariationinanairrowsuchascompressiblerows ,thepathofthewavefront isforcedtochangeorbend.Figure 3-11 illustratesthisphenomenaofabentwavefront passingthroughamediumofdierentdensities.Thelightwi lltravelfasterintheregion oflowdensityforcingthelighttobendwiththeangleofrefr action 2 lessthantheangle ofincidence 1 56

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Figure3-11.Bendingoflightduetodensitygradientoftwom ediums. Theobjectusedtoblockthelightraysistypicallycalleda' knifeedge'becausethe SchlierentestoriginatedwiththeFoucaultKnifeTest.Equ ation 3{7 relatestherelative displacementoftheimagetotherelativechangeinlightint ensityasexplainedby Settles ( 2001 ) I I = h h (3{7) wheretherighthandsideisreferredtoastherelativedispl acementoftheimageandthe lefthandsideisreferredtoastherelativechangeinlighti ntensity. Gladstone&Dale ( 1864 )introducedtheGladstone-Daleequationwhichisusedtoca lculatethederection angleofthelightduetothedensitygradients.Withtheuseo ftheGladstone-Dale equationitcanbeshownthattheSchlierensystemdependsup ontherstderivativeof thedensityinadirectionnormaltothelightpath. Figure 3-12 illustratesthetypicalZ-typeSchlierenarrangementused inthesupersonic windtunnellaboratoryattheREEF.Thepointsourcelightis rerectedoofaparabolic mirror( M 1 )whichcolumnatesthelightprovidingparallelbeamsthrou ghthetest-section. Theparallellightbeamsarethenrefocusedbyasecondparab olicmirror( M 2 ).Arazor bladeisusedinthefocalplaneofthesecondmirrorwithcapa bilitytobevertically adjusted.Theknifeedgeispositionedtoblockapproximate lyone-halfoftheincidentlight allowingauniformlyilluminateddarkerimage.Thelightis thenfocusedusingasimple lensandprojecteddirectlyontoadigitalcameracharge-co upledevice(CCD).TheCCDis connectedtoaPCthroughare-wireportfordigitalimageac quisition. 57

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Figure3-12.Z-typeSchlierenarrangementforqualitative rowassessment. Iftherearenodensityvariationsintherowtheknifeedgewi lluniformlyreducethe intensityoftheimage.Ifthereisadensityvariationbetwe entherstmirrorandsecond mirror,thelightleavingthesecondmirrorthatpassedthro ughthedensityvariationwill notfollowtheparallelpathtothefocusinglens,itwillbeb entasdescribedabove.Dueto thedivergentpathofthelightthelightiseitherdivertedt owardsorawayfromtheknife edge.Thisproducesthealternatinglightspotsanddarkspo tsseeninSchlierenimage giveninFigure 3-13 whichisanexampleofabulletduringsupersonicright. Figure3-13.Schlierenimagingofabulletinmid-right.Rep rintedwithpermissionfrom Davidhazy ( 2009 ). 58

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TheSchlierensysteminthisresearchconsistsoftwo152.4m mparabolicmirrorswith a1232mmfocallength.Lightissuppliedfroma3mmarc-lengt hXenonpulsedrashlamp witha2 spulsedurationcontrolledbyanexternaltrigger.ASonyXC D-V60digital videocamera(640 480VGAResolution)isusedforimageacquisitionandiscapa bleofa samplingrateof90Hz.3.5.2ParticleImageVelocimetry PIVisanon-intrusiveopticaltechniqueusedtomeasurethe velocityeldinaplane. Therowisseededwithtracerparticlesandilluminatedperi odicallybyalightsource thatisspreadintoathinsheetwhichisgeneratedthroughth euseofacylindricallens asshowninFigure 3-14 .Thelightisscatteredfromtheparticlesintwosuccessive laser Figure3-14.ParticleImageVelocimetry(PIV).pulsesandareacquiredonadigitalcamera.Thelagofthetwo laserpulsesgivesthe systemachangeoftimeforsuccessivedigitalimagestobeac quiredfromtheCCD.Each imageinthepairpairisthendividedintosmallsubsections calledinterrogationareas 59

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(IA).Theinterrogationareasfromeachimageframearethen cross-correlatedonapixel bypixelbasistoobtainainstantaneoussnapshotofthevelo cityeld. PIVincompressiblerowswaspioneeredby Kompenhans&H}ocker ( 1988 )and Adrian ( 1991 )whousedphotographicrecordingandimage-shiftingtechn iques.Investigations rangefromsupersonicjetsby Krothapalli etal. ( 1994 )toturbulencemeasurements incompressiblemixinglayersby Vakili&Gauthier ( 2001 ).Theywereabletoobtain instantaneoussnapshotsofplanarvelocitytocomputevort icitythusallowingplanarmaps ofturbulencestatisticstobecomputedfromensembleavera ges.Recently Zhuang ( 2007 ) successfullyappliedPIVtocavityrowwith M 1 =2toinvestigatethesuppressionof pressureructuationsusingleadingedgemicrojets. ThePIVsystemusedinthisresearchissuppliedfromLaVisio nusingthe32-bit DaVisVersion7.xsoftwarepackageforimageacquisition,p rocessingandcontrolofall hardwarecomponents.The2DPIVsoftwarepackageincludesc ross-andauto-correlation imageprocessingalgorithmswithhighspatialresolution, second-ordercorrelation includingdeformedwindowcorrelationandmulti-passcorr elationsforvectorelddisplays. TwocomponentPIVisimplementedforthemeasurementofthei nstantaneous velocityroweldonthecavitycenterline-plane( z =0)andusedconcurrentlywiththe unsteadypressuremeasurementsthroughoutthisresearch. AnaerosolofDi-ethylhexyl sebacate(DEHS)oilisusedforseedwithameanparticlediam eterof1 masreported by Kahler ( 2003 ).TherowseedisintroducedbytheuseofaLaVisionFlowMast er fournozzleheadsubmicrondropletgeneratorcapableofa70 00particlepersecond generationrate.ANd:YAGLitronNanodualhead135MJperpul selasersystemwas usedtoilluminatetheseedparticles.Thelaserlightwasco nvertedfrominfraredlightat awavelengthof1024nmtogreenvisiblelightatawavelength of532nmusingharmonic generatorsandemittedwith135MJofenergyperpulse.Theno minalpulsedurationof thelaserwasapproximately5-10ns.Thefocusedpointenerg ysourcewasconvertedinto adiverginglasersheetusingLaVisionsuppliedallquartza djustablelightsheetoptics 60

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withanti-rerectingcoatingforwavelengthsof532nm.Thel aserpulsesarerelayedtothe test-sectionatarateof7Hzwiththeuseofamirrortorerect thelaser90 Imageswerecapturedwithahighresolution14-bitImagerPr oXcamerawith2048 2048pixelresolutionand512MBofon-boardmemorysupplied byLaVision.Thevendor listedminimuminterframetimeis115nswhichismuchlowert hanrequiredforthese studies.Thescatteredlightpatternrelativetoeachtimei nstantisstoredontoseparate recordingswhichallowsanalyzingtheimageswiththecross -correlation-basedalgorithms. Cross-correlationhasthemajoradvantageofresolvingthe directionalambiguityandhas ahighersignal-to-noiseratioofthecorrelationsignalre sultinginhigheraccuracyand resolution.Figure 3-15 illustratesthesynchronizationbetweenthelaser,camera and Figure3-15.PIVandpressuresynchronization.pressuresignalforagivenPIVimagepair.Theanticipatedn umberofimagepairsperrun willbelimitedbycondensationeectsandpoor-visualizat ionduetotheseedingparticle clingingtothewindowsandnottheruntimeofthetunnel.The minimumtransferrate 61

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isthetimerequiredtotransferoneimagefromthephotodiod earray(pixelarray)toa storagecellarray.Iftheminimumtimeisnotallowedtoclea rthechargesfromthepixel arraytherecouldbedegradationinimagequality.Vectorsw erecomputedusingacross correlationalgorithmusing50%overlapandamulti-pass64 64pixel(rstsweep)and 16 16pixel2-sweepinterrogationregiontoenhancespatialre solutionforvectoreld displays. 3.6ExperimentalSetup AlloftheexperimentswereconductedatafreestreamMachnu mberof1.44with thestandardoperatingconditionsgiveninTable 3-1 .Twocavities,fullandnitespanas previouslydiscussed,wereconsideredinthestudy.Therod centerlinelocation( X cl )inthe streamwisedirectionwasxedforeachcavitymodelandwas3 .2mmfromtheleadingedge foreachcontrolledcavityconguration.Table3-1.Standardoperatingconditions. P 0 Mach Re = U 1 L Q 1 = 1 2 1 U 2 1 U 1 TypicalValues172kPa1.441 : 29 10 6 74.4kPa415m = s Thepressuretransducerswerelocatedonthecavityroorand aftwallcenterlineas showninFigure 3-16 withstreamwiselocationstabulatedinTable 3-2 .Theaftwallsensor wasmountedonthecenterpointoftheaftwall.Thesensorswe repoweredandtheir outputsignalswereconditionedusingEndevco136 TM DCdierentialvoltageampliers. Allsensorsweresampledsimultaneouslyat90kHzwithhighp asslteringat100Hzand lowpasslteringat30kHz.AlldataweredigitizedusingaNa tionalInstruments(NI) PXI447224-bit8-channelDataAcquisitionCard(DAQ)throu ghaLabViewinterface. Thespectrafortheexperimentaldatausesarecordlengthof 8192pointsgivinga frequencyresolutionof11Hz. Forthefullviewcavityparticleimagevelocimetrycasesth ecameraresolutionbased ontheeldofviewfortheexperimentwasroughly18pixelspe rmillimeter.Considering afreestreamrowvelocityof420m = stheseparationtimebetweentwoexposureswasset 62

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K 10 K 9 K 8 K 7 K 6 K 5 K 4 K 3 K 2 K 1 Aftwall Floory Dz Wz Wx L0 0 : 50 1 0 0 : 25 0 : 50 0 : 75 1 0 0 : 50 1 0 0 : 50 1 Figure3-16.PressuresensorlocationsTable3-2.Cavityroorpressuresensorlocations. No x L y D K10.0420K20.1250K30.2500K40.3750K50.5000K60.6250K70.7500K80.8750K90.9580 K100.5000.5 to1.5 stoallowaparticletotravelatleast10pixels(inthefrees treamandimmediately aroundtheshearlayer)betweensuccessiveimagepairs.Ase tofdatawascollectedwhere theeldofviewwasroughlycutinhalffocusedontheleading edgeofthecavityby theuseofa2Xteleconverter.Fortheseexperimentsthetime betweenlaserpulseswas adjustedto0.75 sasthespatialresolutionwasdoubledandtherowvelocityi mmediately aroundthecylinderwasacceleratedbeyondfreestreamvalu es. 3.7MeasurementUncertainty Ananalysisoftheanticipatedmeasurementuncertaintywil lbediscussedinthis section.Thesectionwillrstdiscussuncertaintiescoupl edwithmeasurementsofthe unsteadypressuresandwillconcludewithadiscussionofun certaintyanalysisforparticle imagevelocimetrymeasurements. 63

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3.7.1UnsteadyPressureMeasurement Theexperimentaluncertaintyforthespectralpowerdensit yofthesurfacepressure ructuationsisprimarilyduetothestatisticalconvergenc euncertainty.Thesignals areacquiredthroughtheuseof16and24-bitdigitalacquisi tioncards.Theaccuracy speciedbythevendorofthetransducersisontheorderof0. 5%offullscaleor0.17kPa and1.03kPa.Followingtheanalysisfoundin Bendat&Piersol ( 1986 )thenormalized statisticalconvergenceuncertainty,orrandomerror,isf oundtobeinverselyproportional tothesquarerootofthenumberofensembles.Thesamplingra teintheexperiments is90kHzforaperiodofapproximately11.6s.Theresultingb locksofdataaresplitinto independentdatarecordsof8192pointsresultingin256ens embles.Theseblocksare analyzedandensembleaveragedtoobtainanestimateforqua ntitiessuchas p rms orthe peaktonesfoundinthePSDplots.Theresultingconvergence uncertaintymaytherefore beexpressedasafunctionoftheindependentsamplesandcal culatedasshownbelow. convergence = 1 p #ofEnsembles = 1 p 256 =6 : 25%(3{8) Propagatingthemeasurementerrorduetoselectedinstrume ntationusingtheleast accuratepressuretransducerandemployingarootsquaresu m(RSS)methodyieldsa completeerrorgivenbyEbelow. E = p (6 : 3 2 +0 : 5 2 )=6 : 3%(3{9) 3.7.2PIVMeasurement TheuncertaintyinthePIVmeasurementsislargelydominate dbythreeevents: 1.biaserrorintroducedbysoftwarecorrelations2.statisticalerrorduetotheroweldturbulence3.errorduetotheintroductionandtrackingofseedparticl e Thebiaserrorduetosoftwarecomputationsassuppliedbyth evendorisroughly3% meaningthemappingofthevelocityvectorsistypicallywit hin0.03pixels.Thestatistical 64

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errorislargelycontrollablebyensuringenoughimagepair sareacquiredfordataanalysis. Theerrorisestimatedaccuratelyforroweldswithmoderat etolowturbulentintensities thatareunder30%asexplainedin Grant&Owens ( 1990 ).Equation 3{10 illustratesthis byconsideringtherelativeerrorofthestandarddeviation where Z c istheanticipated condenceintervalandNisthenumberofsamplesorimagepai rs. = Z c p 2 N (3{10) Duetotherandomnessinherentinturbulence,multiplemeas urementsarerequiredto adequatelyobtainthemeanrowproperties.Therequirednum berofensemblesisrelated directlytotheturbulentintensityoftheroweldwithhigh erturbulentlevelsrequiring agreaternumberofaverages.Figure 3-17 illustratesarapidconvergenceoftheerrorfor a95%condenceintervalovertherst1000ensembles.There afterimprovedaccuracy becomesexceedinglyexpensivenearingthelimitofdiminis hingreturns. % N 0 500 10001500 2000 0 5 10 15 20 Figure3-17.PIVuncertaintyforturbulentintensitiestha tarelessthan30%. Theerrorintroducedbytheabilitytotracktheseedingpart icleismoredicultto quantify.Aparticlesabilitytostaytruetoandtrackthero weldwilldependlargelyon itsmassandsizewithsmallerandlighterparticlespreferr ed.Astheparticlesizedecreases theabilityofthecameratoadequatelyvisualizeandtrackt heparticlemotionsbeginsto 65

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dropoduetothexedspatialresolution. Samimy&Lele ( 1991 )discoveredtheratioof theaerodynamicandturbulentcharacteristictimescalesp redictstheabilityoftheparticle totrackcompressibleshearlayerswithinroughly2%errori ftheratioismaintainedunder 20%. a t 0 : 20(3{11) TheaerodynamictimescaleisestimatedusingOseen'scorre ctionsfordragoverasphere inStokesrowwhere: a = f ( u i v i ) @v i @t (3{12) v i isathreecomponenttensorfortheparticlevelocity, u i istherowvelocity,the correlationfactoristypicallytakenas f =1+0 : 15 Re p 2 3 and Re p = j u i v i j d p where d p isthemeanparticlediameterreportedby Chung&Troutt ( 1988 ).Theaerodynamic timeconstantmayalsobeestimatedusing: a = p d p 2 18 (3{13) where p =istheruidviscosityand p istheparticledensity.Thelargestturbulent characteristictimescale, t ,istypicallytakenbythedominantfrequencyorturbulent velocityoftheroweldwhichforthisresearchcanbetakena s t = 1 3100 hz =32ms. t may alsobeestimatedbydimensionalanalysis t = u 0 asshownin Tennekes&Lumley ( 1972 ) where u 0 isthecharacteristicturbulentvelocitytakenas25% U 1 forthisresearch.Athird waytoestimatethecharacteristicturbulenttimescalewas proposedby Samimy&Lele ( 1991 ) t = 10 0 U 1 U 2 whichusestheinitialvorticitythickness 0 oftheshearlayer.Using anyofthesecalculationsfor t theratio a t << 0 : 2indicatingtheseedshouldadequately tracktherow. TheeectoftemperatureandpressureontheDEHSoilparticl esizefrequency distributionisillustratedin 3-18 .Thedependenceoftheparticlesizedistributiononthe viscosityoftheoilisapparent.Whenthetemperatureislow (thestagnationtemperature isexpectedtovaryfromanywherebetween(0 C T 1 30 C)DEHSoiliswellsuited 66

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togenerateasmall-bandparticlesizedistributionwithac ut-odiameterat2 mwhichis withinthedesiredrangetoensuresucientrowtracking. Figure3-18.DEHSparticlesizefrequencydistributionrep rintedfrom Kahler ( 2003 ). ThePIVerror,basedonthesethreecomponents,andusingthe RSSmethod assumingtheacquirednumberofimagepairsisgreaterthan1 000and2%errorfrom seedingisgivenbelow. E = p (4 : 25 2 +2 2 +3 2 )=5 : 6%(3{14) 67

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CHAPTER4 NUMERICALSIMULATIONAPPROACH Thischapterwillpresentthebackgroundforthenumericals imulationsperformed throughoutthisdissertation.Numericalinvestigationso frowoveropencavitieshavebeen conductedforovertwodecades.Acompletelistofnumerical referencesisnotthegoal herebutcomprehensivereviewscanbefoundin Grace ( 2001 )and Colonius ( 2001 )which demonstratethenumericalworkwhichdatesbackto Komerath etal. ( 1987 ). ApplicationofCFDallowsresearcherstoexamineinstantan eousfeaturesofafully three-dimensionalsupersonicrowinatimeresolvedmanner ,whereasthisinformation isdicultifnotimpossibletoobtainfromexperimentalobs ervations.Additionally, applicationofCFDhastheabilitytodemonstratesensitivi tytoturbulencenotcapableof beingexplicitlymeasuredinexperiments.Themajordrawba ckofcomputationalmodeling ofcavityrowsrevolvesaroundaccuracyoftheappropriatec losuremodels,physicalwall clocktimerequiredforsolutions,storageandpost-proces singlargequantitiesofdata. 4.1Turbulence Abriefintroductionintothetopicofturbulenceisappropr iatebeforetackling theintricaciesofturbulencemodelingandcomputationalr uiddynamics.Foramore comprehensivereviewtheinterestedreaderisdirectedtot heworksof Launder ( 1991 ), Pope ( 2000 )and Tennekes&Lumley ( 1972 ).Figure 4-1 illustratesthetransitionfrom laminartoturbulentconditionsforrowoveraratplate.The actuallaminar-turbulent transitionisacomplicatedprocessandisbeyondthescopeo fthisresearchastherowsin thiseortarefullydevelopedturbulentrows. Launder ( 1991 )summarizesthetransition toturbulencenicelybysaying:\AtmoderateReynoldsnumbe rstherestrainingeects ofviscosityaretooweaktopreventsmall,randomdisturban cesinashearrowfrom amplifying.Thedisturbancesgrow,becomenon-linearandi nteractwithneighboring disturbances.Thismutualinteractionleadstoatanglingo fvorticitylaments.Eventually therowreachesanapparentchaoticstatenon-repeatingfor mdescribedbestonlyin 68

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statisticalterms."Turbulenceisinherentlyanunsteadyt hreedimensionalviscous phenomenaassociatedwithmoderatetohighReynoldsnumber rows.Itshouldbe stressedthatturbulenceisanotapropertyoftheruidbutap ropertyinherenttothe row. Figure4-1.Transitionofaturbulentboundarylayeroverar atplate.Reprintedwith permissionfrom Nichols ( 2005 ). Figure 4-2 presentsanidealizedturbulenceenergyspectrumwhichdet ailstheaverage energypereddyinaturbulentrow.Throughoutthisdisserta tiontheuseoftheconcept ofeddiescommonlyusedinturbulencetextbooksisadoptedw hicharemeanttodescribe whirlsintherowofagivenscale. Byconvention,theenergyisplottedagainstwavenumberwhi chisproportionalto theinverseoftheturbulenteddysize.Itiseasilynotedtha tlargeeddiescontributetothe leftsideoftheplotandsmalleddiestotherightside.Theen ergyspectrumcanbebroken intothreedistinguishableregionsasfollows:1.EnergyContainingRange(ECR)whoseturbulentbehaviors accountsforthe majorityoftheturbulentkineticenergy 2.InertialRangewhoseturbulentmotionsareresponsiblef orthetransferofenergy fromlargetosmallscalesthroughdissipation 3.DissipationRangewheretheturbulentenergyisremovedf romtherow Thesmallestscalestructureshavenosignicanttransfera bleenergybecauseviscous dissipationdominatesbyconvertingkineticenergyintohe at.Largescalestructures, 69

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representedontheleftsideoftheplot,extractenergyfrom themeanrowintheenergy productionregionofthespectrum.Intheinertialrange,wh eresmallereddiesfeedo largerstructures,turbulenceisinessenceinequilibrium .Thepeakcontributioninthe energyspectrumoccurswiththelargerscalestructuresgen erallyinducedfromthe geometryoftheroweld,i.e.,themeanrowgradients. Whenconsideringtheappropriatediscretizationtobeused inanumericalanalysis itisusefultoquantifythelengthandtimescalesassociate dwiththeturbulentrow. Turbulentscalesaretypicallydenedintermsofturbulent kineticenergy,turbulent dissipationandtheruidviscosity.Thelargescaleeddiesa ttributedtoturbulent productionhavelengthscalescharacterizedby L = k 3 2 (4{1) where k istheturbulentkineticenergyand istheturbulentdissipation.Theturbulent timescalefortheselargescalestructuresmaybecharacter izedbytheratioofthe turbulentkineticenergyandturbulentdissipation T = k : (4{2) Ontheotherhand,thesmallestscaledisturbancesareassoc iatedwiththeKomlogorov scalegivenin Tennekes&Lumley ( 1972 )as = 3 1 4 (4{3) where isthekinematicviscosity.Equation 4{4 givestherelationshipforthesmallest turbulenttimescaleas = 1 2 (4{4) wherethetheturbulentReynoldsnumbermaybegivenby Re t = k 2 : (4{5) 70

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Consideringtheratioofthesescales,givenbelowforclari ty, L =Re 3 4 t (4{6) T =Re 1 2 t (4{7) itiseasytoseethatrangeofturbulentscalesmayspanmanyo rdersofmagnitudefor increasinglyhighReynoldsnumberrows.Thesmallestturbu lentstructuresaremuch smallerthanthephysicalscalesofinterest,inthiscasedi mensionsofthecavity.Therefore itiseasilyrealizedthatanenormousspatialdiscretizati onisrequiredtoadequately resolvethecompleteturbulentspectrumevenformoderatel yhighReynoldsnumbers. Computationalruiddynamicscanbeclassiedbasedontheir approachtoresolve theturbulenceintheturbulenceenergyspectrumdepictedi nFigure 4-2 .IfaCFDmodel resolvesallruidmotionsthroughthedissipationregionth eyarereferredtoasDNSwhich yieldsexactresultsassumingsucientaccuracyofthedisc retizationscheme.Estimates ofDNSsimulationssuggesttherequiredresolutionscalesb yEquation 4{7 .Thetotal computationalcost(numberofoperations)maybeestimated byconsideringthefactorof spatialandtemporalresolutionsrequired.Table 4-1 summarizestheanticipatednumber ofgridpointsandtime-stepsrequiredtoemploythegivenme thodandtheestimated datethegivenmethodwillbereadyconsideringhighReynold snumberrowovera complexgeometrysuchasanairplane.Theinterestedreader ispointedtotheworkof Spalart ( 2000 )wherethistablewassummarizedfrom.Itisunderstoodtheg eometryin thisresearchismoresimplethanthatconsideredinTable 4-1 ,thoughitstillillustrates thepotentialbottleneckincomputationalresourcesrequi redforfullLESsimulationsof highReynoldsnumberthreedimensionalrows.Ifthelargest turbulentstructuresare Table4-1.Summaryofnumericalstrategiesandcost. MethodGridPointsTimeStepsDateReady DES10 8 10 4 Now LES10 11 : 5 10 6 : 7 2045 DNS10 16 10 7 : 7 2080 71

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resolved,placingthecutointhedissipationrange,thete chniqueisconsideredtobea LESmethod.Largeeddysimulationisanattractivesolution methodasitprovidesamore accuratesolutionforthemostenergeticturbulentstructu resintherow.LESonlymodels (doesnotdirectlycompute)thelessenergetic,moreuniver salsmallerscalestructures. ThistechniqueisonlyrealizableformoderatelyhighReyno ldsnumbersifusingthesame typeofanalysisasabove.LESalsoisfacedwiththechalleng eofnearwallmodeling explainedby Radhakrishnan ( 2007 )wheretheturbulenteddysizesarelimitedbythe distancetothewallsotheinertialrangeofnearwallturbul enceisforcedintosmaller scalesrequiringspecialwallmodelstobeenforced. CFDtechniquesusingRANSmethodologiesrepresenttheothe rsideofthespectrum oftechniquesfromDNS.ThecutoscaleforRANSisatthefarl eftoftheenergy spectrumbecauseRANSdoesnotresolveanyturbulencedirec tlyinsteadrepresents turbulenceeectsinitssubgridturbulencemodel.Thecomp utationaloverheadand Figure4-2.TurbulenceenergyspectrumdepictingRANSthro ughDNSsimulationscales 72

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dicultiesassociatedwiththeuseofthestandardLESmodel s,particularlyinnear-wall regions,coupledwiththeshortcomingsofunsteadyRANShas leadtothedevelopment ofhybridmodels.Thesemodelsattempttocombinethebestas pectsofRANSandLES methodologiesinasinglesolutionstrategycommonlyrefer redtoasDetached-Eddy Simulation(DES)detailedin Radhakrishnan ( 2007 ).DEStechniquesplacethecuto intheenergy-containingrangesomewhereinthemodeledvol umeandattemptstotreat near-wallregionsinaRANS-likemannerwhiletreatingther estoftherowinanLES-like manner. Grace ( 2001 )hascompiledadetailedlistofpublicationswhichhaveill ustratedthat timedependentRANSsimulationsaretypicallyunabletocap turetheessentialfeatures ofcavityrowelds. Rizzetta&Visbal ( 1988 )and Shih etal. ( 1994 )haveevenreported RANSsimulationsmaybeoverlydissipativeintheturbulenc emodelsdrivingthesolution tosteadystateconditions.Henceitisfairlywellestablis hedthatRANSsimulationsare notcapableofadequatelyresolvingtheacousticturbulenc ecouplingeectsaccompanied incavityrows.Ontheotherhand,DNSoersasolutionforres olvingthefullrangeof lengthscalesinturbulentrowthereforerequiringnoturbu lencemodel.However,the computationalresourcesrequirementhaslimiteditsappli cationtolowReynoldsnumbers andtwodimensionalrowsasshownin Colonius etal. ( 1999 ).TheHybridRANS-LES approach,beingutilizedforthisresearch,hasbeenshownt obecapableofproviding amorethoroughinsightintothephysicsofcavityrowsandth eroleofturbulenceas discussedby Lawson&Barakos ( 2010 ).Intheoryitexplicitlyresolvesthemostrelevant andhighestenergycontainingstructuresoftherow.Forhig hReynoldsnumberrows thehybridRANS-LESformulation,commonlyreferredtoasDE S,iscurrentlyaccepted asthemostpracticalmethod.Anexampleofaninstantaneous snapshotillustratingthe densitygradientfromsuchasimulationisillustratedinFi gure 4-3 .Moredetailsofthese researchareasmaybefoundin Rizzetta&Visbal ( 2003 ), Hamed etal. ( 2003 2004 )and Larcheveque etal. ( 2003 )amongothers. 73

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Figure4-3.Instantaneoussnapshotofillustratingnumeri calSchlierenofsupersoniccavity rowinanL/D=6,L/W=1cavity. 4.1.1CompressibleNavier-StokesEquations Thebasicgoverningequationsforthenumericalsimulation sinthisresearcharethe threedimensionalcompressibleNavier-Stokesequations. Theequationsarewrittenin dimensionlessformtogivebetternumericalconditioningb ydecreasingthenumberof parametersandkeepingthemtightlyboundedandaregivenbe low @ Q @t + @ F i + F v @x + @ G i + G v @y + @ H i + H v @z (4{8) wherethesuperscripts i;v denotetheinviscidandviscouscomponentsoftheruxvector s and x;y;z;t representspatialcoordinatesandtimerespectively. Q isthevectorof conservedvariablesdenedas Q =( ;u;v;w;e ) 0 (4{9) where u;v;w arevelocitiesand rho istheruiddensitynon-dimensionalizedbythe freestreamspeedofsoundanddensityrespectively.Thepri medenotesthetranspose matrixoperation.Theinviscidandviscousruxvectorsareg iveninEquation 4{10 and 74

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Equation 4{11 respectively F i = uu 2 + puvuw ( e + p ) u 0 G i = vuvv 2 + pvw ( e + p ) v 0 H i = wuwvww 2 + p ( e + p ) w 0 (4{10) F v = 0 xx xy zx u xx + v xy + w xz + c p P r T x 0 G v = 0 xy yy yz u xy + v yy + w yz + c p P r T y 0 H v = 0 xz yz zz u xz + v yz + w zz + c p P r T z 0 (4{11) wheretheruidpressureisgivenby p ,theabsoluteviscosityisgivenby and P r isthe Prandtlnumber.Thetotalenergyisgivenby e = p r 1 + 2 u 2 + v 2 + w 2 (4{12) where r = c p c v =1 : 4.Thetemperatureisgivenby T andcanbederivedfromtheperfect gasrelation p = RT .Foracaloricallyperfectgasthismayberewrittenas p =( r 1) e 1 2 u 2 + v 2 + w 2 (4{13) whereSutherland'slawisusedtodetermineviscosityinter msoflocaltemperaturesof theruidunderconsideration.Thebulkmodulus = 2 3 issetaccordingtoStokes 75

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hypothesis.Theviscousstressesmaythenbecalculatedby xy = yx = ( v x + u y ) xz = zx = ( w x + u z ) yz = zy = ( w y + v z ) xx =2 u x u x + v y + w z 3 yy =2 v y u x + v y + w z 3 zz =2 w z u x + v y + w z 3 : (4{14) TheNavier-Stokesequationsarerewrittenincurvilinearf ormemployingaspace transformationfromthecartesiansystemtothelocalsyste m.Thedetailsareleftout herebutcanbefoundin Tannehill&Anderson ( 1984 ). 4.1.2Spalart-AllmarasDES ExaminationofthesetofequationsderivedforRANSandLESo nenotestheyare verysimilarinform.Thefundamentaldierenceliesinthes patiallteringoftheLES formulationnotfoundbydenitionintheRANSequations.Th eobviousnextstepinthe evolutionofCFDmodelingwastotakethebestofbothworldsa ndapplyspatialltering totheRANSequationsanddevelopanothersetofequations,n owcommonlyreferredto asDESorHybridRANS-LESmodeling.Theselteredequations maybethoughtofasa subgridmodelforlargevorticaleddiesintherow.Themostc ommonapproach,andthe approachusedthroughoutthisresearch,involvesthemodi cationoftheproductionand dissipationtermsinsidetheexistingRANSturbulencemode l.Themodiedtermsadda localscalingfactortodestroyorproducetheturbulentedd yviscositybasedonlocalgrid density. Zero-equationturbulencemodelssuchastheBaldwin-Lowma xarecommonlyreferred toasalgebraicmodels.Theyarecalculateddirectlyfromth ecomputedrowvariables andthusrequirenotransportequationsforclosure.Thesem odelscannothandlekey 76

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turbulentparameterssuchasconvectionanddiusionoftur bulenceandaretypically overlysimplisticformostproblemsofengineeringinteres t.Oneequationmodelsoerone transportequationtypicallyforturbulentkineticenergy butofteneddy-viscositiesmodels areusedaswell.Therearemoreadvancedturbulencemodelsa vailabletoday,suchas Menter'sShearStressTransportmodelintroducedby Menter ( 1994 )whichcombinesthe bestaspectsofthetwoequationmodels k and k modelsthroughtrigonometric blendingfunctions.ReynoldsStressModels(RSM)discarda transportequationforthe eddyviscosityalltogetherandattemptstodirectlymodelt heReynoldsstressterms.For moreelaboratediscussionsofturbulencemodelingtheinte restedreaderisdirectedto Wilcox ( 2006 ). Theoneequationturbulencemodelconsideredinthisresear chwasbornfrom theRANSimplementationoftheSpalartAllmarasturbulence modelintroducedby Spalart&Allmaras ( 1994 ).Theauthoracknowledgesthelimitations,especiallydue to thelackofaclosuremodelfortheturbulentkineticenergy. Itshouldbenotedthough, thereareagrowingnumberofpaperscitingfairlysuccessfu luseoftheSA(withthe DESimplementationtobediscussedbelow)modeltosimulate cavityrowsexamplesof whichareshownin Nichols ( 2006 )and Boydston etal. ( 2008 ).ThehybridSA-DESmodel wasoriginallyformulatedby Spalart etal. ( 1997 )andfurtherdemonstratedby Strelets ( 2001 )byreplacingthewalldistancefunction, d ,intheoriginalSAmodelwithamodied distancefunction, ~ l ,asillustratedinEquation 4{15 .TheSpalart-Allmarasturbulence modelwasderivedusingempiricalrelationshipswiththeai mofdevelopingaturbulent transportmodelthatwasfast,robust,andaccurateforboth shearlayersandboundary layers.Thetransportedviscosityisgivenby~ andafulldescriptionoftheturbulence 77

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modelandassociatedconstantscanbefoundinAppendix-A. @ ~ @t + U j @ ~ @x j | {z } Convection = C b 1 (1 f t 2 ) ~ S ~ | {z } Production C w 1 f w C b 1 k 2 f t 2 ~ d 2 | {z } Destruction + 1 @ @x k ( +~ ) @ ~ @x k | {z } Dissipation + C b 2 @ 2 ~ @x 2k | {z } Diusion (4{15) ThemodiedlengthscalefortheSA-DESsimulationsis ~ l =min( d;C DES ) =max( x ; y ; z ) (4{16) whereonlytheproductionanddestructiontermsoftheorigi nalSpalart-Allmaras turbulencemodelarescaled.Theconstant C DES istypicallysetto0.65andisthe maximumlengthofthecelledge.Thisformulationensuresth atforwallbounded separatedrowsthestandardRANSSAmodelisusedintheattac hedboundarylayer andthesubgridscalemodelisusedwhere( d>C DES )awayfromthewallwherethe rowseparationoccurs. 4.2FlowSolver TheBeggarCFDcodehasbeendevelopedandmaintainedattheA irForceSeek EagleOce(AFSEO)sincetheearly1990'sby Rizk etal. ( 2002 )andwasthesolver chosenforthisresearch.Thecodeisbasedontheconservati veformofthecompressible RANSequationsaspresentedinAppendix-A.Thecodeusesblo ckedandoversetgrid techniquestoallowformappingcomplexgeometricalcongu rations.Beggaremploysa secondorderrowsolutionschemewhichrequiresatwolayeri nterpolationfringeateach overlappingboundary.Beggar'srowsolverprovidestime-a ccurateinviscidandviscous roweldsolutionsusinganitevolumecellcentereddiscre tization.ThehybridSA-DES modelasdescribedabovewaschosenforunsteadysimulation susingthiscodebecauseitis themostappropriateturbulencemodeltostudyunsteadyshe arlayerrowavailableinthe code. 78

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Theturbulencemodelandthegoverningrowequationsarewea klycoupled.Ateach timesteptheturbulentequationsaresolvedandthentherem ainingvegoverningrow equationsaresolved.ArstorderEulerimplicittimediscr etizationisemployedwhere thenumericalruxesareobtainedusingtheRoeruxnumerical schemeasdetailedin Rizk etal. ( 2002 ).Fortimeaccuracyateachtime-stepNewton'smethodisapp liedto satisfyconvergencecriteria.Thelefthandsideoftheresu ltingdiscretizedrowequations ateachcellisderivedusingrstorderSteger-Warmingruxe sforstabilityasexplainedby Rizk etal. ( 2002 ).Theincrementalcorrectiontotherowvariableiscalcula tediteratively byusingasymmetricGauss-Seidelrelaxationiterativepro cedure. Thenumericalsimulationspresentedinthisresearchweres electedtomatchtherow conditionsoftheREEFsupersonicwindtunnelasdescribedi n Dudley etal. ( 2008 ).To reducethegridpointsandalleviateissuesarisingduetoth enozzlestart-upshock,the nozzlewasnotmodeledineachsimulation.AseparateRANSso lutionofthenozzle wasperformedwherethesteadyboundarylayerprolesfor u;v;w;;e;p were extractedatthenozzleexit.Theonedimensionalprolesfr omtheRANSsimulations weretheninterpolatedontothedesiredgridinletrowdomai nforaprescribedinlet boundarycondition.Thesolutionwasslowlytimerampedfor stabilitypurposestothe nalprescribedvalues.Theinletfacesolutionwassubsequ entlyfrozentoachievethe desiredboundarycondition. Inthesubsequentchapterswheretheresultsofthenumerica lsimulationswillbe presentedmoredetailswillbeincludedsuchasappliedboun daryconditions,spatialand temporalresolutionanddetailsrelevanttothepost-proce ssingofthedata. 79

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CHAPTER5 CYLINDERIMMERSEDINASUPERSONICBOUNDARYLAYER Flowaroundacircularcylinderhasbeenstudiedextensivel yasitrepresentsa canonicalrowofsignicantengineeringimportance.Atsu cientlyhighenoughReynolds numbers,theboundarylayerseparatesfromthesurfacealon gthecylinderandformsan unstablefreeshearlayer.Theshearlayerdetachesfromthe cylinderupperandlower surfaceandrollsintodiscretevorticesinaprocessreferr edtoasvortexshedding.A regularvortexpatterninthecylinderwakeemergesasthesh earlayervorticesshed alternatelyfromthetopandbottomofthecylindersurfacea ndinteract.Thebehavior androwpatternsoftheserecirculationregionsareknownto bewellcorrelatedwiththe freestreamReynoldsnumber.AbovethecriticalReynoldsnu mber,anasymmetrydevelops inthewakethatleadstogrowthoftherecirculationregiona ndeventualsheddingforming thewellknownVon-Karmanvortexstreet.Ingeneraltheab ovedescriptionisvalidfor cylindersimmersedinsubsonicrows.Previousexperimenta linvestigationscitedby Bashkin etal. ( 2002 )showedqualitativelythatrowsatsubsonicandtransonics peedsthe rowpatternintheneighborhoodofacircularcylinderchang eswiththeReynoldsnumber muchinthesamewayaswithanincompressibleroweld.WhenM achnumbersexceed M 1 > 0 : 9afairlyextensivesupersonicrowzoneisformednearthecy linderbody.This leadstoashortenedseparationzoneandandalackofunstead ynear-wakerowregimes. Therowstructurewhichformsaroundacylinderandthecylin derwakevary dramaticallywhenthecylinderisplacedincloseproximity toplanarwallsasreported by Buresti&Lanciotti ( 1979 ), Gotkun ( 1975 )and Roshko etal. ( 1975 ).Thepresenceof thewallboundarylayerinruencesthecylinderwakeandther owbehaviorissensitiveto factorssuchas:(1)theboundarylayershapefactor( H ),(2)theReynoldsnumberbased onroddiameter Re d = U 1 d ,(3)thescalingoftheroddiameter( d )relativetothe boundarylayerheight( ),(4)thegapheight( G )betweentherodandthewall. 80

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Bearman&Zdravkovich ( 1978 )suggestedthatvortexsheddingissuppressedfor smallgapratios( G d ).Theyalsonoticedthatseparationbubbleswereformedont he wallbothupstreamanddownstreamofthecylinderwhosesize isdependentonthegap ratio.Thistrendisconsistentthroughoutmuchofthepubli shedliteratureandhasbeen veriedintheexperimentalmeasurementsconductedby Buresti&Lanciotti ( 1979 1980 ) and Zdravkovich ( 1985 1986 ).Theymeasuredthemeanforcesandvelocityructuations detailingtheinterferenceofaplanewallonacylinder.The yeachindicatevortexshedding persistsalmostunaltereddowntogapheightswhere0 : 3 G d 0 : 4. Buresti&Lanciotti ( 2003 )measuredthemeanandructuatingforcesactingon acylinderincross-rowplacednearawallwithboundarylaye rtoroddiameterratio d between0 : 1 d 1 : 1.TheReynoldsnumberbasedontheroddiameterranged from0 : 86 2 : 77 10 5 .TheauthorsindicatethatthisisrepresentativeoftheRey nolds numberrangewheretheregularvortexsheddingofanisolate dcylinderinfreestream progressivelydisappears.Theresultsindicateaperiodic liftingforcewasinducedby thevortexsheddingirrespectiveoftheboundarylayerthic knessforgapheightsdown to G d 0 : 4.Theynotedthemeanliftcoecientrapidlydecreasedfori ncreasinggap heights.TheyalsofoundtheStrouhalnumberdecreasedfori ncreasinggapheightsfrom 0 : 20 G d 0 : 80withthinboundarylayersandconversely,increasedfort hickboundary layerswithincreasinggapheightsbetweenthesamerange.F orrangesoutsideof G d > 0 : 80 thevariationinStrouhalnumberwasnegligible.Itshouldb enoted,themaximum variationinthemeasuredStrouhalnumberdidnotexceed5%. Price etal. ( 2002 )conductedParticleImageVelocimetry(PIV)andhotlm anemometryexperimentsforcylindersnearaplanewallwith 1200 Re d 4960. Theyfoundforsmallgapratios,where G d 0 : 125,therowisvirtuallysuppressedbetween thecylinderandwallandnovortexsheddingofthecylindero ccurred.Largergapheights where0 : 25 G d 0 : 375theinteractionbetweenthecylinderinnershearlayera nd separatedwallboundarylayerincreasewithlimitedvortex shedding.Vortexsheddingwas 81

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initiatedwhen0 : 5 G d 0 : 75andtheupstreamseparatedboundarylayerwasshownto reduceinsize.Forthelargergapratios,where G d 1,therewasnoseparationofthewall boundarylayerandtherowaroundthecylinderwasessential lyequivalenttotherowof anisolatedcylinder. Therowaroundacircularcylinderfor Re d 10 4 orgreaterisstillachallenging problemformoderncomputationalruiddynamics.RANSsimul ationscannotaccurately reproducethesmallscalestructuresfoundinthecylinderw akeandtendtobeoverly dissipative.TheapplicationofDNSislimitedduetotheove rbearingcomputationalcost asexplainedinChapter4.SomepromisingLESresultshavebe enobtainedforsimilar rowsbutarepredominatelyfoundatlowerReynoldsnumbers. Anexampleofsuch isfoundintheworkof Sarkar&Sarkar ( 2010 )whoperformedLESforlowReynolds numbersubsonicrowaroundacylindernearaplanewallfollo wingtheexperimentof Price etal. ( 2002 ).Theyfoundthegapratiohasastronginruenceonthecylind ershear layerinstabilityandthusthedynamicsofthecylinderwake .Whentheinnershearlayer lieswithintheboundarylayer( G )theynoteastrongcouplingbetweenthecylinder innershearlayerandapproachingboundarylayerwithsuppr essionofvortexshedding. Theyalsoreportthatthefrontstagnationpointshiftstowa rdsthewallasthegapratiois decreasedandaliftingforceisgeneratedbytherowacceler atingunderneathcylinderand asymmetryofthebasepressurecoecient. Nishino etal. ( 2008 )performedunsteadyRANSandDESsimulationsusing theSA-DESclosuremodelforacylindernearaplanewallat Re d =4 10 4 .The detached-eddysimulationscorrectlycapturedthecessati onofthevortexsheddingwhereas theRANSsimulationspredictedthecessationatmuchsmalle rgapratioswhencompared withexperimentalresults.Thewakestructurespredictedb ythesimulationswereingood agreementwithexperimentsforbothsmallandlargegaprati os. Travin etal. ( 2000 )also performeddetached-eddysimulationsofrowaroundacylind erincludingthesub-critical (characterizedbylaminarseparation)andpost-critical( characterizedbyturbulent 82

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separation)rowregimes.Theresultswereingoodagreement withlarge-eddysimulations andavailableexperimentaldata. Thereexistsawealthofresearchfocusingonsubsonicrowsw ithcylindersnear planewallsbuttherelacksasubstantialeorttostudythis geometryforapproaching supersonicfreestreamrows.Thoughnotexplicitlydetaile dinalloftheabovementioned works,itshouldbenotedthat d 1.Thisgeometricallimitrestrictsthecylinder frombeingfullyimmersedintotheboundarylayerwhen G 0.Incontrast,theresults presentedinthisworkincludescompressibilitythroughas upersonicboundarylayerand thesizingofthecylinderrelativeto isbetween2 d 5.Thecylinderistherefore exposedtoaupstreamnon-uniformvelocitygradient U u overitsentiresurfaceand thelocalvelocitytypicallyexceedsthelocalspeedofsoun d.Thisconditionleadsto theformationofaleadingedgebow-shockwhosestrengthdep endsonthegapheight andthustheapproachingMachnumber.Therealsoexistsacom plexwavepatterndue tothealternatingexpansionandcontractionsoftherowint hewakeofthecylinder. Theinteractionofthesewavesleadstoadversepressuregra dientsthattherowcannot negotiateandtherowseparatesbothupstreamandimmediate lydownstreamofthe cylinder.Theseasymmetricconditionscoupledwiththeten dencyoftherowtoaccelerate inthegapbetweenthecylinderandwallleadstowardameanfo rcedirectedawayfrom thewall.Theinteractionofthevorticalstructuresshedby thecylinderandtheseparated roorboundarylayerdecreasewithlargergapheightsindica tingforclosewallproximity theinteractionexcitesthecylinderwake. 5.1ComputationalGrids Green etal. ( 2008 )havedemonstratedtheselectedsolverhastheabilitytoac curately predictthesheddingcharacteristicsofanisolatedrodatl owerspeeduniformrowandhas beencomparedsuccessfullywithpublicationsby Nichols&Heikkinen ( 2005 ).Thisisan importantconsiderationduetothelackofavailableexperi mentaldatawiththecurrent investigationdescribednext.Thesimulationswereselect edtomatchtherowconditionsof 83

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theREEFSWTasdescribedinChapter4andin Dudley etal. ( 2008 ).Tworoddiameters ( d )wereplacedatvariousgapheights( G )abovethewalltostudytheboundarylayer andcylindersheddinginteraction.Theroddimensionsandg apheightswereselected baseduponaliteraturereviewconcerningthesuppressiono fsupersoniccavityrowusinga rod-spoilerdeviceasillustratedin Stanek ( 2005 )and Ukeiley etal. ( 2004 a ). Theleadingandtrailingstreamwiselengthsofthecomputat ionaldomainextended atleast20diametersfromthecylindercenterlineasillust ratedinFigure 5-1 .Thewall normalboundariesextendedaminimumof20diametersabovet hecylinderwhilethe spanwisecomputationalboundariesextendedaminimumof2d iameters.Thesimulations utilizedathreedimensionalstructuredblocktoblockgrid systemasillustratedinFigure 5-2 .Theroorandcylinderusedano-slipboundarycondition.Th eouterextentsof theZ-planesweredenedusingsymmetryboundarycondition swithanon-rerective characteristicboundaryconditionlocatedontheupperand outerstreamwisedomains. Figure5-1.Sketchofcylinderimmersedinasupersonicboun darylayer. Nishino etal. ( 2008 )conductedarelativelythoroughgridsensitivitystudyfo r Re d =4 10 4 detached-eddysimulationusingtheSA-DESclosuremodelfo rrowaround cylindernearaplanewall.Inordertoresolvetheviscoussu blayerthegridmustbe 84

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y x d (to5d) 0.04dMax 1 : 5 d 2 1 0 12 3 1 : 50 1 : 00 0 : 50 0 Figure5-2.Gridtopology.appropriatelyresolvedinthenearwallregion.They-plus( y + )valueiscommonlyusedto determinetherstgridspacingnormaltothewallandisgive nby y + = u ? y (5{1) where u isthefrictionvelocitycalculatedby u ? = q w .Theshearstressatthewallfor threedimensionalrowsisgivenby w = s du dy 2 + dw dy 2 (5{2) whichisthemagnitudeofthesurfacestressvector. Nishino etal. ( 2008 )foundthe cylinderwakewascapturedwellwhen y + 1atthewallandthemaximumgridspacing inthewakeregionofthecylinderwasnogreaterthe0 : 05 d Theprimaryregionofinterestinthecurrentstudyisupto 5 d fromthecylindercenterlineinthestreamwiseandwallnormaldirections .Followingtheeorts of Nishino etal. ( 2008 ),asimilargridrenementandtime-stepstudywith= 0 : 080 d; 0 : 040 d; 0 : 025 d wasconducted.Thevariabledepictsthemaximumcelledge lengthinthex,yandzcoordinatedirections.Theeectofth egridresolutiononthe pressurespectraasmeasuredonediameterdownstreamofthe rodisillustratedinFigure 85

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5-3 .Thendingsareingoodagreementwith Nishino etal. ( 2008 )whostudiedgridrenementbasedontheconvergenceofseparationanglesandthedr agcoecient.Thespectra Fine ;dt 1 Med ;dt 1 Coarse ;dt 1p h (^ p p) i Q1f ( kHz ) 1025 50 100 200 500 10 4 10 3 10 2 10 1 1 5 (a) G =0 : 11 Fine ;dt 1 Med ;dt 1 Coarse ;dt 1p h (^ p p ) i Q 1f ( kHz ) 1025 50 100 200 500 10 4 10 3 10 2 10 1 1 5 (b) G =0 : 21 Med ;dt 2 Med ;dt 1p h (^ p p ) i Q 1f ( kHz ) 1025 50 100 200500 1000 10 4 10 3 10 2 10 1 1 5 (c) G =0 : 11 Figure5-3.Gridandtime-stepsensitivityonspectralresu ltsfor d =0 : 40. werecomputedwithaminimumof10(negrid)averagedensemb leswithalength50 periodsassuggestedby Cummings etal. ( 2008 )using75%overlappingwhichresultedin afrequencyresolutionof976hertz.InFigure 5-3 itisapparentthatthesolutionishighly griddependentat G =0 : 11.Atthislocationboththeamplitudesandfrequencyconte nt, mostnotablyathigherfrequencies,aresensitivetothegri drenement.Themedium andnegridsconvergednicelyasthegapratioisincreaseda sshownwhere G =0 : 21. Thetime-stepstudywasconductedafterselectionofthemed iumgridandconducted 86

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at G =0 : 43wherethestrengthofthevortexsheddingpeaked.Itwasco ncludedthat thelargertime-stepadequatelyresolvedthespectraofint erestfortheproblemathand wherelessthan5%deviationisnotedinthefrequencyorampl itudesofthespectraplots. Itshouldbenotedthefrequencyresolutionofeachplotwasi denticalbutthenumberof ensembleswerelessfortheneresolutiongridandsmallert ime-stepsimulations.The variationonbroadbandlevelsathigherfrequenciespoints totheabilityofthesmaller time-steptoadequatelyresolvehigherfrequenciesasexpe cted. Theviscouswallsusedarstpointwallspacing4 10 4 d resultingina y + 1.The cellgrowth,measuredfromthersttotheadjacentcell-cen ter,inthenormaldirection waslessthan20%insidetheboundarylayer.Thecomputation algrid(s)consistedof roughly2 10 6 pointswith201equidistantpointsaroundthecylinder.The meanvelocity roweldandmeanturbulentvelocityroweldsdeviatedless than5%inapproximately 50periodsconsideringthedominantsheddingfrequencyofe achcongurationbutthe solutionwastypicallyrunanadditional300periodstoincr easetheaccuracyofthe spectralanalysis. Table 5-1 outlinesthegeometryandcomputationaldetailsconsidere dduringthis study.TheReynoldsnumberbasedontheroddiameteristabul atedconsideringthe velocityattherodcenterline( U c )andthefreestreamvelocity( U 1 ).Basedonthe ndingsofthespatialandtemporalresolutionstudies,the mediumgridconsistingof approximately2 10 6 cell-centeredgridpointsandatime-stepof5 10 07 swereselected. 5.2Results Theresultsofthemeanroweldvelocitiesandtheinstantan eousvorticityroweld arediscussednext.Meanpressureeldsandmeanturbulents tatisticsarealsopresented todeterminetheeecttheroddiameterandgapheighthaveon thecylindernearbody rowdynamics. 87

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Table5-1.Summaryofcomputationaldetails d G dRe d 10 4 ( U 1 ;U c ) dt 10 7 (s)Notation 0.20 0 : 110 : 0401.9,1.455d1G1, dt 1 0 : 210 : 0401.9,1.605d1G2, dt 1 0 : 53 + 0 : 0401.9,1.755d1G3, dt 1 1 : 00 0 : 0401.9,1.855d1G4, dt 1 0.40 0 : 110 : 0803.8,2.95d2G1,Coarse, dt 1 0 : 110 : 0403.8,2.95,2.5d2G1,Med, dt 1 dt 2 0 : 110 : 0253.8,2.95d2G1,Fine, dt 1 0 : 210 : 0403.8,3.25d2G2, dt 1 0 : 64 + 0 : 0403.8,3.55d2G3, dt 1 1 : 29 0 : 0403.8,3.75d2G4, dt 1 + topofrodneartopof bottomofrodneartopof 5.2.1MeanFloweld Thetimeaveragedresultspertainingtothecylinderwakear ediscussedinthis section.Figure 5-4 presentsthetimeaveragedstreamwisevelocitycontoursfo rallofthe congurationsbeingstudiedinthiseort.Ingeneral,asth ecylinderismovedawayfrom thewall(lookingtoptobottominthegure)thelengthofthe cylinderwakeincreases inthestreamwisedirectionduetotheacceleratedrowinthe gap.Thisalsoleadstothe wakebeingmoresignicantlyderectedtowardsthewall.The separationbubbleonthe rooraftofthecylinderalsobecomeslargerinbothstreamwi seandwallnormaldirections. Astrongcouplingexistsbetweenthesideoftheshearlayerc losesttothewall,which willbereferredtoastheinnershearlayerhereafter,andth eboundarylayerwhenthe gapratioisverysmall.Asthegapheightisincreasedthecyl inderwakeaccentuates thederectionoftheboundarylayerawayfromthewallformin galargedownstream separationbubbleforbothroddiameters.Thelocationofth isseparatedregionappears tobecorrelatedwiththelocationofthecylinder.Thisregi onmovesdownstreamasthe rodismovedfurtherawayfromthewallandastherodwakeleng thensinthestreamwise direction.Themeancontoursalsodepicttheformationofas eparationbubbleupstreamof thecylinderforallcases. 88

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y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 2 0 2468 0 0 : 5 1 1 : 5 2 (a) d =0 : 21 ;G =0 : 11 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 2 0 2468 0 0 : 5 1 1 : 5 2 (b) d =0 : 41 ;G =0 : 11 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 2024 6 8 0 0 : 5 1 1 : 5 2 (c) d =0 : 21 ;G =0 : 21 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 2024 6 8 0 0 : 5 1 1 : 5 2 (d) d =0 : 41 ;G =0 : 21 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 202468 0 0 : 5 1 1 : 5 2 (e) d =0 : 21 ;G =0 : 53 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 202468 0 0 : 5 1 1 : 5 2 (f) d =0 : 41 ;G =0 : 64 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 202468 0 0 : 5 1 1 : 5 2 (g) d =0 : 21 ;G =1 y/ x/d 0 : 25 00 : 250 : 500 : 7511 : 25 4 202468 0 0 : 5 1 1 : 5 2 (h) d =0 : 41 ;G =0 : 1 : 29 Figure5-4.Meanstreamwisevelocitycontours. 89

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Theevolutionofthemeanstreamwisevelocityprolesforea chrodisillustratedin Figure 5-5 .Each h u i U 1 proleonthex-axisisconstantlyosetforclarity.Thevel ocity decitduetothepresenceofthecylinderisfairlyshortliv edforeachroddiameterand appearstobeindependentofgapratio.Foreachcasethemaxi muminrectedvelocity isroughly50%oftheoriginalvalueaftertravelingvecyli nderdiametersdownstream. Theseparatedboundarylayeraftofthecylinderbecomesmor eprominentastherods aremovedawayfromthewalluptothepointwheretherodissti llfullyimmersedin theboundarylayerasevidentby d =0 : 40 ;G =0 : 64and d =0 : 20 ;G =0 : 53.The largerroddiameterproducesalargerseparatedrowinthecy linderwake.Thesizeofthe separatedregioninthestreamwiseandnormaldirectionsar ealsoenhancedforthelarger diameterrod.Thestreamwisevelocityincreasesasthegapo fthecylinderisincreased (uptothepointwheretherodisstillfullyimmersed)therow acceleratesaroundthe bodyofthecylindergeneratingalargeradversepressuregr adientwhichresultsinthe rowbeingturnedmoreaggressivelyawayfromthewall.Asthe rodismovedawayfrom thewalltheleadingedgebowshockisstrengthenedandthero wspeedinthecylinder gapisincreasedduetothecylinder'sexposuretoincreased approachingboundarylayer rowvelocities.Therowexpandsmoreontheupperportionoft hecylinderasthegap heightisincreased.Thiscombinedbehaviorforcesthewake ofthecylindertoderect downwardeventuallyleadingtoaseriesofexpansions(asev idencedbytheaccelerated rowseenin 5-4 )andshocks(notclearlyvisiblebutappearasthetailderec tsupwards causingacompressioncorner)nearthetrailingedgeofthew akewhichturnsthewake moreparallelwiththeroor.Thevelocitydecitrecoversqu icker(intermsofphysical downstreamdistance)forthesmalldiameterrodthanthelar gercylinder.Thisinturn weakenstheinteractionbetweenthecylinderwakeandthesu rfaceoftheplanarwall.It shouldalsobenotedthatwhenusedasasuppressiondevicein cavityrows,rodshave typicallybeenplacedveryneartheleadingedgeofthecavit y.Astherodisraisedabove theroorthecylinderwakebecomeselongatedinthestreamwi sedirectionandtheregion 90

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G=1 : 00 G=0 : 53 G=0 : 21 G=0 : 11y/ x/d0 1 2 34 5 6 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 (a) d =0 : 20 G=1 : 29 G=0 : 64 G=0 : 21 G=0 : 11y/ x/d0 12 3 4 56 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 1 : 75 2 : 00 (b) d =0 : 40 Figure5-5.Meanstreamwisevelocityprolesatgivenaxial locations. ofseparatedrowontheroorispushedfurtherdownstream.Th eseparatedregiononthe roorismoveddownstreamofthecavityleadingedgeastherod israisedabovethewall andthewakelengthens.Thismayhelpexplainthereducedee ctivenessoftherodsto suppressthepressureructuationsofsupersoniccavityrow swhentherodisplacedoutside theboundarylayerasreportedin Ukeiley etal. ( 2004 a ). Figure 5-6 illustratesthemeanpressuredistributiononthecylinder surfaceasa functionofthetaandgapheight.Theanglethetaismeasured withzeroattheforward 91

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moststreamwiselocationandcontinuallyincreasingonthe surfaceofthecylinderinthe clockwisedirectionupto360degrees.Whentherodiscloset othewallthepressure initiallyrisesonthecylinderandaftertherowstagnatest hepressurefallsastheair acceleratesoverthetopofthebodyuntiltheseparationpoi ntisreachedforeachcase. Wheneitherrodismovedfurtherawayfromthewallthepressu redrasticallyfallsuntil theseparationpoint.From90to270degreesthepressuredis tributionexhibitsnoticeable valleysthatarestronglyevidentforsmallgapratiosindic ativeoftheupperandlower shearlayerregionsofthecylinderwake.Forhighergaprati osthesevalleysoccurat substantiallylowerpressuresindicatingthepresenceofs trongerrecirculationzones. Foreachrodtheforwardstagnationpointonthecylindershi ftstowardsthewallfor G 0 : 21asthegapheightisincreasedbutremainsonupperhalf(wh ere 0)of therod.Forhighergapratios,withtherodstillfullyimmer sedintheboundarylayer, theforwardstagnationpointshiftstothelowerhalfofthec ylinder.Thelowersurface separationpointshiftstowardsthewallasthegapheightis increasedandlevelsowhen thetopoftherodisbroughtnearthetopoftheboundarylayer .Thetopseparation pointissensitivetothepositionoftherodinsidethebound arylayerandappearsto shifttowardsthewallforincreasinggapheight.Astherodr emainsimmersedinside theboundarylayertherowseparatesatanglesof120 and90 forthesmallandlarge diameterrodrespectively. Theintricatedetailsofthevortexsheddingarediculttoi nterpretfromtheaforementionedtimeaveragedrowelds.Thereexistsastrongper iodicitytothepressure signalassociatedwiththesheddingofthecylindershearla yermostnoticeablyatlowgap ratios.Aspectralanalysiswasperformedaftercollecting thepressuresignalontheroor atonediameterdownstreamfromthecylindercenterline.Ea chensemblewascollected overatimeperiodof300Twheretheperiod(T)correspondsto thedominantvortex sheddingfrequency.Therewereaminimumof8averagedensem blesintheresultingpower spectraillustratedinFigure 5-7 whicharenon-dimensionalizedbythefreestreamdynamic 92

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LowerSurface UpperSurface G =1 : 29 G =0 : 64 G =0 : 21 G =0 : 11h p i Q 1 0 60 120180 240 300 360 0 0 : 50 1 : 00 1 : 50 2 : 00 (a) d =0 : 20 PSfrag LowerSurface UpperSurface G =1 : 29 G =0 : 64 G =0 : 21 G =0 : 11h p i Q 1 0 60120 180 240 300360 0 0 : 50 1 : 00 1 : 50 2 : 00 (b) d =0 : 40 Figure5-6.Timeaveragedpressuredistributiononcylinde rsurface. pressure.Therodplacedatthetopedgeoftheboundarylayer wasomittedfromeach plotastherowsolutionconvergedtoanon-oscillatorystea dystate. Placingtherodincloseproximitytothewallproducedthehi ghestpeaklevels observedintheructuatingpressurespectra.Thedominantp eakforthevortexshedding shiftedtoahigherfrequency(near100kHzforeachrod)when thegapheightwasraised to G =0 : 21thougharstlowerfrequencypeakwasstillevidentinsom eofthespectra. Whenthetopoftherodwasbroughtnearthetopoftheboundary layertherstpeak inthespectrawasnearlycompletelydissipatedforthesmal ldiameterrod.Itshouldbe notedfortheeitherrodthecessationofthevortexshedding occurredwhentherodwas 93

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G=0 : 53 G=0 : 21 G=0 : 11p h (^ p ^ p ) i Q 1f ( kHz )10 25 50 100200 500 1000 10 510 410 310 210 11 10 (a) d =0 : 20 G=0 : 64 G=0 : 21 G=0 : 11p h (^ p ^ p ) i Q 1f ( kHz )10 25 50 100200 500 1000 10 510 410 310 210 11 10 (b) d =0 : 40 replacements G=0 : 53 G=0 : 21 G=0 : 11p h (^ p ^ p ) i Q 1St =fL U 1 0 : 05 0 : 100 : 25 0 : 50 1 10 510 410 310 210 11 10 (c) d =0 : 20 G=0 : 64 G=0 : 21 G=0 : 11p h (^ p ^ p ) i Q 1St =fL U 1 0 : 05 0 : 100 : 25 0 : 50 1 10 510 410 310 210 11 10 (d) d =0 : 40 Figure5-7.CylinderPSDasafunctionoffrequencyandStrou haltakenat x d ; y =(1 ; 0). broughtoutsideoftheboundarylayerconsistentwithwhato newouldexpectbasedonthe visualizationspresentedin Van-Dyke ( 1982 ). Eachrodexhibitedpressureructuationswithdominantpeak sforeachgapratio wheretherodwasfullyimmersedintheboundarylayer.Aslig htincreaseinshedding frequencywasapparentwithincreasedgapheightthatisass ociatedwiththeincreased velocityoftheexposedrod.Thesmalldiameterrodonlyexhi bitedstrongcoherent sheddingwhenthegapratiowasverysmallalbeitatamuchred ucedoverallpowerwhen comparedwiththelargediameterrod.TheVon-Karmantype ofvortexsheddingfor eachrodbecamelessapparentwhentherodwasmovedfurthera wayfromthewall.Gap heightswhere G > 0 : 21,where M 1 1 : 2appearstobethecriticalgapheightwherethe organizedcylindersheddingbecomesweak.Atthisposition ,thebottomoftherodwas broughttothetopofthetransonicregime,i.e.where M 1 1 : 2,andeachroddiameter 94

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showedsignicantdropointhebroadbandlevelsandonlyth epresenceofsmallpeaks wereevidentinthefrequencyspectra.Itshouldbenotedtha tpeaksarestillpresentin thespectraathighergapratios,mostnotablyforthelarged iameterrod,butislikelydue totheoscillatingnatureoftheseparatedregionontheroor .Thisisfurtherdemonstrated withinstantaneousvorticitycontoursshowninFigure 5-10 throughFigure 5-13 Figure 5-8 representstheStrouhalnumberforthedominantpeakofthep owerspectra plottedwithboththefreestreamvelocityandcylindercent erlinevelocity(boundary layervelocityatcylindercenterlinewithnorodpresent)b eingusedasthevelocityscale. Buresti&Lanciotti ( 1979 1980 )foundthatfor Re d =0 : 86 10 5 theStrouhalnumber convergestoapproximately0.19for G greaterthan0.45.Whenplottedasafunction ofthefreestreamvelocitythecurrentsimulationsagreere asonablywellwiththeresults ofthepreviousauthors,atleastforthesmallerroddiamete rtested.Somecareneeds tobetakenwiththiscomparisonasoneshouldnotethereares ignicantdierencesin theexperiments.BurestiandLanciottiusedrodsthatwhere 0 : 1 < d < 1 : 1andthe rowwaspurelysubsonicwithvelocitieslessthan30m = s(comparedto400 + m = sinthe currentstudy).TheReynoldsnumberrangeswereachievedpr imarilybymodication ofthetunnelvelocityandalteringtheroddiameter.Itisbe lieved,forsupersonic d =0 : 40 ;U c d =0 : 40 ;U 1 d =0 : 20 ;U c d =0 : 20 ;U 1St = n fd U G 0 0 : 200 : 400 : 600 : 80 1 : 00 1 : 25 0 : 10 0 : 15 0 : 20 0 : 25 0 : 30 0 : 35 Figure5-8.Strouhalnumberbasedonthefreestreamvelocit y U 1 andthecylinder centerlinevelocity U c 95

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boundarylayersthatexhibithighvelocitygradientsandwi tharodsizedsothat d > 1 fullyimmersedinsidetheboundarylayer,thecylinderfree streamvelocityisstillthe appropriatevelocitytobeusedintheStrouhalnumbercalcu lation.Thecenterlinevelocity tendstobesignicantlylowerthantheapproachingboundar ylayervelocityleadingto overestimatedStrouhalvalues. Figure 5-9 illustratesthemeannormalvelocity,meannormalturbulen tvelocityand meanpressureeldsforselectedgapheights.Thelargerdia meterrodplacednearthetop edgeoftheboundarylayershowsastrongermeannormalveloc itywithsimilarnormal turbulentructuations.Themeannormalvelocityforthelar gediameterrodwasmore pronouncedwhencomparedtothesmalldiameterrod.Thelarg erdiameterrodwithtop oftherodplacednearthetopoftheboundarylayerexhibited thelowestpressuresaft ofthecylinderresultinginalargerseparatedregiondowns treamofthecylinderwhich inturnturnstherowmoreaggressivelyawayfromthewall.It shouldbenotedthatthe geometryconsideredhereisnotidealasinapplicationther odwillbeplacedjustahead ofbackwardfacingstep.Onewouldtendtobelievetheratpla tegeometrywouldbemore restrictiveandmayleadtodiscrepancieswhencomparedtot heactualcavitygeometry consideredlaterthoughtrendsareexpectedtobequitesimi lar. 5.2.2InstantaneousVorticityFloweld Abriefdescriptionofthevortexformationandcessation,i nteractionwiththe boundarylayerandeventualbreakdowntosmallereddiesare describedinthissection. Tovisualizethevortexdynamicsinproximitytoawalldurin gthesheddingcycle, instantaneouscontoursofspanwisevorticity( z )onthecenterlinearepresentedinFigure 5-10 throughFigure 5-13 foreachgapratio.Theinstantaneousguresaredrawnatfou r dierentpointsovertime-periodTofvortexsheddingthath asbeencalculatedassuming sheddingataStrouhalnumberof0.20whichrepresentsanapp roximateaveragevalueover therangeof G consideredinthepresentstudy.Itshouldbenotedinthese guresthatthe phaseoftheinitiationofthiscycleisnotthesameforeachc asepresented. 96

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y/ x/d 0 : 25 0 : 125 00 : 250 : 125 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 (a) y/ x/d 0 : 25 0 : 125 00 : 250 : 125 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 (b) y/ x/d 0 : 25 0 : 125 00 : 250 : 125 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 (c) y/ x/d 0 : 25 0 : 125 00 : 250 : 125 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 (d) y/ x/d0 : 250 : 500 : 751 4 2 02 4 6 8 0 0 : 5 1 1 : 5 2 (e) y/ x/d0 : 250 : 500 : 751 4 2 02 4 6 8 0 0 : 5 1 1 : 5 2 (f) Figure5-9.Smalldiameterrodmeanroweldcontoursfor G =0 : 11.Column(a)is d =0 : 20 ;G =0 : 11andColumn(b) d =0 : 40 ;G =0 : 64.Row(a)ismean normalvelocity,Row(c)meannormalturbulentvelocity,Ro w(d)mean pressureroweld. 97

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d=0 : 20 ;G=0 : 11 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 11 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 11 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 11 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 replacemen d=0 : 20 ;G=0 : 11 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 11 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 11 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 11 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 Figure5-10.Instantaneousz-vorticityfor G =0 : 11. 98

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d=0 : 20 ;G=0 : 21 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 21 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 21 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 21 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 replacemen d=0 : 20 ;G=0 : 21 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 21 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 21 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 21 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 Figure5-11.Instantaneousz-vorticityfor G =0 : 21. 99

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d=0 : 20 ;G=0 : 53 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 64 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 53 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 64 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 replacemen d=0 : 20 ;G=0 : 53 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 64 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=0 : 53 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=0 : 64 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 Figure5-12.Instantaneousz-vorticityfortopofrodneart hetopofboundarylayer. 100

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d=0 : 20 ;G=1 : 29 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=1 t=0(s)y/ x/d 10 5 0510 4 2 0 2 46 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=1 : 29 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=1 t=5e-007(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 replacemen d=0 : 20 ;G=1 : 29 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=1 t=1e-006(s)y/ x/d 10 5 0510 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 d=0 : 20 ;G=1 : 29 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 d=0 : 40 ;G=1 t=1.5e-006(s)y/ x/d 10 5 0510 4 20 2 4 68 0 0 : 5 1 1 : 5 2 Figure5-13.Instantaneousz-vorticityforrodoutsidebou ndarylayer. 101

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Strongvorticalfeaturesarepresentwhensubstantialmixi ngwiththeroorboundary layerispresent.Thisgenerallyoccursfor G 0 : 25witheachrod.CessationofVonKarmantypesheddingwasobservedwhentherodcenterline velocitynearedthesonic velocityforeachdiameterindicatingthepresenceofacrit icalgapratiolinkedtothe boundarylayersonicline.Whentherodsarenearthewallthe positivevortexstreetis derectedawayfromthewallduetotheinteractionwiththese paratedroorboundary layerasshowninFigure 5-10 andFigure 5-11 .Thenegativevortexstreet,whichislocated onthetopoftherod,rollsupinaclassicalVon-Karmansty leresultinginlargevortical featuresthatareconvecteddownstream. As G isincreasedabove0.5foreachroddiameterthecylinderwak eisderected towardsthewall.Theinnerandoutershearlayersofthecyli nderwakeremainessentially steadyuptotheseparationbubbleaftofthecylinder.Thein teractionwiththeroor boundarylayerallowsthesemi-periodicmotionnoticedint heaft-endofthecylinder wake.Asthecylinderisbroughtoutsidetheboundarylayert hereisnoappreciable motionevidentinthecylinderwakeandtheseparatedroorbo undarylayerispushed furtherdownstream.Whentherodiscompletelyoutsidethea pproachingboundary layertheseparatedregionaftofthecylinderontheroorexp eriencesnegligibleturbulent ructuations. Figure 5-14 andFigure 5-15 illustrateaninstantaneousiso-surfaceofthespanwise vorticitycomponentcoloredbyvelocitymagnitudeandarei ntendedtodemonstrate thethreedimensionalcharacteristicsofthecylinderwake .Therowtendstoexhibitan elongatedvorticalstructureinthespanwisedirectionfor G =0 : 50forbothroddiameters. Theouterpartoftheshearlayershedfromthetopofthecylin derappearstoremain smoothmuchfurtherdownstreamasthegapratioisincreased asevidentinFigure 5-14 andFigure 5-15 .Forlowergapratiosforeachrod,theshearlayerinstabili tyarisesand quicklybreaksdownthecoherentspanwisetypestructure.A spreviouslystated,the derectionoftheboundarylayerishighlyevidentforeachga pratioandeachcylinder. 102

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Examiningeachgureitisevidentthatthestrongcouplingo fthisseparatedboundary layerwiththecylinderwakediminishesquicklyasthegapra tioisincreased. VelocityMagnitude[ Plot 3 D ]VorticityIsoSurface30 0 : 250 : 500 : 751 : 001 : 251 : 50 (a) G =0 : 11 VelocityMagnitude[ Plot 3 D ]VorticityIsoSurface30 0 : 250 : 500 : 751 : 001 : 251 : 50 (b) G =0 : 21 VelocityMagnitude[ Plot 3 D ] VorticityIsoSurface30 0 : 250 : 500 : 751 : 001 : 251 : 50 (c) G =0 : 53 VelocityMagnitude[ Plot 3 D ] VorticityIsoSurface30 0 : 250 : 500 : 751 : 001 : 251 : 50 (d) G =1 : 00 Figure5-14. d =0 : 20Instantaneousiso-vorticitycoloredbyvelocitymagnit ude. 5.2.3TurbulentStatistics Tounderstandtherelativeeectsofthegapratioonthewake andboundarylayer interactioncontoursoftheresolvedReynoldsshearstress andlineprolesillustratingthe evolutionoftheresolvedturbulentkineticenergyarepres entedinFigure 5-16 andFigure 5-17 .Acylinderinsubsonicfreestreamrowisknowntoexhibitsy mmetricturbulent stressesaboutthecylindercenterlineinitswakewithpeak valuesinthecenterofboth theupperandlowershearlayerswhichisnotevidentforeith erroddiameterinthisstudy. Theturbulentkineticenergyprolesexhibitpeaksatthein nerandoutershearlayermost notablyatelevatedgapheightswhere G 0 : 21.Theprolesremainlargelyasymmetrical evenwhentherodisplacedcompletelyoutsidetheboundaryl ayer.Thecompressionand downwardderectionofthecylinderwakeduethepresenceofa wellestablishedshockon thefreestreamsideofthewakeareevidentasthegapratiois increasedwhichchangesthe wakesdevelopmentasdiscussedabove.Itisalsoworthwhile noting,theenhancementof 103

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VelocityMagnitude[Plot3D]VorticityIsoSurface500 : 250 : 500 : 751 : 001 : 251 : 50 (a) G =0 : 11 VelocityMagnitude[Plot3D]VorticityIsoSurface500 : 250 : 500 : 751 : 001 : 251 : 50 (b) G =0 : 21 VelocityMagnitude[Plot3D]VorticityIsoSurface500 : 250 : 500 : 751 : 001 : 251 : 50 (c) G =0 : 64 VelocityMagnitude[Plot3D]VorticityIsoSurface500 : 250 : 500 : 751 : 001 : 251 : 50 (d) G =1 : 29 Figure5-15. d =0 : 40Instantaneousiso-vorticitycoloredbyvelocitymagnit ude. theturbulentstressesoccursalongthetrajectoriesofthe innerandoutercylindershear layer.Asthegapratioisincreasedabove G =0 : 21forthesmalldiameterrodthisleaves relativelylowvaluesofshearatthewall.Thelargediamete rrodwakeresultsinelevated shearstresslevelsatthewallforgapheightsashighas G =0 : 64. Theevolutionoftheturbulentkineticenergyisillustrate dinFigure 5-17 tohighlight theinruenceofroddiameterandgapratio.Theenhancemento ftheturbulentructuationsoccursalongthetrajectoriesofthewakeofthecylind erasexpected.Thepeaklevels forall G valuesaresimilarcomparingeachroddiameter.Theproles mixoutquickly forlower G valuesandthedoublehumpsurviveslongerforcaseswhereth erodismoved furtherawayfromthewall.Wheneitherrodisplacednearthe walltheturbulentenergy tendstobequicklyspreadoutindicatingthesmallturbulen tstructuresinducedfromthe rodsheddingmaybeshortlived. 104

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y/ x/d 0 : 025 00 : 0250 : 05 4 20 2 4 68 0 0 : 5 1 1 : 5 2 (a) d =0 : 20 ;G =0 : 11 y/ x/d 0 : 025 00 : 0250 : 05 4 20 2 4 68 0 0 : 5 1 1 : 5 2 (b) d =0 : 40 ;G =0 : 11 y/ x/d 0 : 025 00 : 0250 : 05 0 4 20 2 46 8 0 0 : 5 1 1 : 5 2 (c) d =0 : 20 ;G =0 : 21 y/ x/d 0 : 025 00 : 0250 : 05 4 20 2 46 8 0 0 : 5 1 1 : 5 2 (d) d =0 : 40 ;G =0 : 21 y/ x/d 0 : 025 00 : 0250 : 05 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 (e) d =0 : 20 ;G =0 : 53 y/ x/d 0 : 025 00 : 0250 : 05 4 2 0 24 6 8 0 0 : 5 1 1 : 5 2 (f) d =0 : 40 ;G =0 : 64 y/ x/d 0 : 025 00 : 0250 : 05 4 2 02 4 6 8 0 0 : 5 1 1 : 5 2 (g) d =0 : 20 ;G =1 : 29 y/ x/d 0 : 025 00 : 0250 : 05 4 2 02 4 6 8 0 0 : 5 1 1 : 5 2 (h) d =0 : 40 ;G =1 Figure5-16.Meanresolved RSS = u 0 v 0 U 1 2 onz=0plane. 105

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G=1 : 00 G=0 : 53 G=0 : 21 G=0 : 11y/ x/d0 1 2 34 5 6 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 (a) d =0 : 20 G=1 : 29 G=0 : 64 G=0 : 21 G=0 : 11y/ x/d0 12 3 4 56 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 1 : 75 2 : 00 (b) d =0 : 40 Figure5-17.MeanresolvedTKEprolesatgiven x d locations. 5.3Summary Thenumericalsimulationspresentedinthischapterindica tedthatastherodwas raisedabove G =0 : 21thereappearedtobecessationofthevortexshedding.Evi dence waspresentedconcurrentlythattheseparatedregionofthe cylinderincreasedinthe streamwiseandverticaldirectionastherodisplacedneart hetopoftheboundarylayer forcingtherowtoturnmoreaggressivelyawayfromthewall. Thissuggeststherodmay moreeectivelyloftthenearbodyroweldofacavityrow.Th eturbulentkineticenergy 106

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prolesmixoutquicklywhentherodisnearthewall.Theturb ulentvelocityructuations survivelongerforthelargediameterrodwhentherodismove dfurtherawayfromthe wall.Wheneitherrodisplacednearthewalltheturbulenten ergytendstobequickly spreadout.Couplingthiswithsnapshotsoftheinstantaneo usvorticitycontoursitis believedthesmallturbulentstructuresinducedfromthero dsheddingatlowgapheights areshortlived. Strongvorticalfeaturesarepresentwhensubstantialmixi ngwiththeroorboundary layerispresentwhichgenerallyoccursfor G 0 : 25witheachrod.Cessationofwell organizedVon-Karmantypesheddingwasobservedwhenthe bottomoftherodneared theboundarylayersoniclinewhichindicatesthepresenceo facriticalgapratio. 107

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CHAPTER6 UNSTEADYPRESSUREMEASUREMENTS Inthischapteradetailedanalysisoftheexperimentallyme asuredsurfacepressurefor boththefullandnitespancavitywillbeperformedusingte chniquesoutlinedinChapter 2.Thestreamwisedirectionistakentobethex-directionwh ilethey-directionisnormal totheroorofthewindtunnelforallofthediscussions.Thez eropointinthecoordinate systemislocatedontheleadingedgeofthecavityatthespan midpoint.Thefreestream Machnumberwas1.44forallcasesandthepertinentrowcondi tionsarelistedinTable 3-2 6.1FullandFiniteSpanBaselineCavityComparison Thefullandnitespancavityunsteadypressuremeasuremen tsarecomparedinthis section.Themajoradvantageofthefullspancavitywasthea bilitytoacquirehigher delityPIVmeasurementsandmaythereforebeveryusefulto studyifthesuppression mechanismsarethesame.Previousresearchby Ahuja&Mendoza ( 1995 )and Block ( 1976 )havesuggestedthattheroweldforcavitieswhoselengtht owidthratioisless thanoneispredominatelytwodimensionalinnature.Thelen gthtowidthratioforthe nitespancavityinthisresearchisthreesooneexpectsthe rowtobehighlythree dimensional. Ahuja&Mendoza ( 1995 )havereportedthatthelengthtowidthratiohas littleeectontheresonantfrequencieseventhoughtherow patternsmightbedierent. Theyfurtherdemonstrateduptoa15dBdecreaseintheoveral lsoundpressurelevels onthenitespancavitieswhencomparedtothefullspanmode ls.Theyconcludeditis possibletocomparetwodimensionalcomputationalaeroaco usticanalysiswithexperiments conductedonathreedimensionalcavity. Thebaselinespectraoftheructuatingsurfacepressuremea suredontheaftwall androorofeachcavityarepresentedinFigure 6-1(a) throughFigure 6-1(d) .ThePSD isplottedasafunctionoffrequencyandStrouhalnumberres pectivelyfortheaftwall measurements.Thespectraexhibitmultiplepeakswhichare reasonablywellestimated 108

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usingthemodiedRossiterequationwithdefaultconstants k c =0 : 57and =0 : 25and aretabulatedinTable 6-1 .Thelargestdierencebetweentheobservedresonanttones andestimatedfrequenciesusingthemodiedRossiterequat ionwas8.6%foundatthe rstresonanttone.Thedierencesarenotalarmingduetoth esomewhatgenericnature oftheconstantsintheempiricalequationthatwerederived fromsubsonicexperiments andmayneedtobeslightlyreadjustedtopreciselyestimate thefrequenciesfortheserow conditions.ThespectraarealsoplottedagainstStrouhaln umberwheretherstresonant peakoccursat St =0 : 28foreachcavity.Thedominantpeakoccursat St 0 : 60forthe threedimensionalcavityandat St 0 : 97forthefullspancavitywhichcorrespondtothe secondandthirdresonanttonesrespectively.Table6-1.Rossiterresonanttonepredictions. Tone(kHz)12345 Experiment1.513.295.277.218.56 Rossiter1.383.225.066.908.73 TheRMSvaluesoftheructuatingsurfacepressureonthecavi tyroorareplotted inFigure 6-2 andisrepresentativeoftheructuatingpressurelevelsint egratedover allfrequencies.Theunsteadysurfacepressuresbehavease xpectedwiththehighest ructuatingpressuresonorneartheaftwallandconstantlyt aperingotoauntil x L = 0 : 25.Thepressuresriseagainneartheforwardwall.Theobser vedreductioninpressure ructuationsbetweentheniteandfullspancavityisconsis tentwiththendingsreported by Ahuja&Mendoza ( 1995 )and Block ( 1976 ). Figure 6-3(a) andFigure 6-3(b) showthecorrelationcoecientsplottedagainst timelagfortheaftwallsensor(thereferencesignal)corre latedwithitselfandlocations x L =0 : 25 ; 0 : 50 ; 0 : 75onthecavityroorforbothbaselinecavities.Theplotsha vebeen truncatedtoinvestigatesmalltimeseparations.Thefullt imehistoryplotwouldshow steadilydecayingoscillations.Theseplotsshowincrease dcorrelationmagnitudesforthe fullspancavityateachpositioninsidethecavitywithlitt ledampingofthepeaks.The 109

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FiniteSpan FullSpanp h (^ p ^ p ) i Q 1f ( kHz ) 1 35 10 20 10 2 10 1 1 2 3 5 (a)PSDvs.f FullSpan FiniteSpanp h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 0 : 25 0 : 501 5 10 2 10 1 1 2 3 5 (b)PSDvs.St x L=0 : 042x L=0 : 250x L=0 : 500x L=0 : 750x L=0 : 875p h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 : 50 1 3 5 10 2 10 1 1 5 (c)FullSpan x L=0 : 042x L=0 : 250x L=0 : 500x L=0 : 750x L=0 : 875p h (^ p ^ p) i Q1St =fL U1 0 : 10 : 50 1 3 5 10 2 10 1 1 3 (d)FiniteSpan Figure6-1.FullandnitespanbaselinePSDcomparison.Aft wallmeasurementsare givenintherstrowtakenat y D = 0 : 5asafunctionoffrequencyand Strouhalnumberrespectively.ThesecondrowgivesthePSDm easuredon eachcavityrooratvariousaxiallocationsplottedagainst Strouhalnumber. increaseddampingobservedinthenitespancavityislikel yduetotheincreasedcrossrowfoundinthedownstreamportionofthecavitywhichwillb efurtherdemonstratedin Chapter7andChapter8.Thehighlyresonantphenomenaofthe cavitywhichmanifests itselfasthenearlysinusoidaloscillationsarealsoappar entintheseplots.Thecoherence function,asdenedbyEquation 2{15 ,isillustratedinFigure 6-3(c) and 6-3(d) forboth baselinecavities.Theplotsindicateahighlycorrelatedp ressuresignalnearthetrailing 110

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FiniteSpan FullSpan x Lp rms Q 10 0 : 20 0 : 40 0 : 600 : 80 1 0 0 : 025 0 : 050 0 : 075 0 : 100 0 : 125 0 : 150 Figure6-2.Fullandnitespanbaseline p rms onthecavityroor. edgeofthecavityroorwiththereferencesignaltakenfromt heaftsensor.Thelarge amplitudesofthecoherencefunctionoccuridenticallytot hefrequenciesassociatedwith thepeaksobservedintheaftwallspectra,i.e.,theRossite rresonantfrequencies.This wouldindicatethatthegenerationofthedisturbanceinsid ethecavityandtheresonant tonesobservedinthecavityevolveastheresultofdisturba ncespropagatingupstream fromtheaftwall.Thesignalsbecomeslightlylesscorrelat ed,asnoticedbythedecreasing peaks,movingfurtherupstreamonthecavityroor.Thelowco herenceobservedato peakfrequenciessuggestthebroadbandlevelsmaybeinsens itivetotheresonance-coupling phenomena.Thedistinctresonanttonesobservedinthecavi tyaretightlycoupledwith theresonantconditionscreatedbytheshearlayersinterac tionwiththeaftwall.The broadbandlevelsexhibitpoorcorrelationwhichislikelyd uetotherandomturbulent featuresofthesupersonicroweld. Itisevidentfromthepresentedspectraandcoherencefunct ionsthattheresonant tonesexistatdierentfrequenciesandthedistributionof thepoweracrossthefrequenciesdiers.Onelimitationoftimeaveragedfrequencyspec traisitisnotcapableof distinguishingwhethertheresonanttonescoexistorexper iencemodeswitchingwhere thedominanttoneshiftsinfrequencywithtime.Jointtimefrequencyanalysisprovides informationcontainingfrequency,timeandamplitude.Thi sallowsonetovisualizethe 111

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aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)FullSpan aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b)FiniteSpan aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz )r xyf ( kHz ) AftWallPSD 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (c)FullSpan p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 13 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (d)FiniteSpan Figure6-3.Baselinecavitysurfacepressurecross-correl ationandcross-coherence coecients.Theaftwallsensoristhereferencesignal. time-evolutionofthefrequencycontentandhasbeenusedsu ccessfullyby Kegerise etal. ( 2004 )and Murray&Ukeiley ( 2007 )forsubsoniccavityrow.Theresultingspectrograms fortheunsteadyaftwallpressuresacquiredfromthefullan dnitespancavityareplotted inFigure 6-4(a) and 6-4(b) respectively. ThespectrogramsareillustratedinFigure 6-4 wherethemagnitudeisnormalized by PSD = PSD max( PSD ) .Examiningthegureitisnoticedthatsixresonanttonesex ist foreachcavity.Thedominanttoneshiftsbetweenthesecond andthirdtonewhen 112

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comparingtheniteandfullspancavityrespectively.The rstresonanttoneofeach cavityappearsintermittently.Thesecondtone'spresence isstrongduringthecomplete timehistorywhilethethirdispresentforthewholetimehis toryinthefullspancavity onlyandappearsintermittentlyforthenitespancavity.T hefourthtoneexhibitsa strongpresenceforthemajorityofthetimehistoryforthef ullspancavityonly.Thefth andsixthresonanttonesshowaweakintermittentpresencef oreachcavity.Thereisno evidenceofmodeswitchingofthedominanttonewhenexamini ngthetimehistoryofeach cavityasitisalwayspresent.Allotherresonanttonesappe artobepresentwithvarying degreesofintensitywithtime. PSD 4 3 2 1 Tonef ( kHz )t ( s ) 06 0 3 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (a)FullSpan PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 3 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 (b)FiniteSpan Figure6-4.Fullandnitespanspectrogramsmeasuredonthe aftwall. 6.2FiniteSpanCavitywithSuppression Theexperimentstosuppresstheructuatingsurfacepressur einsidethenitespan cavitywererunwithveroddiametersandmultipleheightso thewallasillustrated inTable 6-2 .Athickwall25.4mmlonghollowcylindermadeof304stainle sssteelwas selectedforeachroddiameter.Thehollowcrosssectionwas selectedtoensurethenatural frequencyoftherodswereabovethedominantpeaksrealized inthebaselinespectra.The detailedcalculationsandgeometricandmaterialproperti esoftherodcanbefoundin Appendix-B. 113

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Table6-2.Finitespancavitycongurations. d G 0.210,0.18,0.36,0.54,0.71,0 : 86 ++ ,0 : 96 0.430,0.18,0.36,0.54,0 : 64 ++ ,0 : 86 0.570,0.18,0.36,0 : 54 ++ ,0 : 79 0.790,0.18,0 : 29 ++ ,0 : 68 0.890,0 : 18 ++ ,0 : 63 ++ topofrodattopofboundarylayer centerlineofrodattopofboundarylayer 6.2.1EectsofRodMounts Inordertoisolatetheeectsofthephysicalhardwareneces sarytoholdtherodin placeaseriesofexperimentswereconductedwithonlythero dmountsinplace,i.e.,no circularrodpresent.Themountingstructureswereplacedi nthetunnelwiththecavity andtwocaseswereconsidered.Intherstcasethetopofthem ountingstructurewas placedasclosetotherooraspossible,correspondingtoaro dwith d =0 : 21and G =0. Thesecondcasewasrunwherethestructurewasfullyextende dleadingtoamaximum gapheightwitharodofdiameter d =0 : 43.Figure 6-5 illustratestheaftwallPSDand roor p rms forthebaselinecavityandthecavitywithonlythemounting structurepresent. ExaminingFigure 6-5 itisclearthattheabilityoftheisolatedrodmountsto suppressthecavitypressureructuationsdoesnotexist.At eachpositiontherodmounts actuallyincreasedboththepeakandbroadbandlevelsinthe sespectra.Whenthemounts werelocatedneartheroorthebroadbandlevelswereslightl yhigherwithanoticeable jumpinthedominantpeaktonewhilethehigherfrequencypea ksremainedlargely unaected.Whenthemountswerefullyextendedthepeaksatt hehigherfrequencieswere increasedandthebroadbandlevelsweresignicantlyraise d. Examiningthe p rms prolesonthecavityroorfromFigure 6-5(c) therewaslittle evidencesupportingthemountsalonehadthecapabilitytor educetheaverageructuating surfacepressuresandactuallytendedtoincreasetheructu ationsforeachposition. 114

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Floor Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 25 0 : 5 1 23 4 5 10 2 10 1 1 2 5 (a)MountsLowered Extended Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 25 0 : 5 1 23 4 5 10 2 10 1 1 2 5 (b)MountsExtended Extended Floor Baseline x Lp rms Q 10 0 : 2 0 : 40 : 6 0 : 8 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 (c)Floor Figure6-5.Eectofrodmountpositionwithnorodpresenton thenitespancavity ructuatingsurfacepressures.AftwallPSDtakenat y D = 0 : 5androor p rms atvariousstreamwiselocations. Figure 6-6 illustratesthemeanSchlierenimageforeachcasepresente dandqualitatively demonstratesthatthereislittledierenceonthebroadeni ngoftheshearlayeroverthe cavity.Notingtheexcitationofthepeaktonesandincrease inbroadbandlevelswithonly themountspresent,thesuppressionlevelsandecacyofthe rodappeartobeevenmore impressive. 115

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Figure6-6.Qualitativeassessmentoftheeectoftherodmo untpositiononthemean shearlayerthicknessusingmeanSchlierengradients. 6.2.2EectsofRodDiameterandGapHeight Theeectsofroddiameterandgapheightontheructuatingsu rfacepressureswhich includepowerspectraldensitymeasurement, p rms prolesandcorrelationanalysisare presentedinthissection.Figure 6-7(a) and 6-7(b) illustratestheeectoftheroddiameter ontheructuatingsurfacepressuresmeasuredontheaftwall .Eachdiameterisplotted withthetopoftherodplacednearthetopoftheboundarylaye rwhichwasfoundtobe themosteectiverodplacement. Withrodssizedgreaterthan40%oftheboundarylayerheight theoverallpeaktones measuredontheaftwallwereeectivelyreducedandingener al,thelargertherodthe betterthesuppressionthoughthedierencesweresmall.Th emostnotabledierence intheeectofroddiameterwasontheructuatingsurfacepre ssuremeasuredonthe cavityaftwallandroor.Themosteectivesizedrodwasfoun dtobe d =0 : 43when consideringthecombinessuppressionontheaftwallandroo rmeasurements.Rodswith largerdiameterswerenotaseectiveatsuppressingtheruc tuatingpressuresonthecavity roor.Figure 6-8(a) through 6-7(j) areaddedforcompletenesswherethepowerspectral densityand p rms plotsforeachroddiameterateachplacementintheboundary layerare illustrated. 116

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Thendingthatarodsizedapproximately50%oftheboundary layerheightand placedsuchthatthetopoftherodisneartheedgeofthebound arylayer,provides themosteectivesuppressionisconsistentwiththesubson icexperimentsby Stanek ( 2005 ), Smith etal. ( 2002 )andbothsubsonicandsupersonicexperimentsconducted by Ukeiley etal. ( 2004 a ).Theyeachconcludedthatarodontheorderof50% was recommendedforeectivesuppressioncharacteristics.Ea chrod(otherthanthesmallest rod)examineddemonstratedtheabilitytoeectivelylower theresonanttonestonearthe broadbandlevelsasshowninFigure 6-7 .Theoverallsurfacepressurereductiononthe aftwallreachedover60%forthemosteectivecaseswhileth eleasteectivecasesstill exhibitedreductionsofjustover50%. d =0 : 89 d =0 : 79 d =0 : 57 d =0 : 43 d =0 : 21 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 0 : 25 0 : 50 13 5 10 2 10 1 1 3 5 (a)PSD d =0 : 89 d =0 : 79 d =0 : 57 d =0 : 43 d =0 : 21 Baselinep rms Q 1x L 0 0 : 200 : 40 0 : 60 0 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (b) p rms Figure6-7.Fluctuatingsurfacepressuresbasedonroddiam eterwithxedgapheight withthetopoftherodat y =1.PSDmeasuredontheaftwallat y D 0 : 5and p rms takenonthecavityrooratvariousaxialpositions. FurtherexaminationoftheplotspresentedinFigure 6.2.2 allowformoretrendstobe observed.Ingeneral,astheroddiameterincreasedthepeak tonesweremoresuciently suppresseduptothepointwherethetopoftherodreachedthe topoftheboundarylayer. Astherodswerebroughtpartiallyoutsidetheboundarylaye rthereductionsmeasured ontheaftwallbegantotapero.Theoverallsuppressioninc reasedsharplybetweenrod diametersof20-40%oftheboundarylayerthickness.Theee ctivenessoftherodsfor 117

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reducingtheructuatingsurfacepressuresontheaftwallre mainedlargelyinsensitiveto roddiameterthereafter.Thenotabledierencewasmeasure donthecavityroorwhere rodsexceeding50%oftheboundarylayerthicknesswerenotc apableofreducingtheroor pressureructuationsaseciently.Itshouldbenotedthatd uetothelimitedsampling rateoftheA/Dconverterthepressuresignalwasnotrecorde datafrequencywhere pressureructuationsassociatedwiththerodsheddingcoul dberesolved.Thismayslightly biastheresultsastheselfnoiseofthecontrolactuationis notincluded.Howeverone wouldnotexpectthedistincttonesfromthesheddingtohave astrongpresenceinthe spectraallthewaytotheaftwallandisdemonstratedwithth enumericalsimulations foundinChapter8. Correlationplotsforthesuppressedcasesarepresentedin Figure 6-8(a) through Figure 6-8(f) .Thebaselinecorrelationisalsorepeatedforconvenience .Examiningthe autocorrelationforeachcaseitiseasilynotedthatthemax imumcorrelationcoecient occursatzerotimelagwhereasforothertimelagsthecorrel ationismuchweaker, particularlyforrodsgreaterthan d =0 : 21.Themoderatenegativecorrelationthat existsforthe d =0 : 21rodnearthecavityleadingedgeindicatedthepressuresi gnalsare inverselyrelated.Thisindicatesthatforapressureriseo nthecavityaftwalltheretends tobeadropinpressuretowardstheleadingedgeofthecavity andvice-versa.Forrods where d 0 : 43thecorrelationbecomespoorandtheoscillationshavene arlycompletely dampedexceptatzerolagfortheautocorrelation. Thecoherencefunctionofthepressuresignalsforthebasel ineandsuppressedcavities arepresentedinFigure 6-9(a) throughFigure 6-9(f) .Theaftwallspectraissuperimposed oneachgureforconvenience.Thecoherenceplotforthe d =0 : 21caseshowsrelatively strongcoherencewithcoecientsexceeding0.80formostre sonanttonesevidentinthe spectraevenasoneprogresstowardsthecavityleadingedge .Thisisconsistentwith whatwaspreviouslyillustratedwiththecross-correlatio nplots.Thetonesmeasuredon theaftwallwerenoteectivelyreducedforthiscongurati onwhichsuggestslittlewas 118

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G =0 : 96 G =0 : 86 G =0 : 71 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 0 : 250 : 50 1 3 5 10 2 10 1 1 3 5 (a) d =0 : 21 G =0 : 96 G =0 : 86 G =0 : 71 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep rms Q 1x L 0 0 : 20 0 : 40 0 : 600 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (b) d =0 : 21 G =0 : 86 G =0 : 64 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 100 : 25 0 : 50 1 3 5 10 2 10 1 1 3 5 (c) d =0 : 43 G =0 : 86 G =0 : 64 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep rms Q 1x L 0 0 : 20 0 : 400 : 60 0 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (d) d =0 : 43 G =0 : 79 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 100 : 25 0 : 50 1 3 5 10 2 10 1 1 3 5 (e) d =0 : 57 G =0 : 79 G =0 : 54 G =0 : 36 G =0 : 18 G =0 Baselinep rms Q 1x L 0 0 : 20 0 : 400 : 60 0 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (f) d =0 : 57 119

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G =0 : 68 G =0 : 29 G =0 : 18 G =0 Baselinep h (^ p ^ p ) i Q 1St =fL U 1 0 : 10 0 : 25 0 : 50 13 5 10 2 10 1 1 3 5 (g) d =0 : 79 G =0 : 68 G =0 : 29 G =0 : 18 G =0 Baselinep rms Q 1x L 0 0 : 200 : 40 0 : 60 0 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (h) d =0 : 79 G =0 : 63 G =0 : 18 G =0 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 0 : 25 0 : 50 13 5 10 2 10 1 1 3 5 (i) d =0 : 89 G =0 : 63 G =0 : 18 G =0 Baselinep rms Q 1x L 00 : 20 0 : 40 0 : 60 0 : 80 1 0 0 : 02 0 : 04 0 : 06 0 : 08 0 : 10 0 : 12 (j) d =0 : 89 Figure6-7.Fluctuatingsurfacepressuresbasedonroddiam eterandgapheight.First columnisaftwallPSDmeasuredat y D = 0 : 5.Secondcolumnis p rms taken onthecavityroor. donetodisturbthefeedbackmechanism.Astheroddiameteri ncreasedto d 0 : 43the magnitudesofthecoherencefallsignicantlycoincidingw ithabroadeningandloweringof thepeaksmeasuredonthecavityaftwallspectra.Incontras ttothesmalldiameterrod, thisstronglysupportsthatfeedbackmechanismoreventher eceptivityoftheshearlayer itself,hasbeendisturbedtherebyreducingtheamplicati onoftheresonanttonesrealized ontheaftwall. 120

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aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)Baseline aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b) d =0 : 21 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (c) d =0 : 43 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (d) d =0 : 57 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (e) d =0 : 79 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 1 0 12 3 4 56 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (f) d =0 : 89 Figure6-8.Cross-correlationcoecientfortopofrodnear topofboundarylayerat selectedaxialpositionsonthecavityroor.Referencesign alistakenfromaft wallsensor. 121

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p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 13 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (a)Baseline p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 13 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (b) d =0 : 21 p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 13 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (c) d =0 : 43 p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 13 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (d) d =0 : 57 p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 1 3 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (e) d =0 : 79 p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSD aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25r xyf ( kHz ) 123581220 1 3 5 8 1215 0 : 05 0 : 25 0 : 50 1 2 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (f) d =0 : 89 Figure6-9.Cross-coherencecoecientfortopofrodnearto pofboundarylayerat selectedaxialpositionsonthecavityroor.Referencesign alistakenfromaft wallsensor. 122

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Thespectrogramforthemosteectiveroddiameterisplotte datselectedgapheights inFigure 6-10(a) 6-10(d) .Thepresenceoftherodloweredthehighestresonanttoneto nearthebroadbandlevelsthroughouttheentiretimedomain independentofgapheight. Resonanttonesthreethroughsixweredrasticallyloweredw hentherodwasrestingonthe roor.Thesecondtonestillwasstillapparentforthesuppre ssedcongurationsatlower magnitudeandslightlymoreintermittentlywithtime.Exam iningthespectrogramatany gapheight,thepowerappearstobedispersedbetweenfreque ncybandswiththepresence oftherod.Basedontheseresults,howeectivelythepoweri sdispersedappearstobea functionoftherodgapheight. 6.3FullSpanCavitywithSuppression Thecontrolledfullspancavityexperimentswererunwithtw ogapheightsfortworod diameterscorrespondingto d =0 : 21with G =0 : 08 ; 0 : 79and d =0 : 43withgapheights G =0 : 08 ; 0 : 59.Thetwogapheightsselectedwerebasedontheleastandmo steective placementsoftherodsfromthenitespanpressuremeasurem ents.Therstpositionwas suchthatthebottomoftherodwasneartheroorandthesecond wasplacedsuchthat thetopoftherodwasnearthetopoftheboundarylayer.Somec arewastakenwhen consideringtheselectionofrodlengthandmaterialforthe fullspancavity.Themethod determiningtheselectionoftherodmaterial,cross-secti onandlengthfollowedsuitwith whatwaspreviouslydoneforthenitespancavity.Thelengt hoftherodwasincreased by12.7mminanattempttomaintainthefullspaneectasbest aspossible.Thisreduced thederectionresistanceoftherodwhencomparedwiththen itespancases.Athickwall 38mmlonghollowcylindermadeof304stainlesssteelwassel ected.Amaximumlength rodthatminimizedbendingwaspreferredtoreducemounting andderectioneects. Theminimumderectionbasedonelasticbeambendinganalysi sforthesmallestrodwas < 25%of (roughlyoneroddiameter)andunder5%of forthelargediameterrod basedonfreestreamloadingconditions.Thehollowcrossse ctionwasselectedtoensure thenaturalfrequencyoftherodswereabovethedominantpea ksrealizedinthebaseline 123

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PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 3 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 (a)Baseline PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 3 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 (b) d =0 : 43 ;G =0 PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 36 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 (c) d =0 : 43 ;G =0 : 36 PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 36 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 (d) d =0 : 43 ;G =0 : 64 Figure6-10.Finitespancavityaftwallspectrogramfor d =0 : 43atselected G values. spectra.Thefullcalculations,equations,geometricandm aterialpropertiesofeachrod canbefoundinAppendixB. Theructuatingsurfacepressuredmeasuredonthefullspanc avityaftwallandroor arepresentedinFigure 6-11(a) andFigure 6-11(b) respectively.Theresultsareanalogous tothenitespancavitypresentedabovewherethemosteect iveplacementoftherod wasatthetopoftheboundarylayerandthelargerdiameterro doutperformedthesmaller rod.Bothpeakandbroadbandsuppressionlevelswereincrea sedwiththefullspancavity 124

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whichislikelyaneectofabetterestablishedfeedbackmec hanisminthebaselinecavity. d =0 : 08 ;G =0 : 57 d =0 : 21 ;G =0 : 79 d =0 : 43 ;G =0 : 08 d =0 : 43 ;G =0 : 57 Baselinep h (^ p ^ p ) i Q 1St = fL U 1 0 : 10 0 : 25 0 : 50 13 5 10 2 10 1 1 2 3 5 (a)PSD d =0 : 08 ;G =0 : 57 d =0 : 21 ;G =0 : 79 d =0 : 43 ;G =0 : 08 d =0 : 43 ;G =0 : 57 Baseline x Lp rms Q 10 0 : 20 0 : 40 0 : 600 : 80 1 0 0 : 025 0 : 050 0 : 075 0 : 100 0 : 125 0 : 150 (b)Floor Figure6-11.Fullspanbaselineandcontrolledructuatings urfacepressuresmeasured wherethePSDismeasuredontheaftwallat y D = 0 : 5and p rms ismeasured atvariousstreamwiselocationsonthecavityroor. Plotsofthecross-correlationforthesuppressedcasesare presentedinFigure 6-12(a) throughFigure 6-12(e) .Comparingthebaselineandcontrolledcasesthendingsar e nearlyidenticalwiththenitespancavity.Thefullspanca vityshowedincreasedcorrelationlevelsateach x L positioninsidethecavitylikelyduetotheabsenceofthero w enteringthecavityfromthesidewalls.Thecorrelationand coherenceplotsindicatea largerdisruptionofthefeedbackloopastherodincreasesi nsize. Plotsforthecross-coherencecoecientareillustratedin Figure 6-13(a) through Figure 6-13(e) .Comparingthebaselineandcontrolledcasesforthecrosscoherence coecientthendingsareagainnearlyidenticalasreporte dforthenitespancavity.The baselineplotshowswellcorrelatedpeaksateachresonantt onefoundontheaftwallPSD gureasexpected.Thesplittonefoundatthefthpeak(loca tedat9 10kHz)fromthe PSDplotsstandsoutclearlyinthecoherenceplot.Largerdi ameterrodsagainappear todisruptthefeedbackloopwithdecreasedcorrelationswi thincreasingdiameter.There wereonlyminorchangesbasedontherodgapheight. 125

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aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)Baseline aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b) d =0 : 21 ;G =0 : 08 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (c) d =0 : 21 ;G =0 : 79 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (d) d =0 : 43 ;G =0 : 08 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25 aft aftC xy ( s ) 10 1 2 34 5 6 x 10 4 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (e) d =0 : 43 ;G =0 : 57 Figure6-12.Fullspancavitycross-correlationatselecte daxialpositionsonthecavity roor.Referencesignalistakenfromaftwallsensor. 126

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aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz )r xyf ( kHz ) AftWallPSD 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (a)Baseline aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSDr xyf ( kHz ) 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (b) d =0 : 21 ;G =0 : 08 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSDr xyf ( kHz ) 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (c) d =0 : 21 ;G =0 : 79 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSDr xyf ( kHz ) 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (d) d =0 : 43 ;G =0 : 08 aft x L =0 : 75 aft x L =0 : 50 aft x L =0 : 25p h (^ p ^ p ) i Q 1f ( kHz ) AftWallPSDr xyf ( kHz ) 123581220 1 3 5 812 15 20 0 : 01 0 : 1 1 5 0 0 : 20 0 : 40 0 : 60 0 : 80 1 (e) d =0 : 43 ;G =0 : 57 Figure6-13.Fullspancavitycross-coherenceatselecteda xialpositionsonthecavityroor. Referencesignalistakenfromaftwallsensor. 127

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Thespectrogramofeachrodandgapheightforthefullspanca vityareplottedin Figure 6-14(a) throughFigure 6-14(e) .Thesametendsnoticedforthenitespancavity arerealizedinthefullspancavity.Thisindicatesthesupp ressionmechanismsarelikely identicalineachcavity.Itisapparentthelargerdiameter bettersuppressesallresonant tonesinthecavity.Theeectofthegapheightontherodsupp ressionecacyseemsto besecondarytotheselectionoftheroddiameterifonlycons ideringaftwallreductions. Asindicatedpreviously,theeectofgapheightseemstopla yamoredominantroleinthe suppressionoftheructuatingsurfacepressuresonthecavi tyroor.Thisagainseemsto pointthattherelikelyexistsatleasttwosuppressionmech anismsforthecontrolledcavity. 6.4Summary Inthissectiontheresultsofthesuppressionontheructuat ingsurfacepressureswill besummarizedforboththefullandnitespancavities.Figu re 6-15 summarizesthereductionof p rms measuredontheaftwallforeachcavityandroddiameterandg apheight. Thefullspancavityexhibitedsuppressionlevelsthatwere roughly15%greaterthanthat ofthenitespancavityatsimilargapheights.Thisisnotco mpletelyunexpecteddueto thendingsofpreviousresearchsuchasthatpresentedby Ahuja&Mendoza ( 1995 ).A rodsizedroughly40%oftheapproachingboundarylayerthic knesswasthemosteective atsuppressingtheructuatingsurfacepressuresonboththe aftwallandroorwhenplaced appropriatelyinsidetheboundarylayer.Thesmallestrode xhibitedthegreatestsensitivitytoplacementwithintheboundarylayerandwastheleaste ectiverodatanyposition. Themosteectiveo-wallplacementoftherodneglectingst reamwiselocationvariation, appearstooccurwhenthetopoftherodisplacednearthetopo ftheboundarylayer. Similarresultswerepreviouslyreportedin Ukeiley etal. ( 2004 a ). Figure 6-15(a) andFigure 6-15(b) clearlyindicatethatastherodisbroughtat leastpartiallyoutsidetheboundarylayer(forthenitesp ancavitythelastpointis locatedwiththerodcenterlinenearthetopedgeofthebound arylayer)thesuppression ecacybeginstodropo.Themosteectivecongurationsel ectedwas d =0 : 43 ;G = 128

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PSD 4 3 2 1 Tonef ( kHz )t ( s ) 03 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (a)Baseline PSD 4 3 2 1 Tonef ( kHz )t ( s ) 03 6 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (b) d =0 : 21 ;G =0 : 08 PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 : 6 0 3 69 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (c) d =0 : 21 ;G =0 : 79 PSD 4 3 2 1 Tonef ( kHz )t ( s ) 0 3 69 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (d) d =0 : 43 ;G =0 : 08 PSD 4 3 2 1 Modef ( kHz )t ( s ) 0 36 9 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 0 : 8 0 : 9 1 0 : 05 0 : 10 0 : 15 0 : 20 0 : 25 (e) d =0 : 43 ;G =0 : 57 Figure6-14.Fullspancavityaftwallspectrogramforgiven d and G values. 129

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0 : 64whichplacesthetopoftherodjustbelowthereportedboun darylayeredgeeven thoughtheaftwall p rms reductionswerenotthehighestasillustratedinFigure 6-15(b) Consideringthattheboundarylayerwillexhibitslightlyd ierentthicknessesbasedon theoperatingtemperature(thereisnotemperaturecontrol inthefacility)thecritical pointisthatthebestoverallsuppressionwasachievedwhen thetopoftherodwasnear thetopoftheboundarylayer.Thenumericalsimulationspre sentedintheprevious d=0 : 43 d=0 : 21 G %Reductionof p rms0 0 : 2 0 : 4 0 : 60 : 8 1 56 58 60 62 64 66 68 70 (a)FullSpan d=0 : 89 d=0 : 79 d=0 : 57 d=0 : 43 d=0 : 21 G %Reductionof p rms00 : 2 0 : 4 0 : 6 0 : 81 20 25 30 35 40 45 50 (b)FiniteSpan Figure6-15.Percentreductionof p rms measuredontheaftwallat y D = 0 : 5. chapterindicatedthatastherodwasraisedabovethesonicl ineintheboundarylayer thereappearedtobecessationoftheorganizedvortexshedd ing.Evidencewaspresented concurrentlythattherowwasmoreaggressivelyturnedaway fromtheroorasthe separationbubbledownstreamofthecylindergrew.Thiswas achievedasthetopofthe rodwasbroughtnearthetopoftheboundarylayer.Thissugge ststherodmaybeableto loftthenearbodyroweldawayfromthewallwhichcouldintu rnaltertheshearlayer aftwallimpingementpoint.Anupstreampropagatingpressu redisturbanceisgenerated bytheshearlayerrappingintothecavityimpingingontheaf twall.Thefactthatthe correlationtimescalesarethesameandthecorrelationstr engthsareloweredlikely indicatesthattheoriginatingwaveattheaftwallisweaker 130

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CHAPTER7 FLOWFIELDMEASUREMENTS Thischapterisdividedintotwomainsectionsbasedonthefu llandnitespancavity whichbothwillincluderesultsfromthebaselineandcontro lledcases.Detailedanalysis oftheroweldassociatedwiththefullspancavityincludin gresultswheretheeldof viewwasfocusedontheleadingedgeofthecavitywillbepres entedrst.Thiswillbe followedbyamorelimitedanalysisofthenitespancavityd uetodicultiesencountered withobtaininghighqualityrowvectorsthroughouttheshea rlayer.Thenitespancavity analysiswillbefurtherexpandeduponwithcomputationalr uiddynamicspresentedin thefollowingchapter. Thecongurationsexaminedherewereselectedbaseduponth endingsofthe pressuremeasurementspresentedinthepreviouschapter.T heseincludeabaseline congurationforeachcavityandcongurationswheretheco ntrolwasthemostandleast eectivewereselectedfordetailedanalysis.Themosteec tivecasecorrespondedtoarod d =0 : 43 ;G =0 : 57wherethetopoftherodwasplacednearthetopofthebounda ry layer.Theleasteectivecasewas d =0 : 21 ;G =0 : 08wherethesmallestrodwas placedneartheroor.Alloftheexperimentswereconducteda tafreestreamMachnumber of1.44withthestandardoperatingconditionsasdescribed inChapter3.Thedetailed descriptionofthePIVarrangementandvectorprocessingca nalsobefoundinChapter3. 7.1FullSpanCavity Inthissectionadetailedanalysisofthefullspancavitywi thandwithoutcontrolis presented.Meanroweld,meanturbulentroweld,twopoint spatialvelocitycorrelations andProperOrthogonalDecompositionresultswillbediscus sed. 7.1.1MeanFloweld Theconvectivevelocityoftheshearlayerasdenedby Papamoschou&Roshko ( 1987 )isgivenbyEquation 7{1 .Theconvectivevelocityis U c ,thefreestreamvelocityis givenby U 1 a isthespeedofsoundandthesubscripts 1 and2indicatethefreestream 131

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andlowspeedstreamsrespectively. U c = a 1 U 1 + a 2 U 2 a 1 + a 2 ( U 1 + U 2 ) 2 (7{1) Assumingnegligiblevariationofthelocalandfreestreams peedofsound(temperature variationisnegligible)theequationsimpliestotheappr oximaterelationgivenin Equation 7{1 .DetailedexaminationofthemeanrowillustratedinFigure 7-1 forthe baselinecavityshowedthattheshearlayerrowpenetratedr oughly20%ofthecavity depth.Theconvectivevelocitywasfoundtoberoughly45%of thefreestreamvelocity. Thelowerstreamvelocity U 2 wasevaluatedat x L =0 : 875and y D =0 : 25.Incontrast,the y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-1.Fullspanmeanstreamwisevelocitycontourswit hvectors. suppressedcavitymeanstreamwiseroweldindicatesmuchl owerstreamwisevelocities impingingontheaftwall.Theshearlayerandimpingementpo intisclearlyraisedabove theaftwall.Comparingthetwocontrolledcasesitisnotedt helargerdiameterrodmore 132

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eectivelyspreadtheshearlayercausingslowerrowtoimpi ngeontheaftwall.Thiscases alsoexhibitedlargerrecirculationvelocitiesinsidethe cavity.Themainrecirculationinside thecavityalsoexhibitedslightlyelevatedupstreamveloc ities. Figure 7-2 showstheresultsofanexperimentconductedwiththe2Xtele converter narrowingtheeldofviewtoapproximatelytherst25%ofth ecavityasshowninFigure 7-2 .Theexperimentwasconductedtoincreasethespatialresol utionofthecylindernear eldandbetterunderstandtheinteractionofthecylinderw akeandcavityshearlayer. Theintroductionoftherodleadstoacavityshearlayerwhic hissubstantiallyspread andmoreeectivelyderectedasevidentbycloseexaminatio nofthevelocityvectors.The resultspresentedinChapter5indicatedthecylinderwakew asnotderectedawayfrom thewallasshownintheexperimentalobservationshere.The simulationsdidillustrate thatthenormalvelocitywasincreasedwithalargerseparat ionbubbledownstreamof thecylinderwhichderectstherowmoreaggressivelyawayfr omthewallforthemost eectivecases.Eachoftheseconditionsmayleadtoashearl ayerthatisderectedaway fromthewall,asshownhere,especiallynearthecavitylead ingedge.Itisalsoimportant toconsidertheeectthealternategeometrymayhaveonther esultsofChapter5where therodwasplacedaboveaextendedratplatecomparedtotheb ackstepinthecurrent study.Examiningthetwocontrolcasesitisevidentthelarg erdiameterrodexhibiteda largernormalvelocitycomponentespeciallyatthecavityl eadingedge. Figure 7-3 illustratethestreamlinesforeachcavity.Thebaselineca vityisdominated byalargeclockwiserotatingrecirculationregionthatstr etchesthewholecavityandis centeredat x L ; y D =(0 : 85 ; 0 : 50).Themaximumrecirculationvelocitywasfoundto benearly25%ofthefreestreamvelocitywhichwaslocatedne arthecavityroor.The cavitycenter-point(bothaxiallyandnormal)showedrecir culationvelocitiesnear10% ofthefreestreamvalues.Thestreamlinesclearlyindicate asecondarycounter-rotating recirculationwhichismuchsmallerandlocatedintheleadi ngedgecornerofthecavity. Thesendingsareconsistentwithsubsonicexperimentscon ductedby Murray etal. ( 2009 ) 133

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y/Dx/L 0 : 125 0 : 250 : 500 : 7510 0 : 10 0 : 200 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)Baseline y/Dx/L 0 : 125 0 : 250 : 500 : 7510 0 : 10 0 : 200 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 : 125 0 : 250 : 500 : 7510 0 : 10 0 : 20 0 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (c) d =0 : 43 ;G =0 : 57 Figure7-2.Zoomedcavitystreamwisemeanvelocitycontour swithvectors. foracavitywithalengthtodepthratioof6.Thesuppressedr ecirculationvelocities neartheroorontheaftendofthecavitywheresomewhatinsen sitivetothepresenceof therod.Figure 7-4 plotsthevelocityprolesasafunctionof y D forvariousstreamwise locationsalongthecavitylength.Examiningthisgureiti sseenthattherecirculation velocitywasnotablyincreasedatthecavitycenter-pointa ndneartheleadingedge.The eectoftherodontheinteriorcavityrowisevidentwithexa minationofthestreamline plots.Theprimaryclockwiserotatingrecirculationwasmo vednearthecenter-pointofthe cavity.Thelargerdiameterrodexhibitedamoreorganizeda ndbroader(attachedtothe cavityroorfurtherupstream)recirculationregionthanno ticedwiththesmallerdiameter rod.Thecounter-rotatingbubbleatthecavityleadingonth eroorwasstillpresentfor 134

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y/Dx/L 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-3.Fullspancavitystreamlines.eachrod.Thelargerrodstreamlinesinthisregionindicate rowthatdoesnotimpinge ontheleadingedgeuntilnearthecavitylip.Thesmalldiame terrodstreamlinesatthe cavityleadingedgeareverysimilartothebaselinerow.The intricaciesofthetrailing edgestreamlinesarenotfullyunderstoodandisbelievedto beanartifactofthepartial spanwisecoverageoftherodspoilerbytheintroductionofe ndeects.Therodonlyalters theportionoftheshearlayerwhichitspans.Theareatherod spanslikelyliftstheshear layercreatinganoutrowatthecenterandaninrownearthewa lls. Figure 7-4 illustratesthevelocityprolesatvariousaxialposition salongthecavity length.ExaminingFigure 7-4(a) andFigure 7-4(b) itisapparenttheprolehasbeenderectedawayfromthewallmoreaggressivelyforthelargedia meterrod.Thegureshows clearevidencetheshearlayerisbroadenedinthecontrolle dcavitiesasoneprogresses downstream.Theshearlayerappearstospreadmoreeective lywiththelargerdiameter rod.Thecontrolledcavitiesalsoexhibitedincreasedleve lsofreversedrowinsidethe cavity. 135

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D 2 G 3 D 1 G 1 Baseliney Dh u i h U 1 i 0 : 25 0 0 : 25 0 : 500 : 751 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (a) x L =0 : 125 D 2 G 3 D 1 G 1 Baseliney Dh u i h U 1 i 0 : 25 0 0 : 25 0 : 500 : 751 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (b) x L =0 : 25 D 2 G 3 D 1 G 1 Baseliney Dh u i h U 1 i 0 : 25 0 0 : 250 : 500 : 751 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (c) x L =0 : 50 D 2 G 3 D 1 G 1 Baseliney Dh u i h U 1 i 0 : 25 0 0 : 250 : 500 : 751 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (d) x L =0 : 75 D 2 G 3 D 1 G 1 Baseliney Dh u i h U 1 i 0 : 2500 : 250 : 500 : 75 1 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (e) x L =0 : 875 Figure7-4.Fullspancavitymeanstreamwisevelocityprol es. 136

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7.1.2TurbulentFloweld Theeectoftherod'spresenceontheturbulentrowelddete rminedbythevelocity ructuationsarepresentednext.Figure 7-6(a) andFigure 7-7(a) showthatforapproximatelytherst10-15%ofthecavityleadingedgeunreliable vectorswerecalculated.This islikelyattributabletotheaccumulationofoilonthetest sectionsidewallsandcavity interior.Thiswasconsistentlyobservedwiththebaseline cavityPIVexperiments. Themeanturbulentstreamwisevelocitycontoursandprole sareillustratedinFigure 7-5 andFigure 7-6 respectively.Theturbulentructuationshadpeaksapproxi mately 30-35%ofthefreestreamrowforthebaselinecavitywithsli ghtlyelevatedructuations witheachcontrolledcavity.Eachrodincreasedthestreamw isevelocityructuations y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 30 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 30 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 30 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-5.Fullspancavitymeanstreamwiseturbulentvelo citycontours. nearthetrailingedgebutmostnotablyshiftedthepointwhe rethemaximumructuation levelsoccurredabovethecavitylip.Thesmallerdiameterr odraisedthemeanstreamwise 137

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D 2 G 3 D 1 G 1 Baseliney Dh U rms i h U 1 i 0 0 : 05 0 : 100 : 15 0 : 20 0 : 250 : 300 : 35 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (a) x L =0 : 125 D 2 G 3 D 1 G 1 Baseliney Dh U rms i h U 1 i 0 0 : 05 0 : 100 : 15 0 : 20 0 : 250 : 300 : 35 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (b) x L =0 : 25 replacemen D 2 G 3 D 1 G 1 Baseliney Dh U rms i h U 1 i 0 0 : 050 : 10 0 : 15 0 : 200 : 250 : 300 : 35 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (c) x L =0 : 50 D 2 G 3 D 1 G 1 Baseliney Dh U rms i h U 1 i 0 0 : 050 : 10 0 : 15 0 : 200 : 250 : 300 : 35 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (d) x L =0 : 75 D 2 G 3 D 1 G 1 Baseliney Dh U rms i h U 1 i 00 : 05 0 : 10 0 : 150 : 200 : 250 : 30 0 : 35 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (e) x L =0 : 875 Figure7-6.Fullspancavitymeanstreamwiseturbulentvelo cityproles. 138

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turbulentructuationsslightlymorethanthelargediamete rrodatthetrailingedge.It shouldbenotedthesedierencesarewithinmeasurementunc ertaintyrangesthatwere discussedinChapter3.Thelargediameterrodexhibitedagr eaterdisplacementofthe peakRMSvalueswhichbecomesmoreevidentasoneprogresses downstreaminthecavity. Thissuggeststherodisabletoaltertheaftwallimpingemen tpointlikelybyliftingthe shearlayer(asshowninFigure 7-4 )orbyalteringtherappingmotionoftheshearlayer. Theleadingedgemagnitudeanddisplacementexhibitedonly slightvariationsupwhere 0 x L 0 : 50. Themeannormalturbulentvelocitycomponentisillustrate dinFigure 7-7 where theturbulentructuationswereroughly15-20%ofthefreest reamrowforthebaseline cavity.Thecontrolledcavitiesagainexhibitedelevatedr uctuationsnearthetrailingedge. Thezoomedeldofviewforthecavitiesnormalructuatingve locitiesarepresentedin y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-7.Fullspancavitymeannormalturbulentvelocity contours. 139

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Figure 7-8 .Thewakeofthecylinderisonlypresentintheplotsforthel argediameter rodclearlyindicatingthenormalvelocityoftherowisgrea tlyenhancedbytheincreased roddiameterandgapheight.Thiswasexpectedandalsorepor tedinthenumerical simulationsofChapter5.Theshearlayerisintensiedands preadmoreeectivelywith thepresenceofthelargediameterrod. y/Dx/L 00 : 0380 : 0750 : 1130 : 1500 : 10 0 : 20 0 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)Baseline y/Dx/L 00 : 0380 : 0750 : 1130 : 1500 : 10 0 : 20 0 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 : 100 : 20 0 : 30 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (c) d =0 : 43 ;G =0 : 57 Figure7-8.Zoomedcavitynormalturbulentvelocitycontou rs. Theshearcomponent( u 0 v 0 )oftheReynoldsstresscontoursandprolesaregivenin Figure 7-9 andFigure 7-10 respectively.Theshearstressattheleadingedgeofthecav ity wasslightlyelevatedforeachcontrolledcaseasonemighte xpectduetotheinteraction withthecylinderwake.From0 : 125 x L 0 : 5thepeakRSSdecreasedforthesuppressed cavitieswithlittlechangeinit'slocationrelativetothe cavitylipline.Asoneprogresses 140

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furtherdownstreamthelocationofthepeakRSSbeginstoshi ftabovethecavitylip lineandnearthecavitytrailingedgethereisanotableincr easeinpeakRSSlevels. y/Dx/L 0 : 02 0 : 015 0 : 010 0 : 005 00 : 0050 : 010 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 : 02 0 : 015 0 : 010 0 : 005 00 : 0050 : 010 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 : 02 0 : 015 0 : 010 0 : 005 00 : 0050 : 010 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-9.MeanRSSrowcontoursforthefullspancavity.Theoveralleectofthecylinderonthemeanturbulentrowe ldarequitesimilarfor eachcontrolledcavitywhichvariedmostlyinmagnitudes.T hequantitiesconsidered exhibitedthesametendencieswithashiftinthepeaklocati onbythecavitiestrailing edgeandmarkedincreaseinmagnitudes.Thelargerdiameter rodseemedtoexhibitthe largestdisplacementofpeakmagnitudeswhichisanindicat ionofit'scapabilitytomore eectivelyloftoraltertherappingoftheshearlayer. Shearlayersdevelopedbyrowovercavitiesdierslightlyf romthatofaconventional singlestreamfreemixinglayer.TheKelvin-Helmholtzinst abilitiesareconstantlyexcited bythefeedbackmechanismandtheentrainmentofruidismodi edbythepresenceofthe cavity.Thespreadingratehasbeenreportedusingboththem omentum( )andvorticity 141

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D 2 G 3 D 1 G 1 Baseliney Dh u 0 v 0 i h U 2 1 i 0 : 03 0 : 02 0 : 010 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (a) x L =0 : 125 D 2 G 3 D 1 G 1 Baseliney Dh u 0 v 0 i h U 2 1 i 0 : 03 0 : 02 0 : 010 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (b) x L =0 : 25 D 2 G 3 D 1 G 1 Baseliney Dh u 0 v 0 i h U 2 1 i 0 : 03 0 : 02 0 : 01 0 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (c) x L =0 : 50 D 2 G 3 D 1 G 1 Baseliney Dh u 0 v 0 i h U 2 1 i 0 : 03 0 : 02 0 : 01 0 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (d) x L =0 : 75 D 2 G 3 D 1 G 1 Baseliney Dh u 0 v 0 i h U 2 1 i 0 : 03 0 : 02 0 : 01 0 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 : 00 1 : 25 1 : 50 (e) x L =0 : 875 Figure7-10.FullspanmeanRSSproles. 142

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thicknessparameters( w ).Theprimarylimitationwhenusingthemomentumthickness as ameasureisthatthevariationcausedbytherecirculatingr egioninsidethecavityleads tosomearbitraryboundsontheintegrationdomain.Thevort icitythickness,asgivenin Equation 7{2 ,isalocalmeasureofthemaximumshearwhichmaybearguedto better determinetheinstabilitypropertiesoftheshearlayer.To thisend,thevorticitythickness ischosenastheparametertomeasurethegrowthoftheshearl ayerinthisresearch. w = U 1 @U @y max (7{2) Ithasbeenfoundthatshearlayersovercavitiescloselyres embleturbulentfreeshear layersinthateachexhibitnearlylinearspreadingrates.T hespreadingratesofshear layerswithupstreamlaminarboundarylayerswasstudiedby Sarohia ( 1975 )andfound tobenearlylinearandincreasedas L 0 increasedwhere o istheinitialmomentum thicknessoftheshearlayer.Turbulentmixinglayersgener allyexhibitspreadingrates where d w dx =0 : 162asreportedby Brown&Roshko ( 1974 ).Cavityrowwithaturbulent boundarylayeratsubsonicspeedswasreportedtoexhibitli nearspreadingratesby CattafestaIII etal. ( 1997 )withvaluesnearthosereportedby Brown&Roshko ( 1974 ). Toestimatetheeecttherodshadonthegrowthandmeanshear rateofthecavity thevorticitythicknessisplottedasafunctionofcavityle ngthinFigure 7-11(a) .The ordinateplotisnormalizedbythevorticitythicknesscomp utedasneartotheleadingedge aspossible.Thecavitylengthtomomentumthicknessratioi sapproximately140.The x-axisscaleslinearlyfrom0 x 0 40where0represents x L =0and40is x L =0 : 875. Gharib&Roshko ( 1987 )conductedexperimentswithincompressiblecavityrowswi th 85 L 0 > 130andreportedthatthemomentumgrowthratewasnearlycon stantat 0.032. Rowley ( 2002 )gavethevorticitygrowthrateasfourtimesthemomentumgr owth rateimplyingthevorticitygrowthratefortheworkconduct edby Gharib&Roshko ( 1987 )isapproximately0.12.Thebaselinecavityshearlayergro wthrateislowerthan thespreadingrateofturbulentfreeshearlayersandwasfou ndtoequal0.10.Thesmall 143

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diameterrodexhibitedagrowthrateslightlylessthantheb aselinecavityat0.09but startedwithathickervorticitythicknessduetothepresen ceoftherod.Thelarger diameterrodspreadmorerapidlyatarateof0.37untilthesh earlayercouldnolonger maintainthatgrowthrateandsubsequentlycrossedbackove rthebaselineratenearthe cavitytrailingedge.Incidentally,thisisalsowherethem agnitudeoftheRSSforthe suppressedcongurationsdipsbelowthebaselinecase.Nea rthecavitytrailingedgethe RSSincreasesdrasticallyforthesuppressedconguration swhichisalsonoticedinthe vorticityspreadingrate.Theelevatedgrowthrateofthesh earlayerisalsodemonstrates therod'sabilitytoalterthemeanshearthroughoutthecavi tyshearlayer.Themaximum meanshearexperiencedbytheshearlayer,asestimatedbyth edenominatorofEquation 7{2 ,isalsoshowntobemoreeectivelyreducedbythelargerdia meterrod. Figure 7-11(b) illustratedthelocationofthemaximumshearasafunctiono fvertical displacementabovethecavitylipmarkedby y D =0.Thelocationandtherateofchange ofthemaximumshearrisesabovetheaftwallwiththelargerd iameterrod.Itappearsas theshearlayerisloftedabovetheaftwallthemaximummeans heardecreaseswiththe addedbenetofraisingthelocationoftheremainingmaximu mshearfarawayfromthe aftwall.7.1.3EvolutionofTwo-PointStatistics Inthissectionthetwo-pointturbulentvelocitystatistic sthroughoutthemeasurement regionarepresented.Thetwopointstatisticsareincluded tohelpdescribepropertiesof thehighReynold'snumberrowwherethemagnitudeofthefeed backsourceinthecavity isbelievedtobelinkedtothehighlycorrelatedructuating velocityregionsinteracting withtheaftwall.Inalloftheplotspresentedinthissectio nthecorrelationcoecients willbepresentednormalizedbytheappropriateRMSvelocit y.InFigure 7-12 through Figure 7-15 ,thebaselinecavityisrepresentedintherstcolumnofima gesfollowedby thesmalldiameterrodandlastlythelargediameterrod.Rea dingupanddownthepage 144

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L 0 140 dw dx =0 : 10( Baseline ) dw dx =0 : 09( d =0 : 21 ;G =0 : 08) dw dx =0 : 37( d =0 : 43 ;G =0 : 57) d =0 : 43 ;G =0 : 57 d =0 : 21 ;G =0 : 08 Baseline w 0x 0 05 101520253035 0 1 2 3 4 5 (a)Vorticitythickness d=0 : 43 ;G=0 : 57 d=0 : 21 ;G=0 : 08 Baseline Baseline :dy dx=0 : 21 d=0 : 21:dy dx=0 : 29 d=0 : 43:dy dx=0 : 57y Dx L 00 : 250 : 500 : 751 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 0 : 6 0 : 7 (b)Locationofmaximumshear Figure7-11.Vorticitythicknessofthefullspancavity.Da shedlinesrepresentcalculation for d w dx and dy dx correspondstoadvancingspatiallyeitherincreasing x L orincreasing y D .Theexactoriginof thecorrelationisgivenineachgurecaption. Thenormaldirection(constantstreamwisedirection)spat ialevolutionnearthe cavitytrailingedgeareillustratedinFigure 7-12 andFigure 7-13 forthenormaland axialturbulentvelocitycorrelationsrespectively.Only onelocationispresentedassimilar behaviorswereobservedateachstreamwiseposition.Exami ningthisgureitbecomes apparenttheareaofhighlycorrelatedruiddecreasesforth ebaselinecaseastheorigin ismovedawayfromthecavitylip.Thecontrolledcavitiesma intainhigherlevelsof correlationasonemovesawayfromthecavitylip.Thelarged iameterrodexhibitedthe bestoverallcorrelationat y D =0 : 5suggestingthecavityhasbeenalteredsuchthatit nowlieshigherabovethelip-line.Ithasbeenpreviouslyde monstratedthelargerdiameter rodmoreeectivelyspreadtheshearlayerthanthesmallerd iameterrod.Therefore,this correlationbehaviorisexpectedasonewouldexpectthecor relationstofollowthespread oftheshearlayerfairlyclosely. Thecorrelationofthestreamwiseturbulentvelocityindic atesaskewedcorrelation regionwhichmaybeduetoastrainingeect(duetoraisedshe arstressesintheaftofthe 145

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cavity)ofthelargescaleturbulentstructuresinthisregi on.Asimilartrendisobserved forthenormalvelocitybasedon v 0 illustratedinFigure 7-13 .Thecorrelationsbecame elongatedinthedirectionofthevelocityforeachgurewhe rethesuppressedcavities exhibitedmuchhigheraspectratioregionsthanthebaselin ecavity.Thelargerdiameter rodagainmaintainedfairlywelldenedcorrelationshighe rabovethelipofthecavity whichisbelievedtobeduetothespreadingoftheshearlayer aspreviouslymentioned. y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (l) Figure7-12.Evolutionoftwo-pointspatialaxialturbulen tvelocitycorrelationcenteredat y D =( 0 : 5 ; 0 ; 0 : 5 ; 0 : 75).Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 43 ;G =0 : 57. x L positionrunstoptobottomingurewithaxialorigin locatedat x L =(0 : 25 ; 0 : 50 ; 0 : 75 ; 0 : 875) 146

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y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (l) Figure7-13.Evolutionoftwo-pointspatialnormalturbule ntvelocityspatialcorrelation centeredat y D =( 0 : 5 ; 0 ; 0 : 5 ; 0 : 75).Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 43 ;G =0 : 57. x L positionrunstoptobottom ingurewithaxialoriginlocatedat x L =(0 : 25 ; 0 : 50 ; 0 : 75 ; 0 : 875) 147

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Thestreamwisespatialevolutionofthe u 0 u 0 and v 0 v 0 componentofthenormalstress nearthecavitytrailingedgeareillustratedinFigure 7-14 andFigure 7-15 respectively. Examiningthesecorrelationsthedevelopingelongationwi thcavityspanbecomesapparent.Forthecontrolledcavitiesitisapparentthecorrelat ionregionistiltedwithrespect tothecavitylip-linenearthecavityleadingedgewhichisb elievedtobeanindicationof therodsabilitytolofttheshearlayer.Thesmallandlarged iameterrodexhibitsimilar behaviorateachspatiallocationthoughthelargerdiamete rrodtendstoexhibitslightly moreelongatedwithlargertiltanglesforthecorrelatedre gions.Thenormaldirection correlationexhibitssimilaroverallbehaviorateachspat iallocation.Thecorrelation becomesmoreelongatedinthenormaldirectionasoneprogre ssestowardsthecavityaft wallwherethiseectismagniedforthecontrolledcavitie s.Thecorrelatedregionsfor thelargerdiameterrodtendtobeslightlylargerthanthose ofthesmallerdiameterrod. Thecontrolledcavitiesshowagreaterpropensityfortheco rrelatedregiontoevolveinthe directionofthestresswhencomparedwiththebaselinecavi ty. Theelongationofthecorrelatedregionsinthedirectionof thenormalvelocityis expectedforaturbulentmixinglayer.Thegeneralshapesof thecorrelationareasare quitesimilarwithonlyslightalterationsofthemaximumco rrelationanglesforthe controlledcases.7.1.4ProperOrthogonalDecomposition Itcanbeadvantageoustoextractthemostenergeticfeature softherowfromthe backgroundturbulenceintheshearlayerinordertogainins ight.Severaltechniquesexist foridentifyinglargescalestructuresinaturbulentrowe ldwhicharesummarizedby Bonnet etal. ( 1998 ).TheuseofProperOrthogonalDecomposition(POD)iswells uited forsuchanalysissincethePODmodesinthisresearcharewri ttensuchthattheoptimal decompositionisfoundfortheturbulentkineticenergy.Ty picallyonlytherstfewmodes areneededtocaptureamajorityofthelargescalebehaviora ndenergy. 148

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y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (l) Figure7-14.Evolutionoftwo-pointspatialaxialturbulen tvelocitycorrelationcenteredat y D =0.Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 43 ;G =0 : 57. x L positionrunstoptobottomingurewithaxialorigin locatedat x L =(0 : 25 ; 0 : 50 ; 0 : 75 ; 0 : 875) 149

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y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 0 : 10 0 : 250 : 500 : 751 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 0 : 10 0 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (l) Figure7-15.Evolutionoftwo-pointspatialturbulentvelo citycorrelationcenteredat y D =0.Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 43 ;G =0 : 57. x L positionrunstoptobottomingurewithaxialorigin locatedat x L =(0 : 25 ; 0 : 50 ; 0 : 75 ; 0 : 875) 150

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Studieswereconductedtoensurethatenoughsnapshotswere usedinordertoverify thatthedominantmodesweresucientlyresolved.Thiswasa ccomplishedbyapplying avaryingnumbersnapshotsasillustratedinthecontourplo tsofselectedPODmodesin Figure 7-16 andFigure 7-17 toaccessthesensitivityofthePODcharacteristics.Figur e 7-17 showstheconvergenceoftheturbulentkineticenergycaptu redperPODmodeand thecumulativecapturedturbulentkineticenergyasafunct ionofthenumberofsnapshots. Thoughallavailablesnapshotswereusedthroughoutthisre searchitisnotedthatit appears300-500wouldbesucientifonewereinterestedinr econstructingtheroweld. Theamountofturbulentkineticenergyrepresentedbyeachm odeisdeterminedby themagnitudeoftheeigenvaluesandisplottedforeachcon gurationinFigure 7-18 Itisseenthattherewaslittlechangeinthersttwentymode sbetweenbaselineand suppressedcavitieswiththeexceptionofthersttwomodes .Thefairlyrapidconvergence ofthebasissetisrealizedwherethecaptureof50%ofthetot alturbulentkineticenergy forthecontrolledcavitiesrequiredlessthan2%ofthetota lsnapshotswhilethebaseline requiredslightlymorearound4%.Thelagofthebaselinecav ityisduetotheamount ofenergycapturedintherstPODmode.Therstmodeofeachc avitycaptured between13-17%ofthetotalturbulentkineticenergywheret hesuppressedcasescaptured morekineticenergyintherstmodewhichisconsistentwith thenumericalresultsof Arunajatesan etal. ( 2009 ). TherstvePODmodesassociatedwiththestreamwiseandnor malcomponentof velocityforeachcongurationisdisplayedinFigure 7-19 and 7-20 respectively.Ineachof theseguresthecolumnsrepresentthecavityconguration whiletherowsrepresentthe PODmode.Thereareseveralinterestingqualitativefeatur esofthesemodesthatwillbe highlighted.ForeachcomponentofthePODmodesthepresenc eoftheroddidnothave aprofoundeectonthestreamwiseorganizationoftheresol vedmodes.Therstmodeof eachcavityexhibitedoneelongatedfeaturewhichisconsis tentwiththesubsonicresults 151

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y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (l) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (m) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (n) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (o) Figure7-16.NormaldirectionPODmodes1,3and5convergenc ebasedonthenumberof snapshotsforthefullspancavity.Columns:(a)Baseline(b ) d =0 : 21 ;G =0 : 08(c) d =0 : 43 ;G =0 : 57.Rowsrepresentdiering numberofsnapshotsinascendingorder(Snapshots=50,200, 350,500,1000) 152

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N =1000 N =500 N =350 N =100 N =50% E05101520 0 5 10 15 20 (a)EnergyperPODMode N =1000 N =500 N =350 N =100 N =50E kP k0510152025 0 0 : 2 0 : 4 0 : 6 0 : 8 1 (b)CumulativeEnergyDistribution Figure7-17.ConvergenceofPODenergyateachmodeandconve rgenceofcumulative energybasedonnumberofsnapshots. d =0 : 43, G =0 : 57 d =0 : 21 ;G =0 : 08 Baseline% E k 0 5 1015 20 0 2 4 6 8 10 12 14 16 18 (a)EnergyperPODMode d =0 : 43, G =0 : 57 d =0 : 21 ;G =0 : 08 BaselineE kP k 1 10100 1000 0 : 15 0 : 25 0 : 375 0 : 50 0 : 75 1 (b)CumulativeEnergyDistribution Figure7-18.FullspancavityPODtemporalconvergenceande nergydistribution. of Murray etal. ( 2009 )Theorientationandverticallocationofthismodewaslarg ely inruencedbytheselectionofroddiameter.Themodeisalsow ellorganizedunderthe cavitylipforthebaselinecavity.Incontrast,thesuppres sedcavitieshavebeenfurther elongatedinthestreamwisedirectionandbecomemorecompa ctinthenormaldirection. Themodehasbeenliftedoutsidethecavityandturnedinacou nterclockwisemanner whichisconsistenttherecirculationregionattheaftwall clearlyshowninthestreamlines ofthesuppressedcavitiesillustratedinFigure 7-3 .Ingeneral,thesuppressedcavities exhibitedlessenergeticregionsfoundunderthecavitylip asseeninPODmodes2,3and 153

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5.ItshouldbenotedthatingeneralthemodesfromthePODana lysisseemalignedwith theoutlineofthecavitymeanshearlayerasonewouldexpect sincethisiswherethe dominantamountofturbulentenergyresides. Itisinterestingtonotethesuppressedcavitiesseemtoexh ibitatendencytoswap behaviorsbetweenthethirdandfourthmodes.Anwellorgani zedstructureisfoundunder thecavitylipforthefourthmodeofthesmalldiameterrodwh ereasasimilartrendis observedforforthethirdmodeofthelargerdiameterrod.Th edominantmodeofthe cavityseemstoestablishitselfwhentherowswitchesbetwe enthemultipleresonantpeaks observedinthepressurespectra.Thespectrogramspresent edinChapter6indicatethe dominantresonanttonewaspresentnearlyallthetimewhere asthesmallerresonant peakoccurredmoreintermittently.Thisviewisconsistent withthatofthePODmodes extractedfromtherowwhichdemonstraterowfeaturesconsi stentwithmultipleRossiter modes. InordertoquantitativelyassessthesimilarityofthePODm odesforeachcavitya methoddenedby Rempfer&Fasel ( 1994 )wasemployedwherethesimilarityvariable wascalculatedaccordingtoEquation 7{3 r ij = ~ i1 ~ j2 (7{3) Thesubscriptsoneandtworefertoalternatecavitycongur ationswhilethesuperscript i,jrefertothePODmode.Thesimilaritywillbeequaltoonei fthecomparedmodes areidenticalorzeroifcompletelydissimilarduetotheort hogonalityofthePODmodes. Figure 7-21 illustratesthesimilaritycomputedforthebaselineandsu ppressedcases alongwiththesuppressedcasescomparedtooneanother.Com paringtheunsuppressed andsuppressedmodesitisevidentthebestsimilarityoccur sattherstmodewiththe largerdiameterrodcorrelatingsignicantlybetter.Ther eisevidenceofrelativelyweak correlationothediagonalsparticularlyundertherstv emodesforeachsuppressed case.Formodesgreaterthanvethesimilarityisweakoreve nnon-existentinmostcases. 154

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y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (l) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (m) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (n) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (o) Figure7-19.FirstvePODmodesbasedonthestreamwiseruct uatingvelocityforthe fullspancavity.Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 57 ;G =0 : 57.PODmodesaresequentialfromtopofpagedown. 155

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y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) y/Dx/L 1 0 : 5 00 : 51 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (i) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (j) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (k) y/Dx/L 1 0 : 5 00 : 51 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (l) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (m) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (n) y/Dx/L 1 0 : 5 00 : 51 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (o) Figure7-20.FirstvePODmodesbasedonthenormalructuati ngvelocityforthefull spancavity.Columns:(a)Baseline(b) d =0 : 21 ;G =0 : 08(c) d =0 : 57 ;G =0 : 57.PODmodesaresequentialfromtopofpagedown. 156

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Acomparisonofthesimilaritybetweenthelargeandsmalldi ameterrodshowsabetter establishedcorrelationpatternonthediagonals.Thisind icatestheresolvedfeaturesexist innearlythesamespatialreferencebetweenthetworods.Th isimplieseachrodeects theshearlayerinasimilarmanner. d =0 : 21 ;G =0 : 08ModeNumberBaselineModeNumber 00 : 250 : 500 : 751 15101520 1 5 10 15 20 (a) d =0 : 43 ;G =0 : 57ModeNumberBaselineModeNumber 00 : 250 : 500 : 751 15101520 1 5 10 15 20 (b) d =0 : 43 ;G =0 : 57ModeNumberd =0 : 21 ;G =0 : 08ModeNumber 00 : 250 : 500 : 751 15101520 1 5 10 15 20 (c) Figure7-21.FullspanPODspatialcorrelationmapforther st20modes. 7.2FiniteSpanCavity Thissectionwillpresentroweldmeasurementswhichwillb efocusedprimarilyon thenitespancavitycongurations.Thebaselinecavityan dtwosuppressedcongurationswhichareidenticaltothefullspancavitywillbedi scussed.Abriefdiscussion regardingtherowelddierencesintheniteandfullspanc avityisincludedtofurther contrastthetwocavities. Itshouldbenotedthatforallplotspresentedthevectorsfa llingincloseproximity totheedgeofthecavitysidewallhavebeenremoved.Thiswas necessaryduetoglare atthesidelipofthecavity.Figure 7-22 showsarawimageofaninstantaneousseeded roweldforthebaselinecavityhighlightingtheilluminat edtracerparticleandtheedgeof thecavitysidewall.Themaskedregionsweredeterminedbya veragingthemeanintensity oftheunprocessedtracer-images.Appropriatethresholdv alueswhereappliedtoclearly identifyhighintensitypixelsandsubsequently"masked"f orthePIVvectorcalculations. 7.2.1FullandFiniteSpanFloweldComparison Acomparisonofthefullandnitespancavitymeanroweldsa representedinFigure 7-23 .Thenitespancavityshearlayerpenetratesmuchdeeperin sidethecavitywhich 157

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y Dx L00 : 250 : 50 0 : 75 1 1 0 1 Figure7-22.Exampleofseededroweldillustratingedgeof cavitysidewall. isapparentbythemoreaggressivelyturnedvelocityvector s.Therowislikelyforced intothecavityduetotherowspillageoversidewalls.Thepr imaryrecirculationregion hasbeenforcedtowardsthecenterofthecavitywiththeorig inlocatedapproximatelyat themidpoint.Theincreasedcrossrowinsidethenitespanc avityisbelievedtobethe primaryfactorleadingtothereducedpressureructuations y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)FullSpan y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b)FiniteSpan y/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c)FullSpan y/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d)FiniteSpan Figure7-23.Comparisonoffullandnitespancavitystream wisevelocitycontours. 158

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7.2.2MeanFloweld Figure 7-24 illustratesthecontoursofthemeanstreamwiseroweldfor thebaseline andtwocontrolledcases.Thebaselinecavityrowshowsashe arlayerthatpenetrates deepinsidethecavityroughly y D =0 : 75withvelocitiesontheorder45%ofthefreestream velocity.Astheroddiameterandgapheightareincreasedth estreamwisevelocityinside thecavitydecreaseswhilethereversedrowbecomesmorepro minentandiseasilyseen whenexaminingthevelocityvectorsforeachconguration. Thestreamlinesandvelocity vectorsinandimmediatelysurroundingtheshearlayerrema innearlyparalleltothe streamwiserowinitiallythenturnintothecavityaround x L =0 : 5.ExaminingFigure 7-24 itiseasytoseetheshearlayerhasbeenbroadenedbythetrai lingedgeandthe penetrationofthestreamwisevelocityhasbeenlargelyred ucedforthesuppressedcavities. Astheroddiameterandgapheightisincreased( d =0 : 43 ;G =0 : 57)italsobecomes apparentthattheshearlayerisfurtherbroadenedandlower streamwisevelocityrow impingesontheaftwall.Figure 7-25 illustratesthestreamlinesforeachcavity.Dueto themissingdataaroundthecavitiesliplinetheseplotsapp eardisjointanditishardto followsomeoftherecirculationpatterns.Thebaselinecav ityinteriorrowisdominated byasingleclockwiserotatingrecirculationbubblethatsp ans75%ofthecavitylength andiscenteredat x L ; y D =(0 : 5 ; 0 : 5).Theprimaryclockwiserotatingrecirculation bubbleinsidethecavityismovednearthecavitytrailinged geforthelargediameter rodwhencomparedwiththebaselineorsmallrodcavity.Thev erticallocationofthe originofthisrecirculationregionremainsinsensitiveto thebaselineorsuppressedcavity conguration.Themaximumrecirculationvelocitywasfoun dtobenearly10%ofthefree streamvelocityforthebaselinecavityandupto20%forthes uppressedcavitieswhich waslocatednearthecavityroor.Thecavitymidpointshowed recirculationvelocitiesnear 5%and10%ofthefreestreamvaluesforthebaselineandsuppr essedcavitiesrespectively. Thestreamlinesindicatewhatappearstobeasecondarycoun ter-rotatingrecirculation 159

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y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 d =0 : 43 ;G =0 : 57 d =0 : 21 ;G =0 : 08 Baseliney/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 2 (d) u U 1 Figure7-24.Finitespancavitymeanstreamwisevelocityco ntoursandproles. regionthatspans0 x L 0 : 25ofthecavityforeachcavitywhichismoreprominentthan whatwaspreviouslyshownforthefullspancavity.7.2.3TurbulentFloweld Theeectoftherodspresenceontheturbulentroweldforth enitespancavities arepresentedinthissection.Figure 7-26 throughFigure 7-28 showthecontoursand prolesforthemeanandturbulentroweldsrespectively.T heructuationsareonthe sameorderasthefullspancavitynearly35%and15%ofthefre estreamrowforthe streamwiseandwallnormalructuatingvelocitiesrespecti velyforthebaselinecavity. Examiningtheprolesitisseenthattheestimatedpointofm aximumstreamwise ructuationappearstoshiftdownwardintothecavity.Eachr odseemedtohavelittleeect ontheoverallmagnitudeofthestreamwiseornormalRMSvelo cityructuationsthoughit shouldbenotedthiseectcouldbehiddenbythemissingvect orsthroughtheshearlayer. 160

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y/Dx/L 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 Figure7-25.Fullspancavitymeanstreamlines.Mostnotablytherodsexhibitedthecapabilitytoshiftthep ointwherethemaximum ructuationabovethecavitylipconsistentwithfullspanca vitythoughnotasdrastically. Forthenitespancavitythepotentialliftingoftheshearl ayerisinconstantbattlewith therowspillingintothecavityfromthesidewallstendingt oinhibittheoveralllofting eect.Thelargerdiameterexhibitedagreaterdisplacemen tofthepeakRMSvalueswhich becomesmoreevidentasyouprogresstowardstheleadingedg eofthecavity. Instantaneousimagesofthedensitygradientforthebaseli neandsuppressedcavity areillustratedinFigure 7-29 .Theimagesclearlyshowtheshearlayerliftedabovetheaft wallwhichresultsinreducedgradientsinsidethecavity.I tshouldbenotedthatalthough thisisonlyonesinglerepresentationofthisphenomena,ea chsnapshotillustratedsimilar behavior.Thissupportsthenotionofthefeedbackmechanis mbeingdisruptedinside thecavitybyalteringtheaftwallimpingementlocation.Fu rthermore,thissupportsthe discussionbasedonthepressurecorrelationanalysispres entedinChapter6. Arunajatesan etal. ( 2002 2003 )performedhighdelitynumericalsimulationsusing arodasasuppressiondeviceforsubsonicrow.Throughanana lysisoftheturbulent 161

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y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 300 : 35 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 300 : 35 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 00 : 050 : 100 : 150 : 200 : 250 : 300 : 35 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 d =0 : 43 ;G =0 : 57 d =0 : 21 ;G =0 : 08 Baseliney/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 2 (d) h u rms i proles Figure7-26.Finitespanmeanstreamwiseturbulentvelocit ycontours. kineticenergybudgettheyreporttheadditionofnegativep roductionaltersthebalance ofproductionanddissipationcommonlyfoundinfreeshearr ows(orinthiscase,similar tobaselinecavityrows).Theyexplainedtheenergybudgetw asdominatedbydissipation ratesduetothesmallscalestructuresintheshearlayer.Th ereexistedlowerproduction ratesduetotheadditionofthenegativeproductionadirect artifactoftheturbulence generatedbytheoscillatingrodwake.Thesecoupledevents ,theyargue,arelikely thedrivingmechanismforthereductionoftheresonanceint hecavity.Althoughthe energybudgetscouldnotbeextractedfromthemeasurements thedecreasedmeanshear throughouttheshearlayer @U @y isconsistentwiththisnotion. 7.3Summary Floweldmeasurementsforthefullandnitespancavitiesw ithandwithoutsuppressionwerepresentedinthischapter.Therowspillingin tothecavityforthenite 162

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y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 500 : 75 1 0 : 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 00 : 0380 : 0750 : 1130 : 150 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 d =0 : 43 ;G =0 : 57 d =0 : 21 ;G =0 : 08 Baseliney/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 2 (d) h v rms i proles Figure7-27.Finitespanmeannormalturbulentvelocitycon tours. spancaseisbelievedtoresultinadierentbaselineinteri orcavityrow.Theprimary recirculationregionisshortenedandshiftedtowardsthec enterofthecavityforthenite spancavity.Themeanturbulentroweldsforthesetwocavit iessharesimilarintensities albeittherowpenetrateddeeperintothenitespancavity. Theenhancedmixingofthe nitespancavityleadstoslightlyelevatedructuatingvel ocitiesandcounter-intuitively, lowersthepressureructuationsinsidethecavity.Theenha ncedmixingonthecavityroor likelyhindersanddampensthepropagationofdisturbances insidethecavitywhichleads toadisruptedfeedbackmechanismthusloweringthemagnitu desoftheresonanttones.To thisend,thesuppressionlevelsrealizedappeartobedrive nbythesamemechanismfor eachcavityastheobservedturbulenteldreductionsandtr endswereverysimilar. Theuseoftwo-pointstatisticsrevealedthenormaloraxial turbulentvelocity evolutionbecameelongatedinthedirectionexamined.Thes eregionswerealignedwith 163

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y/Dx/L 0 : 02 0 : 010 00 : 010 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 : 02 0 : 010 00 : 010 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) d =0 : 21 ;G =0 : 08 y/Dx/L 0 : 02 0 : 010 00 : 010 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) d =0 : 43 ;G =0 : 57 d =0 : 43 ;G =0 : 57 d =0 : 21 ;G =0 : 08 Baseliney/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 2 (d)RSSproles Figure7-28.FinitespanmeanstreamwiseRSScontours.orslightlyturnedintothecavityforthebaselinecavitywh ereasthesuppressedregions werederectedawayfromthecavitylipimmediatelyatthelea dingedgeofthecavityand liftedabovetheaftwallbythetrailingedge.Theexaminati onofPODmodesfurther supportedthatthespatialstreamwiseorganizationofthet urbulentstructureswasaltered whencontrolled.Thewellestablishedstructuresthatwere identiedinsidethecavityfor thebaselinecavitywheretypicallynotpresentwiththecon trolledcavitiesandtendedto beliftedhigherabovethecavitylipline. Thepresenceoftherodwasshowntodecreasethemeansheargr adient(more eectivelyforthelargerodplacedatthetopoftheboundary layer)intheshearlayer. Therodwasalsoabletoshiftthepointofmaximumshearawayf romtheaftwall. Aweakerwaveoriginatingattheaftwallasexplainedinthes ummaryofChapter6 coupledwiththeroweldvisualizationpresentedinthisch apter,clearlydemonstratethe 164

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y Dx L 0 0 : 250 : 500 : 751 2 1 0 1 (a)Baseline y Dx L 00 : 250 : 500 : 751 2 1 0 1 (b) d =0 : 41 ;G =0 : 57 Figure7-29.SelectedinstantaneousSchlierenimagesforb aselineandsuppressed ( d =0 : 43 ;G =0 : 57)congurationsfromthenitespancavity. impingementpointontheaftwallofthecavityhasbeendrast icallyaltered.Itisplausible toassociatethereducedpressureructuationswiththesupp ressionoftherappingmotion oftheshearlayerortheloftingoftheshearlayerwhichwoul dinturnrapabovethe cavitylipline.Thiswouldresultinlowerspeedrowimpingi ngonthecavityaftwall relativetothebaselinecase.Thesuppressionoftherappin gmotioncouldalsobelinked totheincreasedthicknessoftheshearlayer.Theincreased thicknesscouldbetiedto increasedturbulencelevelsintheapproachingboundaryla yerattheleadingedgeas reportedin Ukeiley etal. ( 2004 b ). 165

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CHAPTER8 SIMULATIONOFBASELINEANDCONTROLLEDCAVITY ThischapterwillfocusondiscussingtheresultsoftheCFDs imulationsusingthe methodsoutlinedinChapter4.Thesimulationswererunfort hebaselineandcontrolled cavitywith d =0 : 43 ;G =0 : 58.Thiscongurationwaspreviouslyshown(Chapter6) tobethemosteectivecontrolarrangement.Thechapterwil lstartwithadiscussion onthevalidationeortforthesimulationsandwillthenpro gresssimilarlythroughthe dataanalysisaswasperformedinChapter6andChapter7.Res ultswillbepresented forructuatingpressuremeasurements,meanroweldandmea nturbulentroweldin conjunctionwithtwo-pointstatisticsandPOD. Figure8-1.Geometryandboundaryconditionsforthenumeri calsimulations. ThegeometryandboundaryconditionsareillustratedinFig ure 8-1 withthephysicalparameterslistedinTable 8-1 .Allsolidsurfacesusedaclassicalno-slipboundary conditionexceptthetest-sectionsidewallswhereaslipbo undaryconditionwasapplied. Moredetailsoftheprescribedboundaryconditionarefound inChapter4.Theremaining boundaryconditionsappliedwerenon-rerectingcharacter isticboundaries.ArepresentationoftheoversetstructuredmeshisgiveninFigure 8-2 withtheapproximatepoint 166

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Figure8-2.Structuredoversetgrid.countforthegridsystemgiveninTable 8-2 .Thebaselinegridsystemconsistedofthree blocktoblocksuperblocks.Thecavitygridwasoversetwith thetest-sectiongridwhich requiredacollargridforpropercommunication.Thecontro lledcongurationaddeda oversetcylindersuperblockwhichfollowedthegridrenem entandtopologyoutlinedin Chapter5.Thesolutionwasrunon60Opteron2.8GHzCentralP rocessingUnits(CPU) andeachsimulationrequiredonaverage8 : 6 10 4 CPUhoursofcomputationtime. Theselectedbaselinecavitymesh(negrid)consistedofte nmillionnodes(16million forthecontrolledconguration)includingthecollargrid .Thedistributionofpoints weremaximizedinsidethecavityandshearlayerregioninho pesofloweringtheoverall eddyviscosityintroducedbytheturbulencemodel.Themaxi mumlengthofacellinany coordinatedirection,givenby,didnotexceed 0 : 10 L .Thisconstraintwasapplied withinthecavityandforadistance D outsideandaroundthecavitytoadequatelyresolve theshearlayer.Therewereatotalof51pointslocatedinapp roachingboundarylayer witharstwallspacingof y =4 10 4 leadingtoa y + < 2.Thepointdistributioninthe normaldirectionwasspeciedusingahyperbolictangentpr oleandthecellstretching ratioinanycoordinatedirectionwaskeptunder20%.Table8-1.Listofdimensions(mm)referringtoFigure 8-1 LDW ` w L e h O dG 76.212.725.4152.476.238.150.83.83.21.52.0 167

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Table8-2.Computationalgridsizesforthebaselinecavity GridPoints ijk Cavity(Fine)7 : 2 x 10 6 322121111 Cavity(Coarse)3 : 5 x 10 6 1515151 Test-Section2 : 8 x 10 6 25171151 Eachsimulationwasrunataconstantphysicaltimestepof dt =5 10 7 swhich wascorrespondedtothephysicaltimestepusedinthesimula tionspresentedinChapter5. Thesolutionwasmarchedforwardsixteenperiodsbasedonth elowestpredictedRossiter tonetoallowtherowtoachievestatisticalstationarity.P SDplotswerecomputedwith arecordlengthof2 14 using75%overlapwhichresultedinaPSDfrequencyresoluti onof f r = f S nfft =122Hz.Ensembleaveragingwasusedformeanrowandturbule ntroweld calculationswherecompletesolutionleswereoutputever y20iterationsoverthenext16 rowperiods. 8.1ValidationofCavitySimulations Alimitedgridrenementstudywasconductedtoevaluatehow cellsizeeectedthe performanceofboththesolverandselectedturbulencemode l.Asaresultofthisstudy, itwasalsofoundthattheoriginalprescribedboundarylaye r( BL 1 )velocitieswerenotin goodagreementwiththetypicalexperimentalboundarylaye r.Theoriginalnozzlesolution waschokedatstagnationconditionsthatwerenotconsisten twithwhatwasobservedin theexperimentalobservations.Thisledtovelocitiesinth eboundarylayerthatwereon theorderof20%higherthanexpectedandisconsistentwitht hehigherresonanttones observedinFigure 8-4(a) .Thiserrorwasresolvedforthenegridsimulationsonly( BL 2 ). Thisleadstothegridresolutionstudiesbeingconductedwi thdierentinletprolesfor thedierentgridshowevertheresultsofthegridresolutio nstudiesarestillbelievedtobe valid. Figure 8-3 illustratesrandominstantaneoussnapshotsoftheZ-vorti citycomponent forthebaselinecavitywithboththecoarseandnegrids.Th enegridresolvedstructuresonasmallerspatialscaleindicativeoftheenhancedo verallspatialresolution.Figure 168

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8-4 illustratesthesensitivityoftheructuatingsurfacepres suresmeasuredonthecavity roorbasedontheselectedapproachingboundarylayerandle velofgridrenement.As expected,Figure 8-4(a) indicatesthesolutionisequallysensitivetoboththeleve lof gridrenementandtheapproachingboundarylayer.Theaftw allspectraiscompared inFigure 8-4(b) wherethemostnotabledierencesarefoundinthefrequenci esofthe resonanttones.Highervelocitiesintheapproachingbound arylayerresultedinhigher resonanttonesduetotheincreasedconvectivevelocityint heshearlayer.Themagnitudes ofthehigherfrequencyresonanttoneswereslightlyexagge ratedwhencomparedwith thecorrectedboundarylayer.Itshouldbenotedthatonlyth eaftwallandroorwere consideredbuttheforwardwallexhibitedsimilarbehavior saswell.Subsequently,itwas decidedtogowiththethenerofthetwogridsforallofthean alysispresentedhere. (a)Coarse( BL 1 ) (b)Fine( BL 2 ) Figure8-3.Instantaneoussnapshotsof z vorticitycontoursillustratingthecombinedgrid renementandcorrectedboundarylayereects. Figure 8-5 illustratesaqualitativeassessmentoftheroweldbasedo nthecontrolled cavityleadingedgenear-eld.Figure 8-5(b) illustratesarandominstantaneoussnapshot ofthetracerparticleintheroweldforthe d =0 : 40 ;G =0 : 57whileFigure 8-5(a) illustratesa z vorticitycontourplottakenfromarandomsnapshotofthesi mulation.Each imageillustratesthecomplexinteractionoftheseparated boundarylayeraftofthecylinderandthewakeofthecylinder.Thegeneralbehaviorofthec ylinderinteractionwith theshearlayerappearstobeinfairagreementbetweentheex perimentalobservationsand simulations. 169

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CoarseBL 1 FineBL 2 Expp h (^ p ^ p) i Q1f ( kHz )1 35 10 15 25 10 210 11 2 5 (a)PSD CoarseBL 1 FineBL 1 FineBL 2 Expprms Q1x L 0 0 : 20 : 4 0 : 6 0 : 8 1 0 0 : 05 0 : 10 0 : 15 0 : 20 (b) p rms Figure8-4.Eectofgridrenementoncavityructuatingsur facepressures.PSDplot measuredontheaftwallat y D = 0 : 5and p rms takenonthecavityroorat variousaxialpositions y Dx L 0 0 : 040 : 08 0 : 125 0 : 17 0 0 : 25 0 : 50 (a)CFD y Dx L 0 0 : 040 : 08 0 : 125 0 : 17 0 : 50 0 : 25 0 (b)Experiment Figure8-5.Instantaneoussnapshotoftracerparticleinth ecylinderneareldandCFD vorticityillustratingtheneareldcylinderrowdynamics Thesimulationswerefurthervalidatedwithexperimentalp ressureandvelocityeld measurementsthatwerepresentedinChapter6andChapter7. Themeanstreamwise velocitycontourswithvectorsarequalitativelycompared inFigure 8-6(a) andFigure 8-6(b) respectively.Thereisgoodagreementbetweenthesimulati onandexperimental measurementsforthebaselinecavitywhichisillustratedh ere.Themeanstreamwise velocityintheshearlayerandthemeanstreamwiseturbulen tvelocityalongwiththerow penetrationdepthareingoodagreementasdemonstratedbyF igure 8-6 andFigure 8-7 170

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ThemeanstreamlinesfortheCFDsimulationandexperimenta lmeasurementsareshown inFigure 8-6(c) andFigure 8-6(d) .Eachgureshowsaprimaryrecirculationbubble locatednearthecenterofthecavitythoughthenumericalsi mulationshavethisbubble locatedslightlyfartherdownstream.Itshouldbenotedtha tsomeerrorisanticipatedin theexperimentalstreamlinesneartheleadingedgeduetooi lbuildupanddicultyin adequatelyseedingthisregion. y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Numerical y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Experimental y/Dx/L0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c)Numerical y/Dx/L 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d)Experimental Figure8-6.Experimentalandnumericalmeanstreamwisevel ocitycontoursandmean streamlines. Meanstreamwisevelocityprolesandmeanstreamwiseturbu lentvelocitiesare quantitativelycomparedinFigure 8-7 atselectstreamwiselocationsforthebaseline cavity.Thesimulationsandexperimentalobservationsare ingoodagreementwiththe dierencesfoundnearthecavityroornearthetrailingedge .Theincreasederrorin 171

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theseregionsarelikelyattributabletothedicultiesenc ounteredintheexperimentwith measurementsclosetothewall(laserbloom)andinadequate seeding. CFD Exp u/ U 1y/Dx/L 00 : 25 0 : 50 0 : 75 1 1111111 1 0 : 50 0 0 : 50 1 1 : 50 (a)Meanstreamwisevelocity PSfrag CFD Exp u rms =U 1y/Dx/L 0 0 : 25 0 : 50 0 : 751 0000000 1 0 : 50 0 0 : 50 1 1 : 50 (b)Meanstreamwiseturbulentvelocity Figure8-7.Experimentalandnumericalmeanstreamwiseand turbulentvelocity comparison. Acomparisonoftheaftwallspectraforthebaselineandcont rolledcavityare presentedinFigure 8-8(a) andFigure 8-8(b) respectively.Thecomputationalspectra agreesfavorablywiththeexperimentaldataforbothcaviti es.Thelargestdiscrepancy isfoundinthemagnitudeofthedominantresonanttoneandat decreasedbroadband accuracyathigherfrequencies.Theresolutionofthepeakt onemagnitudeisafunction ofthefrequencyresolutiondierencebetweenthedatawher etheexperimentalresolution was11Hzandthenumericalcouldonlyberesolvedto122Hz.Th edivergenceofthe solutionathigherfrequenciesislikelyattributabletoth elackofspatialresolution, turbulencemodellimitationsandimplicitmethodsinthenu mericalscheme.Inparticular, 172

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tofullyresolvethebroadbandbehaviorathigherfrequenci es,onewouldneedtoapproach resolutionontheorderoftheKolomogorovscaleandRANSsol utionsarenotcapable ofthisduetothelimitationsdiscussedinChapter4.Figure 8-9(a) againshowsgood CFD Expp h (^ p ^ p) i Q1f ( kHz ) 1 3 5 1015 25 10 210 11 2 3 (a)Baseline CFD Expp h (^ p ^ p) i Q1f ( kHz ) 1 3 5 1015 25 10 210 11 2 3 (b)Controlled Figure8-8.Comparisonofcomputationalandexperimentala ftwallspectratakenat y D = 0 : 5. agreementinbothtrendandmagnitudewiththeructuatingsu rfacepressuresmeasured onthecavityroorforbothbaselineandsuppressedcavities 8.2InstantaneousFloweldandPressureMeasurement Unsteadypressuremeasurementsextractedfromthecavityc enterlineat z =0are presentedinFigure 8-9 .Theeectivenessoftherodasapassivesuppressiondevice is clearlyevidentwherebothresonanttonesandbroadbandlev elswerereducedbysimilar levelstothoseobservedintheexperimentalmeasurements. Thespectraontheaftwall forthecontrolledcaseindicatedabroadpeaknear50kHzass ociatedwiththeshedding oftherodwhichisingoodagreementwithPSDpeaksillustrat edinChapter5forthis sizerodandgapheight.Itshouldbenotedthispeakhasanamp litudeanthatisan orderofmagnitudelowerthanthedominantresonantfrequen cyofthebaselinecavity. ThissuggeststheA/Dlimitation(45kHz)presentedinChapt er6willonlyslightly eecttheresultsofpercentreductionsfortheexperimenta lresults.Crossandauto173

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CFDSupp CFDBaseline ExpSupp ExpBaselineprms Q1x L 0 0 : 20 0 : 40 0 : 600 : 80 1 0 0 : 050 0 : 10 0 : 15 (a)Floor p rms Suppressed Baseline p rms Q 1y D0 0 : 05 0 : 10 0 : 150 : 20 0 : 25 0 : 30 1 0 : 75 0 : 50 0 : 25 0 (b)AftWall p rms Suppressed Baseline p rms Q 1y D00 : 01 0 : 02 0 : 03 1 0 : 75 0 : 50 0 : 25 0 (c)ForwardWall p rms RodShedf 50( kHz )CFDSupp CFDBaselinep h (^ p ^ p ) i Q 1f ( kHz ) 1 25 100 10 2 10 1 1 2 (d)AftWallPSD Figure8-9.Baselineandcontrolledcavityructuatingsurf acepressuresontheaftwall, leadingedgewallandroor.Spectrameasuredontheaftwalli sat y D = 0 : 5 correlationsofthepressurefortheaftwallatselectlocat ionsontheroorandupstream wallareillustratedinFigure 8-10 .Boththebaselineandcontrolledcavityareinclose agreementwiththecorrelationspresentedinChapter6fort henitespancavity.Table 8-3 containsthetimelags,distancebetweencorrelatedpoints andthecalculatedspeed ofthedisturbancepropagatingupstreaminsidethecavity. Thecalculateddisturbance propagatesupstreamnearlyatthelocalspeedofsoundimply ingthatthecorrelationis pickinguptheacousticfeedbacktravelingthroughthecavi tyforboththebaselineand controlledcongurations. 174

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Aft Fwd Aft x L =0 : 25 Aft x L =0 : 50 Aft x L =0 : 75 Aft AftC xy ( s ) 1 012 3 4 5 678 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (a)Baseline Aft Fwd Aft x L =0 : 25 Aft x L =0 : 50 Aft x L =0 : 75 Aft AftC xy ( s ) 1 012 3 4 5 678 0 : 5 0 : 25 0 0 : 25 0 : 50 0 : 75 1 (b)Controlled Figure8-10.Baselineandcontrolledcavityautoandcrossc orrelations. Table8-3.Cavitywavespeed,timelagsanddistances. Case (s) 10 4 x L a(m = s) CFDBase2.041374 CFDSupp2.001381 ExpBase2.110.75361 ExpSupp2.110.75361 Thoughthecorrelationsclearlydemonstratetherod'sabil itytoeectivelydisrupt thefeedbackmechanismofthecavityrowasexplainedinChap ter6,therewaslittle eectonthespeedatwhichthesedisturbancesoccur.Thehig hlyresonantphenomenaof thebaselinecavitywhichmanifestsitselfasfairlysmooth nearlysinusoidaloscillationsis apparentintheseplots.Thereisanotablelackofsmoothnes stothecorrelationproles whichtendtoexhibitsharperpeakswithnerscalenoisefor thecontrolledcavity.The levelsofthecorrelatedsignalwhicharepropagatingupstr eamarebeingreducedcloserto thebackgroundturbulencelevelshencethecorrelationsar enoisier. Theshearlayerofthecontrolledcavityisdominatedbysmal lscalestructureswhich isqualitativelyveriedbyexaminationofthebaselineand controlledcavityinstantaneous vorticitycontourspresentedinFigure 8-11 .Itisevidentfromtheseguresthatthewake ofthecylinderappeartoberollingupafterapproximatelyt wodiametersandisclearly pullinginsomeoftheruidfromthecavityshearlayer.Thisi nteractiontendstotearo 175

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clockwiserotatingstructuresfromthecylinderuppershea rlayer.Thereisnoapparent alternatesheddingfromtheupperandlowercylindershearl ayer.Thesestructuresare introducedintothecavityshearlayerinaperiodicfashion similarlytothatpredictedby thevorticitycontoursillustratedinChapter5forthe d =0 : 41 ;G =0 : 64simulation. Theadditionofthesestructuresintotheshearlayercurbst hespanwiseorganizationofthe shearlayer. y Dx L 15 10 5 051015 00 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 y Dx L 15 10 5 051015 00 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 y Dx L 15 10 5 051015 00 : 250 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 y Dx L 15 10 5 051015 00 : 250 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 y Dx L 15 10 5 051015 00 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 y Dx L 15 10 5 051015 00 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 Figure8-11.Instantaneousnon-dimensionalvorticitycon toursforthebaselineand controlledcavitytakenat5 10 7 sintervals.Thebaselinecavityisshownin therstcolumnandthecontrolledcavityinthenextcolumn. 176

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8.3MeanFloweld Inthissectiontheensembleaveragedmeanroweldwillbean alyzedanddiscussed. MeanstreamlinesforbothcavitiesarepresentedinFigure 8-12 .Thebaselinecavityis dominatedbyalargeclockwiserotatingrecirculationregi onwhichcoversfrom0 : 35 x L 1.Amuchsmallerlocalizedcounter-rotatingrecirculatio nbubblesisfoundonthecavity roorattheleadingedge.Thesuppressedcavityexhibitsasi ngledominantrecirculation bubbleaswellwhichiscenterednear x L ; y D =(0 : 95 ; 0 : 375).Thecounter-rotating leadingedgebubblehasbeenenlargedandthethestreamline sinsidethebayindicate anaggressiveupwardtrendothecavityroornotrealizedin thebaselinecavity.The freestreamstreamlinesareslightlyderectedawayfromthe cavityinthenormaldirection forthecontrolledcase. y/Dx/L 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Controlled Figure8-12.Baselineandcontrolledmeanstreamlines. 177

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EachcomponentofthemeanvelocityeldisillustratedinFi gure 8-13 .Referring toFigure 8-13(a) and 8-13(b) theshearlayerisclearlyspread,mostobviouslynearthe leadingedge,andliftedabovetheaftwall.Thepresenceoft herodresultsinastronger recirculationregioninsidethecavity.Examiningthemean streamwisevelocitycontours andprolesthewakeofthecylinderintheneareld(blueint hecontourplots)appears tobeslightlyderectedtowardsthewallsimilartothesimul ationsofChapter5. Examiningtheverticalvelocitycomponentitisimmediatel ynoticedthattherod reducestheamountofinrowinthedownstreamportionofthec avity.Theliftingofthe shearlayerresultsinameanpositivenormalvelocityeldt hroughoutthemajorityof thecavitynotobservedinthebaselinecavity.Figures 8-13(e) and 8-13(f) showthemean span-wisevelocitycontourswhereitisapparentthatthepr esenceoftherodenhancesthe three-dimensionalityoftheroweldinsideofthecavitymo stnotablyattheleadingand trailingedges. Themeanstreamwiseandnormalvelocityprolesareillustr atedinFigure 8-14 at selectaxialpositions.Thepresenceoftherodwakeisstill clearlyvisibleintheinrected velocityprolefoundneartheleadingofthecavity.Theinr ectionclearlyoccursbelowthe rodcenterlineandisstillhasastrongpresenceat3ddownst ream.Thisisconsistentwith resultsfromChapter5wherethemeanstreamwisevelocitypr oleswerealsoderected towardsthewall.Theprolesstillhadapresenceroughly3d downstreamwiththe inrectionoccurringatroughly y =0 : 60ornearthebottomofthecylinderforeach simulation.Themeanstreamwisevelocityproleiseectiv elyspreadoutthroughthe entireshearlayerwhiletherecirculationvelocityinside thecavityinthedownstream portionisincreased.Theeectofthecontrolonthemeannor malvelocityisvisible wheretherowdirectioninthedownstreamportionofthecavi tyisreversedbetween 0 : 50 x L 0 : 875.Therowinruxatthetrailingedgeisvastlyreducedwith rowcontrol. Thereisanotableincreasethemeannormalvelocityimmedia telydownstreamofthe 178

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y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a) h u i U 1 y/Dx/L 0 : 25 00 : 250 : 500 : 751 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) h u i U 1 y/Dx/L 0 : 15 0 : 10 0 : 05 00 : 050 : 100 : 15 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) h v i U 1 y/Dx/L 0 : 15 0 : 10 0 : 05 00 : 050 : 100 : 15 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) h v i U 1 y/Dx/L 0 : 05 0 : 025 00 : 0250 : 05 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (e) h w i U 1 y/Dx/L 0 : 05 0 : 025 00 : 0250 : 05 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (f) h w i U 1 Figure8-13.Meanvelocityroweldcontours.Column(a)and (b)arethebaselineand controlledcavityrespectively. 179

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cylinder,consistentwithndingsfromChapter5,whichisb elievedtoinitiallyderectthe shearlayerawayfromthecavitylipline. y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 (a) h u i U 1 y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 250 : 50 0 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 (b) h v i U 1 Figure8-14.Meanstreamwiseandnormalvelocityprolesat variousaxiallocations. Contoursofthemeanverticalvelocitycomponentwithveloc ityvectorsatselect streamwiselocationsonthe yz planeareillustratedinFigure 8-15 andFigure 8-16 for thebaselineandcontrolledcavityrespectively.Thethree dimensionalityofthecavity rowbecomesclearerwhenexaminingthesegures.Thecavity rowisnotonlydominated bythemainstreamwiserecirculationbubbleasillustrated bythestreamlinesofFigure 8-12 butalsoexhibittwoopposingrotatingregionsdividednear thecavitycenterline. Theseregionsbecomemoreintenseastherowprogressestowa rdsthetrailingedgeof thecavityleadingtomorerowbeingpulledinsidethecavity andincreasedmixing.The existenceandconsequenceoftheserecirculationregionsi swhatisbelievedtobethe majordierencewhencomparingthefullandnitespancavit ies.Thecontrolledcavity showsasimilartrendwherethesetwoopposingrecirculatio nregionsgainintensityasone progressestowardstheaftwall.Themajordierenceisthed irectionofrotation,i.e.,the presenceoftherodreversesthedirectionofeachrecircula tionregion.Whenthecavity iscontrolledtheshearlayerisliftedabovethecavitylipline.Thisresultsinanaltered pressuregradientwhichdrawstherowintothecavityoverth esidewallsandforcesitout overthecavitylipnearthecenterline. Thevorticitythicknessandlocationofmaximumshear,calc ulatedinasimilar mannertotheexperimentaldata,isplottedinFigure 8-17(a) and 8-17(b) foreachcavity. 180

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y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 00 : 25 0 : 50 05 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (a) x L =0 : 25 y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 00 : 25 0 : 50 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (b) x L =0 : 50 replacements y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 0 0 : 250 : 50 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (c) x L =0 : 75 y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 0 0 : 250 : 50 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (d) x L =0 : 875 Figure8-15.Baselinecavitymeannormalvelocitycontours onthe yz planeatvarious axiallocationsshowingopposingrecirculationregions. Thegrowthrateisseentobehavelinearlywithvaryingslope sthatseemdependenton axiallocation.Thebaselinecavityexhibitedgrowthrates of0.22overtherst5%of thecavityanddroppedoto0.08overthenext30%ofthecavit ylength.Theshear layerappearstogrowveryaggressivelyoverthenext60%oft hecavitywhererates approachedtheinitialgrowth.Thecontrolledcavityexhib itedsimilarbehaviorwiththree fairlydistinctlineargrowthraterangesidentiablealth oughtheirextentandrateswere signicantlydierent.Therangeswereconsiderablylonge rthanthebaselinecavityand regionsIandIIgrowmoreaggressivelythanthebaselinecav ity.Thetrendsobservedare 181

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y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 00 : 25 0 : 50 05 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (a) x L =0 : 50 y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 00 : 25 0 : 50 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (b) x L =0 : 75 y/Dz/W 0 : 125 0 : 0625 00 : 06250 : 125 0 : 50 0 : 25 0 0 : 250 : 50 1 0 : 75 0 : 50 0 : 25 0 0 : 25 0 : 50 0 : 75 1 1 : 25 (c) x L =0 : 875 Figure8-16.Controlledcavitymeannormalvelocitycontou rsonthe yz planeatvarious axiallocationsshowingopposingrecirculationregionsro tatinginopposite directionofbaselinecavity. ingoodagreementwiththeexperimentaldatawherethecontr olledcavitygrowthrate exceedsthebaselinerateuntilthestreamwiseposition x L 0 : 50.Thegrowthratesthen continuallytaperotowardsthetrailingedgeofthecavity eventuallyfallingbelowrates observedinthebaseline.Thelocationofmaximumshearalso risesabovetheaftwall consistentwiththeexperimentalobservations.Thisisacl earindicationthemaximum shearisdecreasedandliftedabovethecavitylipwhenthero dispresent.Thereduced shearisbelievedtobeakeycontributortothesmallerlesso rganizedstructurespresented inFigure 8-11 182

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0=0 : 23 mm III II Idw dx I=0 : 22dw dx II=0 : 08dw dx III=0 : 18 d =0 : 43 ;G =0 : 57 Baseline w 0x 0 III II Idw dx I=0 : 35dw dx II=0 : 10dw dx III=0 : 06 050100150200250300 5 10 15 20 25 30 35 40 45 (a)VorticityThickness Baseline : dy dx =0 : 25 Suppressed : dy dx =0 : 51 d =0 : 43 ;G =0 : 57 Baseliney Dx L 00 : 250 : 500 : 751 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 (b)Locationofmax dU dy Figure8-17.Baselineandcontrolledcavityvorticitygrow thrateandlocationofmaximum shear. Thegrowthrateoftheshearlayerisfurthersubstantiatedw ithaqualitativeassessmentofthemeanvorticitycontoursshowninFigure 8-18 foreachcavity.Itisapparent thatthemeanvorticityovertherst25%ofthecavityshearl ayerisdrasticallyreduced andbroadened.Themagnitudeofthemeanvorticityinsideth ecavityhasalsobeen reducedmostnotablyatthetrailingedge. y/Dx/L 15 10 5 051015 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Baseline y/Dx/L 15 10 5 051015 0 0 : 25 0 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Controlled Figure8-18.Meanvorticity( z )contoursforthebaselineandcontrolledcavity. 8.4MeanTurbulentFloweld Theturbulentfeaturesofthecavityroweldwillbediscuss edinthissectionand contourplotsofthemcanbefoundinFigure 8-19 .Theturbulentvelocitiesfromthe 183

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simulationsareingoodagreementwithwhatwasobservedint heexperimentswith thepeakturbulencelevelsgenerallywithin5%.Therodsabi litytopullthestreamwise velocityructuationsabovethecavitylipareseeninFigure 8-19(b) .Asimilartrendis observedforthenormalandspanwiseturbulentvelocitiesw heretherodwasableto reducethevelocityructuationsinsidethebay. Theturbulentvelocities,RSSandTKEprolesareplottedat variousaxiallocations inFigure 8-20 .Examinationofthestreamwiseandnormalcomponentsofthe turbulent velocityprolesillustratesthespreadingoftheshearlay eringreaterdetailforthe controlledcavity.Thepointwherethemaximumturbulentru ctuationexists(foreach component)israisedabovethelipofthecavityasoneprogre ssesdownstreamwhichis consistentwithexperimentalobservations.Theprolesal soclearlyshowthereduced turbulentructuationsinsidethecavityforboththeaxiala ndnormalvelocities.Withthe liftingandbroadeningoftheshearlayertheturbulentkine ticenergyandturbulentshear stressatthetrailingedgeofthecavityhavebeensubstanti allyreduced.Therodcauseda similarreductionofmaximumructuationlevelsforeachvel ocitycomponent.Thisisagain anindicationoftherodsabilitytointerferewiththecoher enceofthebaselineshearlayer leadingtosmallerlessenergeticturbulentstructures.8.4.1EvolutionofTwo-PointVelocityCorrelations Inthissectiontheevolutionofthetwo-pointturbulentvel ocitystatisticsarepresented.Inalloftheplotspresentedinthissectionthecorr elationcoecientswillbe presentednormalizedbytheappropriateturbulentvelocit yasdenedinChapter2.For Figure 8-21 andFigure 8-22 thebaselinecavityisrepresentedintherstcolumnofimag es followedbythesuppressedcavity.Forallofthesegurespr ogressingdownthecolumn correspondstoadvancinginthestreamwisedirection. Figures 8-22(a) 8-22(j) illustratesthestreamwiseevolutionofthestreamwise componentoftheturbulentvelocitycorrelationsfortheba selineandcontrolledcavity withtheoriginlocatedat y D =0.Thehighlycorrelatedregionsforeachcavityare 184

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y/Dx/L00 : 050 : 100 : 150 : 200 : 25 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (a) h u rms i U 1 y/Dx/L 00 : 050 : 100 : 150 : 200 : 25 0 0 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (b) h u rms i U 1 y/Dx/L00 : 050 : 100 : 150 : 200 : 25 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (c) h v rms i U 1 y/Dx/L00 : 050 : 100 : 150 : 200 : 25 0 0 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (d) h v rms i U 1 y/Dx/L00 : 050 : 100 : 150 : 200 : 25 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e) h w rms i U 1 y/Dx/L00 : 050 : 100 : 150 : 200 : 25 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f) h w rms i U 1 y/Dx/L00 : 050 : 100 : 15 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (g) h u 0 2 + v 0 2 + w 0 2 i U 1 2 y/Dx/L00 : 050 : 100 : 15 00 : 25 0 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (h) h u 0 2 + v 0 2 + w 0 2 i U 1 2 Figure8-19.Meanturbulentroweldcontours.Column(a)an d(b)arethebaselineand controlledcavityrespectively. 185

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y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 25 0 : 50 0 : 751 1 0 : 50 0 0 : 50 1 1 : 50 (a) h u rms i U 1 y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 25 0 : 50 0 : 751 1 0 : 50 0 0 : 50 1 1 : 50 (b) h v rms i U 1 y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 25 0 : 500 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 (c) h u 0 v 0 i U 1 2 y/Dx/L d =0 : 43 ;G =0 : 57 Baseline 0 0 : 25 0 : 500 : 75 1 1 0 : 50 0 0 : 50 1 1 : 50 (d) h u 0 2 + v 0 2 + w 0 2 i U 1 2 Figure8-20.Meanroweldandmeanturbulentroweldprole s. elongatedinthestreamwisedirection.Thebaselinecavity exhibitslargerwellcorrelated regionswhilethecontrolledcaseexhibitsmorecompactcor relatedregionsmostnotablyat x L =0 : 5.Uponinspection,thecorrelationsclearlyindicatethat theturbulentlengthscale (ifoneweretointegratethecorrelationregion)islengthe nedintheaxialdirectionforthe baselinecavity.Thissuggeststhatthecontrolledcavitys hearlayerwillonlybecapableof containingsmallerturbulentstructures. Figure 8-22 illustratestheevolutionofthenormalcomponentofthetur bulent velocitycorrelationforthebaselineandcontrolledcavit ywiththeoriginlocatedat y D =0. Thehighlycorrelatedregionsareagainelongatedinthedir ectionoftheturbulentvelocity, i.e.,theverticaldirection.Thebaselinecavityexhibits largerwellcorrelatedregionswhile thecontrolledcavityexhibitsmorecompactcorrelatedreg ions.ExaminingFigure 8-22 one canseeanalternatingpatternoflowandhighcorrelationar easthroughoutthebaseline 186

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y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Originat x L ; y D =(0 : 125 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Originat x L ; y D =(0 : 125 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (c)Originat x L ; y D =(0 : 25 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (d)Originat x L ; y D =(0 : 25 ; 0) replacemen y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (e)Originat x L ; y D =(0 : 50 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (f)Originat x L ; y D =(0 : 50 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (g)Originat x L ; y D =(0 : 75 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (h)Originat x L ; y D =(0 : 75 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (i)Originat x L ; y D =(0 : 875 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j)Originat x L ; y D =(0 : 875 ; 0) Figure8-21.Evolutionofstreamwisespatialvelocitycorr elation.Columns(a)and(b)are thebaselineandcontrolledcavityrespectively. 187

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shearlayerwhichdonotexistforthecontrolledcavity.Thi sbehaviorisindicativeofisa periodicnatureoftheturbulentstructuresincavityshear layerforthebaselinecase. Figure 8-23(a) andFigure 8-23(b) illustratespatialcorrelationplotsoftheructuating surfacepressureasafunctionofseparationwiththeorigin locatedatthecenterpoint oftheaftwallforeachcavity.Similartothespatialveloci tycorrelationsreportedin Figure 8-22 ,thepressurecorrelationregionstendtobeelongatedinth enormaldirection foreachcavity.Thebaselinecavityexhibitsamuchlargerh ighlycorrelatedregionthat extendsfromjustbelowtheorigintothetopoftheaftwall.T hecontrolledcavity'swell correlatedregionisnoticeablymorecompactwhichsuggest sthatsmallerscalestructures haveimpactedtheaftwall.Thisisconsistentwiththeargum entthatthecontrolis breakinguptheorganizedturbulentfeaturesimpactingthe aftwallthusreducingthe sourceoftheupstreampropagatingwave.8.4.2POD ResultsfromtheapplicationoftheProperOrthogonalDecom positiontovelocity eldsnapshotsfromthenitespancavityispresentedinthi ssectioninthesamemanner aswasdoneforthefullspancavityinChapter7.Asmentioned inChapter7,itis advantageoustoextracteventswhichcontainlargeamounts ofturbulentkineticenergy. Figure 8-24(a) andFigure 8-24(b) representtheturbulentkineticenergydistribution andcumulativeenergyfortheextractedPODmodesrespectiv ely.Thebaselinecavity exhibitedslightlymoreenergyintherstvemodeshowever therewaslittledierence thereafteruptothersttwentymodesconsidered.Thefairl yrapidconvergenceofthe basissetisobservedwhere50%ofthetotalturbulentkineti cenergyofforthebaseline cavityrequiredapproximately2%ofthetotalsnapshotswhi lethesuppressedcavity requiredslightlymoreatapproximately2.5%.Theenergyca pturedintherstmode wasroughly8%foreachcavitywhilethenextthreemodesdie redbyasmuchas1.5% butcapturedunder5%ofthetotalturbulentkineticenergyp ermode.Onlytherst vemodesareconsideredinthefollowingdiscussionfortwo reasons.Therstisfor 188

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y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Originat x L ; y D =(0 : 125 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Originat x L ; y D =(0 : 125 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (c)Originat x L ; y D =(0 : 25 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (d)Originat x L ; y D =(0 : 25 ; 0) replacemen y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (e)Originat x L ; y D =(0 : 50 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (f)Originat x L ; y D =(0 : 50 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (g)Originat x L ; y D =(0 : 75 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (h)Originat x L ; y D =(0 : 75 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (i)Originat x L ; y D =(0 : 875 ; 0) y Dx L 0 : 50 0 : 25 00 : 250 : 500 : 75 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j)Originat x L ; y D =(0 : 875 ; 0) Figure8-22.Evolutionofnormalspatialvelocitycorrelat ion.Columns(a)and(b)arethe baselineandcontrolledcavityrespectively. 189

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y Dz W 0 : 25 00 : 250 : 500 : 751 0 : 50 0 : 2500 : 250 : 50 1 0 : 75 0 : 50 0 : 25 0 (a)Baseline y Dz W 0 : 25 00 : 250 : 500 : 751 0 : 50 0 : 2500 : 250 : 50 1 0 : 75 0 : 50 0 : 25 0 (b)Controlled Figure8-23.Fluctuatingsurfacepressurecorrelationont heaftwallwiththeorigin locatedatthecenterpoint. conciseness,ashighermodesdemonstratedsimilartrends. Thesecondreasonisbecause forincreasingmodenumberstheenergyshouldoccuratsmall erspatialscalesanditis likelytheywillbecomeunimportant(saybeyondtherst1020modes)formodelingor reconstructingtheoverallglobalbehaviorofthecavity.T othisend,itisunderstoodthat itisnotclearthatthemodeswiththemostkineticenergysho uldbeuniquelytargetedfor rowcontrolpurposes. d=0 : 43 ;G=0 : 57 BaselineE k 0 5 10 1520 0 2 4 6 8 10 (a)TKEDistributionperMode d=0 : 43 ;G=0 : 57 BaselineE kP k 1 10 1001000 0 : 15 0 : 25 0 : 375 0 : 50 0 : 75 1 (b)CumulativeTKE Figure8-24.EnergydistributionofPODmodes. 190

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Figure 8-25(a) andFigure 8-25(j) representtherstvePODmodesbasedonthe normalcomponentoftheturbulentvelocityforthebaseline andcontrolledcavities.It shouldbenotedthatonlythePODmodesassociatedwiththeve rticalvelocitycomponent isshownasthoseassociatedwiththestreamwisevelocityco mponentexhibitedsimilar behavior.Therstmodeofthebaselinecavityexhibitedone elongatedfeaturewhich isconsistentwiththesubsonicresultsof Murray etal. ( 2009 )andthosepresentedin Chapter7.ThePODmodeructuationsforthebaselinecavitya rein-phaseacrossthe lengthofthecavityandrepresentstheoscillationofthesh earlayer.Thesuppressedcavity PODmodeructuationsarenolongerinphaseasillustratedby thealternatingnegative andpositiveregionslocatedthroughouttheshearlayerreg ion.Incontrasttothefullspan cavityPODresultsdiscussedinChapter7,thepresenceofth eroddidhaveaprofound eectonthestreamwiseorganizationoftheresolvedmodesf orthenitespancavity.The largescalewellorganizednatureofthebaselinecavityshe arlayerisrerectedinthePOD modeswherealternatingpositiveandnegativeregionsappa rentlyresemblelocationswhere highlyenergeticturbulentstructurescouldreside.Thisi sclearlymissinginthesuppressed cavityPODmodes,particularlyevidentintheoddnumberedm odes.Here,smallcompact regionsofhighlyenergeticrowarelocatedinthecylindern eareldandisexpecteddue totheturbulentructuationsofthecylinderwake.Theregio nswithhighturbulentkinetic energyfoundunderthecavitylipofthebaselinecavityform odesthreeandfourarenot presentinthesuppressedcavitymodes.Theevennumberedmo desforthesuppressed cavityaregenerallyelongatedinthestreamwisedirection whilecompactedinthevertical directionwhencomparedtothebaselinecavitymodes.Compa ringthemodesforthe baselineandcontrolledcases,primarilyforevennumbered modes,oneseesthehighly energeticregionsarebetteralignedinthestreamwisedire ctionforthesuppressedcavity. Thisisbelievedtobeaconsequenceofthealteredbehavioro ftheshearlayeroscillations wherethebaselinecavityshearlayerexhibitsastrongtend encytorapintoandoutof thecavityatthetrailingedge.ThecontrolledcavityPODmo desseemtoindicatethis 191

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rappingnatureoftheshearlayerhasbeendrasticallyalter ed.Ingeneral,thereisgood agreementintheobservedbehaviorofthePODmodeswiththee volutionofthetwopoint statisticsdiscussedabove.Thecavityshearlayerconsist sofsmallerlessorganizedregions inspacewherehighlyenergeticrowstructureswouldbefoun d. Recentworkby Arunajatesan etal. ( 2009 )hasfoundthatthespanwisevelocity componentmodesshowsmallerscalestructureswiththecavi tycontrolledbymicrojets.Theystatethemajoreectofcontrolisthechange,ord isorganization,inspanwise structureoftheshearlayerwhencontrolispresent.Thefou ndthatatwo-dimensional (streamwiseandnormalvelocities)PODofthevelocityeld didnotcorrelatewellwith themodesofpressure.Athree-dimensionalPOD,ontheother hand,correlatedwell. Theyobservedthatbyneglectingthespanwisevelocityinth ePODanalysisthephysical interpretationofwhatthePODmodesrepresentarelikelysk ewedespeciallyinthecase ofthecontrolledcavitybecauseoftheincreasedthreedime nsionalityoftherow.They furthersurmisethatsincethepressureeldinthecavityis moreorlesstwo-dimensional, thetwo-dimensionalPODofthepressurecanbeagoodreprese ntationoftherow.In lightofthisobservation,itisacknowledgedthisisapoten tialshort-comingofthePOD workinthisresearchandwillwarrantmoredetailedinvesti gationinthefuture. 8.5Summary Simulationsusingasecond-ordertimeandspacesolverutil izingaDESapproachwith aoneequationturbulencemodelwereperformed.Therewasgo odagreementbetweenthe simulationsandexperimentalobservationsinexaminingov eralltrendsandmagnitudes. Thecomparisonsincludedructuatingsurfacepressuresmea surements,meanroweld analysis,turbulentructuationintensitiesandthegrowth rateofthecavityshearlayer. Theabilitytovisualizeallthreedimensionsgaveinsighti ntotheeecttherodhadonthe interiorcavityrow.Lookingatthe z =0planerevealedadominantclockwiserotating recirculationpatternwiththeoriginlocatednearthecavi tycenterpoint.Therewerealso smallercounterrotatingbubbleslocatedontherooratthec avityleadingandtrailing 192

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y Dx L 0 : 50 00 : 501 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (a)Mode1 y Dx L 0 : 50 00 : 501 0 0 : 250 : 50 0 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (b)Mode1 y Dx L 1 0 : 50 00 : 501 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (c)Mode2 y Dx L 1 0 : 50 00 : 501 00 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (d)Mode2 y Dx L 1 0 : 50 00 : 501 00 : 250 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (e)Mode3 y Dx L 0 : 10 00 : 10 00 : 250 : 500 : 75 1 1 0 : 5 0 0 : 5 1 1 : 5 (f)Mode3 y Dx L 1 0 : 50 00 : 501 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (g)Mode4 y Dx L 1 0 : 50 00 : 501 00 : 25 0 : 50 0 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (h)Mode4 y Dx L 1 0 : 50 00 : 501 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (i)Mode5 y Dx L 0 : 10 00 : 10 0 0 : 250 : 500 : 751 1 0 : 5 0 0 : 5 1 1 : 5 (j)Mode5 Figure8-25.PODmodesbasedonthenormalructuatingveloci ty.Columns(a)and(b) arethebaselineandcontrolledcavityrespectively. 193

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edges.Thecontrolledcavityshiftedthedominantrecircul ationregioncenternearthe trailingedgeofthecavity.Theinteriorrowofthecavitywa smoreaggressivelyswept upwardswhencontrolled.Anexaminationofthemeannormalv elocityontheselected yz planesindicatedtwoopposingrecirculationregionssplit bythecavitycenterlineforthe eachcavity.Theseregionsintensiedasoneprogressedtow ardsthetrailingedge.The baselinecavityrotatedinsuchawayasthecenterlineofthe cavitywasdrawingruidinto thecavityandruidwasexpelledoutsidethecavityatthesid ewalls.Thepresenceofthe rodreversedtherotationoftheseregionswherethemeannor malvelocityatthecenterline wasnowoutofthecavityandruidwasdrawnintothecavityatt hesidewalls.Thisis believedtobetheconsequenceofthepressuregradientgene ratedastheshearlayeris liftedabovethecavitylip. Theuseoftwo-pointstatisticsshowedtheevolutionofhigh lycorrelatedregions becameelongatedinthedirectionofthecomponentofthetur bulentvelocityconsidered. Theseregionsweremuchmorecompactforthecontrolledcavi ty.Coupledwiththeresults fromthePOD,itwascleartheorganizationalbehaviorofthe sestructureswashighly sensitivetothepresenceofrowcontrolwiththerod.ThePOD modesclearlyillustrated muchsmallerregionsofhighturbulentkineticenergywhere onlysmallerscalestructures couldresidecomparedtothebaselinecavity.Instantaneou svorticityimagesrevealedthe complexinteractionoftherodwakeandtheseparatedbounda rylayerdownstreamofthe rod.Thisinteractionresultedinsmallscaleturbulentstr ucturesinjectedintotheshear layerinaperiodicfashiontypicallyontheupstrokeofthec ylinderwakeeventhoughthere wasnostrongVon-Karmantypesheddingpresent. Crossandauto-correlationanalysisrevealedidenticaltr endsasobservedwiththe experimentalobservations.Thecorrelationlevelsdroppe dwiththepresenceofthe rodandisastrongindicationofadisruptedfeedbackmechan ism.Thespeedatwhich thedisturbancepropagatesinsidethecavityappearstobei ndependentoftherod's presence.Theloweredcorrelationvaluesalsosuggesttheu pstreampropagatingwave 194

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emanatingfromtheshearlayerimpingementontheaftwallis weakenedwhenthecavity iscontrolled. Thepresenceoftherodinthisresearchleadstoathickershe arlayer(mostnotably atthecavityleadingedge)withanelevatedshearlayerspre adingrate.Thewakeofthe rodpullsattheshearlayerleadingtohighershearstresses andathickershearlayer closetothecavityleadingedge.Theseconditionslikelyin hibittheabilityoftheshear layertoroll-upintolargescaleorganizedstructuresbyde creasingthereceptivityand growthrateofthedisturbancesintheshearlayer.Themeanr oweldandmeanturbulent roweldexaminationalsorevealedtheshearlayerwaslifte dwhencontrolled.Thelifting oftheshearlayerandtherappingmotionoftheshearlayerma ybeduallyresponsiblefor alteringtheaftwallimpingementpoint.Thepenetrationof theshearlayerintothecavity wouldbereducediftheshearlayerrapsabovetheaftwallorr apslessaggressively.It isfurtherpostulatedthattherappingmotionwouldlikelyb ealteredbythepresenceof thethickershearlayer.Lastly,itwasobservedthatthemea nshearwasdecreasedinthe controlledcasewhilethepointofmaximumshearwasshifted awayfromtheaftwall. 195

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CHAPTER9 CONCLUSIONSANDFUTUREWORK Acoupledexperimentalandnumericalinvestigationofthee ectsofcontrolona M 1 =1 : 4supersonicrowoveraopenrectangularcavitywithalength -to-depthratioof sixwasconducted.Thealteringoftherowwasachievedbypla cingacylindricalrodin boundarynearthecavitiesleadingedge.Measurementsincl udedtheructuatingsurface pressureonthecavitycenterlineforveroddiametersandv ariousgapheights.Itwas determinedthatarodsizedapproximately40%oftheapproac hingboundarylayerheight placedsuchthatthetopoftherodisnearthetopofthebounda rylayer,resultedin thebestoverallsuppression(bothbroadbandandtonalcomp onents)oftheructuating pressuresonthecavitysurfaceshowevermanyothercasesde monstratedsignicant reductionsaswell.Thebroadbandlevelsofthebaselinecav ityarelikelyattributableto theapproachingboundarylayerturbulencelevelsandthefa cttherodwasabletoreduce theselevelsinthecontrolledcasessuggeststheturbulenc ethroughouttheshearlayer wasslightlyaltered.Themeanvelocityandturbulentrowe ldstatisticsforthebaseline andselectedcontrolcaseswereacquiredandanalyzedonthe cavitycenterlineusingdata acquiredfromparticleimagevelocimetry.Computationalr uiddynamicsusingadetached eddysimulationapproachwasconductedtogainfurtherinsi ghtformeasurementsoof thecavitycenterlineandforthenearbodyroweldofthecyl inder. Thisnalchaptersummarizesthemajorndingsandcontribu tionsbroughtoutin thisworkwhichwerediscussedthroughoutthisdissertatio n.Itstartswithasummaryof theniteandfullspancavityresultswhichisfollowedbyad iscussionofthedemonstrated accuracyoftheCFDapproachemployedthroughoutthisresea rch.Thisisfollowedwith asummaryofthemechanismsbelievedtobepresentforthecon trolofthecavityrow. Finally,thechapterconcludeswithrecommendationsforim provementsandwherethis workcouldbeextendedupon. 196

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9.1FiniteandFullSpanCavities Floweldmeasurementsforthefullandnitespancavitiesw ithandwithoutcontrolwerepresented.Thespectrameasuredontheaftwallexh ibitedresonanttonesthat occurredatthesamefrequencies.Forbothcongurationsth esefrequencieswerereasonablyrepresentedbyRossiter'sequationwiththedefaultco nstants.Thedominanttone switchedbetweenthesecondandthirdlowestresonanttonef ortheniteandfullspan cavitiesrespectivelyandwassubstantiallygreaterforth efullspancavity. Thecenterlinevelocityeldshadsimilaroverallfeatures althoughthereweremany dierences.Themostsignicantdierenceobservedinther oweldmeasurementswas thatthenitespancavityallowsforruidexchangefromthes idewallswhichcausesa dierentinteriorrowforthebaselinecase.Theprimaryrec irculationregionisshortened andshiftedtowardsthecenterofthecavityforthenitespa ncavity.Themeanturbulent roweldsforthesetwocavitiessharesimilarintensitiesa lbeittherowpenetrateddeeper intothenitespancavity.Theincreasedcross-rowlikelyd ecreasesthestrengthofthe disturbancesinsidethecavitywhichleadstolowermagnitu desoftheresonanttones. Whencontrolled,thefullspancavityshearlayerappearedt obeliftedslightlyhigher abovethecavityliplinewithdierencesintheshearlayerg rowthrate.Thedierences observedbetweenthecontrolledcavityroweldsarebeliev edtobeaconsequenceofthe partialcoverageoftherodwiththefullspancavityandthed ierencesintheinteriorrow asexplainedabove.Tothisend,thesuppressionrealizedap pearstobedrivenbythesame mechanismforeachcavitythoughexaggeratedwiththefulls pancavity. 9.2DemonstrationofDES Asecondorder(timeandspace)detached-eddysimulationth atusesReynolds averagingwithaoneequationturbulencemodelwasfoundtob ereasonablyaccurate for M 1 =1 : 4cavityrow.Theabilityofthecurrenttechniquetodecentl ycapturethe relevantphysicsofthecontrolledandbaselinecavityrows wasdemonstrated.Specically, itwasshownthattherewasgoodagreementbetweenexperimen talandsimulatedvalues 197

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oftheructuatingsurfacepressureonthecavitysurfaceswh entheappropriateboundary layerwasprescribedinthesimulations.Additionally,the meanvelocityroweldand turbulentstatisticswereingoodagreement.Theshearlaye rgrowthrateswereingood agreementwhenconsideringtrendsbetweenthebaselineand controlledcases.The simulationsaccuratelypredictedhigherspreadingratesf orthecontrolledcavitythat taperoat x L =0 : 5eventuallyfallingbelowthevaluesrealizedinthebaseli necavity. Themagnitudeofthegrowthratesexhibitedsomevariancebu titshouldbenoted thiswaswhencomparinganexperimentalfullspancavitywit hthesimulatednite span.Itisnotentirelyclearwhetherthesedierencesarea ttributabletothegeometry, numericalapproachorcombinationthereof.Morespecical ly,itisnotedthatmost compressibilitycorrectionstoturbulencemodelshavebee nappliedtoimprovecorrelations withexperimentsforthespreadingrateoffreeshearlayers atlowerspeedsthanthe focusofthisresearchandfurthermoretheshearlayersprea dingrateforthenitespan cavitywasnotabletobecomputedaccuratelyfromtheexperi ment.Furtherinvestigation iswarrantedtodeterminethecost/benetsofincludingcom pressibilitycorrections, two-equationturbulencemodels,numericalschemesandful lLESascomputational resourcescontinuetogrow.Nonetheless,itisthisauthors opinionthatanalysisusing theguidelinesposedinthisresearchwouldbebenecialtot hecommunityandare immediatelyavailableinmostmodernCFDsolvers.Itshould benotedthatthea-priori knowledgeoftheapproachingboundarylayerdetailswerepi votalintheachievedaccuracy ofthesimulationsandhencethisrepresentsakeyneedtomat chexperimentaldata. 9.3CavityControl Theuseoftwo-pointturbulentvelocitycorrelationsandPO Danalysisillustrated thestreamwiseorganizationoftheturbulentstructureswa salteredbythecontroleorts. Thewellestablishedstructuresthatwereidentiedinside thefullornitespanbaseline cavitywheretypicallynotpresentwiththecontrolledcavi tiesandwereshowntobe typicallyliftedabovethecavitylip.ThePODmodesclearly illustratedmuchsmaller 198

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regionsofhighintensityturbulentkineticenergyforthec ontrolledcaseswheresmaller scalestructureswouldlikelyresidewhencomparedtotheba selinecavity. Theabilitytovisualizeallthreevelocitycomponentsbyth euseofCFDgaveinsight intotheeecttherodhadontheinteriorcavityrow.Conside ringthe xy planelocated atthecenterofthecavityitwasseenthatthereexistedadom inantclockwiserotating recirculationwithit'soriginlocatednearthecavitycent erpoint.Thepresenceoftherod shiftedthedominantrecirculationpatternclosertothetr ailingedgeofthecavityandthe interiorrowwasmoreaggressivelysweptupwards.Anexamin ationofthemeannormal velocityontheselect yz planesindicatedtwoopposingrecirculationregionssplit bythe cavitycenterlinefortheeachcavitythatincreasedininte nsityasyouprogressedtowards thetrailingedge.Therowinthebaselinecavityrotatedins uchawayasthecenterlineof thecavitywasdrawingruidintothecavityandexpelleditat thesidewalls.Thepresence oftherodreversedtherotationoftheseregionswerethemea nnormalvelocityatthe centerlinewasnowoutofthecavityandruidwasdrawnintoth ecavityatthesidewalls. Itislikelythattheliftingoftheshearlayerduetotherodc reatedthepressuregradient suchthattherowisdrawninsidethecavityatthesidewallsl eadingtothechangesin therowpattern.Inlightofthisobservation,itisnotyetde terminediftheructuating pressureontheentirecavityiseectivelysuppressed.Ino therwords,hastheoverall loadingonthecavitywallsbeensuppressedorjustre-distr ibutedwherefurtheranalysisof theresultsfromthesimulationswillberequiredtodetermi nethis. Previousresearchershaveattemptedtostudythelinearsta bilityanalysisofthe cavityshearlayerwhenarodintheapproachingboundarylay erwasusedasthecontrol device.However,as Freymuth ( 1966 )showedforaseparatedlaminarboundarylayer,the growthratesareindependentoftheamplitudeofforcingwhi chdenitelyhasananalogy tothecavityrowbeingstudiedhere.Directnumericalsimul ationswereperformedby Guillaume&Colonius ( 2008 )toinvestigatethethree-dimensionalglobalinstability of compressiblerowoveropencavities.Theyappliedthesocal ledbi-globallinearstability 199

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originallyusedby Theolis&Colonius ( 2002 )totakeintoaccountnon-paralleleects andpossiblecouplingoftheshearlayerwiththeacousticsc atteringandtherecirculated rowinsidethecavity.Theyidentiedthree-dimensionalin stabilitieswhichtookthe formofdisturbancesgrowingintherecirculationregionwi thinthecavity.Theresults indicatetheinstabilityistightlycoupledwiththestreng thoftherecirculationregion inthedownstreamportionofthecavity.Theynotetheseinst abilitieswouldnotbe accessibletoclassicallinearstabilityofparallelrows. Furthermore,theclassicalstability argumenthasbeenutilizedtosaythatcontrolisachievedth roughthebreakdownofa pairofoppositesignvorticalstructures(alternatetop/b ottomsheddinginthecaseofthe cylinder)whichwasnotevidentinthesimulationspresente dinChapter8severelylimiting theapplicabilityofthisnotion. Thecouplednumericalandexperimentalanalysisofthenit espancavityseemto pointconsistentlytothesamerowelddierenceswhenarod isplacedattheleading edgeofthecavity.Thecontrolledcavityleadstoathickers hearlayer(mostnotablyat thecavityleadingedge)withanelevatedspreadingrate.Re ferringtothesimulations inChapter8,thestructuresinthewakeoftherodinteractwi ththecavityshearlayer withatimeperiodicexcitationwhichpullsupontheshearla yer.Additionally,these structurestendtoincreasetheentrainmentofruidinsidet hecavityandspreadoutthe shearlayermorerapidly.Theseconditionsinhibittheabil ityoftheshearlayertoroll-up intolargescaleorganizedstructuresthatimpingeontheaf twall.Thisisillustratedin theturbulentvelocitycorrelationsandPODanalysiswhere theorganizationalnatureof highlyenergeticstructureswasshowntobedrasticallyalt ered.Thisleadstosmallerless organizedturbulentstructuresthroughouttheshearlayer asdemonstratedinthevorticity plotsfoundinChapter8.Thisisfurtherdemonstratedthede creasedfootprintonthe aftwallructuatingpressurecorrelationsfromthesimulat ionspresentedinChapter8. Thethickershearlayerwithsmallerscaleturbulentstruct uresisbelievedtodecreasethe receptivityandgrowthrateofthedisturbances. 200

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Themeanroweldandmeanturbulentroweldexaminationsal sorevealedtheshear layerwasliftedwhencontrolled.Theliftingoftheshearla yerandtherappingmotionof theshearlayermaybeduallyresponsibleforalteringtheaf twallimpingementpoint.The penetrationoftheshearlayerintothecavitywouldbereduc edifeithertheshearlayer isliftedorifitrapsabovethecavityliplineorlessaggres sively.Itisfurtherpostulated thattherappingmotionwouldlikelybealteredbythepresen ceofthethickershearlayer. Thisisbelievedtoleadtoaweakenedupstreampropagatingd isturbancedueinpartto thedecreasedrowvelocityimpingingonthecavityaftwall. Thisinturn,resultsinthe reducedpeaksobservedintheresonanttonesforthecontrol ledcases.Thekeypoints regardingthecontrolofthecavityaresummarizedbelow. Therodleadstoathickershearlayerthatinitiallyspreads morerapidlybecause thestructuresinthewakeoftherodinteractwiththecavity shearlayerwitha timeperiodicexcitationwhichliftstheshearlayernearth ecavityleadingedge.The structuresaresmallerandlessorganizedwhichisbelieved toleadtoashearlayer thatislessreceptivetotheupstreamdisturbancesinsidet hecavity. Therodisabletoaltertheaftwallimpingementpointduetot heliftingandaltered rappingnatureoftheshearlayer.Thisresultsinlowerspee drowimpingingonthe aftwall. Theupstreampropagatingdisturbanceemanatingfromtheaf twallisweakenedin partduetotheloweredvelocityrowimpingingontheaftwall .Thesecoupledevents leadtodrasticallyreducedructuatingpressuresmeasured onthecavitysurfacesand thusloweredmagnitudesoftheresonanttones(nearbroadba ndlevels).Thefact thatbroadbandlevelsareslightlylowered,indicatesthec ontrolwasabletoalterthe turbulencelevelsthroughouttheshearlayer. 9.4FutureWork Baseduponthendingsinthecurrentworkthereareseverala reasofresearchoered aspossiblesuggestionsforfurtherinquiryandarebrokeni ntonumericalandexperimental sectionsaccordingly.9.4.1ComputationalFluidDynamics Duetothegoodagreementofthenumericalsimulationswitho bservedexperimental observationsfurtherinvestigationhasalreadybeenunder takenthoughthecompletion 201

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isfurtherdowntheroad.SimilartothezoomedPIVeorts,nu mericalsimulationshave beeninitiatedusingblocktoblockgrids(eliminatingover setboundaries)toinvestigate thecylinderwakeneareldrowfeaturesatvariousgapheigh tsandroddiameters.The useofhigherordernumericalschemesandturbulencemodels isalsobeinginvestigated inattempttoquantifyturbulentproductionanddissipatio nrates.Thestreamwise locationoftherodwilllikelybeinvestigatedtostudythee ectsonthecavityshearlayer. Investigatingtheeectsofturbulentintensitiesintheap proachingboundarylayerusing techniquessimilartothatpresentedin Comte etal. ( 2008 )and Levasseura etal. ( 2008 ) couldbepursued.AsdiscussedbrieryinChapter8,thePODan alysiscouldbefurther expandedtoincludeallthreevelocitycomponentsoverthee ntirecomputationaldomain. Additionally,thePODanalysiscouldbeperformedbasedont hepressureeldasan input.Compressibilityeectscouldbeinvestigatedusing techniquessuchdensityweighted averagingforthecurrentdatasetorcodesthatincludecomp ressibilitycorrectionsinthe turbulencemodeloruseFavreaveraging.9.4.2Experimental FurtherworkisrecommendedinthePIVanalysisofthenites pancavitytominimizethebloomrealizedinthiswork.Averysimilaranalysi swouldbeproposedbut thedesignofthenitecavitywouldbealtered.Morespecic ally,oneshouldensureall surfacesoftheglasscavitysidewallwouldbepolished.Iti sbelievedthisalonewould drasticallyreducethebloomexperiencedinthisresearch. Furthermore,theuseofthreecomponentPIVcouldbeusedtoextractusefulinformationre gardingthespan-wisecavity roweld.Synchronizedsurfacepressuremeasurementsandv elocityeldmeasurementscan beobtained.Notonlywouldthisallowonetoinvestigaterec onstructionoftheroweld usingreducedordermodelsobtainedviaPODbutwouldalsoal lowonetomorethoroughlyinvestigatethecorrelationbetweenstructuresobs ervedintheshearlayerandthe surfacepressuremeasurements.Anextensiveliteraturere viewofsuppressedcavitiesusing arodsuppressiondevicealsorevealsexperimentalroweld measurementsarelacking 202

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atsubsonicandtransonicMachnumbers.Apurelysubsonicro weldcouldbestudied withminimaleortrequiringonlythedesignofanewnozzlet oachievethedesiredrow conditions.Adetailedinvestigationoftheturbulentinte nsitiesmeasuredintheapproachingboundarylayerandare-characterizationofthetest-se ctionrowconditionswouldbe required.TransonicMachnumbersmightrequireamoreintri catelydesignedtestsection tobleedtheboundarylayeraswell.9.4.3FutureConsideration Theresearchpresentedcouldbefurtherextendedbystudyin gthefundamentalrow phenomenaregardlessofwhetheritisnumericalorexperime ntal.Somekeytopicsto befurtherinvestigatedcouldincludeenergybalancesfrom theturbulentkineticenergy equationandamoredetailedanalysisofpressurecorrelati onsonallofthecavitysurfaces toquantifythereductionofmagnitudeofthesources.Lastl y,onecouldconsiderthe dynamicsphenomenatofurthertrytoidentifywhattrulyist hesourceintheroweldfor boththesuppressedandbaselinecavities. 203

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APPENDIXA TURBULENCEMODELFORMULATION AdiscussionofthetransitionfromthecompressibleNavier -Stokesequationstothe RANSequationsandsubsequentoneequationSAturbulencemo delareoeredinthis appendixforconveniencetotheinterestedreader.Theappe ndixbeginswithadetailed reviewofthecompressibleNavier-Stokesequationsandisf ollowedbythederivation oftheRANSequations.Thediscussionisendedwithadetaile dexplanationoftheSA turbulencemodelwithallofthevariablesandcoecientsli stedfortheselectedCFD solver. A.1CompressibleNavier-StokesEquations Thegoverningequationsforthecurrentresearcharederive dfromthegeneral instantaneouscompressibleNavier-Stokesequationswrit tenincartesiancoordinates.The continuityequationmaybewrittenas @ @t + @ ( u i ) @x i =0(A{1) where istheruiddensity, u i isthetensorformofthevelocityvectorand t istime.The momentumequationmaybewrittenas @ ( u i ) @t + @ ( u i u j ) @x j = @ ij @x j @p @x i + f i (A{2) where p istheruidpressureand f i representsbodyforces.Theviscousstresstensoris givenas ij = @u i @x j + @u j @x i 2 3 ij @u k @x k (A{3) where isthemolecularviscositywhichiscalculatedusingSuther land'slaw = ref T T ref 3 2 T ref + S T + S : (A{4) Theunderscore'ref'isthereferencevalueofeitherthetem perature,givenby T ,orthe viscosity.ThevalueoftheSutherlandtemperatureistaken as S =110 : 4K.TheKronecker 204

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deltafunctionisgivenby ij = 8><>: 1if i = j 0if i 6 = j: (A{5) Theconservationofenergymaybewrittenas @ ( E ) @t + @ ( u i E ) @x j = @ @x j ( u i ij q i ) @ ( pu i ) @x i (A{6) whereheatruxisgivenby q i = k c @T @x i = c p Pr @T @x i : (A{7) Theratioofmomentumdiusivity(kinematicviscosity)and thermaldiusivity( )is givenbythenon-dimensionalPrandtlnumber Pr= = c p k c (A{8) where c p isthespecicheatatconstantpressureand k c istheruidthermalconductivity. Thetotalenergyperunitmassisexpressedas E = c v T + 1 2 u i u i = e + k (A{9) where e isthespecicinternalenergyand k representsthekineticenergyperunitmass. Thetotalenthalpyperunitmass( H )andpressure( p )arewrittenas H = E + p = E + RT (A{10) p =( r 1) e (A{11) where R istheuniversalgasconstant.Theratioofthespecicheats ,alsoknownasthe heatcapacityratiooradiabaticindex,istakenasaconstan tforacaloricallyperfectgas r = c p c v =1 : 4(A{12) where c v istheratioofspecicheatatconstantvolume. 205

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A.2CompressibleReynoldsAveragedNavier-StokesEquatio ns Directsolutionoftheseequationsasdescribedaboveisnot yetpracticallyfeasibledue tothespatialandtemporalresolutionrequirementsforsol vingthesmallscalestructures foundwithinaturbulentboundarylayer.Therefore,aformo flteringisappliedto captureonlythescalesofinterestandanewsetofequations isderivedandsubsequently solved.In1895OsbourneReynoldssuggestedtherowvariabl es,namelytheinstantaneous velocity(~ u i )intheNavier-Stokesequations,couldbedecomposedintoa ructuatingand meancomponent ~ u i ( x;t )= u i ( x )+ u 0i ( x;t )(A{13) wherethemeanandructuatingcomponentsaregivenbytheove rbarandprimenotation respectively.Theresultingvelocitiescouldfurtherbete mporallyaveragedoveralarge enoughtimescaletoensurestatisticalstationarity u i = 1 2 T Z T T u i dt: (A{14) wherethemeanoftheructuatingvelocityisdenedas u 0i = 1 2 T Z T T u 0i dt =0 : (A{15) Decomposingandaveragingthecompressiblecontinuityequ ationresultsin @ @t + @ u i @x i =0 : (A{16) TheReynoldsaveragedcompressiblemomentumequationas @ u i @t + @ u i u j @x j + @ @x j u 0i u 0i = @ ij @x j @ p @x i (A{17) wherethedeviatoricstresstensorisexpressedby ij =2 S ij 1 3 S kk ij : (A{18) 206

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TheReynoldsaveragedenergyequationis @ E @t + @ @x j u i E + H + @ @x j u 0j e 0 + 1 2 u 0i u 0i u 0j = @ @x j ( u i ij )+ @ @x j u 0i 0 ij @ @x j k c @ T @x j (A{19) where k c istheheattransfercoecient.Thestrainratetensorisgiv enas S ij = 1 2 @ u i @x j + @ u j @x i (A{20) andtheturbulentkineticenergyisexpressedas k = 1 2 u 0i u 0i : (A{21) Thetotalenergyandenthalpyareexpressedby E = e + 1 2 u i u i = e + k (A{22) H = E + p : (A{23) Asetofveunknowncorrelations,givenbelow,wasaddeddue totheapplicationofthe Reynoldsdecomposition.Thesecorrelationsaretypically determinedempiricallywhenever possibleandappliedtoachieveproperclosureforthesetof partialdierentialequations. 1. u 0i u 0j whenmultipliedby givesthetheReynoldsStresstensor R ij andisassociated withthetransportofmomentumduetotheturbulentmotionof theruid 2. u 0i e 0 whenmultipliedby givestheturbulentheatrux 3. u 0i u 0i u 0j isatriplecorrelationrepresentingturbulentdiusion 4. u 0i 0 ij istheviscousdissipationduetotheructuatingvelocities 5. k istheturbulentkineticenergy Themainprobleminturbulencemodelinginvolvescalculati ngthe R ij fromtheknown meanquantities.Typically,theBoussinesqapproximation isappliedwhichisbasedonthe assumptionthatthemomentumtransfercausedbyturbulente ddiescanbemodeledby 207

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eddyviscosity.Thisideaissimilartohowmolecularviscos itydescribesthemomentum transfercausedbymolecularmotioninagas.TheBoussinesq hypothesisthereforestates thattheReynoldsstressescanbecalculatedastheproducto fdynamiceddyviscosityand thestrain-ratetensor( S ij )ofthemeanrowi.e., R ij = u 0i u 0j = t @u i @x j + @u j @x i 2 3 k ij : (A{24) TheeddyviscosityisascalarandconsequentlytheReynolds stresscomponentsare linearlyproportionaltothemeanstrain-ratetensor. Morkovin ( 1962 )statedthatcompressibilityoftherowisnotimportantunlessthemotionof theeddiesaretravelingat supersonicspeeds.Hefurtherstatesthatthelocalmeanden sityisimportanttothedeterminationoftheturbulentstructurenottheructuatingcomp onentofthedensity. Rumsey ( 2009 )statesthatcompressibilityhasasmalleectonturbulenc eforwallboundedrows under M 1 5.However,itisnoted,mostcompressibilitycorrectionsh avebeenappliedto improvecorrelationswithexperimentforthespreadingrat eoffreeshearlayerssuchasin theworkof Bardina etal. ( 1997 ). A.3Spalart-AllmarasTurbulenceModel Aone-equationturbulencemodel,suchastheSpalart-Allma rasmodel,canbederived toaccountfortheconvectionanddiusionofturbulence.Th isturbulencemodelwas derivedusingempiricalrelationshipswiththeaimofdevel opingaturbulenttransport modelthatwasfast,robust,andaccurateforbothshearlaye rsandboundarylayers.This wasaccomplishedbyderivingatransportequationforthetu rbulenteddyviscosity t = t where and t arethedensityandabsoluteturbulentviscosityoftheruid respectively. Oneequationmodelshavenomechanismtocomputetheturbule ntlengthscaleandthus mustincludeonetoaccountforthegeometriclengthscale(s uchasdistancetothewall). TheeddyviscosityisrelatedtotheReynoldsstressesby t = u 0 v 0 du dy (A{25) 208

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andissubjecttoconversionlawdescribedby D t Dt = @ t @t +( u r ) t =Diusion+Production Destruction : (A{26) Toconstructthefullmodelforaturbulentrow,eachtermont herighthandsideneedto bedened.Deningthesetermsinnon-dimensionalformresu ltsinadditionalconstants andnon-dimensionalfunctionsforeachterm.TheSAmodelus esexperimentsfrom incompressiblefreeshearrowtocorrectlevelsofshearstr essintwodimensionalmixing layersandwakes.Thevaluesforthepeakshearstresswereta kenas0 : 01 U 2 and 0 : 06 U 2 forthemixinglayerandwakerespectivelywheretheterm U isthepeak velocitydierence.Thetransportedquantityinthismodel istheundampededdyviscosity (~ )eddyviscosityandisrelatedtotheeddyviscosityby t =~ f v 1 (A{27) wherethesubscript v impliesthetermisviscousfortheremainderofthediscussi on.The function f v 1 isdenedas f 1 = 3 + c v 1 3 (A{28) where c v 1 isaconstant.FortheviscoussublayerSAintroducedtwovar iables~ whichis equalto t exceptintheviscousrange(nearthewall)andtheratioofth eeddyviscosity andruidviscosity = ~ : (A{29) ForthenalformofthemodelSAconsideredtheclassicallog layeranddevisednear-wall dampingfunctionsdeterminedfromempiricaldatasothat~ = yu allthewaytothe wallwhere y isthedistancetothewall, u isthefrictionvelocityand istheKarman constanttypicallytakenas0.41.Themodelthereforesolve sthefollowingtransport 209

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equation @ ~ @t + U j @ ~ @x j | {z } Convection = c b 1 (1 f t 2 ) ~ S ~ | {z } Production h c w 1 f w c b 1 k 2 f t 2 i ~ d 2 | {z } Destruction + 1 @ @x k ( +~ ) @ ~ @x k | {z } Dissipation + c b 2 @ 2 ~ @x k 2 | {z } Diusion (A{30) wherethesubscripts b;w;t standforbasic,wallandtriprespectively.Theterm ~ S isthe turbulentstrainratetensorandisrelatedtothevorticity magnitude( S )by ~ S = S + ~ k 2 d 2 f v 2 (A{31) where d isthedistancetothewalland k istheturbulentkineticenergytakentobea constant.Thesecondviscousfunctionisrelatedto f v 1 by f v 2 =1 X 1+ Xf v 1 : (A{32) Thefunction f w isdenedby f w ( r )= g 1+ c 6w 3 g 6 + c 6w 3 1 6 (A{33) where g r and f t 2 aregivenby g = r + c w 2 r 6 r (A{34) r = ~ t ~ Sk 2 d 2 (A{35) f t 2 = c t 3 e c t 4 2 : (A{36) Thetripfunction f t 1 andthefunction f t 2 areexpressedas f t 1 = c t 1 g t e c t 2 t 2 U 2 ( d 2 + g t 2 + d t 2 ) (A{37) f t 2 = c t 3 e c t 4 2 (A{38) where d t isthedistancefromthegridpointtothetrip, t isthevorticityatthewall, U isthedierencebetweentheeldpointvelocityandvelocit yatthetrip.Thewall 210

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TableA-1.SA-DESturbulencemodelcoecients c b 1 c b 2 kc w 1 c w 2 c w 3 c v 1 c t 3 c t 4 0.13550.6220.413.23910.327.11.20.5 2 3 boundaryconditionisnaturallysatisedwhen~ =0.Thefunction f w approachesa constantvaluewhenthevalue r maybetruncatedto10.Thefunction g t is g t =min 0 : 1 ; U t x (A{39) where x isthegridspacingalongthewallatthetrip.Lastly,thefun ction c w 1 isgivenby c w 1 = c b 1 k 2 + 1+ c b 2 (A{40) where istheturbulentPrandtlnumber.Alloftheconstantsusedin theCFDrowsolver aregiveninTable A-1 211

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APPENDIXB VIBRATIONANDDEFLECTIONOFSELECTEDRODS Thisappendixgivesthedetailsoftheanalyticalapproachf orestimatingtherod derectionandvibration.Considerationhadtobegiventoth istopicduetothesizing oftherodsconsidered,i.e.,because d .Itwasfoundthatneithertheanticipated longitudinalortransversenaturalfrequencyoftherodwer eofmajorconcern,especially forthemosteectiveroddetailedthroughoutthisresearch ThegeometryofeachrodusedinthisstudyisgiveninTable B-1 where d o and t w representtherodouterdiameterandwallthicknessrespect ively.Eachrodwasmadeof 304stainlesssteelwiththematerialdensityandmodulusof elasticityof =8000kg = m 3 E =193GParespectively. B.1RodFundamentalFrequency Thenaturalfrequencywasfoundfromasimplepin-pinconstr ainedhollowcircular crosssectionrodasshowninFigure B-1 Inman ( 2007 )givesthelongitudinalnatural frequencyforthisdeviceaccordingto n = ( nc ) L f = n 2 (B{1) where c = q E and n =1 ; 2 ;::: representsanintegervalueforharmonicconsiderations. Thefundamentalfrequencyoftherodisexpectedtobe f 97kHzforthesmalldiameter rodwhichiswelloutsidetheresonantfrequenciesfoundint hecavitiesandtheanticipated rodsheddingfrequency.TableB-1.Rodgeometry d 0.210.430.570.790.89 d o (mm)0.751.532.022.813.16 t w (mm)0.150.250.250.300.50 212

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FigureB-1.Suppressiondeviceproperties.TableB-2.Transversefundamentalvibrationofrod d d o (mm) d i (mm) f n (kHz) 0.210.910.764570.431.471.27751 Thetransversevibrationalfundamentalfrequencywascalc ulatedusing n =( 1 L ) r c L 4 (B{2) where( 1 L )=4 : 73, c = EI A A istherodcrosssectionalareaandthemomentofinertia is I = L 4 ( d 4o d 4i ).Table B-2 illustratesthecomputedfundamentalfrequencyforthe twosmallestrodsconsideredthroughoutthisresearch.Its houldbenotedthelargerrods consideredwouldexhibitprogressivelyhigherfundamenta lfrequenciesandlessderection duetotheincreasedstinessandwallthickness. B.2DerectionofRod Aconservativeestimateoftheexpectedderectionoftherod basedonthefreestream velocity U 1 wascalculatedusingthecontinuityequation.Theloadingw asassumed uniformlydistributedoverthecylinder X F x = U 2 1 A (B{3) where A = d o L 2 .Thederectionwasthencomputedfrombeambendingtheorywh ere = 5 384 FL 3 EI : (B{4) 213

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TheresultsareshowninFigure B-2 where ,forthepurposeofthisappendix,represents thecalculatedderectionand 0 isthethicknessoftheapproachingboundarylayer.The largestanticipatedrodderection(forthesmallestdiamet erandlongestrod)wasonthe orderof50%oftheboundarylayerheightandlessthan5%fort hemosteectiverod. d =0 : 40 d =0 : 20L(mm)=00510 15 2025 30 3540 0 0 : 1 0 : 2 0 : 3 0 : 4 0 : 5 FigureB-2.Estimatedrodderectionbasedondiameterand L foragiven U 1 214

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BIOGRAPHICALSKETCH JonathanG.DudleywasborninWesteld,NewYork.Hereceive dhisbachelor's degreein2000fromtheRochesterInstituteofTechnologyin MechanicalEngineering withaconcentrationinAerospaceEngineering.Jonathanbe ganworkatValeoEngine Coolingasaperformanceengineerutilizingniteelementm ethodsandcomputational ruiddynamicsforautomotiveandheavytruckcoolingsystem design.In2003hereceived hismaster'sdegreewhilestudyingtheruid-structureinte ractionofintracranialaneurysms fromtheUniversityatBualo.HebeganworkfortheAirForce SeekEagleOce (AFSEO)in2003asacomputationalruiddynamicsengineer.I n2005hemarriedhis wifeMeganDudleyandin2006heenrolledinthedoctoralprog ramattheUniversity ofFlorida(UF)wherehestudiedundertheadvisementofDr.L awrenceUkeiley.In 2010JonandMeganwelcomedintotheirlivestheirrstnewbo rndaughter,Malia Dudley.HecurrentlyworksfortheAirForceResearchLabora tory(AFRL/RWAVM)as abasicresearchscientist.HeisamemberoftheAmericanIns tituteofAeronauticsand Astronautics(AIAA). 225