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An Experimental Investigation of Circulation Control Acoustics

Permanent Link: http://ufdc.ufl.edu/UFE0042488/00001

Material Information

Title: An Experimental Investigation of Circulation Control Acoustics
Physical Description: 1 online resource (308 p.)
Language: english
Creator: WETZEL,DREW
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: 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: Underwater vehicle maneuverability is currently limited by conventional appendage technology. However, maneuverability can be greatly enhanced with the use of circulation control. A circulation control airfoil uses blowing over a rounded trailing and/or leading edge to increase circulation and generate lift at levels far greater than traditional control surfaces. In addition, large lift forces can be generated at very low speeds. One of the issues preventing the application of circulation control to underwater vehicles is noise. Thus, circulation control can only be applied to underwater vehicles if it improves maneuverability without a substantial noise level increase. The purpose of this investigation is to experimentally identify and characterize the flow field and corresponding noise sources of a circulation control airfoil. Flow and acoustic data are obtained for a dual-slotted, elliptic circulation control airfoil in an anechoic wind tunnel. The effect of single-slot blowing on the aerodynamic characteristics of the airfoil in an open and closed tunnel test section is evaluated. Measurements of the trailing edge curved wall jet flow using particle image velocimetry (PIV) reveal similarity of the outer region mean streamwise flow, and the length and velocity scales required for similarity, as well as the separation location, scale with the product of the chord Reynolds number and momentum coefficient. Streamwise vortices theorized to promote curved wall jet separation are also measured using PIV. Low and high frequency tones, generated by vortex shedding from the round trailing edge and finite slot lip, respectively, are detected. Broadband noise sources are identified for a variety of test conditions using a nested phased acoustic array. At low momentum coefficients, contaminating noise sources, like sidewall scrubbing noise and diffuser flow impingement noise, dominate. At higher momentum coefficients, an interaction between the high-speed trailing edge jet and the sidewall boundary layer flow appears to cause significant noise. The presence of multiple sources violates the assumptions required for using far-field microphone methods. Instead, a spectrum is obtained from the array data and compared to an existing analytical model of circulation control acoustics.
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 DREW WETZEL.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Cattafesta III, Louis N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042488:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042488/00001

Material Information

Title: An Experimental Investigation of Circulation Control Acoustics
Physical Description: 1 online resource (308 p.)
Language: english
Creator: WETZEL,DREW
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: 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: Underwater vehicle maneuverability is currently limited by conventional appendage technology. However, maneuverability can be greatly enhanced with the use of circulation control. A circulation control airfoil uses blowing over a rounded trailing and/or leading edge to increase circulation and generate lift at levels far greater than traditional control surfaces. In addition, large lift forces can be generated at very low speeds. One of the issues preventing the application of circulation control to underwater vehicles is noise. Thus, circulation control can only be applied to underwater vehicles if it improves maneuverability without a substantial noise level increase. The purpose of this investigation is to experimentally identify and characterize the flow field and corresponding noise sources of a circulation control airfoil. Flow and acoustic data are obtained for a dual-slotted, elliptic circulation control airfoil in an anechoic wind tunnel. The effect of single-slot blowing on the aerodynamic characteristics of the airfoil in an open and closed tunnel test section is evaluated. Measurements of the trailing edge curved wall jet flow using particle image velocimetry (PIV) reveal similarity of the outer region mean streamwise flow, and the length and velocity scales required for similarity, as well as the separation location, scale with the product of the chord Reynolds number and momentum coefficient. Streamwise vortices theorized to promote curved wall jet separation are also measured using PIV. Low and high frequency tones, generated by vortex shedding from the round trailing edge and finite slot lip, respectively, are detected. Broadband noise sources are identified for a variety of test conditions using a nested phased acoustic array. At low momentum coefficients, contaminating noise sources, like sidewall scrubbing noise and diffuser flow impingement noise, dominate. At higher momentum coefficients, an interaction between the high-speed trailing edge jet and the sidewall boundary layer flow appears to cause significant noise. The presence of multiple sources violates the assumptions required for using far-field microphone methods. Instead, a spectrum is obtained from the array data and compared to an existing analytical model of circulation control acoustics.
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 DREW WETZEL.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Cattafesta III, Louis N.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042488:00001


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ANEXPERIMENTALINVESTIGATIONOFCIRCULATIONCONTROLACOUSTICS By DREWA.WETZEL ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011DrewA.Wetzel 2

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ACKNOWLEDGMENTS First,Iwanttoacknowledgemyadvisorandcommitteechair,Dr.LouisCattafesta, forhisguidanceandmentorshipduringmytimeatUF.Itrulybelievethatheislargely responsibleformytransformationintoatrueresearchscientist.Second,Iwouldlike tothankDr.MarkSheplakforalwayspushingmetowanttolearnmore.Iwantto alsothankmyremainingcommitteemembers,Dr.AndreasHaselbacherandDr.David Arnold,fortheirparticipationandfeedback. Dr.FeiLiu,Dr.ChrisBahr,andJohnGrinwereindispensablecolleagueswho alwaysgraciouslyhelpedguidemethroughthemostchallengingproblems.Brian Rosenberg,NikZawodny,TarikYardibi,JustinRackley,andAdamHartprovided crucialhelpinimportantareasofthiswork.Ialsothankalloftheundergraduates-and thereweremany-whoprovidedahelpinghandinexperimentation.Finally,Iowealotof gratitudetoeveryoneinIMG,particularlyMatiasOyarzun,MiguelPalaviccini,Brandon Bertolucci,MattWilliams,AlexPhipps,JanhaviAgashe,etc.formakinggradschool particularlyenjoyable. RobinImberandErnieRogersbothgraciouslyprovidedtheirvastknowledgeof circulationcontrolandaidedinexperimentaldesign.Manufacturingoftheairfoilwas absolutelysuperb,thankstoDr.DavidWilliamsandCraigJohnsonattheIllinois InstituteofTechnologyandKenReedofTMREngineering.Ialsograciouslythankthe OceofNavalResearchONRandDr.RonaldJoslinforthenancialsupportthatmade thisresearchpossible. Finally,Iwanttoacknowledgethoseinmypersonallifewhohavemadeinvaluable contributionstomyupbringingandcontinueddevelopment:myparents,Richardand Janice,whowerelargelyresponsibleforfosteringmyinterestsinscienceandengineering asayouth,mythreeolderbrothers,Todd,Eric,andCraig,whowereandcontinueto beoutstandingpersonalandprofessionalrolemodels,andmyanceDanielle,forher constantsupportandencouragement. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................3 LISTOFTABLES.....................................8 LISTOFFIGURES....................................9 ABSTRACT........................................16 CHAPTER 1INTRODUCTION..................................18 1.1Motivation....................................19 1.2CirculationControlFundamentals.......................20 1.3PreviousResearch................................21 1.3.1OriginsofCirculationControl.....................21 1.3.2CirculationControlFluidDynamics..................23 1.3.3UnsteadyCirculationControl.....................31 1.3.4NumericalStudies............................31 1.3.5CurvedWallJetFlows.........................33 1.3.6CirculationControlAcoustics.....................39 1.4UnresolvedTechnicalIssues..........................43 1.5ResearchObjectives...............................44 1.6TechnicalApproach...............................44 2EXPERIMENTALSETUP.............................52 2.1CirculationControlAirfoil...........................52 2.1.1DesignandFabrication.........................52 2.1.2PlenumTreament............................53 2.2AirDeliverySystem..............................54 2.3UniversityofFloridaAeroacousticFlowFacility...............55 2.4StaticMeasurements..............................55 2.4.1SlotFlowUniformity..........................55 2.4.2LipDisplacement............................57 2.5FlowMeasurements...............................57 2.5.1SteadyPressureandLift........................57 2.5.2SlotJetVelocityandMomentumCoecient.............59 2.5.3ParticleImageVelocimetry.......................59 2.5.3.1Imageacquisition.......................60 2.5.3.2Vectorcomputation......................61 2.5.3.3Additionalcalculations....................62 2.5.3.4Uncertainty..........................64 2.6AcousticMeasurements.............................64 4

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2.6.1Free-StandingMicrophones.......................64 2.6.1.1Singlemicrophone......................65 2.6.1.2Coherentoutputpower....................66 2.6.1.3Three-microphonemethod..................67 2.6.1.4Uncertainty..........................67 2.6.2PhasedArray..............................67 2.6.2.1Beamforming.........................68 2.6.2.2Arraysetup..........................69 2.6.2.3Calibration..........................70 2.6.2.4Uncertainty..........................74 2.6.3DataAcquisition............................74 3FLUIDDYNAMICS.................................93 3.1AirfoilCharacterization............................93 3.2FreestreamFlowCharacterization.......................93 3.2.1SurfacePressureMeasurements....................93 3.2.2TestSectionInuence..........................94 3.2.3ClosedTestSectionBehavior......................95 3.2.4PotentialFlowAnalysis.........................97 3.2.5PIVResults...............................101 3.3CurvedWallJetFlow.............................103 3.3.1GeneralSimilaritySolution.......................103 3.3.2DimensionalAnalysis..........................106 3.3.3FlowCharacteristics..........................107 3.3.4FlowSimilarity.............................108 3.3.5LengthandVelocityScales.......................111 3.3.6FlowPrediction.............................114 3.3.7SeparationandStability........................114 3.4Summary....................................118 4ACOUSTICS.....................................141 4.1Tones......................................141 4.1.1LowFrequencyTones..........................141 4.1.2HighFrequencyTones..........................142 4.2AssessmentofNoiseSources..........................142 4.2.1IntheAbsenceofaFreestream.....................143 4.2.2WithaFreestream...........................144 4.3BroadbandNoise................................149 4.3.1Free-StandingMicrophones.......................149 4.3.2Array...................................150 4.4Howe'sModelofCirculationControlAcoustics................151 4.4.1Assumptions...............................152 4.4.2EstimatesofFlowScales........................152 4.4.2.1Curvaturenoise........................153 5

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4.4.2.2Passiveslotnoise.......................154 4.4.2.3Slot-jetinteractionnoise...................155 4.4.3ComparisonwithMeasurement.....................155 4.5Summary....................................157 5CONCLUSIONSANDFUTUREWORK......................189 5.1KeyFindings..................................189 5.1.1FluidDynamics.............................189 5.1.2Acoustics.................................191 5.2ResearchImpact................................193 5.3RecommendationsforFutureWork......................194 APPENDIX AAIRFOILNOTESANDTECHNICALDRAWINGS...............196 A.1FabricationPicturesandNotes........................196 A.2AirfoilSealing..................................197 A.3TechnicalDrawings...............................197 BIDEALFLOWOVERANELLIPSE........................243 B.1StreamFunctionandVelocityPotential....................243 B.2ComplexVariableTheory...........................243 B.3LiftingFlowOveraCylinder.........................244 B.3.1VelocityPotentialandStreamFunction................244 B.3.2VelocityComponents..........................245 B.3.3StagnationPoints............................246 B.3.4PressureCoecient...........................246 B.3.5Drag...................................247 B.3.6Lift....................................247 B.4ConformalMapping...............................248 B.5ResultsandAnalysis..............................249 B.6InuenceofBoundaries.............................250 CVENTURIMETEREQUATIONS.........................265 DJETVELOCITYCALCULATION.........................269 D.1JetVelocity:GeneralCase...........................269 D.2PlenumVelocity.................................270 D.3InuenceofNonzeroPlenumVelocityonJetVelocityEstimate.......273 EPIVUNCERTAINTYANALYSIS..........................275 E.1Samplecountandconvergence.........................275 E.2Biasuncertainty.................................275 E.2.1Velocitymagnitudeanddirection...................275 6

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E.2.2Tangentialandnormalvelocitycomponents..............276 E.2.3Turbulenceintensities..........................278 E.2.4Reynoldsstress.............................278 E.3Randomuncertainty..............................279 FCURVEDWALLJETSIMILARITYSOLUTION.................282 F.1GoverningEquations..............................282 F.1.1Continuity................................282 F.1.2 y -Momentum..............................283 F.1.3 x -Momentum..............................285 F.1.4AdditionalAssumptions........................286 F.2Self-Preservation................................287 F.3Self-Preservation:ModiedSimilarityFunctions...............292 REFERENCES.......................................301 BIOGRAPHICALSKETCH................................308 7

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LISTOFTABLES Table page 1-1Summaryofcirculationcontroluiddynamicsstudies...............48 1-2Airfoilgeometryspecications............................49 1-3Ellipticairfoilgeometryspecications........................49 1-4SummaryofCoandaeectandcurvedwalljetstudies...............49 1-5Summaryofcirculationcontrolacousticsresearch.................50 1-6Noisesourcesassociatedwithacirculationcontrolairfoil.............51 2-1Normalizedupper/lowersurfacepressuretapchordwisecoordinates.......87 2-2SummaryofPIVlaseroptics............................87 2-3SummaryofPIVimageacquisitionsetup......................88 2-4Summaryofowseedersandparticlediameterestimates.............88 2-5SummaryofPIVprocessingparameters......................88 2-6Arrayspecications..................................88 2-7Phasedarraymicrophonedesigncoordinatesandsensordetails..........89 2-8Measuredarraymicrophonecoordinateswithtotaluncertainty..........90 3-1Scalingoflengthandvelocityscalesfromsimilaritysolutionofacurvedwalljet withnofreestream..................................139 3-2Scalingoflengthandvelocityscalesfromgeneralsimilaritysolutionofacurved walljetinafreestream................................139 3-3Testcasespresented.................................140 4-1LengthandvelocityscalesusedtoevaluateHowe'smodel.............188 4-2ExampleofscalesrequiredforHowe'smodeltomatchmeasurement.......188 B-1Complexpotentialfunctionsforcomponentsofliftingowoveracylinder....255 B-2MATLABcodeparameters..............................255 D-1Resultsofplenumvelocityanalysis.........................274 E-1Randomuncertaintyestimates............................281 8

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LISTOFFIGURES Figure page 1-1Typicalunderwatervehicleappendages.......................45 1-2Circulationcontrolairfoil...............................45 1-3TheCoandaeectonacirculationcontrolairfoil.................45 1-4Denitionofcirculation...............................46 1-5Flowswithandwithoutcirculationaroundacirculationcontrolairfoil......46 1-6RepresentationofHowe'scirculationcontrolacousticspectrum..........47 1-7Circulationcontrolairfoilnoisesources........................47 1-8Desiredliftperformancerangesforaircraftandnavalapplications........48 2-1Circulationcontrolairfoilgeometry.........................75 2-2Zig-zagturbulatortriptape..............................76 2-3Photographofplenumtreatment...........................76 2-4Airdeliverysystem..................................76 2-5CirculationcontrolairfoilinstalledintheUFAFF.................77 2-6PIVtrailingedgemeasurementregions.......................78 2-7Vectorcomputationsteps...............................78 2-8Coordinatesystemsandnomenclatureattrailingedge...............78 2-9Microphoneexperimentalsetup............................79 2-10Phasedarrayexperimentalsetup...........................79 2-11Scanningregionof l =1 ;:::;L points.........................80 2-12Equal-aperturelog-spiralarraydesigns.......................80 2-13Theoreticalarrayresponsecomputedusingdesignmicrophonecoordinates...81 2-14Picturesofthephasedarray.............................82 2-15Stepsfordeterminingmicrophonelocationsusingphotogrammetry.......83 2-16PictureoflaserpulsearraycalibrationtestsetupinUFAFF............84 2-17Averagedlaserpulsetimeseriesofarraycentermicrophone............84 9

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2-18Theoreticalarrayresponsecomputedusingmeasuredmicrophonecoordinates.85 2-19Plotsof3dBbeamwidth...............................86 3-1Lipdisplacementasafunctionofplenumpressure.................119 3-2Airfoilsurfacepressure C p Re c =6 : 5 10 5 h=c =0.0019...........120 3-3Liftcoecient c l asafunctionofmomentumcoecient, Re c =6 : 5 10 5 h=c =0.0019........................................120 3-4Photographofclosedtestsection...........................121 3-5Airfoilsurfacepressure C p C =0, h=c =0.0019................121 3-6Airfoilsurfacepressure C p h=c =0.0019.....................122 3-7Liftcoecient c l asafunctionofmomentumcoecient, Re c =6 : 5 10 5 h=c =0.0019........................................122 3-8PhotographofclosedtestsectionforPIVmeasurements.............123 3-9Airfoilsurfacepressure C p inclosedtestsection, Re c =6 : 7 10 5 h=c =0.0019123 3-10Airfoilsurfacepressure C p inopentestsection, Re c =6 : 5 10 5 h=c =0.0019124 3-11Potentialow C p ona20%ellipse..........................125 3-12Airfoilsurfacepressure C p inclosedtestsectioncomparedwithpotentialow theory.........................................125 3-13Leadingedgemeanowstreamlines.........................126 3-14Comparisonofprolesatthelipedge, C =0...................127 3-15Comparisonofprolesatthelipedge, C 0.014.................128 3-16Comparisonofprolesatthelipedge, C 0.055.................129 3-17Logarithmicspiralsurface...............................130 3-18Similaritylengthandvelocityscales.........................130 3-19Coandasurface C p ..................................130 3-20Circulationcontrolcurvedwalljetproles, Re c =6 : 5 10 5 h=c =0 : 0019....131 3-21Lengthandvelocityscalesbasedonzeroshear...................132 3-22Circulationcontrolcurvedwalljetproles, Re c =6 : 5 10 5 C =0 : 015, h=c = 0 : 0019.........................................132 3-23Outerregionsimilarityof U .............................133 10

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3-24Lackofinnerregionsimilarityof U .........................133 3-25Velocityandlengthscalesdescribingdecayandspreadofjet, Re c =6 : 5 10 5 C =0 : 015,and h=c =0 : 0019.............................134 3-26Rateofspreadof y max =h andcollapseddatawithbesttline..........134 3-27Rateofspreadof y m; 1 = 2 =h andcollapseddatawithbesttline..........135 3-28Rateofdecayof U max =U jet andcollapseddatawithbesttline.........135 3-29Rateofdecayof U min =U jet forallcases.......................136 3-30Comparisonofmeasuredandpredictedvelocityandlengthscales, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019............................136 3-31Comparisonofmeasuredandpredictedvelocityandlengthscales, Re c =1 : 3 10 6 C =0 : 014, h=c =0 : 0029............................137 3-32Meanspeed j V j =U 1 contoursandmeanvelocityvectors, Re c =6 : 5 10 5 and h=c =0.0019.....................................137 3-33Approximateseparationlocationasafunctionof C Re c .............138 3-34Instabilityrange,indicatedbyGortlernumber, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019.....................................138 3-35Instantaneousspanwisevorticity x h=U jet distribution, Re c =5 : 6 10 5 C = 0 : 014, h=c =0 : 0019..................................138 3-36Instantaneousspanwisevorticity x h=U jet distributionandinstantaneousvelocity vectors.........................................139 4-1PowerspectraldensitymeasuredbymicrophoneM1abovethetrailingedge, Re c =6 : 5 10 5 h=c =0 : 0019.............................159 4-2AllcaseswithtrailingedgevortexsheddingtonesmeasuredbymicrophoneM1.159 4-3SlotlipvortexsheddingtonesmeasuredbymicrophoneM1, C =0.014, h=c = 0.0029..........................................160 4-4SlotlipvortexsheddingtonesmeasuredbymicrophonesM1andM4, Re c = 1 : 3 10 6 C =0.014, h=c =0.0029.........................160 4-5Spectrameasuredbythecentermicrophoneabovethetrailingedge,SSB, U 1 =0, h=c =0.0019..................................161 4-6Samplebeammapwithairfoilandtunnelcomponentslabeled...........161 4-7OuterarraybeammapsdBat992Hz,SSB, U 1 =0, h=c =0.0019......162 11

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4-8OuterarraybeammapsdBat2kHz,SSB, U 1 =0, h=c =0.0019.......163 4-9OuterarraybeammapsdBat4kHz,SSB, U 1 =0, h=c =0.0019.......164 4-10OuterarraybeammapsdBat8kHz,SSB, U 1 =0, h=c =0.0019.......165 4-11InnerarraybeammapsdBat16kHz,SSB, U 1 =0, h=c =0.0019.......166 4-12InnerarraybeammapsdBat32kHz,SSB, U 1 =0, h=c =0.0019.......167 4-13InnerarraybeammapsdBat64kHz,SSB, U 1 =0, h=c =0.0019.......168 4-14Spectrameasuredbythecentermicrophoneabovethetrailingedge,SSB, Re c = 6 : 5 10 5 h=c =0.0019................................169 4-15OuterarraybeammapsdBat992Hz,SSB, Re c =6 : 5 10 5 h=c =0.0019..170 4-16OuterarraybeammapsdBat2kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..171 4-17OuterarraybeammapsdBat4kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..172 4-18OuterarraybeammapsdBat8kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..173 4-19InnerarraybeammapsdBat16kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..174 4-20InnerarraybeammapsdBat32kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..175 4-21InnerarraybeammapsdBat64kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019..176 4-22OuterarraybeammapsdBat992Hz, Re c =6 : 5 10 5 h=c =0.0019.....177 4-23OuterarraybeammapsdBat2kHz, Re c =6 : 5 10 5 h=c =0.0019......178 4-24OuterarraybeammapsdBat4kHz, Re c =6 : 5 10 5 h=c =0.0019......179 4-25OuterarraybeammapsdBat8kHz, Re c =6 : 5 10 5 h=c =0.0019......180 4-26InnerarraybeammapsdBat16kHz, Re c =6 : 5 10 5 h=c =0.0019.....181 4-27InnerarraybeammapsdBat32kHz, Re c =6 : 5 10 5 h=c =0.0019.....182 4-28InnerarraybeammapsdBat64kHz, Re c =6 : 5 10 5 h=c =0.0019.....183 4-29Comparisonoffree-standingmicrophoneprocessingtechniquesforSSB, Re c = 6 : 5 10 5 C =0.11,and h=c =0.0019.......................184 4-30Comparisonoffree-standingmicrophoneprocessingtechniquesforOSB, C = 0.10, Re c =6 : 5 10 5 ,and h=c =0.0019......................184 4-31SinglemicrophoneandintegratedarrayspectraforSSB, C 0 : 065, Re c = 6 : 5 10 5 ,and h=c =0.0019.............................185 12

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4-32Meantangentialvelocityprolealongtrailingedgeat r=c =0.0129, Re c = 6 : 5 10 5 C =0.057, h=c =0.0019withone-seventhpowercurveinset.....185 4-33Meantangentialvelocityproleatslotlipedge, Re c =6 : 5 10 5 C =0.057, h=c =0.0019.....................................186 4-34Meantangentialvelocityproleatslotexitwithone-seventhpowercurve, Re c = 6 : 5 10 5 C =0.057, h=c =0.0019.........................186 4-35Howe'smodel,evaluatedfor Re c =6 : 5 10 5 C =0.057, h=c =0.0019,providing onlyairfoilgeometrydetailsandtestconditions..................187 4-36Howe'smodel,evaluatedfor Re c =6 : 5 10 5 C =0.057, h=c =0.0019using lengthandvelocityscalesestimatedfromPIVdata................187 4-37BesttofHowe'smodeltomeasuredspectrum...................188 A-1Airfeedassemblycomponentsprintedinaselectivelasersinteringmachine...198 A-2Topandbottomleadingedgepieces.........................198 A-3Airfoiltakingshapeaftercompletionofleadingedge,surfaceplates,dividerplate, andtrailingedgeassembly...............................199 A-4Blanktrailingedgeinstrumentrings.........................199 A-5Supportrodandairfeedmountingschemes....................200 A-6Pressuretapsandtubing...............................201 A-7PictureofnishedairfoilupondeliverytoUF....................202 A-8Pictureofabeadofahardeningsilicone-basedsealantappliedalongtheseam betweentheleadingedgeandsurfaceplate.....................202 A-9Pictureofanon-hardeningrubbergasketsealantappliedalongmetal-to-metal contactsurfacesforanon-permanentseal......................203 A-10Airfeedcomponentsandcorrespondinggaskets...................203 A-11Arubbergasketsealantisappliedtoeachsurfaceofthegaskets..........204 A-12Circulationcontrolairfoilexteriorassemblyview..................205 A-13Circulationcontrolairfoilinteriorassemblyview..................206 A-14Midspanpressuretaps.................................207 A-15Topleadingedge....................................208 A-16Bottomleadingedge..................................209 13

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A-17Topsurfaceplate....................................210 A-18Bottomsurfaceplate..................................211 A-19Topsideplatelipassembly..............................212 A-20Topleftsideplatelip.................................213 A-21Topcentersideplatelip................................214 A-22Toprightsideplatelip................................215 A-23Bottomsideplatelipassembly............................216 A-24Bottomleftsideplatelip...............................217 A-25Bottomcentersideplatelip..............................218 A-26Bottomrightsideplatelip..............................219 A-27Dividerplate......................................220 A-28Trailingedgeassembly.................................221 A-29Dividerplateextensionassembly...........................222 A-30Dividerplateextension..............................223 A-31Dividerplateextension..............................224 A-32Dividerplateextension..............................225 A-33Dividerplateextension..............................226 A-34Longtrailingedge...................................227 A-35Shorttrailingedge...................................228 A-36Trailingedgeinstrumentplug.............................229 A-37Trailingedgeinstrumentplugwithpressuretaps..................230 A-38Sideplateassembly..................................231 A-39Leftsideplate...................................232 A-40Leftsideplate...................................233 A-41Rightsideplate..................................234 A-42Rightsideplate..................................235 A-43Airfeed.........................................236 14

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A-44Airfeedtransition...................................237 A-45Airfeedconnectorplate................................238 A-46Supportrod......................................239 A-47Supportrodextension.................................240 A-48Supportrodmountingbracket............................241 A-49Supportrodconnectingplate.............................242 B-1Streamlinesforliftingowoveracylinder.....................252 B-2Streamlinesand C p distribution, U 1 =20m/s, C l =0...............253 B-3Streamlinesand C p distribution, U 1 =20m/s, C l =0.5..............253 B-4Streamlinesand C p distribution, U 1 =20m/s, C l =1...............254 B-5Streamlinesand C p distribution, U 1 =20m/s, C l =3...............254 B-6Schematicofmethodofimagesappliedtoliftingowoveranellipse.......255 C-1Venturimeterschematic...............................268 D-1Circulationcontrolairfoilplenummodeledasasimple,one-dimensionalnozzle..274 E-1Opentestsectiondataset, Re c =0.65M, C =0.014, h=c =0.0019.......280 E-2Curvedwalljetdataset, Re c =0.65M, C =0.015, h=c =0.0019........280 E-3Curvedwalljetseparationdataset, Re c =0.65M, C =0.015, h=c =0.0019..280 E-4Illustrationofvectordirectionbiaserror.......................281 15

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy ANEXPERIMENTALINVESTIGATIONOFCIRCULATIONCONTROLACOUSTICS By DrewA.Wetzel May2011 Chair:LouisN.CattafestaIII Major:MechanicalEngineering Underwatervehiclemaneuverabilityiscurrentlylimitedbyconventionalappendage technology.However,maneuverabilitycanbegreatlyenhancedwiththeuseofcirculation control.Acirculationcontrolairfoilusesblowingoveraroundedtrailingand/orleading edgetoincreasecirculationandgenerateliftatlevelsfargreaterthantraditionalcontrol surfaces.Inaddition,largeliftforcescanbegeneratedatverylowspeeds.Oneof theissuespreventingtheapplicationofcirculationcontroltounderwatervehicles isnoise.Thus,circulationcontrolcanonlybeappliedtounderwatervehiclesifit improvesmaneuverabilitywithoutasubstantialnoiselevelincrease.Thepurpose ofthisinvestigationistoexperimentallyidentifyandcharacterizetheoweldand correspondingnoisesourcesofacirculationcontrolairfoil.Flowandacousticdataare obtainedforadual-slotted,ellipticcirculationcontrolairfoilinananechoicwindtunnel. Theeectofsingle-slotblowingontheaerodynamiccharacteristicsoftheairfoilin anopenandclosedtunneltestsectionisevaluated.Measurementsofthetrailingedge curvedwalljetowusingparticleimagevelocimetryPIVrevealsimilarityoftheouter regionmeanstreamwiseow,andthelengthandvelocityscalesrequiredforsimilarity, aswellastheseparationlocation,scalewiththeproductofthechordReynoldsnumber andmomentumcoecient.Streamwisevorticestheorizedtopromotecurvedwalljet separationarealsomeasuredusingPIV. 16

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Lowandhighfrequencytones,generatedbyvortexsheddingfromtheroundtrailing edgeandniteslotlip,respectively,aredetected.Broadbandnoisesourcesareidentied foravarietyoftestconditionsusinganestedphasedacousticarray.Atlowmomentum coecients,contaminatingnoisesources,likesidewallscrubbingnoiseanddiuserow impingementnoise,dominate.Athighermomentumcoecients,aninteractionbetween thehigh-speedtrailingedgejetandthesidewallboundarylayerowappearstocause signicantnoise.Thepresenceofmultiplesourcesviolatestheassumptionsrequiredfor usingfar-eldmicrophonemethods.Instead,aspectrumisobtainedfromthearraydata andcomparedtoanexistinganalyticalmodelofcirculationcontrolacoustics. 17

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CHAPTER1 INTRODUCTION Circulationcontrolisapromisingtechnologyinmanyindustries.Circulationcontrol airfoilsemploytheCoandaeectviablowingoveraroundedtrailingedgetomovethe stagnationpointsandgenerateliftCoanda1938.Withtheapplicationofcirculation control,appendagese.g.wings,hydrofoils,rotors,etc.cangeneratehighamountsoflift withoutcontrolsurfaces,suchasapsorslats,thatincreaseappendageweight,noise,and complexity. Asaresult,mostresearcheortshavefocusedontheapplicationofcirculationcontrol technologytoaircraftandseavessels.However,manyotherapplicationsofcirculation controlhavebeenproposedandtested.Day2006suggestedtheuseofcirculation controlforavarietyofitems,includingahovercraft,windturbine,vacuumcleaner, andevenachickenshedforeectivelyexhaustingfumes.TheGeorgiaTechResearch InstituteGTRIstudiedthebenetsofcirculationcontrolforavarietyofautomotive applications.Englar2006demonstratedinscaledwindtunnelandfull-scaleteststhat tractor-trailerandSUVfueleciencycanbeimprovedbyblowingovertherounded edgesofthevehicles'rearends.Englaralsotestedthepotentialforcirculationcontrolto improvetheperformanceandsafetyofstreamlinedautomobilesLane1999.Gaeta etal. 2006designedandtestedanaerodynamicheatexchangerimbeddedinawingthatrelied oncirculationcontroltogeneratethesucientpressuredierencebetweentheupper andlowersurfacesofthewingtodrivecooling.Suchaheatexchangercouldbeusedto decreaseautomobilefrontalareaand,hence,dragaswellasimprovehandlingfromthe downforcegeneratedbythewing. Althoughcirculationcontrolhasanassortmentofpotentialapplications,theresearch presentedhereinisfocusedontheapplicationofcirculationcontroltoxedwings.The motivationforthisresearchdirectionisdiscussedinthefollowingsection. 18

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1.1Motivation Navyresearchershavestudiedcirculationcontrolforyears,initiallyconcentratingon itsapplicationtorotorcraft,butrecenteortshavefocusedonapplyingcirculationcontrol technologytounderwatervehicles.Underwatervehiclemaneuverabilityiscurrentlylimited byforcesontraditionalappendagesthatscalewiththesquareoffreestreamvelocity.With theuseofcirculationcontrol,maneuverabilitycanbegreatlyimprovedaslargeappendage forcescanbegeneratedatverylowspeedsandevenwhiletheunderwatervehicleis stationary.Largerliftforcescanalsobeproducedbycirculationcontrolappendages. Forexample,Rogers&Donnelly2004showedthatalowaspectratioratioofspan tochordcirculationcontrolhydrofoilcouldproducedoubletheliftforcesgeneratedby traditionalappendages.Circulationcontrolcouldpotentiallybeappliedtoanyunderwater vehicleappendage,includingbowplanes,sternplanes,andthesail,orbridgefairwater,as illustratedinFigure1-1.Infact,Imber etal. 2007alreadytestedtheperformanceofa scaledunderwatervehiclemodelwithacirculationcontrolbridgefairwater. However,therearemanyissuespreventingcirculationcontrolfrombeingappliedto underwatervehicles.Joslin2005summarizedtheseatthe2004NASA/ONRCirculation ControlWorkshop.Theyincludethedesignoftheinternalowdeliverysystemintake, pump,etc.,theeectofcirculationcontrolonperformanceofallunderwatervehicle systems,costsmaintenanceandenergy,acoustics,safetyandreliability,andthe environmentfoulingandcorrosion.Quietperformanceisnecessarybeforecirculation controlcaneverbeappliedtounderwatervehicles.Lownoiseimprovesvehiclestealth,the performanceofthevehicle'sownsonarsystem,andultimatelythevehicle'ssurvivability. Unfortunately,Howe2002predictedthathigherfrequencynoisefromacirculation controlairfoilcouldbe20dBormoregreaterthantrailingedgenoisefromaconventional airfoil.Therefore,thisstudyfocusesonidentifyingandcharacterizingthenoisesourcesof acirculationcontrolairfoil,sothatthenoiseemittedfromacirculationcontrolhydrofoil canultimatelybereducedtoacceptablelevels.Withpropernoisereduction,oneofthe 19

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hurdlespreventingtheapplicationofcirculationcontroltounderwatervehiclescanbe overcome. 1.2CirculationControlFundamentals Beforebeginningareviewofpreviousresearch,thefundamentalsofcirculationcontrol areintroduced.AtypicalcirculationcontrolairfoilisshowninFigure1-2.Thetrailing edgeisrounded,andablowingslot,suppliedwithuidfromaninternalpressurized plenum,dischargesawalljetwithnominalspeed U jet tangenttothetrailingedgesurface. Thejetwrapsaroundthetrailingedge,asillustratedinFigure1-3,duetothebalance betweencentrifugalforceandtheradialpressuregradient.Thisbalanceisoftenreferredto astheEuler-nequation,whichstatesthatpressure p increasesasradiusofcurvature @ ^ n increases, @p @ ^ n = U 2 jet r : {1 Hence, p increasesasradialdistance r movesoutward,helpingthejetadheretothe surface.Asthejetmovesaroundthetrailingedge,itentrainsuidfromthegenerally slowermovingfreestream.Thejetslowsandeventuallydetachesinpartfromtheadverse pressuregradientalongtheroundedsurface.Themomentumofthejetdelaysseparation andmovestherearstagnationpoint,increasingcirculation,whichisdenedastheline integralofthetangentialvelocityaroundaclosedcontour C )]TJ/F20 11.9552 Tf 10.635 0 Td [( I C ~ V d ^ l: {2 Figure1-4isaschematicillustratingEquation1{2.Figure1-5demonstratesthe productionofcirculationachievedwithtrailingedgeblowing.InFigure1-5A,anelliptical airfoilisplacedinafreestreamatzeroincidenceresultinginzerocirculation.InFigure 1-5A,trailingedgeblowingisemployed,movingthestagnationpointsandproducing circulation.CirculationisdirectlyrelatedtoliftbytheKutta-Joukowskitheorem,which statesthattheliftgeneratedperunitspanbyincompressible,inviscid,irrotationalow aroundabodyisequaltotheproductofthefreestreamdensity,freestreamvelocity,and 20

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circulation. L = 1 U 1 )-14710({3 Therefore,ascirculationincreases,sodoestheliftgenerated.Liftperunitspanis typicallypresentedinnon-dimensionalformastheliftcoecient, c l = L q 1 c ; {4 where q 1 = 1 2 1 U 2 1 isthedynamicpressureand c istheairfoilchord. Theperformancecharacteristicsofacirculationcontrolairfoilatagivenangleof attackaredescribedbythemomentumcoecient,whichistheratioofjetandfreestream momentumlevels.Themomentumcoecientisdenedas, C = mU jet q 1 S ; {5 where_ m isthejetmassowrateand S = sc istheplanformareaoftheairfoilwithspan s .Foranincompressiblejetow,Equation1{5canbereducedto C =2 h c U jet U 1 2 ; {6 where h istheslotheight. 1.3PreviousResearch Thissectionprovidesareviewofcirculationcontrolliterature.First,theoriginsof circulationcontrolarediscussed.Asummaryofuiddynamicsexperimentsoncirculation controlairfoilswithsteady,subsonicblowingfollows.Unsteadycirculationcontroland numericalstudiesarethenbrieyintroduced.Next,researchoncurvedwalljetsis presented.Finally,priorstudiesofcirculationcontrolacousticsarereviewed. 1.3.1OriginsofCirculationControl Circulationcontrolairfoilsdierfromconventionalairfoilsintwosignicantways: thepresenceofablowingslotandaroundedtrailingedge.Although,totheauthor's knowledge,therstexperimentalcirculationcontrolstudywasnotpublisheduntilthe 21

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1960s,thefoundationforcirculationcontroltechnologywasestablishedinprioryears, mostnotablyintheformofpatents. In1929,theFifteenthAnnualReportoftheNationalAdvisoryCommitteefor Aeronauticsdescribedasetofexperimentsonwhatcloselyresemblesacirculationcontrol airfoilNACA1929. ...Stillgreaterimprovementseemstobepossiblebytheuseofaproperly locatednarrowslotopeningintothewingandsoarrangedthatairmaybe dischargedfromthewingorsuckedintoitbysuitablemeans.Aninvestigation ofthepossibilitiesofthistypeofslottedwinghasbeencarriedoutinthe atmosphericwindtunnelduringthepastyear.Amodelequippedwithaslot adjustablebothinpositionandwidthwasused.Arrangementsweremadefor applyingeithersuctionorpressure,andthepowerrequiredtoproducetheow wasmeasured.Largeincreasesinliftwereobtainedwithmoderatepressures, andtheminimumdragwasreduced. TheNACAreportreferredtoasetofexperimentsperformedbyKatzmayr1929atthe ViennaAeromechanicalLaboratory.Katzmayrmodiedthewell-establishedLachmann andHandleyPageslottedwingsbysupplyingtheairusedinuppersurfaceblowingfrom anexternalsourceasopposedtothepressuresurfaceofthewing.Katzmayrconcluded thatthesenozzle-slotted"wingsgeneratehigheramountsofliftthantraditional slottedwingsatthesameangleofattack.LikeKatzmayr,Haus1931alsosuggested choosingnozzle-slotted"wingsoverslottedwingsduetotheirsimplicityandcomparable performance. In1938,Coanda1938patentedhiswell-knownpropellerdevice.Compressedgas isreleasedinanarrowgapformedbetweenthepropellerfaceandaconcentricring.The gasadherestothecurvedpropellerface,producingalowpressureregion.Atmospheric pressureisventedtotherearofthepropellerthroughthehollowpropellercenter.The resultingpressureimbalancedrivesthepropellerforward.TheCoandaeect"wasnamed forCoandaandhispropeller,eventhoughYoung1800wasthersttoobserveand documenttheadhesionofaowtoacurvedsurfacenearly140yearsearlier.Regardless, circulationcontrolissynonymouswiththeCoandaeect. 22

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Griswold1959,1960ledtwopatentsinthemid-1950sfordirectlift"and circulatory-jet"airfoils.Griswold'sdirectlift"airfoilincludedaleadingedgeblowing circulationcontrol"slotandaporoustrailingedgesuctionboundarylayercontrol" surface.Theuseofbothelementspermittedcontrolofthestagnationpoints'positions andeliminatedowseparationonthesuctionsideoftheairfoil,thusprovidinghighlift andreduceddrag.Griswoldextendedhisoriginalpatentwiththeintroductionofhis circulatory-jet"airfoilthatreplacedthedirectlift"airfoiltrailingedgesuctionwithone ormoretrailingedgeblowingjets. In1962,Davidson1962patentedanaerofoilboundarylayercontrolsystem" comprisedofanellipticalwingwithopposingupperandlowerblowingslotsoneitherside ofaroundedtrailingedge.Davidsonproposedvaryingtheratesofeachblowingslotso thatthecirculationaroundtheairfoilandhenceliftcouldbecontrolled.Thisdual-slotted congurationisthefocusofthepresentstudy. 1.3.2CirculationControlFluidDynamics Sincethe1950s,therehavebeennumerousexperimentalstudiesontheuiddynamics ofcirculationcontrolairfoilsandwings.Thoseinvestigationsthatfocusedonsteady, subsonicblowingaresummarizedinthissectionandarelistedinTable1-1. Oneoftheearlyinvestigationsofacirculationcontrolwingwasperformedby Lehnert&Hazen1956fortheStroukoAircraftCorporation.Citingthedicultyto adequatelysupplyathincirculationcontrolairfoilwithblowingair,theauthorsproposed theadditionofanuppersurfacesuctionslotupstreamoftheblowingslottoprovide additionalmassow.Thisconceptwastestedona12%thicksweptwingwithahinged apandaileron,anuppersurfacesuctionslotlocatedatmidchord,andablowingslot positionedat73.5%chord.Testswereconductedinthe1.22mby1.52mSubsonicWind TunnelatPrincetonUniversity.LehnertandHazenobservedanincreaseinliftwith anincreaseinblowingforallmodel,ap,andaileronanglestested.Whentheapand 23

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aileronwerenotdeected,dragwasfoundtodecreaseasblowingincreasedandproduced aneectivethrust. Totheauthor'sknowledge,therstpublishedcirculationcontrolstudywas printedin TheAeronauticalQuarterly byKind&Maull1968.Theytesteda20% thickness-to-chordratioellipticalcirculationcontrolairfoilwithupperandlowerblowing slotsatCambridgeUniversity.KindandMaullobservedsomeimportantcharacteristics, includingtheproductionofhighliftwithmodestsingle-slotblowingandanose-down pitchingmomentatzerogeometricangle-of-attackresultingfromthelargetrailingedge suctionpeak.Theyalsonotedtheadvantagesofcirculationcontrol,likeitsability togeneratehighliftatzeroincidence,lowblowingratescomparedtoajet-ap" conguration,andtheabsenceofmechanicalpartsrequiredforsimilarliftproduction onconventionalliftingsurfaces.Slotheighteectswerefoundtobeminimal,and simultaneousdual-slotblowingwasdeemedinecienteciencyreferstothelift-drag ratioatavarietyofsecondaryslotblowingratescomparedwithsingle-slotblowing. NavyresearchersbeganstudyingcirculationcontrolattheNavalSurfaceWarfare Center,CarderockNSWCCD;thenknownastheDavidTaylorModelBasinwithafocus towardsrotaryaircraftin1967Imber2005.Williams1969presentedinitialndings fromaninvestigationofcirculationcontrolforastowed-rotoraircraft.Thestudyfocused onslotposition,slotheight,andtrailingedgegeometryeectsusingthreesingle-slotted 20%thickellipticalcirculationcontrolairfoils,whoseattributesarecomparedinTable 1-2.Foraconstantmomentumcoecient,higherliftwasproducedwhentheslotwas locatedclosertothetrailingedgeoftheairfoilCM1.ContrarytoKind&Maull1968, Williams1969observedincreasingliftwithdecreasingslotheightforagivenmomentum coecient.HealsonotedthattheairfoilwiththelargesttrailingedgeradiusCM1 producedhigherliftforaconstantmomentumcoecient.Finally,separateleadingand trailingedgestallphenomenonwereobserved. 24

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Williams&Howe1970continuedstudyingcirculationcontrolforrotaryaircraft withaninvestigationofa20%thickellipticalcirculationcontrolairfoilwith5%camber. Theuppersurfaceblowingslotwaslocatedatthenormalizedchordwiseposition x=c = 0 : 973,andtheslotheight-to-chordratiowas h=c =0 : 00128.Theairfoilwasdesignedfor maximumeciencyatzeroangleofattackandaliftcoecientbetween1.0and2.0.In thisdesiredliftcoecientrange,theairfoilgeneratedhigherlift-dragratiosthanother highliftsystems.Theauthorsalsoaddedatrailingedgeaptotheairfoiltodetermine thefeasibilityofacirculationcontrolwingforxed-wingaircraft.Theydeterminedafree oating"ap,unlessplacedanimpracticaldistancefromtheCoandasurface,reducedlift augmentationandinsteadproposedusingaretractableap. Circulationcontrolresearchforrotaryaircraftpersistedintothe1970swithEnglar's subsonictestoftwothincirculationcontrolairfoilsdesignedforandpreviouslytested intransonicowEnglar1970,1971.Bothairfoils,whosespecicationsareincludedin Table1-3,weretestedatavarietyofincidenceangles,momentumcoecients,andslot heightsatchordReynoldsnumbersbetween5 : 2 10 5 and5 : 5 10 5 ,wherethechord Reynoldsnumberisdenedas Re c = 1 U 1 c 1 ; {7 where 1 isthefreestreamviscositycoecient.Englarobservedquiteafewsignicant performancedierencesduelargelytothemodels'dieringthickness-to-chordratios andtrailingedgeradii.Theroundellipse"wascapableofgeneratingfarmoreliftthan thepureellipse,"whichduetoitssmallertrailingedgeradiussueredfromearlierjet detachment.Inaddition,thepureellipse's"forwardslotpositionmeantenergylevelsin thejetwereweakeratsimilarazimuthaltrailingedgepositionscomparedtotheround ellipse"atidenticalblowingrates.However,thisforwardslotpositionallowedgreater controlovertheuppersurfaceboundarylayerandhenceowseparation,makingthe pureellipse"abetterchoiceforoperationatpositiveanglesofattack.Foridentical momentumcoecientsandanglesofattack,liftwasgenerallyaugmentedasslotheight 25

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decreased.Theonlyexceptionoccurredwhentheslotheightwastoosmallandthe Coandajetowqualitywaspoor.ComparedtoaconventionalNACA0012rotorsection, thecirculationcontrolairfoilsproducedsubstantiallymoreliftandlessdrag,thelatter causedbyexcessmomentuminthedetachedjetsheetthatproducedthrust.However, theNACA0012wasapproximatelytwiceasecientatanglesofattackgreaterthanfour degrees. Englar1972alsoconductedexperimentsona30%thickellipticalcirculation controlairfoilwith1.5%camberforuseasahelicoptermidspanbladesection.The blowingslotwaslocatedat x=c =0 : 964andthenon-dimensionaltrailingedgeradius was R=c =0 : 0601.Testswereperformedatavarietyofslotheights,anglesofattack, momentumcoecients,andfreestreamvelocities.Asinhispreviousstudy,Englar observedanincreaseinliftasslotheightdecreasedforaconstantmomentumcoecient. At8 angleofattack,thehighestincidenceangletested,uppersurfaceseparationresulted inreducedperformancecomparedtoloweranglesofattackforarangeofmomentum coecientsbetween0and0.14.Thisowseparationpreventedthejetfromadequately entrainingtheuppersurfaceow,thuseliminatingthegradualaftincreaseinsuction typicallyobserved.Atmomentumcoecientsgreaterthan0.14,blowingwassucientfor properentrainment. Abramson1975performedaseriesoftestsonanon-cambered20%thickelliptical airfoilwithanupperblowingslotplacedat x=c =0 : 97.Theslotheight-to-chordratiowas xedat h=c =0 : 00126.DatawerepresentedforachordReynoldsnumberof3 : 4 10 5 Abramsonnotedapreviouslyunseendiscontinuityintheliftcurveslopeatmomentum coecientsthatshetheorizedwereindicativeoftheamountofblowingnecessarytoshift theuppersurfaceseparationpointdownstreamoftheblowingslot. Inthemid-1970s,Navycirculationcontrolresearchshiftedfocustoxed-wing aircraft.Englar1975 a publishedareportin1975onacirculationcontrolairfoilfor shorttakeoandlandingSTOLaircraft.Englarperformedaseriesoftestsona 26

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hingedapcirculationcontrolwingwiththreedierentoperatingcongurations:a conventional"cruise apangle,blownap"fortakeo,andCoandatrailingedge" forlanding apangle.Hereplaceda37%chordsingleslottedapthatspanned 49.5%ofthewingsemispanona1/5scaleNavyT-2Caircraftwitha15%chordhinged apcirculationcontrolwingofthesamespan.ThelandingCoandatrailingedge" congurationgeneratednearlytwicethemaximumliftcoecientand240%moredrag thantheconventionalaircraftmodel.Thecirculationcontrolcongurationalsoproduced just20%lessliftwhenunblown,provingthatthecirculationcontrolcongurationcould provideadequateliftincaseblowingfails.Theresultsfromthisexperimentdemonstrated thatcirculationcontrolcouldworkonxedwingSTOLaircraft. Inthesameyear,Englar1975 b releasedndingsfromanextensivestudyonthe eectofhighjetvelocitiesoncirculationcontrolperformance.Englartestedanairfoilwith a20%thickellipticalprolefromtheleadingedgetomidchordandconstantthickness frommidchordtotheblowingslot.Theslot,locatedat x=c =0 : 91,wasadjustablefrom h=c =0to0.0182andemittedajettangentiallyoverarotatabletrailingedgecylinderof radius R=c =0 : 0909.Inadditiontomidspanstaticpressuretapsontheairfoilsurface, thetrailingedgecylinderhad30staticpressuretapsstaggeredintworowsseparatedby 0.635cmandpositionedazimuthallyin6 incrementsovera180 sectionofthetrailing edge.Otherinstrumentsincludedastatictapplate"mountednormaltothetrailing edgecylindertomeasurethestaticpressureacrossthejetandaushmountedhot-lm sensorformeasuringshearstressanddeterminingthejetseparationpoint.Englarshowed thattheboundarylayerapproximationdoesnotholdfortheCoandajetsincethestatic pressureacrossthejetwasnotconstantatmultipleangularlocationsbetween18 and90 fromtheslot.Furthermore,bymeasuringthejetseparationpoint,hewasabletooera plausibleexplanationforthereasonforimprovedliftaugmentationatsmallerslotheights foraconstantmomentumcoecient.Astheslotheightdecreases,jetvelocity,andhence 27

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jetkineticenergy,increases,movingthestagnationpointfartherawayfromtheslotalong thetrailingedgesurfaceandincreasingcirculation. Todeterminetheeectofthetrailingedgeshapeoncirculationcontrolperformance, Abramson1977testeda15%thickellipticalcirculationcontrolairfoilwith1%camber andaninterchangeabletrailingedge.Twotrailingedgegeometriesweretested:therst maintainedapureellipticalshape,whilethesecond,labeledasaspiral"trailingedge, hadaradiusofcurvaturethatincreasedwithangularpositionfromtheslot.Abramson testedeachcongurationatavarietyofmomentumcoecients,anglesofattack,and slotheights,andobservedsimilarperformanceintermsoflift,drag,andeciency.The spiraltrailingedgecongurationproducedalowermagnitudenegativepitchingmoment improvedcontrollabilityandamorepositiveminimumpressurehighercriticalMach number. Intheearly1980s,Englar&Huson1983investigatedtrailingedgeshapesfor STOLaircraftcirculationcontrolwingsinordertoreducecomplexity,size,andweight ofapriorcirculationcontrolwingsuccessfullytestedonaNavy/GrummanA-6test plane.Amongthedesiredimprovementswasasmallertrailingedgeradiustoreduce cruisedrag.Asupercriticalcirculationcontrolairfoilwithatrailingedgeradiusone quarterthesizeoftheA-6circulationcontrolwingproducedgreaterliftatlowerangles ofattackandmomentumcoecientsthantheA-6circulationcontrolwing.Unblown draglevelswerealsosimilartothebaselinecirculationcontrolwing.Inaddition,the largeleadingedgeeliminatedtheneedforaowseparation-preventingslatrequired fortheA-6conguration.AsecondgroupofairfoilsderivedfromtheoriginalA-6 congurationmaintainedtheleadingedgeslat.Avarietyoftrailingedgesweretested withthisconguration,includingsomethatincorporateddualradius"apsupperap surfacedenedbytwoarcsofdierentradiiorsimilarly,aretractablearc."Thedual radius"apsprovedtobesuperiorinperformancecomparedtotheothercongurations testedandtheoriginalA-6circulationcontrolwing.Incruise,thenearlysharptrailing 28

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edgeofthedualradius"congurationreduceddrag,andwiththesmallapextended, theairfoilwascapableofproducinghighlift. NovakandhiscolleaguespresentedresultsfromalaserDopplervelocimetryLDV studyoftheoweldsurroundingthetrailingedgeofacirculationcontrolairfoilinthe late1980sNovak&Cornelius1986;Novak etal. 1987.Theairfoilhadasupercritical leadingedge,cylindricaltrailingedge R=c =0 : 0667,andamostlyatsurfacebetween thetwo.Flowchracteristicslikesteadysurfacepressure,separationpointlocation,jet mixingandgrowth,turbulenceintensity,andturbulenceshearstressalongtheCoanda surfacewerepresentedforafewcases.ScalingsuggestedbyLaunder&Rodi1983and Wilson&Goldstein1976wereusedtonormalizemeanvelocityandturbulenceintensity proles,respectively.Totheauthor'sknowledge,thiswastherstpublishedstudythat providedadetailedglimpseofacirculationcontroltrailingedgeoweld. TherstNavytestofadual-slottedcirculationcontrolairfoilwasperformedby Abramson2004from1986to1987,althoughtheresultswerenotreporteduntil2004. ThepurposeofAbramson'sexperimentwastodetermineiftheinclusionofalower opposingblowingslotcouldexpandtheoperationalliftrangeoftheairfoilwithout interferingwiththecurvedwalljetow.Testswereconductedona17%thickelliptical circulationcontrolairfoilwith1%camber.Theupperslot,locatedat x=c =0 : 968,and thelowerblowingslot,locatedat x=c =0 : 970,utilizedseparateairsupplyplenumsand weretestedindividually.Experimentsshowedthatthepresenceofthelowerslotdidnot inhibitupperblowingperformanceandvice-versa,andlowerslotblowingwassuccessful indoublingtheliftrangeoftheairfoil. Rogers&Donnelly2004publishedresultsfromastudyofadual-slotted,low aspectratio,sweptnitewingforunderwatervehicles.Thewinghadaprolebasedon a20%thickness-to-chordratioellipseandwastestedintheNSWCCDLargeCavitation Channelatamean-chordReynoldsnumberof2 : 1 10 6 .Singleslotblowingrevealed anunanticipatedliftroll-o"beginningat C =0 : 12,whichRogersandDonnelly 29

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attributedtoexcessivejetturning"thatoccurswhentheCoandajetseparatestoofar upstreamalongtheairfoilsurfaceoppositeoftheblowingslot.Theaddedmomentum ofthejetalongthepressure-sidesurfacelocallyreducedthestaticsurfacepressureand, asaresult,lift.However,asmallamountoflowerslotblowing%to5%oftheupper slotmomentumuxpreventedthejetfromreachingtoofarupstreamandeliminated theliftlimit.Thecirculationcontrolwingwasfoundtoproducedoublethemaximumlift generatedbyaconventionalappendagewiththesameaspectratio.Anothersignicant ndingwasthattherewerenouniquenitewingeectsforthecirculationcontrolwing comparedtoconventionalliftingsurfaces. Jones etal. 2006studiedthetrailingedgeowcharacteristicsofatwo-dimensional supercriticalcirculationcontrolairfoilusingparticleimagevelocimetryPIVtohelp verifycomputationaluiddynamicsCFDmodelsofcirculationcontrolow.The modiedGAW-1airfoilwastestedintheNASALangleyBasicAerodynamicResearch TunnelBARTatMachnumbersrangingfrom0.08to0.1.Theairfoilwascongured withtwointerchangeabletrailingedges:a9%chordhingedapforaxedseparation pointanda2%chordcirculartrailingedge.Thenormalizedslotheightforboth congurationswasapproximately h=c =0 : 00106.UsingatwocomponentPIVsystem,the owaroundthetrailingedgeandnear-wakeregionwasstudiedandcomparedtoresults fromCFD. Circulationcontroluiddynamicshavebeenextensivelystudiedthroughexperiments overthelast50years.Flowandperformancecharacteristicshavebeenmeasuredfora largecollectionofairfoilgeometriesandvaryingcongurations,includingdierentslot heights,slotpositions,numberofslots,trailingedgeradiussizeandshape,camber,and airfoilthickness.Flappedcirculationcontrolairfoilshavebeeninvestigated,andnite wingeectshavebeenassessed.Circulationcontrolhasbeenstudiedforapplications includingrotaryaircraft,STOLaircraft,andunderwatervehicles.Renewedinterestin STOLaircraftisdrivingresearchinpulsedblowing,orunsteady,circulationcontrol. 30

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1.3.3UnsteadyCirculationControl Oneofthemajorobstaclesimpedingthepracticalapplicationofcirculationcontrolto aircraftisthemassowrequirement.Thus,somestudieshavefocusedonpotentialmass owreductionsincurredbypulsingtheCoandajet. Ina1972report,Walters etal. 1972usedarotaryvalvetopulsetheCoandajetat frequenciesupto100Hzona20%thickellipticalairfoilwith5%camber.Attheobserved optimumblowingfrequencyof40Hz,massowwasreducedbyasmuchas25%compared tosteadyblowingatthesameliftcondition.Similarly,liftincreasedby15%versussteady blowingatthesamemassowratewhenthejetwaspulsedattheoptimumfrequency. Morepromisingresultshavebeenpresentedinthepastdecade.Cagle&Jones2002 andJones etal. 2002developedandtestedatwo-dimensionalsupercriticalcirculation controlairfoilwithasmall,roundedtrailingedgeandbothsteadyandpulsedblowing. Twentyhigh-speedpneumaticsolenoidvalveswithstereolithographicdiuserspulsedthe airatfrequenciesupto200Hzanddutycyclesrangingfrom20%to80%.Testsconducted attheNASALangleyResearchCenterrevealedthat50%lessmassowwasrequired toproducealiftcoecientof1.0whenthejetwaspulsedinsteadofsteadilyblown. Likewise,unsteadyblowingproducedasmuchas35%moreliftthansteadyblowingat thesamemassowrate.AdditionalresearchatGeorgiaTechyieldedsimilarresultsfora appedcirculationcontrolwingataliftcoecientof3.0Jones&Englar2002. Pulsedblowingsuccessfullyreducesthemassowrequirementbutintroduces additionalconcernslikecomplexityandnoise.Furthermore,robustandreliableactuators mustbedevelopedbeforepulsedblowingisappliedtofull-scaleappendages.These challengesfurtherhindertheapplicationofcirculationcontroltounderwatervehiclesand arenotconsideredinthecurrentresearchstudy. 1.3.4NumericalStudies Foraconventionalairfoil,theKuttaconditionxesthecirculation)-326(thatisproduced foragivenangle-of-attackandfreestreamvelocity.However,foracirculationcontrol 31

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airfoilwitharoundedtrailingedge,therearstagnationpointisfreetomove,andthere aremanypossiblevaluesof)-327(foragivenangle-of-attackandfreestreamvelocity.Hence, foragivenangleofattack,circulationvarieswiththemomentumcoecient.Furthermore, whenblowingratesaresucient,trailingedgeblowingpreventsviscousowseparation. Thus,idealowsolutionscanprovideafairlyaccurateassessmentoftheowaround acirculationcontrolairfoilanditsliftcharacteristics.Foragivenliftcoecientand angleofattack,thesteadysurfacepressurearoundacircularcylindercanbecomputed andtransformedfortheellipticalairfoilgeometrybeingstudied.AMATLABprogram performingthesecomputationsiswrittenanddescribedinAppendixB.Theidealow solutionisonlyan estimate ,sincethecontributionofthehighspeedblowingjettothe surfacepressuredistributioncannotbetakenintoconsideration.However,researchers havereportedexcellentcomparisonsbetweenexperimentalandanalyticalsurfacepressure distributionsusingidealowsolversEnglar1971,1972;Abramson1975. Forhigherspeedapplications,atwo-dimensionalsubsoniccompressibleowsolver wasdevelopedinplaceoftraditionalcompressibilitycorrectionfactorsRogers1973. Moreadvancedalgorithmswereappliedstartinginthe1980s.Shrewsbury1985studied lowandtransonicspeedperformanceofacirculationcontrolairfoilusingReynolds AveragedNavier-StokesRANSequations.Liu etal. 2001alsousedRANStostudy steadyandpulsedblowingandnotedagoodagreementwithexperimentaldata.Slomski etal. 2002assessedtheeectofdierentturbulencemodelsforavarietyofblowing rates.Morerecentinvestigationshavefocusedonusingthree-dimensionalRANSandlarge eddysimulationLEStostudytheowaroundacirculationcontrolairfoilChang etal. 2005;Slomski etal. 2006. ThecomputationaluiddynamicsCFDcommunityhasbeenhamperedin theireortstomodelcirculationcontrolairfoilsbecauseofthescarcecollectionof experimentalturbulencedataavailable.Thus,asetofhighly-detailed,accurateturbulence 32

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measurementsoftheoweldaroundacirculationcontrolairfoilwouldbeconsiderably usefultotheCFDcommunityforvalidatingturbulencemodelsandcomputationalresults. 1.3.5CurvedWallJetFlows Sincethemechanismresponsibleformovingthestagnationpointonacirculation controlairfoilisathin,curvedwalljet,asdepictedinFigure1-3,researchershave focusedonthefundamentalcharacteristicsofwalljetsovercurvedsurfaces.Acollection ofsignicantworksinthiseldissummarizedinTable1-4.Someoftheseworksare describedinmoredetailinthissection. In1961,Newman1961presentedhisexperimentalndingsonatwo-dimensional, incompressible,turbulentwalljetowingoveracircularcylinder.Thesurrounding uidwasstagnant.Apitotprobewasusedtomeasuremeanvelocityprolesnormal tothecylindersurface.Experimentswereperformedatvariousslotheights.Newman determinedthattheowwasindependentoftheslotheight-cylinderradiusratio.As iscommonforaplanewalljet,henormalizedthemeantangentialvelocitywiththe maximumvelocityoftheprole, U m ,andthenormaldistancefromthesurfacewith y m= 2 thedistancebetweenthesurfaceandtheouterregionlocationwherethemeanvelocityis one-halfthemaximumvelocity.Dimensionlessmeantangentialvelocityprolesmeasured at =45 ; 90 ; 135 ; and180 fromtheslotexhibitedsimilarity,collapsedtoasingle curve,andagreedwellwithplanewalljetmeasurementsandtheory.Newmannoted somejetReynoldsnumbereects.AtlowerjetReynoldsnumbers,thejetspreadquicker, theboundarylayerwasthicker,andseparationoccurredearlier.However,atlargerjet Reynoldsnumbers,theseparationlocationremainedconstant.Newmanalsodevelopedan analyticalexpressionforthesurfacepressuredistribution,butitfailedtoagreewiththe measurementsbecausetheboundarylayerassumption v u ,whichNewmanassumed wasvalidinderivinghisexpression,mayhaveindeedbeeninvalid. SevenyearsafterNewman,Dunham1968publishednumericalandexperimental resultsforacircularcylinderwithoneormultipleblowingslotsinafreestream.Theslot 33

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owwasconsideredone-dimensionalandcompressible,whereasthefreestreamowwas assumedtobetwo-dimensionalandincompressible.ThegeneralprocedureforDunham's numericalapproachfollows: 1.Specifytheliftcoecient. 2.Calculatethepressuredistributionfrompotentialowtheory. 3.Beginningattheforestagnationpoint,computethelaminarboundarylayerusing themethodofThwaitesaroundthelowersurfaceuntilthetransitionpointis reached.Thetransitionpointischosenasthelocationoftherstslotor,when possible,theupstreamlocationwherethevelocitygradientbecomesadverse. 4.Fromthetransitionpoint,computetheturbulentboundarylayerusingthetheory ofSpalding1965untilseparationoccurs.Determinetheseparationpressure. 5.Computethelaminarandturbulentboundarylayerowsfortheuppersurfaceat theslot,assumetheexteriorboundarylayerandwalljetprolesimmediatelymerge intoaSpaldingprole. 6.Iterate,adjustingtheslotjetblowingrateuntiltheuppersurfaceseparationpressure matchesthelowersurfaceseparationpressure. Similarly,iftheblowingratewasspecied,theliftcoecientcouldbedetermined. Dunham'stheorywasabletoaccuratelypredictliftandbasepressurevaluesforselect cases,butonlytrendsforothers.Inparticular,accuracywaslowwhenthedistance betweentheblowingslotorslotsandseparationlocationwaslarge,forhightlift,and whenonlyoneblowingslotwasused. Parks&Petersen1968publishedananalyticalsimilaritysolutionof`Coanda' typeow."Theyassumedsteady,two-dimensional,incompressible,laminarjetowand stagnantsurroundings.Theyalsoconsideredtheslotwidthtobemuchsmallerthanthe radiusofcurvature,justifyingtheuseoftheboundarylayerassumptionsandneglecting theeectofcurvatureinthetangentialmomentumequation.However,inmostapplied casesofcirculationcontrol,thejetisturbulentandtheeectofcurvaturemaynotbe neglected. 34

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Levinsky&Yeh1972soughttoprovideatheoreticalanalysisofcirculationcontrol onacircularcylinderwithtangentialblowingbyincludingcurvatureeects.They followedasimilarproceduretoDunham1968.Theowwasconsideredtwo-dimensional, incompressible,andinviscideverywhereexceptthenear-surfaceregion,whichwastreated asviscousandturbulent.Pressuredistributions,separationlocations,andliftdata predictedfromtheanalysisagreedwellwithdatafromwindtunnelexperiments. Wilson&Goldstein1976experimentallycomparedplaneandcurvedwalljet owstodeterminethespeciceectofsurfacecurvature.Measurementsofaplanewall jetexhaustedfroma0.609cmslotwerecomparedwithmeasurementsofajetemitted froma0.615cmslotoveracircularcylinder h=R =0 : 0605.SlotReynoldsnumbers Re s = hU jet = were13 : 2 10 3 forbothtestcases,andhot-lmsensorswereusedto measurevelocity.Liketheplanewalljetow,meantangentialvelocityprolesappeared tobesimilarforthecurvedwalljetow.However,WilsonandGoldsteinclaimedthe proleswerenotactuallysimilar.Theycomputedtheradialvelocitycomponentusing themeasuredtangentialvelocitycomponentandcontinuity.Theresultingradialvelocity proleswereindeeddissimilar.WilsonandGoldsteinalsoprovidedturbulencedata. Streamwiseandradialturbulenceintensitieswerehigherinthecurvedwalljetcase. Reynoldsstressprolesexhibitedsimilarityonlyfortheplanewalljetcase,whilethe magnitudesofReynoldsstresswerehigherforthecurvedwalljetow,suggesting enhancedturbulencemixing.AnimprovedcollapseoftheReynoldsstressprolesand agreementwithplanewalljetdatawasfoundwhentheReynoldsstresswasnormalized usingturbulenceparametersinsteadofmeanvelocityscales.However,theagreementwas stillpoorintheinnerregion,implyingthattheinnerregionprolesofplaneandcurved walljetscouldnotberepresentedbyasingleequation.Furthermore,thepositionofzero Reynoldsstresswaslocatedat y=y m =0 : 5fortheplanewalljet y m wasthedistancefrom thesurfacetothepositionofmaximumvelocity,butfoundtobesignicantlycloserto thesurfaceforthecurvedwalljet,where y=y m < 0 : 10. 35

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Anexperimentalcomparisonofplane,convex,andconcavewalljetswasconducted byKobayashi&Fujisawa1983.Theirexperimentalapparatusconsistedofaatplate, whichallowedaplanewalljetowingoverittobecomefullyturbulentandsimilar, followedbyconvexorconcaveplatesofdierentradii.Theratioofslot-heightmeasured attheatplatetoradiusofcurvaturevariedbetween3 : 2 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 and8 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 .Hot-wire anemometrywasusedtomeasurevelocity,andStantontubeswereusedtomeasurewall shearstress.TestswereperformedatslotReynoldsnumbersbetween1 : 7 10 4 and 2 : 6 10 4 .Meantangentialvelocityprolesshowedsimilarityforallsurfacetypeswhen velocityanddistancefromthesurfacewerenormalizedusing U m and y m= 2 alsocalled thejethalf-width,respectively.LikeWilson&Goldstein1976,Kobayashi&Fujisawa 1983observedhighervaluesofReynoldsstressandturbulenceenergyproduction comparedtotheplanewalljet.Inaddition,thepositionofzeroReynoldsstress,which waslocatedclosertothesurfacefortheconvexwalljetcomparedtotheplanewalljet, wasfoundtodecreasewithincreasingcurvature. RewandParkcollaboratedontheowoftwoopposingwalljetsoveracircular cylinder,similartothetrailingedgecongurationofadual-slottedcirculationcontrol airfoilRew&Park1988;Park&Rew1991.Twoslotsofequalheightlocated180 apartonacircularcylinder h=R =0 : 1eachemittedawalljet.Oneslotdischarged thepowerjet,"whichhadalargerinitialmomentumuxthantheopposingcontrol jet."Pressuretapsalongthecylindersurface,apitottube,andconstanttemperature hot-wireanemometrywereusedtomeasuretheow.Themomentumuxratioofthe twojetswasvariedbyadjustingthecontroljet"momentumuxwhilemaintaininga constantpowerjet"momentumux.SlotReynoldsnumbersvariedbetween1 : 46 10 4 and3 : 05 10 4 .RewandParkcategorizedtheowintothreeregions:thecurvedwall jetregion,locatedbetweeneachslotandthepositionwherethejetsbegintoseparate, theinteractionregionwherethetwojetscoalesce,andthemergedjetregion wherethetwojetsfullymergedintoasingle,freejetwithamaximumvelocityatthe 36

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centerline.Intherstregion,steadysurfacepressureinitiallydecreasedthenplateaued toaconstantvaluewherethemeanvelocityprolesdisplayedsimilaritywhenvelocity anddistancewerenormalizedby U m and y m= 2 ,respectively.Conversely,turbulence intensityandReynoldsstressprolesdidnotexhibitsimilaritywhennormalizedusing thesameparameters.Theinteractionregion"wascharacterizedbyasharpincreasein surfacepressureandapressurehump"symmetricaboutthemaximumpressurevalue namedtheinteractionpoint."Thelocationoftheinteractionpoint"dependedonthe momentumuxratioofthejets.Whenthepressurebegantoincrease,thenormalized velocityprolesremainedsimilar,butthenormalizedturbulenceintensityandReynolds stressprolesincreasedsubstantiallyduetotheadversepressuregradient.Asthetwo jetsbegantomerge,arapiddecayoflongitudinalandlateralturbulenceintensitywas measured,andtheReynoldsstressquicklychangedsignasdownstreamdistancefrom thecylinderincreased.Thecombinedfreejetthatemergedfromtheinteractionregion" signiedthestartofthethirdandnalregion.Thefreejetinitiallyacceleratedfromthe highpressurealongthecylindersurface.Thispressureslowlydecreasedandapproacheda constantvaluewithincreasingdownstreamdistancefromthecylinder.Wherethepressure wasalmostconstant,thecenterlinevelocitydecreasedbyabout x )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = 2 .Themeanvelocity andturbulenceintensityprolesnormalizedbycenterlinevelocityandjethalf-width showedsimilarityandagreedwithdataforaconventionalfreejet.However,themerged jetreachedsimilarityearlierandspreadfasterthantheconventionalfreejet. Focusedondeterminingthecausesofseparation,Neuendorf&Wygnanski1999 publishedaseminalworkonthecurvedwalljet.Experimentswereconductedona circularcylinderwithasingleblowingslot h=R =0 : 023andstagnantsurroundings. Flowmeasurementswereconductedusinghot-wireanemometry.TheReynoldsnumber, denedas Re N = )]TJ/F21 7.9701 Tf 6.675 -4.977 Td [(1 2 U 2 jet hR= 2 1 = 2 ,was3 : 3 10 4 .Between0 << 40 ,where =0 at theslot,uidwasentrainedandthejetbecamefullydeveloped.Between40 << 120 thestaticsurfacepressureremainedconstant,andself-similarmean,two-dimensional 37

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owdeveloped.Meantangentialvelocityprolesexhibitedsimilarityandagreedwith planewalljetdatawhennormalizedusing U m and y m= 2 .Inaddition,thehalf-widthof thejetwasmuchsmallerthanthecylinderradiusinthisregion,justifyingtheboundary layerapproximation.Anadversepressuregradientexistedbetween120 << 180 increasingturbulenceintensitiesandproducingdissimilarmeanvelocityproles.The outerregionofthevelocityprolesexhibitedsimilaritywhennormalizedusinganew lengthscale, y m= 2 )]TJ/F22 11.9552 Tf 12.707 0 Td [(y m ,where y m wasthedistancefromthesurfacetothelocationof maximumvelocity.However,nolengthscalewasfoundtocollapsetheinnerregiondata. Atlargerangulardistancesfromtheslot,180 << 220 ,themeantangentialvelocity suddenlydecreasedwhilethemeanradialvelocityquicklyincreased.Theboundary layerthickened,increasingthehalf-widthofthejet,and y m increasedrelativeto y m= 2 Turbulenceintensitiesgrewtolevelsmuchlargerthanthoseobservedforaplanewalljet. At =220 ,theowseparatedasthemeantangentialandradialvelocitiesapproached similarvalues.NeuendorfandWygnanskiconcludedthatentrainmentcausestheowto initiallyadheretoandalsoeventuallyseparatefromthecurvedwall. MorerecentworkbyNeuendorfandhiscolleagueshasshednewlightonanother mechanismcausingjetseparation-streamwisevortices.Likhachev etal. 2001werethe rsttoobservesuchstreamwisevorticesusingowvisualizationtechniques.Byxing onehot-wireprobeandtraversinganotheralongthespanofthecylinderinthesame azimuthalplane,thewavelengthsofcounter-rotatingpairsofvorticeswereapproximated andfoundtoincreasewithangulardistancefromtheslot.Neuendorf etal. 2004 conrmedthepresenceofthesevorticesusingPIVmeasurementsofthecross-owplane normaltothejet.Patternrecognitionschemeswereutilizedtocreateaframeofreference xedwiththevortices,whichhadatendencytoslowlymeander"acrossthespan.The dataweredecomposedintosteady,coherent,andrandommotionstoextractthelarge scalevortices.FurthertestsbyHan etal. 2006focusedonstudyingthegrowthofthese vorticesbycontrollingtheiremergenceandspanwiselocationwithvortexgenerators. 38

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Fromthesestudies,thefollowingrevisedpictureofthecurvedwalljetowwaspainted. Centrifugalinstabilityleadstotheproductionofcounter-rotatingpairsofstreamwise vorticesthatdonotaectthemeanspanwiseow,butinsteadcontributetotheReynolds stressesandincreasedturbulenceproduction.Thesevorticeslltheentirejetwidthand slowlymoveacrossthespan.Asthesevorticesconvectdownstream,theystrengthenby combiningwithothervorticesofthesamerotationalsenseandmoveawayfromthewall surface,thickeningthejetandeventuallycausingseparation.Instagnantsurroundings, thecounter-rotatingpairsofvorticesmovelow-momentumuidfromthesurroundings tothesurfaceandsendhigh-momentumuidawayfromthesurfacetotheouter-region. Hanetal.hypothesizedtheoppositecouldoccurinthepresenceofahigh-momentum freestream. Thestudiessummarizedinthissectionhaveprovidedsignicantinsightintothe characteristicsofthecurvedwalljet.However,muchoftheworkwasperformedinthe absenceofafreestream. 1.3.6CirculationControlAcoustics Althoughtherehavebeenamultitudeofresearchinvestigationsoncirculation controlairfoils,onlyahandfulhavefocusedonsoundproduction.Thefewanalytical, experimental,andnumericalstudiesonlow-speedcirculationcontrolacousticsare summarizedinTable1-5anddiscussedinthissection. CirculationcontrolacousticswererstintroducedintheliteraturebyWilliams& Cheeseman1978,whotheoreticallyanalyzedthepotentialnoisesourcesofacirculation controlrotortoshowthatitcouldbesubstantiallyquieterthanaconventionalrotor. Amongthetencirculationcontrolrotornoisesourcessuggestedwerevepotential broadbandnoisesourcesofatwo-dimensionalcirculationcontrolairfoil:classical trailingedgenoise,"laminarboundarylayerinstabilitynoise,"jetnoise," boundarylayernoise,"andincidentturbulencenoise."Therstnoisesource, classicaltrailingedgenoise,"iscausedbyturbulenceinthejetandoutersurface 39

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boundarylayerconvectingpastthetrailingedge.Laminarboundarylayerinstability noise"isgeneratedbyanaeroacousticfeedback"betweenpressureuctuationsinthe wakeandtheinstabilitylocationofthelaminarboundarylayerontheunblownsideof theairfoil.Thethirdnoisesource,jetnoise"isactuallycomprisedofthreecomponents: atheinteractionofturbulenceintheCoandawalljetwiththeairfoiltrailingedge,b theinteractionofturbulenceintheseparatedfreejetwiththeairfoiltrailingedge,and cjet/wakemixing.Boundarylayernoise"isproducedbyturbulenceintheboundary layersalongtheoutersurfacesoftheairfoilandtheseparationoftheboundarylayeron thesideoppositeoftheblowingslot.Finally,incidentturbulencenoise"isgeneratedby freestreamturbulencefromtheatmosphereor,inthecaseofarotor,thewakeofanother blade. Intheearly1980's,whenthefocusofcirculationcontrolresearchwasonrotors, Mosher1983presentedndingsfromanexperimentalcomparisonofthreefull-scale helicopterrotors:aconventionalrotor,theX-Wing"rotorwithleadingandtrailingedge blowingslots,andacirculationcontrolrotorwithasingletrailingedgeblowingslot.The testsdeterminedthatthecirculationcontrolrotorwastheloudestatidenticalforward speedsoradvancingtipMachnumbers.Inaddition,thecirculationcontrolrotorproduced greaterbroadbandnoiselevelsathigherfrequenciesthantheconventionalrotorduetothe trailingedgeblowing. Aspartofaninvestigationofcirculationcontrolwingswithuppersurfaceblowing, Salikuddin etal. 1987measuredthesoundproducedbyacirculationcontrolairfoilin ananechoicwindtunnel.NineBruelandKjrB&K6.35mmdiametermicrophones wereplacedinapolararcbeneaththecirculationcontrolairfoil h=c =3 : 2 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 to 8 : 0 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 R=c =0 : 0219.Datawerepresentedforslotvelocitiesbetween150and341 m/s.Theone-thirdoctavespectrawerecharacterizedbyalowfrequencypeakfroman unknownsourcetheauthorshypothesizeditwasairsupplylinenoiseandabroader highfrequencypeak.Ingeneral,soundpressurelevelsincreasedwithincreasingslot 40

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velocityandaconstantslotheightorincreasingslotheightandaconstantslotvelocity. Whentheslotheightandslotvelocitywerekeptconstantandthefreestreamvelocity wasincreased,onlythelowfrequencysoundpressurelevelsincreasedsignicantly.The directivitymeasurementsrevealedthatsoundlevelswerelargestatshallowerangleswith respecttothemodel'schordline. Munro etal. 2001comparedthesoundproducedbyaconventionalwinganda circulationcontrolwingatasimilarliftcondition.Testswereperformedintheopenjet anechoicightsimulationfacilityatGTRIusingaB&K6.35mmmicrophonetraversed inthey-overplane.Thecirculationcontrolwinghadasupercriticalproleandasmall apforlowdragcruiseight,andtheconventionalwinghadasimilarprolewithalarger Fowlerap.Initially,thecirculationcontrolwingwastestedwithitsaprotatedvertically tothe90 position.Intheabsenceofafreestream,soundlevelswerefoundtoincrease withincreasingjetvelocityandaconstantslotheightorincreasingslotheightanda constantjetvelocity.Withafreestream,alowfrequency,high-amplitudetoneattributed tovortexsheddingfromthebluaptrailingedgewasmeasured.Thevortexshedding tonepersistedwithhighamountsofblowing,sotheapwasdeectedtothelesssevere angleof30 wherethetonewaseliminatedwithjustasmallamountofblowing.Atthis apangle,soundlevelswereobservedtodecreasewithincreasingslotheightataconstant momentumcoecient,asexpected,sincealargerslotheightataconstantmomentum coecientcorrespondstoareducedslotvelocity.However,iftheslotheightwastoolarge, airsupplynoisebecamesignicant.Thecirculationcontrolwingwasultimatelyfound toproducelowersoundlevelsthantheconventionalwingwithamidspanapcut-out. Below10kHz,soundlevelswerereducedbyasmuchas8dB,whereasbetween10and40 kHz,thedierencewasbetween2and4dB. ArguablythemostsignicantworktodateoncirculationcontrolacousticsisHowe's seminalpublicationpresentinghisanalyticalsolutionofthebroadbandnoiseproducedby acirculationcontrolhydrofoilHowe2002.Howeconsideredatwo-dimensional,elliptical 41

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circulationcontrolhydrofoilwitharoundedtrailingedgeplacedatzeroincidencein auniform,lowspeedfreestreamsuchthattheMachnumberwasinnitesimalbutthe Reynoldsnumberwaslarge.Howeidentiedfourpotentialnoisesources:separation noise,curvaturenoise,passiveslotnoise,andslotjetnoise.Separationnoise iscausedbyareactionforcefromtheseparatedow.Howedeterminedseparationnoise wouldbenegligiblerelativetotheotherlowfrequencynoisesource,curvaturenoise. Curvaturenoiseisproducedbytheinteractionofturbulenceintheboundarylayer owpassingovertheroundedairfoiltrailingedge.Passiveslotnoiseisproducedby turbulenceinthefreestreamsurfaceboundarylayerscatteringotheslotlip.Finally, slotjetnoiseisproducedbytheinteractionofturbulenceintheCoandajetwiththe upperandlowerslotsurfaces.Howerepresentedtheacousticpressurespectrumforeach noisesourceasafunctionofthelocalowspeed,frictionvelocity,anddisplacement thickness.HowegatheredestimatesfortheseparametersfromworkbyNovak&Cornelius 1986andNovak etal. 1987toevaluateexpressionsforeachnoisesource.Howe2002 discoveredthatatlowfrequencies,curvaturenoisedominated.Athigherfrequencies non-dimensionalfrequenciesof fh=U jet > 1,slotjetnoise,specically,thenoiseproduced bytheinteractionoftheCoandajetwiththeslotlip,wastheprincipalnoisecontributor. ThenoiseproducedbytheinteractionoftheCoandajetwiththelowerslotsurfacewas negligible.Themiddlefrequencyportionoftheacousticspectrumwasdominatedby passiveslotnoise.Figure1-6isareconstructionofHowe'sacousticpressurespectrumwith eachnoisesourcelabeled. WhileHoweidentiedpotentialbroadbandnoisesources,Slomski2009venturedto determinethecauseofdiscretetonesobservedduringrecentNavytestsofacirculation controlhydrofoil.EarlyanalysisshowedthatthetonesfollowedtraditionalStrouhal numberscalingusingthelipthickness l andslotjetvelocity, St = fl U jet : {8 42

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Slomskiwasabletoconrm,usingLES,thatthetoneswereindeedattributedtovortex sheddingfromtheslotlip.Healsosimulatedtheeectofdierentlipmodicationson thefar-eldacousticspectra.Contouringthelowersurfaceofthelipreducedbutfailedto completelyeliminatethetone.Asawtooth-patternedlipsuccessfullyeliminatedthetone bybreakingupthenoise-producingspanwisevortices. Fewstudieshavefocusedoncirculationcontrolacoustics,andmanyofthepotential noisesourcesofacirculationcontrolairfoilhaveyettobeexperimentallyveried. However,thepreviousstudieshaveprovidedacleardescriptionofpossiblenoisesources andtheirexpectedcharacteristicsi.e.frequencyrange,broadbandvs.narrowband,etc.. AlistofpotentialcirculationcontrolairfoilnoisesourcesisprovidedinTable1-6,andan illustrationofthesemechanismsisprovidedinFigure1-7.Mostofthebroadbandnoise sourcesfollowfromtheanalysisbyHowe2002.Noisesourceslikeincidentturbulence noise"andboundarylayerinstabilitynoise"areneglectedbecauseinthelaboratory environment,tunnelfreestreamturbulencelevelsareextremelylow,andtheairfoil boundarylayersaretrippedtoensureafullyturbulentboundarylayer.Somenoisesources notaddressedintheliteratureareincludedinthelist,likeslotlipvibrationnoiseand plenumresonance. 1.4UnresolvedTechnicalIssues Astheliteratureattests,thereisanabundanceofuiddynamicdataoncirculation controlairfoilsandcurvedwalljets.However,littleisknownabouttheacousticsofa circulationcontrolairfoil.Experimentally,theworkperformedhasfocusedongeneral trendsinsteadofidentifyingthenoisesourcemechanisms.Theoretically,noisesource mechanismshavebeenidentiedbutawaitexperimentalconrmation,and,similarly, Howe'smodel,whichestimatesthatnoisefromacirculationcontrolairfoilcouldbe substantiallylouderthanaconventionalairfoil,hasnotyetbeenveriedorcomparedwith measuredacousticspectraHowe2002. 43

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1.5ResearchObjectives Theobjectivesofthisstudyaretoexperimentallycharacterizetheacousticsanduid dynamicsofanellipticcirculationcontrolairfoil.Particleimagevelocimetrymeasurements provideinsightintotheowphysicsrelatedtothenoisegenerationmechanisms.A varietyofacousticmeasurementtechniquesarecomparedtodeterminewhatmethods maybesuitableformeasuringcirculationcontrolnoise.Noisesourcesareidentied, andtheirfar-eldacousticlevelsarecomparedtodeterminetheprimarysourcesover dierentfrequencyranges.Finally,Howe'stheoreticalmodelofcirculationcontrolnoiseis evaluatedandcomparedwithmeasurementsHowe2002. 1.6TechnicalApproach Atwo-dimensional,dual-slotted,ellipticcirculationcontrolairfoilbasedonthe hydrofoilstudiedbyRogers&Donnelly2004isdesignedandfabricated.Testsare conductedintheUniversityofFloridaAeroacousticFlowFacilityUFAFFatlowand moderatechordReynoldsnumbers Re c =6 : 5 10 5 to1 : 3 10 6 andzeroangleofattack. Focusisplacedonlowblowing,moderateliftperformancetypicalofanunderwater vehiclecontrolsurface.HigherliftcasesappropriateforapplicationtoSTOLaircraft arealsostudied.Desiredliftperformancerangesforaircraftandnavalapplications arequalitativelyillustratedinFigure1-8,alongwiththemaximumliftproducedbya conventionalNACA0012airfoilanda20%ellipseforcomparison.Steadysurfacepressure measurementsareusedtocharacterizethefreestreamowanddetermineliftasafunction ofmomentumcoecient.Thefreestreamandcurvedwalljetowsaremeasuredand analyzedusingPIV.Multiplefar-eldmicrophonesandanestedphasedmicrophonearray areusedfordeterminingsourcelevelsandidentifyingnoisesources.Usingdatafromthe PIVstudy,Howe'smodelofcirculationcontrolacousticsisevaluatedandcomparedwith measurementsHowe2002. 44

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Figure1-1.Typicalunderwatervehicleappendages. Figure1-2.Circulationcontrolairfoil. Figure1-3.TheCoandaeectonacirculationcontrolairfoiladaptedfromEnglar 1975 a 45

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Figure1-4.DenitionofcirculationadaptedfromPanton2005. A B Figure1-5.Flowswithandwithoutcirculationaroundacirculationcontrolairfoilowis fromlefttoright.AUnblown.BWithBlowing. 46

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Figure1-6.RepresentationofHowe'scirculationcontrolacousticspectrumHowe2002. Figure1-7.Circulationcontrolairfoilnoisesources. 47

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Figure1-8.DesiredliftperformancerangesforaircraftandnavalapplicationsNACA0012 data:Abbott&VonDoenho1959;20%ellipsedata:Abramson1975. Table1-1.Summaryofcirculationcontroluiddynamicsstudies AuthorsYear BriefDescription Lehnert&Hazen1956 Earlyexperimentalstudyofacirculationcontrolwing Kind&Maull1968 Firstpublishedcirculationcontrolstudy;notedhigh-lift capabilitiesandotheradvantagesofcirculationcontrol Williams1969 Studiedslotheight,position,andtrailingedgeradiuseects Williams&Howe1970 Assessedfeasibilityofaddingaapforxed-wingaircraft Englar1971 Investigatedslotheight,position,andtrailingedgeradius eects Englar1972 Examinedowseparationatpositiveanglesofincidence Abramson1975 Studiedcharacteristicsofa20%thick,non-cambered circulationcontrolairfoil Englar1975a Studieddierentcongurationsofahingedapcirculation controlairfoil Englar1975b Analyzedowfeaturesalongthetrailingedge Abramson1977 Compareddierenttrailingedgeshapes Englar&Huson1983 TesteddierenttrailingedgeshapesforanA-6testplane Novak&Cornelius1986 LDVstudyoftrailingedgeoweld Novaketal.1987 LDVstudyoftrailingedgeoweld Abramson2004 FirstNavydual-slottedcirculationcontrolairfoiltest Rogers&Donnelly2004 Assessedtheperformancebenetsofadual-slotted congurationandnitewingeects Jonesetal.2006 PIVstudyofaappedcirculationcontrolairfoiloweld 48

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Table1-2.AirfoilgeometryspecicationsWilliams1969 Model TrailingEdge, R=c SlotPosition, x=c A 0.01880.813 C 0.01880.922 CM1 0.03750.934 Table1-3.AirfoilgeometryspecicationsEnglar1971 Model EllipseThickness%TrailingEdge, R=c SlotPosition, x=c PureEllipse" 150.01130.924 RoundEllipse" 15.60.04030.96 Table1-4.SummaryofCoandaeectandcurvedwalljetstudies AuthorsYear BriefDescription Newman1961 Experimentalstudyofthemeanowpropertiesofawalljet owingoveracircularcylinder Wille&Fernholz1965 SummaryofCoandaeectresearchandapplications Dunham1968 Theoreticalanalysisofowoveracircularcylinder Parks&Peterson1968 Similaritysolutionforlaminarowoveracircularcylinder neglectingcurvatureeects Levinsky&Yeh1972 Theoreticalandexperimentalstudyofcirculationcontrolon acircularcylinderincludingcurvatureeects Wilson&Goldstein1976 Experimentalcomparisonofplaneandwalljetows Kobayashi&Fujisawa1983 Experimentalcomparisonofplane,convex,andconcavewall jetows Launder&Rodi1983 Reviewofturbulentwalljetexperiments Rew&Park1988 Experimentalstudyoftwoopposingwalljetsowingovera circularcylinder Park&Rew1991 Extensionof1988studyfocusingonturbulence measurements Neuendorf&Wygnanski1999 Experimentalstudyofcurvedwalljetowseparation Likhachevetal.2001 Firstexperimentalstudytoobservestreamwiseeddies Neuendorfetal.2004 ExperimentalstudyofstreamwiseeddiesusingPIV Hanetal.2006 Experimentalstudyofstreamwiseeddygenerationand growth 49

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Table1-5.Summaryofcirculationcontrolacousticsresearch AuthorsYear BriefDescription Williams&Cheeseman1978 Theoreticallyidentiedvepotentialbroadbandnoise sources Mosher1983 Experimentalcomparisonofthesoundproducedbythree helicopterrotors,includingacirculationcontrolrotor Salikuddinetal.1987 Firstexperimentalacousticsstudyofacirculationcontrol airfoil Munroetal.2001 Comparedsoundlevelsofacirculationcontrolwingwitha conventionalwingatasimilarliftcondition Howe2002 Analyticalinvestigationofbroadbandcirculationcontrol airfoilnoisesources Slomski2009 Computationalstudyoftheslotlipvortexsheddingtone andlipmodications 50

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Table1-6.Noisesourcesassociatedwithacirculationcontrolairfoil NoiseSource MechanismMainCharacteristics Airfoilselfnoise Curvaturenoise trailing-edgenoise Interactionofboundary layerturbulencewith roundedtrailingedge Broadband,mainsourceof lowfrequencynoise Passiveslotnoise Interactionofexterior boundarylayerturbulence withslotlip Broadband,importantat midfrequencies Slot-jetinteractionnoise Interactionofturbulent jetowwithupperslot surface Broadband,signicantat highfrequencies Separationnoise Reactionforcefromow separation Broadband,insignicant accordingtoHowe Boundarylayernoise y Interactionofexterior boundarylayerturbulence withairfoilsurface upstreamoftrailingedge Broadband JetNoise y Interactionofwalljet turbulencewithtrailing edge Broadband Jet/wakemixingBroadband Interactionofseparated freejetturbulencewith trailingedge Broadband Bluntlipedgenoise Vortexsheddingfromblunt slotlip Tonal,highfrequency Blunttrailingedgenoise Vortexsheddingfrom roundedtrailingedge Tonal,lowfrequency Internalownoise Supplyowseparation, compressor,etc. Broadbandandtonal PlenumHelmholtz resonance Tonal Mechanicalnoise SlotlipvibrationTonal Howe2002 y Williams&Cheeseman1978 51

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CHAPTER2 EXPERIMENTALSETUP Thischaptersummarizestheexperimentalsetup.First,thecirculationcontrol airfoilisdescribed,followedbyanoverviewofthepressurizedairdeliverysystem. Next,theanechoicwindtunnelusedthroughoutthisinvestigationisintroduced.Airfoil characterizationexperimentsaredescribedthereafter,followedbyowmeasurementsand, nally,acousticmeasurements. 2.1CirculationControlAirfoil 2.1.1DesignandFabrication Adual-slotted,two-dimensionalcirculationcontrolairfoil,showninFigure2-1A,is designedbasedonthegeometryofthehydrofoilpreviouslystudiedbyRogers&Donnelly 2004.Theairfoil'sproleisa20%thickness-to-chordratioellipsewithacylindrical trailingedgeandnocamber.Detailsregardingthetrailingedgegeometryareprovided inFigure2-1B.Theairfoilhasa0.5207mchordand1.12mspan.Althoughthefrontal arearatiobetweenthemodelandtunnelinletcross-sectionsissomewhathigh%,the largeairfoilsizeischosenforanumberofreasons,including:tosuitanticipatedfull-scale applicationchordReynoldsnumbersandmomentumcoecients,forimprovedspatial resolutioninmeasurements,forreducedcompressibilityeectsintheslotjet,andfor loweracousticfrequencies.Theheightofeachblowingslotisadjustableusingeightsets ofpush-pullscrewsequallyspacedacrossthespan.TheCoandasurfacetrailingedgeand slotlipsareremovabletoallowfuturetestingoftreated"noise-reducingcomponents.The trailingedgeincludesthreeinstrumentringsthatcanberotatedtoprovidemeasurements aroundtheentiretrailingedgesurfacewithonlyafewtransducers.Acompletesetof airfoiltechnicaldrawingsisprovidedinAppendixA. Eachblowingslotissuppliedpressurizedairfromtwoconstantareaairfeeds,one oneachsideofeachplenum,thatsmoothlytransitionairfrom50.8mmdiametersupply pipestotherectangularplenuminlets.Tominimizetheowspeedinsidetheplenum, 52

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theinletareaismadeaslargeaspossible,resultinginaninlet-to-slotarearatioof4.7 atthenominalslotheightof1.0mm.Varioussealantsandrubbergaskets,describedin AppendixA,areusedtopreventairleakage. Themodelisprimarilymachinedfromcastaluminumtoolplate.Thesupportplates androdsaremadefrom1018steel.Thenylonairfeedsarebuiltinaselectivelaser sinteringmachine.MostofthemodelismanufacturedbyCraigJohnsonattheIllinois InstituteofTechnologyDepartmentofMechanical,MaterialandAerospaceEngineering MachineShopunderthesupervisionofDr.DavidWilliams.Thealuminumslotlipsare machinedbyTMREngineeringinMicanopy,FLundermanagementofKenReed. Duringtheinvestigation,onlyasingleblowingslotisusedatanygiventime.The other,unusedblowingslotissettothenominalslotheightof1.0mmandsealedwith tape.Spanwisestripsof12mmwide,0.4mmthick,GlasfaserFlugzeug-Servicezig-zag turbulatortape,showninFigure2-2,areplacedontheairfoilat18%chordtotripthe upperandlowersurfaceboundarylayers.Tripthicknessisdeterminedusingthesteps outlinedbyBraslow&Knox1958. 2.1.2PlenumTreament StripsofporousERGDuocelAluminumFoam,each1.27cmby4.07cmby55cm, areusedforowstraighteningandnoiseattenuation.Together,twostripscanspanthe entire1.10mplenum.Threedierentporedensities,20,and40pores-per-inchor PPIarechoseninitiallyforacomparativestudy.However,maximumsoundabsorptionis providedifallthreefoamstripsareusedinseries.Theinsidewallsoftheplenumarealso linedwith2mmthickadhesive-backedfoamtoreducehighfrequencynoise.Inaddition, thevolumeupstreamoftheDuocelfoamislledwithPoly-lpolyesterllingtofurther attenuatenoisefromtheairsupplyline.Figure2-3showstheDuocelfoamandthinliner plenumtreatments. 53

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2.2AirDeliverySystem Pressurizedairisprovidedbya1000SCFM,300hpcompressorplacedinastructure outsideofandisolatedfromthebuildinghousingtheUFAFF.Afterpassingthrough desiccantdryersandalter,theairisstoredintwo14.4m 3 vessels.A2.54cmglobe valveoperatedbyapneumaticactuatorcontrolstheowratefromthetankstothe circulationcontrolairfoil.InsidetheUFAFF,theairpassesthroughamodularairdelivery system,connectedtothe7.62cmdiametersupplylineviaaunionadapter,thatcan beconvenientlymovedtoneighboringlabsformodelleaktestsandslotowuniformity studiesbetweentunnelentries. Theairdeliverysystem,aschematicofwhichisshowninFigure2-4,includesfour LambdaSquareB-Plus150Venturimeters,oneforeachairfoilsupplyport.AVenturi metergaugesthevolumetricowratebymeasuringthepressuredierenceacrossanozzle ofknowninletandexhaustradii.MoredetailsregardingtheVenturimeters,including howtocomputetheowrate,areincludedinAppendixC.Theairdeliverysystemisalso equippedwithanOmegaPR-20platinumresistancetemperaturedetectorRTDprobe placedupstreamoftheVenturimeters.AsdescribedinAppendixC,assuminganideal gas,thedensityofairinthesupplylinecanbecomputedusingthemeasuredtemperature andthehighpressuremeasurementfromaVenturimeter.Themassowratecanthenbe computedfromtheproductofthedensityandvolumetricowrate.Aircanbeisolatedto individualairfeedsusingavalveoneachVenturimeter. AnadditionalsetofltersdownstreamoftheVenturimetersremovesanyremaining debris.Afterthelters,UniversalSilencerU5-1-1/2straight-throughabsorptivesilencers provideatleast30dBsoundpressurelevelattenuationbetween500and8000Hz accordingtothemanufacturer'sspecicationstoreducenoisecontaminationfrom thecompressormotorandglobevalveowseparation.Flexiblehoses,3.66mlongand 38.1mmindiameter,guidetheairfromthesilencerstosmoothow-reducingnipples connectedtothe50.8mmdiametercircularairfoilairfeeds.Theairdeliverysystemalso 54

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includesinterchangeablepressurereliefvalvestopreventaccidentaloverpressurizationat thevariousslotheightsandvelocitiestested. 2.3UniversityofFloridaAeroacousticFlowFacility TheUFAFFisanopen-returnwindtunnelwithanopen-jettestsectioninstalledin anISO3745-certied100Hzanechoicchamber,whichisusedtosimulateafreeeld,or reection-freeacousticenvironment.Thetestsectionmeasures0.74mtallby1.12mwide andextends1.83mintheowdirection.Testsectionspeedsbetween18and75m/scan bereachedwithturbulentintensitylevelsbelow0.1%atfrequenciesabove10HzMathew etal. 2005.In2010,aowsilencerwasaddedupstreamoftheinlet,andoveronemeter ofthediuserwasremoved.Figure2-5providesschematicsofthecirculationcontrol airfoilinstalledinthepre-andpost-modicationtunnelcongurations.Forillustration purposes,someceilingwedgesareremovedintherenderingstorevealthechamberinterior andtestsection.Theairfoilisinstalled13cmdownstreamoftheinletexitplaneatzero degreesgeometricangleofattack.Themodelisalwaysboundedbysidewallsfortwo dimensionalow.FormoreinformationregardingtheUFAFF,pleaserefertoMathew etal. 2005. 2.4StaticMeasurements 2.4.1SlotFlowUniformity Spanwiseslotowuniformityisassessedusingconstanttemperaturehot-wire anemometry.Thehot-wireprobe,typicallya5 mdiametertungstenorplatinumwire, isoneoffourresistorsofabridgecircuit.Whenplacedinaow,thewireiscooledby theuidthroughforcedconvection.Thebridgecurrentthroughtheprobeisadjustedto maintainaconstantprobetemperatureandhenceresistance.Thegeneralrelationship betweenbridgevoltage E andnormalowspeed U isgivenbyKing'spowerlaw, E 2 = A + BU n ; {1 55

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where A B ,and n areconstantsdeterminedthroughcalibrationBruun1995.Atypical valueof n is0.45. ADantecStreamLineframecontainingaCTAModule90C10isconnectedtoa5 m diameter,1.25mmlong55P11straightminiatureplatinum-platedtungstenwireprobe viaa55H22probesupport.TheprobeiscalibratedusingaDantecStreamLinefreejet calibrator,andafourth-orderpolynomialisttothecalibrationdata, U = AE 4 + BE 3 + CE 2 + DE + F; {2 where A B C D ,and F arethecoecients.Thefourth-orderpolynomialisgenerallyas accurateasEquation2{1Bruun1995,buttemperaturechangesintheowarenottaken intoconsiderationduringthisexperiment. Aftercalibration,thehot-wireprobeissecuredtoaone-axisParkerACR9030 traverse.Usingalaserdisplacementsensor,thetrailingedgeofthecirculationcontrol airfoil,whichisfastenedtoamobile80/20supportstand,isalignedwiththespanwise traverseaxistowithin50 mfromendtoend.Thewireisplacedattheslotexitplane approximatelyequidistantfromtheinnerslotlipandtrailingedgesurfaces.Theow ismeasuredatfteenspanwiselocationscorrespondingtotheeightsetsofpush-pull screwsandthemidpointsbetweenthem.Ateachmeasurementlocation,theprobeis raisedandloweredbyafew matatime,untilthemeanhot-wirevoltagedisplayedona TektronixTDSseriesoscilloscopeisamaximum.Voltagedataarethenacquiredfor5sec atasamplingrateof20kHzusingaNationalInstrumentsNIPXI-4462dynamicdata acquisitioncardinstalledinaPXI-1042Qchassis.Twoplenumpressurescorrespondingto moderate 40m/sandhigh 80m/sslotvelocitiesaretested. Afterthemeasurementsarecompleted,thetime-averagedvelocityiscomputedat eachspanwisedatapoint.Thespanwisemeanvelocityandstandarddeviationofthe 56

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time-averagedvelocitiesarethencalculated. U span = 1 N N X i =1 U i {3 = v u u t 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 N X i =1 U i )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U span 2 {4 Meanspanwiseowuniformityisassessedbydividingthestandarddeviationbythe spanwisemeanvelocity. %deviation=100 U span {5 2.4.2LipDisplacement SlotlipdisplacementismeasuredasafunctionofplenumpressureusingaKeyence LK-G32CCDlaserdisplacementsensorplacedslightlyabovethelipedge.Thesensor emitsaredsemiconductorlaserontoatargetsurfaceandfocusesaportionofthereected lightontoaCCD.ThepositionofthehighestintensitypeakontheCCDismeasuredto sub-pixelaccuracyandmovesasthedistancebetweenthesurfaceandthesensorchanges. Usingtriangulation,thedisplacementisdeterminedfromthetriangleformedbythelaser source,target,andCCDpeakposition. Thesensitivityofthelaserdisplacementsensorissetto0.1mm/Vsincesmall displacementsareexpected.Voltagedataareacquiredfor5secatasamplingrateof50 kHzusingtheNIsystemdescribedinSection2.4.1.Thetime-averageddisplacementis thencomputed. 2.5FlowMeasurements 2.5.1SteadyPressureandLift Steadysurfacepressure, p s ,ismeasuredusing470.711mminnerdiametermidspan staticpressureports,includingninelocatedonthemidspantrailingedgeinstrumentring. Thetapsaresymmetricallydistributedontheupperandlowersurfacesoftheairfoil. Twotapsplacedattheleadingedgeoftheairfoilareslightlyosetfrommidspandue 57

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toinstallationconstraints.Chordwisetaplocations,normalizedbychordandlistedin Table2-1,aredeterminedusinganoptimizationcodedevelopedtominimizeerrorin themeasuredpressuredistributionMoreno2007.Theroutineadjuststhepressuretap locationsuntilthesumofthesquarederrorbetweentheliftcoecientcomputedfroma continuous"pressuredistributionobtainedfromCFDandtheliftcoecientcomputed fromthesamepressuredistributionsampleddiscretelyatthepressuretaplocationsis minimized.Notethatasinglepressureportislocatedat x=c =1.Aschematicofthe pressuretapsisincludedinAppendixA. Three16-channelEsterlinePressureSystemsPSIpressurescannersranges5psi, 1psi,and10inH 2 Omeasure39staticsurfacepressures,fourVenturimeterpressures, andtheblowingslotplenumpressure.Dataareacquiredattwosamplespersecondfor 30seconds.Thepressurescanners'referencepressureportsareconnectedtothetunnel pitot-staticpressureport.Thepitot-staticandpitot-dynamicpressuresaremeasured usingMensor6115digitalpressuretransducerswithrangesof16psiaand0.58psig, respectively.Surfacepressurecoecientsarecomputedbydividingthepressurescanner output, p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 ,bythepitot-dynamicpressure, q 1 C p = p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 q 1 {6 Becauseofthecoarseresolutionofpressuretaps,the C p data,plottedversusnormalized chordwiseposition,aretwithacubicsplinecurvethatisintegratedtodeterminethelift coecientperunitspan. c l = Z 1 0 C p d x=c = Z 1 0 C p;lower )]TJ/F22 11.9552 Tf 11.956 0 Td [(C p;upper d x=c {7 Uncertaintyin c l isestimatedusingaMonte-CarlosimulationColeman&Steele 2009.Givenmean C p values,theirbiaserror,andthestandarddeviationaboutthe mean,eachmean C p valueisperturbedbyarandomly-generatedvaluebasedonauniform distributionofthebiaserrorandarandomdistributionofthestandarddeviationabout 58

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themean.AcubicsplineisttotheperturbeddataandintegratedusingEquation2{7 togive c l .Thisprocedureisrepeatednumeroustimes,providingacumulativeprobability distributionfor c l .Thevaluesof P : 025and P : 975ontheprobabilitydistributionare evaluated,andtheirdierencerelativetothemean c l determinestheupperandlower95% condenceintervalbounds.Basedonaconvergencestudyofthesimulationresults,1000 iterationsisdeemedsuitableforconvergence. 2.5.2SlotJetVelocityandMomentumCoecient Themeanslotjetvelocityiscomputedassuminganisentropicexpansionfromthe plenumtothefreestream, U jet = v u u t 2 RT 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F27 11.9552 Tf 11.956 16.857 Td [( p 1 p 0 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 = # ; {8 where istheratioofspecicheats,and R istheidealgasconstant.Themeasured freestreampitot-staticpressure, p 1 ,isaddedtothemeasuredrelativeplenumpressure todeterminetheabsoluteplenumpressure, p 0 .Theairlinestagnationtemperature, T 0 ,isrecordedusingtheRTDdescribedinSection2.2.Forthecompletederivationof Equation2{8andtheinuenceofanonzeroplenumvelocityonthejetvelocityestimate, seeAppendixD. Themomentumcoecientiscalculatedusingthecombinedmassowratesofthetwo Venturimeters,thejetvelocitycomputedfromEquation2{8,themeasuredfreestream dynamicpressure,andthecirculationcontrolmodelplanformarea, S C = mU jet q 1 S : {9 2.5.3ParticleImageVelocimetry Particleimagevelocimetryprovidesanon-intrusivemeansofmeasuringthevelocity andrelatedpropertiesofaoweldRael etal. 2007.Thelocaldisplacementof tinyparticlesseededwithinaowaredetectedbyimagingtwosnapshotsoftheow 59

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illuminatedbyainstantaneous"lightsource.Thelightsource,whichisalaserbeamthat passesthroughaseriesoflensestoformathinlasersheet,illuminatesaplanarregionof interestintheoweld.Thetwospatiallycoincidentlasersarepulsedataknownrate butseparatedbyaspeciedamountoftime,typicallyontheorderofmicroseconds.For two-dimensionalPIV,acamerapositionedoutsidetheoweldandalignednormalto thelasersheetcapturesapairofimagesforeachpairoflaserpulses.Theimagepairis discretizedintosmallerregionscalledinterrogationwindows,whereeachinterrogation windowisdesignedtocontainseveralcommonilluminatedparticlesforeachframeofthe imagepair.Thespatialcross-correlationofeachcorrespondingpairofwindowsisthen computedusingFastFourierTransformsFFTs.Theaverage x -and y -displacements ofthegroupofparticleswithineachwindowpairaredeterminedfromthelocationof thehighestcorrelationpeak.Thein-planevelocitycomponentsarecalculatedfromthese displacementsandtheknowntimebetweenlaserpulses.Altogether,thediscretized velocitymeasurementsobtainedfromanimagepairprovideasnapshotofthevelocityow eld.Withalargeenoughsetofmeasuredvectoreldssuchthatstatisticalconvergenceis achieved,turbulentowquantitiescanalsobeestimated. FivedierentPIVexperimentsareperformedtocapturedierentowphenomenon. Theowismeasuredinregionsaroundtheleadingedge,behindthetrailingedge,above theslotlip,andalongthecurvedtrailingedgeinthechordwiseplane.Thetrailingedge owisalsomeasuredonceinthespanwiseplane.Theexperimentalsetupsforeachof theseexperimentsaredescribedintheremainderofthissection. 2.5.3.1Imageacquisition ThemeasurementoweldisilluminatedusingaNewWaveResearchSolo120XT Nd:Yaglaser.Thelaserpassesthroughasphericallensandacylindricallenstocreatea lasersheet.TheexactlensesusedforeachexperimentarelistedinTable2-2.Amirror reectsthelasersheetontothesurfaceoftheairfoilattheregionofinterest.Forthe measurementsoftheleadingedgeowandtheowabovetheslotlip,thelasersheet 60

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ispositionedinthechordwiseplaneosetapproximately26cmfrommidspanor27% span.Forthecurvedwalljetandtrailingedgewakemeasurements,thelasersheetis positionedinthechordwiseplanejustafewmillimetersosetfrommidspan.Finally, forthecrossowmeasurements,thelasersheetisrotatedsuchthatitislocatedinthe spanwiseplaneanglednormaltothetrailingedgesurface,intersecting33 downstream fromtheslotexitplane. AsummaryofthecamerasandcameraopticsforeachdatasetarelistedinTable2-3. Figure2-6illustratesthedierentregionsofthetrailingedgeoweldthataremeasured. Theentirecurvedwalljetowiscapturedintwoseparatesetsofimageslabeledasc anddinFigure2-6.Between500and1200imagepairsareacquireddependingonthe experimentandthequalityoffreestreamowseeding. Thefreestreamisseededusingavarietyofsmokemachinesandparticlegenerators placedupstreamofthewindtunnelinlet.Acustom-builtrakeprovidescontrolof particlevolumeandlocationbyadjustingrakeholecountandsize.Whenmeasuring owsincludingthecurvedwalljet,thetrailingedgeblowingslotisalsoseeded.For investigationsofthecurvedwalljetandtrailingedgewake,asmalltubeconnectedto aTSI9302atomizerisinsertedintotheplenumthroughtheendofthemodel.Forthe crossowmeasurements,higherdensityjetseedingisneeded,sothelargerTSI9307-6 atomizerisconnecteddirectlytotheairlineupstreamoftheairdeliverysystem.The seedersusedforeachexperimentarelistedinTable2-3,andparticlesizeestimatesbased oneachseeder'sspecicationsareprovidedinTable2-4.AccordingtoMelling1997,a1 mparticlehasa10kHzfrequencyresponseinair. 2.5.3.2Vectorcomputation VectorsarecomputedusingLaVisionDaVis7.4softwareDaVis2010.Ingeneral, imagesareprocessedfollowingthestepsoutlinedinFigure2-7.Duringpre-processing, eachimagepairisrstshiftedwithrespecttotherstimagetocorrectforcamera vibration.Second,theimagesetaverageissubtractedfromallimagestoimprove 61

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particleclarityandreducesurfacereection.Finally,theairfoilsurfaceandregionsof sparseseedingaremasked.CrossowPIVimagesareprocessedusingaslightlydierent approach.Afteranimageshiftcorrectionisperformed,thetrailingedgesurfaceismasked, anda7pxslidingminimumlterisappliedtoeachimage.Thelteredimagesare subtractedfromtheshift-correctedimages,improvingparticleclarityandreducingnoise. Vectorsarecomputedusingeitheratwo-orfour-iterationrecursivecross-correlation processingscheme.Interrogationwindowsize,overlap,andnalspatialresolutionare giveninTable2-5.Foralldatasets,a1:1Gaussianweightingfunctionisappliedtoeach window.Betweeneachiteration,outliersareremovedusingtheuniversaloutlierrejection techniquedevelopedbyWesterweel1993,1994,a3 3vectorgroupsmoothinglteris applied,andmissingdataareinterpolated.Notethatthesestepsarenotperformedafter thenaliteration.Instead,forallbutthecrossowPIVdata,afterthenaliteration, outliersarerejectedbasedonthevalueofthecorrelationpeakratio,Westerweel's universaloutlierrejection,andthesizeofthesurroundingvectorgroup.Finally,in MATLAB,additionaloutliersareremovedusingthemultivariateoutlierdetection MVODapproachappliedbyGrin etal. 2010toPIVmeasurements.Vectorsarenot interpolatedorsmoothedafterthenaliteration.ForthecrossowPIVdata,nooutlier rejection,interpolation,orsmoothingisapplied. 2.5.3.3Additionalcalculations Aftervectorsarecomputedandoutliersareremoved,furtherprocessingisperformed inMATLAB.Meanandturbulentowquantitiesarecomputedinthelocal^ s -^ n coordinate systemsdenedatthelipedgeandalongthetrailingedge,asshowninFigure2-8.To denetheprolesatthelipedge,rsttheaverageimageforeachtrailingedgedataset isimportedintoMATLAB,anedgedetectionlterisapplied,andtheuserselectsthe lipedgefromthelteredimage.Sincethelocationoftheliprelativetotheoriginofthe ellipticairfoilproleisknown,anellipseisttotheselectedpoint.Toaccountforaslight camerarotationerror,asecondpointonthesurfaceofthelipisselected,andthetted 62

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ellipseisrotatedaboutthelipedgeuntiltheerrorbetweenthettedellipseandsecond pointisminimized.Theslopeoftheellipseatthelipedgeisthencomputed,andasurface normalisdened.Velocitycomponentsareinterpolatedalongthesurfacenormalusing atwo-dimensionallinearinterpolationscheme.Theresolutionoftheinterpolatedproles isdenedtomatchthevectorresolutionofthedataset,whichislistedinTable2-5.An osetof0.043mmisaddedtoeachproletoaccountforthelasersheetsurfacereection, whichisfoundtoeectivelythicken"theimagedsurfaceby4to5px. Todenethelocationsofprolesalongthetrailingedge,anaverageimageis loadedintoMATLAB,anedgedetectionlterisapplied,theouteredgeofthesurface isdetermined,andacircleofthesamediameterasthetrailingedgeisttothedata. Two-dimensionallinearinterpolationisthenusedtodeterminethevelocitycomponents alongeachprole.AsshowninFigure2-8, isdenedrelativetotheslotexitplane.An osetof0.075mmisaddedtoeachproletoaccountforlasersheetsurfacereection. SincevelocityvectorsarecomputedinDaVisintheglobal X Y coordinatesystem,a coordinatetransformationisrequiredtotransformthevelocitycomponentsandturbulent quantities,suchasturbulenceintensityandReynoldsstress,fromthe X Y coordinate systemtothelocal^ s -^ n coordinatesystem.Let representtheclockwiserotationangle betweentheglobal X Y coordinatesystemandthelocal^ s -^ n coordinatesystem.IfV X andV Y arethe X -directionand Y -directionvelocitycomponentsinthe X Y coordinate system,thenthetangentialandnormalvelocitycomponentsinthelocalcoordinatesystem aredenedby U =V X cos )]TJ/F15 11.9552 Tf 11.955 0 Td [(V Y sin {10 V =V X sin +V Y cos : {11 Similarly,theturbulenceintensitiesandReynoldsstressinthelocal^ s -^ n coordinatesystem canbeexpressedasafunctionoftheturbulenceintensitiesandReynoldsstressinthe 63

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global X Y coordinatesystem. u 0 2 1 = 2 = v 0 X 2 cos 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 v 0 X v 0 Y cos sin + v 0 Y 2 sin 2 1 = 2 {12 v 0 2 1 = 2 = v 0 X 2 sin 2 +2 v 0 X v 0 Y sin cos + v 0 Y 2 cos 2 1 = 2 {13 u 0 v 0 = 1 2 v 0 X 2 )]TJETq1 0 0 1 283.523 625.463 cm[]0 d 0 J 0.478 w 0 0 m 17.68 0 l SQBT/F22 11.9552 Tf 283.523 615.199 Td [(v 0 Y 2 sin2 + v 0 X v 0 Y cos2 {14 2.5.3.4Uncertainty AdetaileduncertaintyanalysisforthePIVdataisprovidedinAppendixE. 2.6AcousticMeasurements Acousticmeasurementtechniquesutilizedinthisstudyincludefree-standing microphonesandarray-basedmethods.Botharediscussedinthissection. 2.6.1Free-StandingMicrophones Twodierentsetsoffree-standingmicrophonedataareacquired.Therstis measuredpriortotheshorteningoftheUFAFFdiuserandtheadditionofaninlet silencerbutisincludedinthisanalysisbecausemultipleslotheightsaretested.Forthat experiment,fourG.R.A.S.SoundandVibrationG.R.A.S.6.35mm40BEfree-eld microphonesareplacedoutsidethewindtunnelopenjetshearlayers.Themicrophones areorientedinpairs,twoaboveandtwobelowtheairfoil,asshowninFigure2-9.The G.R.A.S.microphonesarecalibratedusingaBruel&KjrB&K4231SoundCalibrator atafrequencyof1kHz.G.R.A.S.40BEmicrophonesaredesignedforaatfrequency responsefor0 incidencewiththeprotectiongridinstalled.Thefrequencyresponsesofall G.R.A.S.microphonesuseddeviatebylessthan1dBref.20 Pa,sonocorrectionsare appliedtothemeasuredspectra. Thesecondsetofdataarerecordedsimultaneouslywiththephasedarray,which isdescribedinSection2.6.2.Threemicrophonesareplacedabovethetrailingedge, asillustratedinFigure2-10.Thecentermicrophone,labeledM2"inFigure2-10,is aG.R.A.S.40BE,whiletheupstreamM1"anddownstreamM3"microphones displaced10.2cmfromthecentermicrophoneareB&K49396.35mmfree-eld 64

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microphones.Allthreemicrophonesareapproximately1.12mfromthetrailingedge. ThemicrophonesarecalibratedusingaB&K4228Pistonphoneatafrequencyof250Hz. TheB&Kmicrophonesaredesignedforaatfrequencyresponsefor0 incidencewith theprotectiongridremoved.Sincethemicrophonegridsremaininstalled,thefrequency responseofeachB&Kmicrophoneisusedtoadjustthemeasuredspectrumaccordingly. Sincethehighestfrequencyofinterestis80kHz,atmosphericsoundabsorptionmust beconsidered.AccordingtoBlackstock2000,atthisfrequency,soundisattenuated byapproximately4.5dBatadistanceof1.12mfromasource,assumingstandard atmosphericconditionsand100%relativehumidity.Soundat30kHzisattenuatedby 1.1dBatthesamedistanceandconditions.Thedatapresentedarenotcorrectedfor thisattenuation,butifexactlevelsaresoughtfromthemeasurements,theyshouldbe adjusted. Thevariousprocessingtechniquesappliedtothemicrophonedataaredescribedinthe followingsections. 2.6.1.1Singlemicrophone Asinglemicrophonecannotdiscernbetweenmultipleacousticsources.Instead,the powerspectraldensityprovidesanestimateoftheacousticpowerfromallcontributing sourcesatthemicrophone'slocation.Thepowerspectralorautospectraldensityis estimatedfrom ^ G yy f = 2 T E [ Y Y ] ; {15 where Y istheniteFouriertransformofthemicrophoneoutputconvertedtoPascals, Y isthecomplexconjugateof Y ,and T isthelengthofeachrecordBendat&Piersol 2000.Ofallthetechniquesdiscussedinthissection,thesinglemicrophonemethodisby fartheeasiesttoemploy,but,toreiterate,asinglemicrophonecannotdistinguishbetween multipleacousticsourcesorasinglesourceandextraneousnoise. 65

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2.6.1.2Coherentoutputpower ThecoherentoutputpowermethodCOPcanbeappliedbetweenpairsof microphonesandhasbeenusedextensivelyintrailingedgenoisemeasurementsBrooks& Hodgson1981;Hutcheson&Brooks2002;Bahr etal. 2008.Thepowerspectraldensities ofthesignalsmeasuredbythemicrophonesaregivenbythefollowingequationsnotethe superscript`^',whichdenotesanestimate,isdroppedforconvenienceBendat&Piersol 2000. G y 1 y 1 f = G v 1 v 1 f + G n 1 n 1 f {16 G y 2 y 2 f = G v 2 v 2 f + G n 2 n 2 f {17 G v 1 v 1 and G v 2 v 2 arethetruedesiredoutputs,while G n 1 n 1 and G n 2 n 2 representnoise uncorrelatedwiththeinputandeachother.Tocomputethe COP ,rst,theordinary coherencefunctionbetweenthetwooutputsiscomputedBendat&Piersol2000. 2 y 1 y 2 = j G y 1 y 2 j 2 G y 1 y 1 G y 2 y 2 {18 Then,the COP iscalculatedbymultiplyingthecoherencewiththemeasurednoisy outputBendat&Piersol2000. COP 1 = 2 y 1 y 2 G y 1 y 1 = G v 1 v 1 1+1 =c 2 {19 COP 2 = 2 y 1 y 2 G y 2 y 2 = G v 2 v 2 1+1 =c 1 {20 Theaforementionedequationsrevealthat COP i approachesthede-noisedoutput G v i v i as thesignal-to-noiseratio, c i = G n i n i =G v i v i ,oftheadditionalmicrophoneincreases.Thus,in highnoiseenvironmentswherethesignal-to-noiseratioislow, COP i isbiasedlowrelative tothetrueoutput G v i v i Inadditiontotherestrictionsonthesignal-to-noiseratio,COPisappropriatefor situationswherebothmicrophonesaremeasuringasinglehighly-coherentsourceora lumpeddistributionofsourcesthedistancebetweenthesourcedistributionandobserver 66

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ismuchgreaterthanthesizeofthesourcedistributionBahr etal. 2009.However,if themicrophonesmeasuremultiplecorrelatedsourcese.g.airfoilnoise,sidewallscrubbing noise,anddiuserowimpingementnoise,thenade-noisedspectrumwillstillnotbe obtained. 2.6.1.3Three-microphonemethod Thethree-microphonemethodprovidesameans,undercertainassumptions,to extractthetrueoutput G v i v i fromthenoisyoutputsofthreemicrophones.LikeCOP, thethree-microphonemethodassumesthatthemicrophonesareeachmeasuringasingle coherentsourcewithnoisethatismutuallyuncorrelatedbetweenmicrophonesignalsand withthesignalofinterestBendat&Piersol2000.Whentheseassumptionsareapplied, thetrueoutputautospectraoftherstmicrophone,forinstance,canbefoundfromthe measuredoutputoftherstmicrophoneandtheordinarycoherencefunctionscomputed betweenpairsofmicrophones.Similarly,asshowninEquation2{21,thetrueoutputcan alsoberelatedtothecross-spectrabetweenmicrophonepairsBendat&Piersol2000. G v 1 v 1 = G y 1 y 1 + c 1 = j y 1 y 2 jj y 1 y 3 j j y 2 y 3 j G y 1 y 1 = j G y 1 y 2 jj G y 1 y 3 j j G y 2 y 3 j {21 LikeCOP,thethree-microphonemethodisappropriatewhenallthreemicrophones aremeasuringasinglecoherentsourceoralumpeddistributionofsources.Ifmultiple coherentsourcesexistoveralargedomain,thenthismethodwillnotbeabletodiscern betweenthesoundproducedbyeachsource,anditwillunder-predictoverallacoustic levels. 2.6.1.4Uncertainty UncertaintiesfortheCOPandthree-microphonemethodsarecomputedusinga Monte-CarlosimulationdescribedbyBahr&Cattafesta2010. 2.6.2PhasedArray Phasedmicrophonearraysareusedextensivelyfornoisesourceassessmentin aeroacousticapplications.Aphasedmicrophonearrayconsistsofasignicantnumberof 67

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microphonesstrategicallylocatedinaframeorush-mountedinaatplate.Thearrayis focused"toapointofinterestbyappropriatelyweightinganddelayingthemicrophones' signalsinaprocessknownasbeamforming,whichisdescribedinthefollowingsection. 2.6.2.1Beamforming Thesimplestandmoststraightforwardbeamformeristhedelay-and-sumbeamformer. ConsidertheillustrationinFigure2-11ofanarraycomprisedof m =1 ;:::;M microphones andascanningregionof l =1 ;:::;L pointslocatedat ~x l ,eachassumedtobeanincoherent andstatisticallyindependentmonopolesource.Theestimatedacousticpowerateach locationisgivenbythefollowingequation2{22Dougherty2002;Yardibi etal. 2010 a P l = 1 M 2 ~a H l G~a l {22 Theweight,orsteering,vector ~a l compensatesforthedierentpropagationdistances traveledbythesoundwavefromthescanningpoint ~x l tothedierentmicrophonesand isdenedbyEquation2{23.Thesuperscript H inEquation2{22correspondstothe Hermitian,orcomplex-conjugatetransposeofthismatrix. ~ a l = 1 r l; 0 2 6 6 6 6 6 6 6 4 r l; 1 e )]TJ/F23 7.9701 Tf 6.587 0 Td [(jkr l; 1 r l; 2 e )]TJ/F23 7.9701 Tf 6.586 0 Td [(jkr l; 2 r l;M e )]TJ/F23 7.9701 Tf 6.586 0 Td [(jkr l;M 3 7 7 7 7 7 7 7 5 {23 InEquation2{23,thedistancefromeachscanningpoint l toeachmicrophone m is designated r l;m ,andthedistancefromthearraycentertothe l thscanningpointis denoted r l; 0 .Notethatthe1 =r l; 0 factorisusedtoscalethepowermeasuredbyeach microphonetomatchthepowerthatwouldbemeasuredatthearraycenter.Finally, G representstheestimatedcross-spectralmatrixCSMofthemeasuredmicrophonesignals. Integratedspectracanbeobtainedbysummingthepowersofeachscanningpoint andnormalizingbythepoint-spread-functionPSFofthearrayOelermans etal. 2007; 68

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Yardibi etal. 2010 b P L = P l 2 L P l P l 2 L PSF l {24 ThePSFisthetheoreticalarrayresponsetoamonopoleplacedabovethecenterofthe arrayatthesameheightasthescanningplane.IncomputingthepowerviaEquation 2{24,onlyselectedscanningpointsinaparticularregionofinterestneedtobeevaluated, providingameanstoremove"noisesourcesotherthanthesourcesofinterest. Comparedtothepreviouslydescribedmethods,phasedarraymeasurementsare muchmoreexpensivefromcost,time,andcomputationalstandpoints.Inaddition,while relativesoundlevelsfromarrayintegrationcantypicallybetrusted,absolutelevelsmay beerroneouslylowduetocoherencelossbetweenmicrophonesinthearray,particularlyat higherfrequenciesOelermans etal. 2007.However,unliketheconventionalmicrophone techniquesdiscussedearlier,phasedarraysareextremelyusefulinidentifyingnoisesources inacomplexacousticeld. 2.6.2.2Arraysetup A60-elementfree-eldnestedmicrophonearrayisdesignedandbuiltforthe circulationcontrolexperiments.Theinner,7.62cmaperture,25-elementarrayisdesigned forfrequenciesbetween10kHzand80kHzandiscomprisedof10B&K4954Band15 G.R.A.S.40BEhigh-frequencymicrophones.Theouter,73.7cmaperture,40-element arrayisdesignedforfrequenciesbelow10kHzandiscomprisedof35B&K4958array" microphonesinadditiontovesharedG.R.A.S.40BEmicrophonesalsopartofthe innerarray.Thearraypatternsarebasedontheequal-aperture-arealogarithmic-spiral designofUnderbrink1995,2001,2002andshowninFigure2-12.Innerandouterarray specicationsarecomparedinTable2-6.Theoreticalpointspreadfunctionsforapoint source1.12mabovethearraycenterarealsoprovidedinFigure2-13. Themicrophonesareinstalledonastereolithographyframewithbuilt-inmicrophone holders.TheframeiswrappedinNomexandfoamtomitigatereectionscausedbythe frameanditsmount.Thearrayisinstalledintheanechoicwindtunnel1.12mbelowthe 69

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airfoil'strailingedgeandcenteredatmidspan,asshowninFigure2-10.Apictureofthe testsetupalongwithpicturesofthearrayareprovidedinFigure2-14.Designmicrophone coordinatesandtheserialnumbersofallmicrophonesusedarelistedinTable2-7.Note thatanadditionalB&K4954Bisplacedinthearraycenter.Thearrayisinstalledinthe UFAFFsuchthatthe x y coordinatesystemdescribingthemicrophonelocationsinTable 2-7andTable2-8isrotated94 clockwisetodenethearray x y coordinatesystem illustratedinFigure2-10,where+ x isdenedopposingtheow. 2.6.2.3Calibration Thegoalofarraycalibrationistocharacterizethemagnitudeandphaseresponseof eachmicrophonerelativetothearraycenter,orreference,microphone.Thefrequency responsebetweeneachsensorandthereferenceisdierentduetotheindividual microphoneresponsesandscatteringfromneighboringmicrophonesandthesensor's ownprotectivegrid.Thecalibrationprocedureusedinthisinvestigationrstrequires precisedeterminationofallmicrophonelocationsrelativetothearraycentermicrophone. Photogrammetry .Accuratemicrophonelocationsarecrucialifreliablelevelsare soughtfromarraymeasurements,particularlyathighfrequencies.Forexample,at60 kHz,ifamicrophoneisunknowinglymisplacedjust2.9mmfromitsdesignlocation,then thedelayedmicrophonesignalwillbeahalf-wavelength,or180 ,outofphase,causing destructiveinterference.Atwo-stepprocedureutilizingphotogrammetryandapoint sourcemeasurementprovidesmicrophonecoordinatesinallthreedimensions.Thissection describesthephotogrammetrymethodusedtodeterminethe x and y coordinatesofeach microphonerelativetothecentermicrophone. WiththearrayinstalledintheUFAFF,aLaVisionImagerProX4M2112 2072 pxcameraismountedabovethearrayandleveledusingadigitalinclinometer.To photographtheinnerarrayandouterarrays,axed65mmlensandan18-50mm adjustablefocuslensareused,respectively.Twoimagesofeacharrayarecapturedusing 70

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DaVis7.4,thesecondofwhichincludesacalibrationtargetofknownphysicallength.The imageswithtargetsprovideareferenceforcalibratingthearrayimagesinDaVis. Afterimagecalibration,eachimageisloadedinMATLABforprocessing,asoutlined inFigure2-15.First,thephotographismagniedateachmicrophone,andthreepoints alongthecircumferenceofthemicrophonegridarechosen.Acircleisttothesethree points,andtheoriginofthecircleisassignedasthemicrophonelocation,asshownin Figure2-15B.Inspectionofthecircletyieldserrorsontheorderof1px,or0.075mm fortheinnerarrayand0.42mmfortheouterarray.Next,thecentermicrophonelocation issetastheorigin,andtheothermicrophonecoordinatesareadjustedaccordingly. Althoughthecenterofthearrayandthemicrophonecoordinatesrelativetotheorigin areknown,amethodisneededtoaccountforcamerarotation.Ahigh-accuracyreference markonthearrayisidealbutnotprovisionedonthecurrentarray.Instead,oneby one,theanglebetweeneachmeasuredmicrophoneandtheimage x axisiscomputed alongwiththeanglebetweentheeachmicrophone'sdesigncoordinatesandthearray x axis.Thedierencebetweentheseanglesprovidesarotationcorrectionbasedoneach microphone'sposition.Usingacoordinatetransformation,allmeasuredcoordinates aretransformedforeachrotationcorrectionangle,resultinginadistributionofeach microphone'smeasuredposition.Themeanofthedistributionprovidesanestimateof thetruemicrophonelocation,andthestandarddeviationofthedistributionisused tocompute95%condenceintervalsofeachmeanlocation.Thedesignandmeasured coordinatesfortheinnerarrayarecomparedinFigure2-15D. Asmentionedpreviously,thebiaserrorfortheinnerarraycoordinatesis0.075mm duetoerrorinthecirclet.Thebiaserrorfortheouterarraycoordinatesconsistsof thecircleterror.42mmandanadditionalimagedistortionerror.5mmfrom thelensusedtottheentireouterarrayintheeldofview.Hence,thetotalbiaserror fortheouterarraycoordinatesis0.92mm.Thisdistortionbiaserrorisdeterminedby 71

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comparingthepixellengthofcalibrationtargetsneartheimageedgeandimagecenter. Themeasured x y microphonecoordinateswithuncertaintiesarelistedinTable2-8. Laserpointsource .Alaser-generatedpointsourceisusedtodetermineeach microphone'sverticalpositionrelativetothereferencemicrophoneandcomputethe frequencyresponsebetweeneachmicrophoneandthecenterreferencemicrophone.The lasersetupissimilartothatusedbyBahr etal. 2010.ANewWaveResearchSolo 120XTNd:YaglaserisplacedabovetheUFAFFinletanddirectedtowardsthetunnel diuser.Thelaserisalignedwithaseriesoffourlensesthatexpandandrefocusthelaser toapoint,producingapoint-source-likeplasma.Thersttwolenses,2.54cmdiameter plano-concavelenses,areseparatedby2.54cm,producinganeectivefocallengthof -20mm,andcausethelasertospread.Thenaltwolenses,76.2cmdiameterdoublet achromaticlenses,havefocallengthsof250mmand500mm,respectively,andfocusthe lasertoapoint.SincetheNewWavelaserisdesignedforPIVapplications,itconsistsof apairoflasers.Bothlasersareredsimultaneouslytoincreasebeampower.Figure2-16 showsthelaserpulsesetupintheUFAFF. ArraycalibrationdataareacquiredusingfourchannelinputNIPXI-4462cards -bitresolution,118dBdynamicrange,andbuilt-inanti-aliasinglterinstalledin anNIPXI-1045chassis.Allmeasurementsareaccoupledat3.4Hz.Thelaserq-switch initiatesdataacquisitionatasamplingfrequencyof204,800Hzfor6400samples,and500 blocksofdataareacquired.Eachmicrophone'ssignalissplitintoblocksinMATLAB, andeachblockofthereferencemicrophoneischeckedforapulseduetosomevariation inthelasersetup,noteverylaserringproducesaplasma.Typically,around430ofthe 500blocksincludepulses.Thesatisfactoryblocksarethenensembleaveragedandgated toremovereectionsfromtheairfoil,laseroptics,andtunneltestsection,asillustrated inFigure2-17.Innerarraymicrophonescatteringeectsremain.Thespectrumforeach microphoneiscomputedbytakingtheFFTofitsgatedensembleaveragedtimeseries. 72

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Theprocessedlaserpulsedataarerstusedtodeterminemicrophonevertical displacementsrelativetothereferencemicrophone.Thetolerancesofthestereolithography arrayframearenotideal,andthereissignicantvariationinmicrophoneheight.Byrst assumingallmicrophoneheightsareuniform,thephotogrammetry-measured x y z =0 microphonecoordinatesareusedtoestimatethepropagationtimebetweenthesource andeachsensor.Thedierencesbetweenthecomputedpropagationtimesandthe propagationtimetothearraycentermicrophonedenesatimedelaythatisappliedto eachsignal.Thefrequencyresponseofeachdelayedmicrophonesignalrelativetothe centermicrophoneisthencalculated.Itisassumedthatthephaseresponseshouldbe atbelow10kHzandanyphaselag,representedbyaconstantphaseslope,iscaused bydeviationinmicrophoneheight.Theslopeofthephaseresponseisdetermined usingacurvet,andthen,sincethereferencemicrophoneandsourcelocationsare assumedknown,theheightofthemicrophoneinquestionisfound.Theheightsand uncertaintiesforallmicrophonesareprovidedinTable2-8.Arraypoint-spread-functions arere-computedusingthemeasuredmicrophonecoordinatesandplottedinFigure2-18. ComparedwithFigure2-13,therearenegligibledierencesbetweenthearrayresponses. The3dBbeamwidthsofeacharrayarecomputedbasedonthemeasuredmicrophone coordinatesandplottedversusfrequencyinFigure2-19. Aftertheheightsaredetermined,thelaserpulsesignalsareprocessedtodetermine therelativefrequencyresponsebetweeneachmicrophoneandthereferencecenter microphone.Themagnitudeofthecenterreferencemicrophonesignalisadjustedbasedon thenominalsensitivityforthemicrophoneprovidedbyB&Kat251.2Hz.Inaddition,its spectrumisadjustedtoaccountforthegridusingthefree-eld,zero-incidence,with-grid corrected"responseprovidedbyB&K.Whilethisisnotaperfectprocess,sinceinner arraymicrophonescatteringeectsstillremain,itisdeterminedtobethebestoptionfor calibrationwithoutdisturbingthemicrophones.TheFFTsoftheremainingmicrophone signalsarecomputedandadjustedbasedontheirmeasuredlocations x y z relativeto 73

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thecentermicrophonetoremovetheeectofsphericalspreading.Thefrequencyresponses betweeneachmicrophoneandthecentermicrophonearethencalculatedandappliedwhen thecross-spectralmatrixisassembled. 2.6.2.4Uncertainty TheuncertaintyanalysisdescribedbyYardibi etal. 2010 a isappliedtothe integratedarrayspectrum.Specically,aMonte-Carlosimulationisperformedwith perturbationsappliedtothemicrophonelocations,temperature,andindividualmicrophone frequencyresponses. 2.6.3DataAcquisition Allacousticdataareacquiredat204,800Hzfor30secondsusingtheNIPXIsystem describedinSection2.6.2.3.Duringprocessing,thedatafromeachchannelareseparated into480blocksof12,800sampleseach,resultingina16Hzbinwidth.Thedataarethen processedusingaHanningwindowand75%overlap,producing996eectiveaveragesand anormalizedautospectralrandomuncertaintyof3.2%. 74

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A B Figure2-1.Circulationcontrolairfoilgeometry.AOveralldimensions.BTrailingedge geometry l isthelipthickness,and t istheellipsethickness. 75

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Figure2-2.Zig-zagturbulatortriptape. Figure2-3.Photographofplenumtreatment. Figure2-4.Airdeliverysystem. 76

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A B Figure2-5.CirculationcontrolairfoilinstalledintheUFAFF.A2009.B2010. 77

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Figure2-6.PIVtrailingedgemeasurementregions:atrailingedgewake,bowover lip,ccurvedwalljet,anddcurvedwalljetseparation.Airfoilgeometry sketchedforreference. Figure2-7.Vectorcomputationsteps. Figure2-8.Coordinatesystemsandnomenclatureattrailingedge. 78

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Figure2-9.Microphoneexperimentalsetup. Figure2-10.Phasedarrayexperimentalsetup. 79

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Figure2-11.Scanningregionof l =1 ;:::;L points. A B Figure2-12.Equal-aperturelog-spiralarraydesignsUnderbrink1995,2001,2002. Microphonesaredrawntoscale.AOuterarray.BInnerarray. 80

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A B C D E F G Figure2-13.TheoreticalarrayresponsenormalizeddBforapointsource1.12mabove thearraycentercomputedusingdesignmicrophonecoordinates.The scanningregionrepresentsthesamescanningregionusedtobeamform measureddata,where c istheairfoilchord.AOuterarray,1kHz.BOuter array,2kHz.COuterarray,4kHz.DOuterarray,8kHz.EInnerarray, 16kHz.FInnerarray,32kHz.GInnerarray,64kHz. 81

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Figure2-14.Picturesofthephasedarray.AInstalledintheUFAFF,lookingtowardsthe inlet.BMagniedviewofthearray.CAcoustictreatmentremovedto revealthearrayframe. 82

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A B C D Figure2-15.Stepsfordeterminingmicrophonelocationsusingphotogrammetry.AInitial photographofinnerarray.BExampleofcirclettomagniedmicrophone. CAllmicrophonelocationsevaluated.DComparisonofnalmeasured redanddesignblackmicrophonecoordinates. 83

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Figure2-16.PictureoflaserpulsearraycalibrationtestsetupinUFAFF. Figure2-17.Averagedlaserpulsetimeseriesofarraycentermicrophonewithreections labeled. 84

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A B C D E F G Figure2-18.TheoreticalarrayresponsenormalizeddBforapointsource1.12mabove thearraycentercomputedusingmeasuredmicrophonecoordinates.The scanningregionrepresentsthesamescanningregionusedtobeamform measureddata,where c istheairfoilchord.AOuterarray,1kHz.BOuter array,2kHz.COuterarray,4kHz.DOuterarray,8kHz.EInnerarray, 16kHz.FInnerarray,32kHz.GInnerarray,64kHz. 85

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A B Figure2-19.Plotsof3dBbeamwidthforeacharray.AOuterarray.BInnerarray. 86

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Table2-1.Normalizedupper/lowersurfacepressuretapchordwisecoordinates x/c 0 0.0049 0.0122 0.0195 0.0244 0.0293 0.0341 0.0390 0.0454 0.0512 0.0585 0.0683 0.0780 0.0888 0.0976 0.1073 0.2146 0.5093 0.7500 0.9698 0.9817 0.9910 0.9976 1 Table2-2.SummaryofPIVlaseropticsadj.referstoadjustablefocallengthwith maximumlisted Region SphericalLensmmCylindricalLensmm leadingedge 2000adj.-10 owoverlip 2000adj.-10 curvedwalljet 1500-15 trailingedgewake 1500-15 crossow 2000adj.-20 87

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Table2-3.SummaryofPIVimageacquisitionsetup,includingcameraLaVisionImager ProX4MorTSIPowerViewPlus630157,cameraoptics,andowseeders Region CameraLensmmTeleconverterFreestreamSeederJetSeeder leadingedge LaVision105noneTSI9307-6-owoverlip LaVision200noneTSI9307-6-curvedwalljet TSI1051.4,2.0TSI9307-6TSI9302 trailingedgewake TSI60noneLeMautreCLF-4500TSI9302 crossow LaVision2001.4,2.0PeaSoupPhantomTSI9307-6 Table2-4.Summaryofowseedersandparticlediameterestimates Seeder ParticleDiameter m LeMautreCLF-4500 2-10 PeaSoupPhantom 0.2-0.3 TSI9307-6 1 TSI9302 1 Table2-5.SummaryofPIVprocessingparameters,includinginterrogationwindowsizes forrsttwoandnaltwopasses,windowoverlap,nalspatialresolution,and useofoutlierrejection InitialWindowFinalWindowWindowResolutionOutlier Region SizepxSizepxOverlapmm/vecRejection leadingedge 64 6432 3250%0.95yes owoverlip 32 3216 1650%0.076yes curvedwalljet 32 3216 1650%0.13yes trailingedgewake 32 32--50%1.4yes crossow 64 6464 6475%0.081no Table2-6.Arrayspecications Specication InnerArrayOuterArray SensorCount 2540 Aperturecm 7.6273.7 InnerRadiuscm 0.8383.81 SpiralElements 58 CircleElements 55 SpiralAngle 7575 88

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Table2-7.Phasedarraymicrophonedesigncoordinatesandsensordetails ArrayChannel ArrayMicrophoneSerialNumber x design cm y design cm center {B&K4954B27162680.0000.000 1 innerB&K4954B27162630.8380.000 2 innerB&K4954B27125700.2590.797 3 innerB&K4954B2701736-0.6780.493 4 innerB&K4954B2716266-0.678-0.493 5 innerB&K4954B27017340.259-0.797 6 innerB&K4954B2716269-0.6251.297 7 innerB&K4954B2701735-1.427-0.193 8 innerB&K4954B2716267-0.257-1.417 9 innerB&K4954B27162641.268-0.682 10 innerB&K4954B27162651.0410.995 11 innerG.R.A.S.40BEM105283-1.495-1.997 12 innerG.R.A.S.40BEM1043041.437-2.039 13 innerG.R.A.S.40BEM794432.3830.737 14 innerG.R.A.S.40BEM794340.0362.494 15 innerG.R.A.S.40BEM104265-2.3610.805 16 innerG.R.A.S.40BEM794330.984-3.066 17 innerG.R.A.S.40BEM1042533.220-0.012 18 innerG.R.A.S.40BEM1053481.0063.059 19 innerG.R.A.S.40BEM104254-2.5981.902 20 innerG.R.A.S.40BEM105355-2.612-1.883 21 sharedG.R.A.S.40BEM794403.073-2.252 22 sharedG.R.A.S.40BEM1053083.0912.227 23 sharedG.R.A.S.40BEM79469-1.1633.628 24 sharedG.R.A.S.40BEM105281-3.8100.015 25 sharedG.R.A.S.40BEM79454-1.192-3.619 26 outerB&K49582716709-8.767-5.243 27 outerB&K495827167052.277-9.958 28 outerB&K4958271670610.174-0.911 29 outerB&K495827167084.0119.395 30 outerB&K49582716707-7.6956.717 31 outerB&K4958271672815.060-9.286 32 outerB&K4958271674213.48611.453 33 outerB&K49582716743-6.72516.365 34 outerB&K49582716744-17.642-1.339 35 outerB&K49582716745-4.178-17.192 36 outerB&K4958271674621.0328.908 37 outerB&K49582716747-1.97322.756 38 outerB&K49582716751-22.2525.155 Continuedonnextpage 89

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Table2-7{continuedfrompreviouspage ArrayChannel ArrayMicrophoneSerialNumber x design cm y design cm 39 outerB&K49582716752-11.779-19.569 40 outerB&K4958271675414.972-17.250 41 outerB&K4958271675513.94823.149 42 outerB&K49582716734-17.70520.418 43 outerB&K49582716735-24.890-10.529 44 outerB&K495827167362.322-26.926 45 outerB&K4958271673726.326-6.112 46 outerB&K495827167392.24530.562 47 outerB&K49582716741-28.37311.579 48 outerB&K49582716711-19.780-23.406 49 outerB&K4958271671216.148-26.045 50 outerB&K4958271671329.7607.309 51 outerB&K49582716714-10.04932.354 52 outerB&K49582716715-33.8760.441 53 outerB&K49582716716-10.888-32.081 54 outerB&K4958271671727.147-20.269 55 outerB&K4958271671827.66619.555 56 outerB&K49582716719-21.18530.127 57 outerB&K49582716720-35.199-10.838 58 outerB&K49582716721-0.569-36.826 59 outerB&K4958271672234.847-11.921 60 outerB&K4958271672322.10629.458 Table2-8.Measuredarraymicrophonecoordinateswithtotaluncertaintyallunitscm ArrayChannel Array x meas y meas z meas x u y u z u 1 inner0.8980.034-0.0110.0080.0150.160 2 inner0.2360.880-0.0300.0150.0080.160 3 inner-0.7200.465-0.0380.0100.0130.160 4 inner-0.663-0.5960.0000.0110.0120.160 5 inner0.298-0.8160.0400.0140.0090.160 6 inner-0.7091.424-0.1260.0220.0120.160 7 inner-1.478-0.173-0.0550.0080.0230.160 8 inner-0.303-1.399-0.0080.0220.0090.160 9 inner1.240-0.715-0.0370.0130.0190.160 10 inner1.0941.091-0.0680.0170.0180.150 11 inner-1.436-1.858-0.0540.0280.0230.160 12 inner1.322-1.977-0.0320.0300.0200.160 13 inner2.3020.736-0.0470.0130.0350.160 14 inner0.0212.472-0.1040.0370.0080.160 Continuedonnextpage 90

PAGE 91

Table2-8{continuedfrompreviouspage ArrayChannel Array x meas y meas z meas x u y u z u 15 inner-2.2780.852-0.1450.0150.0340.160 16 inner0.835-2.849-0.0200.0430.0140.160 17 inner3.141-0.03670.0600.0080.0460.160 18 inner0.91522.9045-0.0790.0430.0160.160 19 inner-2.3991.924-0.1250.0300.0350.160 20 inner-2.471-1.744-0.1180.0260.0370.160 21 shared2.881-2.1260.0280.0330.0420.160 22 shared3.0382.0530.0460.0300.0460.160 23 shared-1.0233.427-0.1000.0510.0160.160 24 shared-3.5070.152-0.0690.0080.0520.160 25 shared-1.192-3.4340.0060.0500.0200.161 26 outer-9.192-5.576-0.2450.0930.0940.162 27 outer2.422-10.3920.0980.0940.0920.163 28 outer10.725-1.067-0.0940.0920.0940.161 29 outer4.2769.643-0.0800.0940.0920.160 30 outer-8.0716.895-0.3430.0930.0930.160 31 outer15.602-9.407-0.2250.0940.0970.164 32 outer13.91311.6180.0450.0950.0960.163 33 outer-6.90516.646-0.3800.0970.0930.160 34 outer-18.075-1.427-0.6620.0920.0980.164 35 outer-4.297-17.609-0.1770.0980.0920.166 36 outer21.6708.898-0.0550.0940.1010.163 37 outer-1.94323.301-0.2020.1020.0920.161 38 outer-22.5745.146-0.8140.0930.1020.164 39 outer-12.063-19.884-0.4910.1000.0950.169 40 outer15.445-17.498-0.0500.0980.0970.168 41 outer14.34923.543-0.0680.1020.0960.162 42 outer-17.90120.656-0.6270.1000.0980.163 43 outer-25.146-10.751-0.9780.0940.1040.170 44 outer2.358-27.4200.0030.1060.0920.172 45 outer26.813-6.259-0.2830.0930.1050.168 46 outer2.34631.054-0.2300.1090.0920.162 47 outer-28.65711.635-0.6940.0950.1070.167 48 outer-19.987-23.511-0.9560.1020.1000.173 49 outer16.528-26.3930.0610.1050.0970.173 50 outer30.2897.254-0.1860.0930.1090.167 51 outer-10.21732.848-0.3280.1110.0940.164 52 outer-34.1260.3101-0.8780.0920.1130.172 53 outer-10.860-32.339-0.3200.1110.0940.177 54 outer27.550-20.348-0.1720.1000.1060.173 Continuedonnextpage 91

PAGE 92

Table2-8{continuedfrompreviouspage ArrayChannel Array x meas y meas z meas x u y u z u 55 outer28.19319.7850.1960.0990.1070.166 56 outer-21.37530.472-0.6290.1090.1010.166 57 outer-35.299-10.973-1.0060.0940.1140.175 58 outer-0.506-37.079-0.2770.1160.0920.180 59 outer35.231-12.018-0.2720.0950.1140.174 60 outer22.51529.7180.0560.1080.1020.166 92

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CHAPTER3 FLUIDDYNAMICS Eventhoughtheprimarymotivationforthisresearchisacoustics,theuiddynamics oftheproblemareequallyimportant.Knowledgeoftheowbehavioriscrucialto understandingthenatureoftheowandtheoreticalnoisesources,likethoseoutlinedby Howe2002.Surfacepressuremeasurementsareusedtoprovideinsightintotheow aroundtheairfoil.PIVisusedtomeasuredetailsoftheoweld,includingleadingedge stagnationpointmovementandstreamlinecurvature,theinuenceoftheblowingsloton thetrailingedgewake,characteristicsoftheturbulentboundarylayerpassingoverthe slotlip,andcurvedwalljetdevelopmentandseparation.Beforetheoweldisstudied extensively,thecirculationcontrolairfoilischaracterized. 3.1AirfoilCharacterization Spanwiseslotjetowuniformityandslotlipdeectionareinvestigatedatthe nominalslotheight, h=c =0.0019.Meanspanwiseowuniformityisassessedusing constanttemperaturehot-wireanemometry.Meanslotowisfoundtovaryby1.6% when U span =41m/sand2.7%when U span =87m/s.Figure3-1showslipdisplacement asafunctionofabsoluteplenumpressure.Slotheightisinitiallysetat1.0mm h=c =0.0019usingplasticshimstock.Thehighestplenumpressuretestedcorrespondsto approximatelythemaximumslotjetvelocitytested, U jet =100m/s.Thelipisfoundto deectbylessthan0.1mm,or8.8%,atthehighestplenumpressure. 3.2FreestreamFlowCharacterization Thefreestreamoweldischaracterizedusingacombinationofstaticsurface pressureandPIVmeasurements.Theresultsfromthesetestsarepresentedinthissection. 3.2.1SurfacePressureMeasurements StaticsurfacepressuredistributionsareplottedinFigure3-2forthreedierent momentumcoecients, C =0,0.015,and0.057at Re c =6 : 5 10 5 and h=c =0 : 0019. Forclarity,uncertaintyboundsarenotincluded,astheyaretypicallysmallerthan 93

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thedatumpointsize.Theeectofblowingonthe C p distributionisapparentbythe largetrailingedgesuctionpeakwhosemagnitudeincreasessubstantiallywithlarger valuesof C .Aleadingedgesuctionpeaktypicallyobservedincirculationcontrol airfoilexperiments,includingthoseofAbramson1975,isabsent.Furthermore,thelift producedbytheairfoilissubstantiallylowerthantheliftproducedbyAbramson'ssimilar ellipticcirculationcontrolairfoil.Liftcoecientsforavarietyofmomentumcoecients arecomparedinFigure3-3.Itislikelythattheabsenceoftheleadingedgesuctionpeak isthesourceoftheliftdeciencyobservedinFigure3-3. 3.2.2TestSectionInuence Todetermineiftheairfoillocationinthetestsectionisresponsibleforthemissing leadingedgesuctionpeak,theairfoilpositionisvariedby13cmvertically.25 c and38 cm.73 c inthestreamwisedirection.However,the C p distributionsremainunchanged. SincethetestsperformedbyAbramson1975tookplaceinaclosedwindtunnel,the absenceoftheleadingedgesuctionpeakcouldbeduetotheUFAFFopenjettestsection. Toinvestigatethisphenomenonfurther,thewindtunneltestsectionisenclosedwithfoam walls,asshowninFigure3-4. C p datameasuredintheclosedcongurationarecorrectedtoaccountforsolid blockagePope&Rae1984.Specically, C p C ,and Re c arecorrectedusing C p;c = C p +2 t {1 C ;c = C +2 t {2 Re c;c = Re c + t {3 where t =0.0357isone-quarteroftheratiobetweenthemodelfrontalareaandtest sectioncross-sectionalarea.Onceagain,forclarity,uncertaintyboundsarenotincluded, astheyaretypicallysmallerthanthedatumpointsize.Figure3-5comparesmidspan C p distributionsmeasuredinbothopenandclosedtestsectionsfor C =0.Both C p 94

PAGE 95

distributionshaveasimilarshape,buttheclosedtestsection C p distributionisnoticeably osetfromitsopentestsectioncounterpart.Notably,theclosedtestsection C p dataare morenegative,indicativeoftheincreasedowspeedcausedbytheadditionofaoorand ceilingandtheircontainmentofthetunneljet. Figure3-6comparesopenandclosedtestsection C p distributionsfor C =0.057. Both C p distributionshavesimilarprominenttrailingedgesuctionpeaksassociatedwith thetrailingedgejet.However,attheleadingedge,the C p distributionsaredrastically dierent.Theleadingedgesuctionpeakabsentintheopentestsectionconguration appearswhenthetestsectionisfullyenclosed.Liftcoecientsfortheclosedtestsection andopentestsectioncasesarecomparedwithdatafromAbramson1975inFigure 3-7.TheclosedtestsectionliftcurveisincloseagreementwithAbramson'sliftvalues Abramsonappliedthesameblockagecorrectionsusedinthepresentanalysis.Therefore, thewindtunnelopenjettestsectionisresponsibleforthedisappearanceoftheleading edgesuctionpeak. Sincetheprimarymotivationforthisinvestigationisacoustics,itisconcerningthat the C p distributionsmeasuredwithanopenandclosedtestsectiondiersignicantly. Themid-frequencycirculationcontrolnoisesourcetheorizedbyHowe2002,passive-slot noise,isproducedbyfreestreamboundarylayerturbulencescatteringotheslotlip.If thecharacteristicsoftheturbulentboundarylayerneartheslotarehighly-dependenton theowupstreamoftheslot,thensotooisthesoundproduced.PIVisusedtofurther investigatetheinuenceofthetestsectionontheoweld,includingthefreestream boundarylayerpassingoverthelip. 3.2.3ClosedTestSectionBehavior Theleadingedgeoweldandtheboundarylayerowpassingovertheslotlip aremeasuredusingPIV.PleaserefertoSection2.5.3forinformationregardingthe experimentalsetup.Inordertoimagetheoweld,onefoamsidewallisreplacedwitha clearpolycarbonatepanel.ThePIVcamerasaremountedbehindthispaneloutsidethe 95

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ow.Whenthetestsectionisfullyenclosed,thefoamceilingisalsoreplacedwithaclear polycarbonatepaneltopermitthelasersheettopassthroughandilluminatetheow. However,thetunneloorremainsfoam.ApictureofthetestsectionenclosedforPIV measurementsisprovidedinFigure3-8. Becausethelasersheetcanonlyilluminatetheowabovetheleadingedgeashadow regioniscreatedbelowtheleadingedge,thetrailingedgejetisemittedfromthelower blowingslotsothatanystagnationpointmovementoccursontheuppersurfaceofthe leadingedge.Theowpassingovertheslotlipisofinterestaswell,sotestsarerepeated withupperslotblowing,sincethelightsheetalsoilluminatestheowabovetheupper surfacelip.Thus,sincethePIVexperimentsutilizebothupperandlowerslotblowing,it isimportanttocompare C p distributionsforboth. Figure3-9compares C p distributionsforupperandlowerslotblowingintheclosed testsectioncongurations.Surprisingly,therearesubstantialdierencesinthe C p distributionsforupperandlowerslotblowingwhenthetestsectionisenclosedforPIV testing.Withupperslotblowingblowingonthesamesideasthepolycarbonateceiling, the C p valuesaremorenegative.Figure3-9alsorevealsthatthe C p dataforlowerslot blowinginthePIVtestcongurationblowingonthesamesideasthefoamooragree withtheupperslotblowing C p dataobtainedwhenthetestsectionisfullyenclosedwith foam.Lowerslotblowingexperimentsarenotperformedwiththeall-foamclosedtest section. Toruleouttheblowingslotsasthecauseforthesedierences, C p distributionsfor upperandlowerslotblowingintheopentestsectionarecomparedinFigure3-10for similartestconditions.Theslightdierencesbetweenthetwodistributionsareminor, anditcanbeconcludedthatthevariationinthe C p distributionsinFigure3-9isnotthe resultofdissimilarblowingbetweentheupperandlowerslots.Instead,itappearsthatthe boundaryconditionsimposedbythetunnelsurfaceonthesamesideastheblowingslot 96

PAGE 97

haveaconsiderableinuenceontheoweld.Apotentialowanalysisisperformedto provideinsightintothisobservedowbehavior. 3.2.4PotentialFlowAnalysis Afulldescriptionofthepotentialowanalysis,includingMATLABcodes,isprovided inAppendixB.Theapproachisoutlinedinthissection.Potentialliftingowovera cylinderinafreestreamcanbetransformedtoowoveranellipseviaconformalmapping Panton2005;Katz&Plotkin2001.Considerthe z = x + iy planewhoseoriginislocated atthecenterofacylinderofradius R .The plane,whoseoriginislocatedatthecenter oftheellipse,isdenedby = z + a 2 z ; {4 where a isthetransformationconstantwrittenintermsofthethecylinderradius R ,the ellipsesemi-majoraxis A ,andtheellipsesemi-minoraxis B R = 1 2 A + B {5 a = r 1 2 R A )]TJ/F22 11.9552 Tf 11.955 0 Td [(B {6 Thecomplexpotentialforliftingowaroundacylinderofradius R isthesummationof thepotentialsforauniformfreestream,adoublet,andavortex. F z = U 1 z + R 2 z + i )-167(ln z 2 {7 TakingthederivativeofEquation3{7yieldsthecomplexvelocity. W z = dF dz = U 1 1 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(R 2 z 2 + i )]TJETq1 0 0 1 378.777 205.907 cm[]0 d 0 J 0.478 w 0 0 m 18.893 0 l SQBT/F15 11.9552 Tf 378.777 194.717 Td [(2 z {8 Todeterminethesurfacepressureonanellipsewithcirculationinafreestream,the complexvelocity W z mustbetransformedto W usingtheconformalmapping functiongivenbyEquation3{4.Thus,itfollowsthat, W = dF d = dF dz dz d = dF dz 1 d dz ; {9 97

PAGE 98

andthecomplexvelocityontheellipsesurface,where z = Re i ,isgivenby W = U 1 )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + i )]TJETq1 0 0 1 352.828 684.017 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 352.828 676.906 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.115 4.707 Td [(a 2 R 2 : {10 Since W = u )]TJ/F22 11.9552 Tf 12.808 0 Td [(iv ,themagnitudeofthevelocityonthesurfaceandhencethe C p distributionalongthesurfaceoftheellipsecanbecomputed. C p =1 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( j V s j U 1 2 =1 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(u 2 + v 2 U 2 1 {11 Theaforementionedanalysiscanbeextendedtoincluderigidboundaries,likeground andceilingplanes,usingthemethodofimagesKatz&Plotkin2001.Considerpotential liftingowaroundacircularcylinderplacedadistance h c fromaceilingplane.An image"cylinderwithcirculationofopposite-sensefromtherealcylinderisplaced"a distance h c fromtheoppositesideoftheplane.Theoriginoftherealcylindercoincides withtheoriginofthe z 1 = x 1 + iy 1 plane,andtheoriginoftheimagecylindercoincides withtheoriginofthe z 2 = x 2 + iy 2 plane,whichisosetfromthe z 1 planesuchthat z 2 = z 1 )]TJ/F22 11.9552 Tf 10.739 0 Td [(i 2 h c .Themappingtransformationfortherealcylinderis 1 = z 1 + a 2 =z 1 ,andthe mappingtransformationfortheimagecylinderis 2 = z 2 + a 2 =z 2 .Thecomplexpotentials fortherealandimaginarycylindersaregivenbythefollowingequations. F 1 z 1 = U 1 z 1 + R 2 z 1 + i )]TJETq1 0 0 1 357.073 292.64 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 357.073 281.451 Td [(2 ln z 1 {12 F 2 z 2 = U 1 R 2 z 2 )]TJ/F22 11.9552 Tf 13.958 8.088 Td [(i )]TJETq1 0 0 1 335.08 259.735 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 335.08 248.546 Td [(2 ln z 2 {13 Anexpressionisdesiredforthecomplexvelocityoftheowaroundbothellipsesasa functionof 1 .Fortherealellipse,thisisstraightforwardandfollowsthestepsoutlinedin thefreestreamowanalysis. W 1 1 = dF 1 dz 1 dz 1 d 1 = U 1 1 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(R 2 z 2 1 + i )]TJETq1 0 0 1 373.599 140.194 cm[]0 d 0 J 0.478 w 0 0 m 23.092 0 l SQBT/F15 11.9552 Tf 373.599 129.004 Td [(2 z 1 1 1 )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(a 2 z 2 1 {14 98

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Thecomplexvelocityfortheimageellipseis W 2 1 = dF 2 dz 2 dz 2 d 2 d 2 d 1 : {15 Thedistancebetweenthecylindersandthedistancebetweentheellipsesarenotthesame, since h e = h c + a 2 h c : {16 Therefore, 2 = 1 )]TJ/F15 11.9552 Tf 12.325 0 Td [(2 ih e d 2 =d 1 =1,andthecomplexvelocityfortheimageellipseis givenby W 2 1 = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( U 1 R 2 z 2 2 + i )]TJETq1 0 0 1 329.834 513.293 cm[]0 d 0 J 0.478 w 0 0 m 23.092 0 l SQBT/F15 11.9552 Tf 329.834 502.103 Td [(2 z 2 1 1 )]TJ/F23 7.9701 Tf 13.15 4.707 Td [(a 2 z 2 2 : {17 Substituting z 2 = z 1 )]TJ/F15 11.9552 Tf 12.048 0 Td [(2 ih c andthenwritingbothcomplexvelocitiesintermsof z 1 = Re i yieldsthefollowingcomplexvelocities. W 1 1 = U 1 )]TJ/F22 11.9552 Tf 5.479 -9.684 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + i )]TJETq1 0 0 1 358.159 428.643 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 358.159 421.531 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.115 4.708 Td [(a 2 R 2 {18 W 2 1 = )]TJ/F15 11.9552 Tf 44.196 8.088 Td [(1 1 )]TJ/F23 7.9701 Tf 34.027 4.707 Td [(a 2 Re i )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 ih c 2 U 1 R 2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ih c 2 + i )]TJETq1 0 0 1 379.35 384.612 cm[]0 d 0 J 0.478 w 0 0 m 79.391 0 l SQBT/F15 11.9552 Tf 379.35 373.423 Td [(2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ih c {19 Finally,thecomplexvelocityonthesurfaceoftherealellipseisjustthesuperpositionof theimageandrealellipsecomplexvelocities, W 1 = W 1 1 + W 2 1 : {20 C p isagaincomputedusingequation3{11. Ifagroundplaneispresentinsteadofaceilingplane,thentheonlychangetothe previousanalysisisthat z 2 = z 1 +2 ih c and 2 = 1 +2 ih e .Thepresenceofbothground andceilingplanesisaccountedforbyaninnitenumberofimageellipses,sinceeach imageitselfisreectedbytheadditionalplaneKatz&Plotkin2001.Thus,thecomplex velocityonthesurfaceoftherealellipsebetweentwoplanesisgivenbythefollowing 99

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summation. W 1 = U 1 )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 + i )]TJETq1 0 0 1 243.824 684.017 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 243.824 676.906 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.116 4.707 Td [(a 2 R 2 )]TJ/F25 7.9701 Tf 16.355 14.944 Td [(1 X n =1 1 1 )]TJ/F23 7.9701 Tf 36.596 4.707 Td [(a 2 Re i )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 nih c 2 U 1 R 2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 nih c 2 + )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 n +1 i )]TJETq1 0 0 1 374.569 637.978 cm[]0 d 0 J 0.478 w 0 0 m 86.378 0 l SQBT/F15 11.9552 Tf 374.569 626.789 Td [(2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 nih c )]TJ/F25 7.9701 Tf 16.355 14.944 Td [(1 X n =1 1 1 )]TJ/F23 7.9701 Tf 36.596 4.707 Td [(a 2 Re i +2 nih c 2 U 1 R 2 Re i +2 nih c 2 + )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 n +1 i )]TJETq1 0 0 1 374.375 598.767 cm[]0 d 0 J 0.478 w 0 0 m 86.184 0 l SQBT/F15 11.9552 Tf 374.375 587.578 Td [(2 Re i +2 nih c {21 Formoderatecirculation,theliftcoecientcomputedfromtheresultsofEquation3{21 convergestowithinthreesignicantdigitswithwellunder500imageellipsesconsidered. However,thisresultislargelydependentonthespeciedcirculation,asconvergenceis establishedwithasfewas20imagesforcertainliftvalues. C p distributionsforpotentialliftingowarounda20%ellipseinavarietyofow congurationsarecomparedinFigure3-11forafreestream c l =1 : 90.Specically,the impactofagroundplane,ceilingplane,andbothgroundandceilingplanesisassessed. Thegroundandceilingplanesareplacedatthesamedistancefromtheellipse,whose chordisidenticaltotheairfoilusedinexperiments,astheoorandceilinginstalledinthe UFAFF.Figure3-11revealsafewinterestingdetails.First,theadditionofaceilingplane drasticallyaugmentsthelift,whetherornotagroundplaneispresent.Furthermore,the additionofagroundplanealoneactuallyreducestheliftcomparedtothefreestreamcase. Thepotentialowanalysisonlyassumesimpermeable,rigidboundaries,whereas inactualtesting,acombinationofimpermeablepolycarbonateandporousfoam boundariesareused.Forthisreason,whilethepotentialowresultscannotbeexpected tomatchexperimentsexactly,thetrendsrevealedbyFigure3-11canstillbeappliedto themeasuredobservations.Forexample,Figure3-11indicatesthatthepresenceofan impermeableceilingplaneproducesthegreatestsuction-sideandlowestpressure-side C p magnitudes,regardlessofwhetheragroundplaneispresent,absent,or,hence, impermeableorporous.ThisagreeswiththemeasureddatapresentedinFigure3-9, wherethegreatestsuction-sideandlowestpressure-side C p magnitudesareobservedwhen 100

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blowingisutilizedonthesamesideasthepolycarbonatetunnelceiling.Additionally, Figure3-9showslittlevariationinthemeasured C p distributionswhenblowingisutilized onthesamesideastheporousfoamceilingandthepressure-sideboundaryiseither polycarbonateorfoam.ThisagreeswiththetheoryinFigure3-11,whichindicatesthat thepresenceofagroundplanehasaminimalinuenceonthe C p distribution. Tofurtherillustratethesetrends,thepotentialowsolutionsfortheellipsenear agroundplaneandceilingplanealonearecomparedwithmeasurements.Sincethe potentialowsolutionsdonotaccountforthepresenceofthetrailingedgejetandits contributiontothetrailingedgesuctionpeak,measuredandcomputed C p distributions cannotbecomparedusingequivalentliftcoecients.Instead,thetheoreticaland measureddistributionsarematchedattheleadingedgestagnationpoints.Themeasured closedtestsection C p distributionsforblowingonthesamesideasandoppositesidefrom thepolycarbonatetunnelceilingarecomparedtotheoryforowwithaceilingplaneand groundplane,respectively,inFigure3-12.Themeasuredleadingedgestagnationpoints arefoundtomatchwiththeoryforbothdatasetswhenthefreestreamliftcoecientis 1.90.Attheleadingedge,thetheoryagreeswellwiththemeasurements,andthetrends revealedbyFigure3-11areclearlyfollowedbythemeasureddata. Furtherpotentialowanalysisindicatesthedierencesbetweenthecomputedclosed testsectionandfreestreamsurfacepressuredistributionsareinsignicantwhenthetunnel blockageratio,computedfromtheratiooftheairfoilthicknesstotunnelheight,isless than6%.Bycomparison,thetunnelblockageratioforthepresentinvestigationis14%, andtheblockageratiosforstudiesbyAbramson1975andNovak&Cornelius1986 were8%and7%,respectively. 3.2.5PIVResults Althoughtheoryandmeasurementsagreethattherearelargeoveralldierences inthe C p distributionsdependingonthesuction-sideboundarycondition,ingeneral, enclosingthetestsectionclearlyproducestheleadingedgesuctionpeakabsentinthe 101

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opentestsectionmeasurements.Theinuenceofenclosingthetestsectiononthe leadingedgeowiscapturedbyPIV.Meanowstreamlinesforbothopenandclosed testsectioncongurationsatavarietyoflowerblowingslotmomentumcoecientsare showninFigure3-13.Intheopentestsectionconguration,leadingedgestagnation pointmovementandstreamlinecurvatureareminimalas C increases.Conversely,when thetestsectionisenclosed,theleadingedgestagnationpointmovesconsiderablywith increasing C ,andstreamlinecurvatureisgreatlyenhanced. AsdiscussedinChapter1,theprimarymotivationforthisresearchinvestigationis acoustics.Ithasbeenrevealedthat,upstreamoftheslot,the C p distributionsmeasured withthetestsectionopenedandcloseddiersignicantly.Passive-slotnoiseistheorized byHowe2002tobeproducedbyturbulenceinthefreestreamboundarylayerscattering otheslotlip.Ifthecharacteristicsoftheturbulentboundarylayerneartheslotvary withtestsectionconguration,thensotoodoesthesoundproduced.Theturbulent boundarylayerpassingovertheslotlipismeasuredusingPIVinbothopenandclosed tunnelcongurationsforavarietyofupperslotblowingmomentumcoecients.The resultsarepresentedinFigures3-14,3-15,and3-16.Prolesofmeantangentialand normalvelocities,turbulenceintensities,andReynoldsstressareplotted.Forclarity, onlyeveryfourthdatumpointisdisplayed,anduncertaintyboundsarerepresentedby solidlines.WithoutblowingFigure3-14,theprolesarenormalizedusingthelocal freestreamvelocityandtheboundarylayerthickness.WithblowingFigures3-15and 3-16,themeantangentialvelocityprolesappearmorelikewalljetproles,andthe localmaximumvelocityandtheboundarylayerthicknessbasedonwherethevelocityis 0.99 U max areusedtonormalizethedata.Forallcases,thereareminordierencesbetween theproles,andthesedeviationsaretypicallywithinthemeasurementuncertainty.From anacousticsperspective,thedierencesbetweentheprolesshouldbenegligible,andthe soundproducedbytheinteractionoftheturbulentboundaryowwiththeslotlipshould bethesamewhetherornotthetestsectionisopenedorclosed. 102

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3.3CurvedWallJetFlow Theprimarymechanismexploitedbycirculationcontrolisthecurvedwalljet, whichisexpectedtoplayanimportantroleintheproductionofcirculationcontrol noise.CurvaturenoiseistheorizedbyHowe2002tobeproducedbytheinteraction ofboundarylayerturbulencewiththeroundedtrailingedge,andthescalesrequiredto assessHowe'smodelofthisnoise-thedisplacementthickness,meanvelocity,andfriction velocity-areclearlygoingtobedictatedbytheinteractionofthecurvedwalljetwith thefreestream.Thus,thereisaneedforhigh-resolutionowmeasurementsofthecurved walljettodeterminethesescales.Ofgreaterimportanceandinterest,however,isthe potentialforowsimilarity.Ifthecirculationcontrolwalljetexhibitssimilarity,thenit maybepossibletopredicttheevolutionoftheowanditsvariouslengthandvelocity scales.SuchandingwouldhavedirectapplicationtoHowe'smodelandalsothedesign ofcirculationcontrolairfoilsforavarietyofapplications. Unfortunately,theliteratureprovidesevidencethatfullowsimilaritymayonlybe possibleforveryspecicgeometries.Guitton&Newman1977presentedasimilarity solutionforacurvedwalljetintheabsenceofafreestreamandconcludedthatfullow similarityisonlypossibleifthesurfaceisdenedbyalogarithmicspiral,e.g. r / e R=x where R isthelocalradiusofcurvature,and x isthearclengthfromtheorigin.A logarithmicspiralisillustratedinFigure3-17.Theiranalysisisextendedinthenext sectionforthecurvedwalljetwithanexternalfreestream. 3.3.1GeneralSimilaritySolution Usingtheturbulentformsofthegoverningequationsinpolarcoordinates,Guitton& Newman1977developedasimilaritysolutionforacurvedwalljetintheabsenceofa 103

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freestream,assumingthefollowingformsofthesimilarityfunctions f g ,and g 12 u = U max f 0 {22 )]TJETq1 0 0 1 266.063 654.84 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 266.063 645.285 Td [(u 0 v 0 = U 2 max g 12 {23 u 0 2 )]TJETq1 0 0 1 282.446 628.652 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 282.446 618.389 Td [(v 0 2 = U 2 max g {24 = y y 1 = 2 m {25 U max isthelocalmaximumvelocity,and y 1 = 2 m isthelocalnormaldistancefromthe surfacewherethevelocityis 1 2 U max .GuittonandNewmanshowedthatthewalljetow canonlybeself-similarprovidedthesurfaceisdenedbyalogarithmicspiral, r / e R=x Theratio x=R dictatestherateatwhichthelocalradiusofcurvature, R ,grows.The largerthevalueof x=R ,thelargerthelocalradiusofcurvatureatsomearclength x Table3.4liststherelevantlengthandvelocityscalesandhoweachscaleswitharclength x Withanexternalfreestream,thevelocitymayneverdecayto 1 2 U max .Forsuch conditions,Launder&Rodi1983suggestedtheuseofadefectvelocity,denedas U max )]TJ/F22 11.9552 Tf 12.06 0 Td [(U e ,where U e isthevelocitywheretheReynoldsstressdecaystoanegligiblevalue, andthelength y e; 1 = 2 ,denedasthenormaldistancefromthesurfacewherethevelocity is 1 2 U max + U e .ThesescalesareillustratedinFigure3-18.TheanalysisofGuitton &Newman1977isextendedforacurvedwalljetinthepresenceofafreestreamby assumingthefollowingarbitraryformsofthesimilarityfunctions. u = U 1 f 0 + U 2 {26 )]TJETq1 0 0 1 271.7 191.623 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 271.7 182.068 Td [(u 0 v 0 = U 2 1 g 12 {27 u 0 2 )]TJETq1 0 0 1 288.083 165.435 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 288.083 155.171 Td [(v 0 2 = U 2 1 g {28 = y )]TJ/F22 11.9552 Tf 11.956 0 Td [(y 2 y 1 {29 104

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IfthescalessuggestedbyLaunder&Rodi1983areappliedtosimilarityfunctionslisted inEquations3{26to3{29,then U 1 = U max )]TJ/F22 11.9552 Tf 10.284 0 Td [(U e U 2 = U e y 1 = y e; 1 = 2 )]TJ/F22 11.9552 Tf 10.284 0 Td [(y max ,and y 2 = y max TheremainderofthesimilarityanalysisfollowsthesamestepsoutlinedbyGuitton& Newman1977andisincludedindetailinAppendixF.Thesimilarityequationis y 1 U 1 dU 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f +2 g )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(dy 1 dx f 00 f + g 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 12 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(y 1 R y 1 U 1 dU 1 dx [ A ]+ dy 1 dx [ B ]+ g 0 12 +2 g 12 + y 1 U 1 dU 2 dx [ C ]+ dy 2 dx [ D ] + y 1 U 1 dU 2 dx f 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f 00 )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(dy 1 dx U 2 U 1 f 00 + dy 2 dx U 2 U 1 f 00 )]TJ/F15 11.9552 Tf 11.955 0 Td [( g 0 + y 1 U 1 U 2 U 1 dU 1 dx f 0 + dU 2 dx )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 2 R g 0 12 =0{30 where A = f 0 f +2 Z 1 f 0 2 d + U 2 U 1 f +4 Z 1 f 0 d +2 Z 1 U 2 U 1 2 d B = f 0 f + Z 1 f 0 2 d + U 2 U 1 f + f 0 +2 Z 1 f 0 d )]TJ/F27 11.9552 Tf 11.955 20.444 Td [( U 2 U 1 2 + Z 1 U 2 U 1 2 d C = f 0 + U 2 U 1 D = U 2 U 1 f 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 2 2 U 2 1 : When U 2 = y 2 =0,Equation3{30reducestotheformofthesimilaritysolutionprovided byGuitton&Newman1977.Inthegeneralcase,alltermsoutsidetheparenthesesin Equation3{30mustbeconstant.Considertheterm, dy 1 dx f 00 f 0 + g 0 : Since dy 1 =dx mustbeconstant,itfollowsthat y 1 / x .Similarly,notetheterm, y 1 R g 0 12 +2 g 12 : Since y 1 =R mustbeconstant,then R / y 1 / x .Therefore,inorderfortruesimilarityto beachieved,thesurfacemustbealogarithmicspiral,i.e. r / e R=x where R isthelocal 105

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radiusofcurvaturelabeledinFigure3-17.Table3.4liststhelengthandvelocityscales fromthisanalysisandhoweachscaleswitharclength x Sincethetrailingedgeofthecirculationcontrolairfoilhasaconstantradius,complete self-similarityoftheowcannotbeachievedinthepresentinvestigation.Alog-spiral trailingedgeisnotpracticalforunderwatervehicleapplicationswherecontrolsurface symmetryisdesired.Inaddition,alog-spiraldesignwouldlikelyinterferewiththe secondaryblowingslotusedtocontrolexcessivejetattachmentatextremelyhighprimary blowingslotmomentumcoecients. Althoughfullowsimilarityisnotpossibleinthepresentinvestigation,evenpartial owsimilaritycouldstillproveuseful.Forinstance,similarityofthemeantangential velocityalonewouldpotentiallyprovideameansofestimatingthedisplacementthickness, meanvelocity,andperhapsfrictionvelocityrequiredforHowe'smodel.Beforethe curvedwalljetowisexaminedforsimilarity,dimensionalanalysisisusedtoidentifythe dependentparametersinthisinvestigation. 3.3.2DimensionalAnalysis Adimensionalanalysisisperformedtodeterminetheimportantdimensionless parametersassociatedwiththetrailingedgeoweld.Asanexample,thelocation ofmaximumvelocity y max isusedastherelevantdependentlengthscale.Anyother dependentlengthorvelocityscalecouldbeusedinitsplace. First,allsignicantparametersintheproblemareidentied.Notethattheowis assumedtobeincompressiblesince M jet < 0 : 3forallcasesconsidered.Thegeometric propertiesincludethechord c ,slotheight h ,trailingedgeradius r ,andlipthickness l .Ofthese,thechord,trailingedgeradius,andlipthicknessarexedinthecurrent experimentalinvestigation.Flowpropertiesincludedensity ,viscosity ,freestream velocity U 1 ,jetvelocity U jet ,and y max If c U 1 ,and arechosenastherepeatingparameters,thegroupsassembledare: 1 = h=c 2 = r=c 3 = l=c 4 = U 1 c= = Re c 5 = U jet =U 1 ,and 6 = y max =c 5 106

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canberecastintothemomentumcoecient, 0 5 =2 1 5 2 =2 h c U jet U 1 2 = C : NotethatthejetReynoldsnumber,denedas Re jet = U jet h ; {31 isrelatedtothechordReynoldsnumber,momentumcoecient,andslotheight-to-chord ratio. C =2 c h Re jet Re c 2 =2 h c U jet U 1 2 {32 Hence,of C Re jet Re c ,and h=c ,onlythreeareindependent.Therefore,thedimensionless lengthscales,suchas y max =c andvelocityscales,like U max =U 1 ,aredependentonthe followingparameters. y max c = f h c ; r c ; l c ;C ;Re c {33 Asnotedearlier, r=c and l=c areconstantinthepresentinvestigation.However, h=c C ,and Re c arevariable.Thus,twoofthesethreeparameterscanbekeptconstant,and theeectofvaryingthethirdparameteron y max andotherlengthandvelocityscalescan bestudied. Table3-3summarizesthesixdierentcasesthatareincludedinthisstudyofthe curvedwalljet.Casesonethroughthreerepresenttestswhere Re c and h=c arexed,but C changes.Incasesfourandve, C and h=c arexed,but Re c varies.Finally,incases fourandsix, C and Re c arexed,but h=c diers.Inthefollowingsections,resultsfrom thesesixcasesarepresentedanddiscussed. 3.3.3FlowCharacteristics BeforestudyingthePIVmeasurementsofthecurvedwalljet,thesurfacepressure alongtheCoandasurfaceisexamined.Trailingedge C p dataforthecaseslistedinTable 3-3areplottedinFigure3-19.Initially,afavorablepressuregradientacceleratestheow forallcasesexcept C =0.0039,where U jet U 1 .Shortlythereafter,near r=c =0.02, 107

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astrongadversepressuregradientdevelops.ThetrendsdisplayedinFigure3-19follow similarndingsreportedbyNovak etal. 1987.Trailingedge C p providesinsightinto theexpectedsimilarity,orlackthereof,ofthecurvedwalljetow.Intheirstudyofa curvedwalljetintheabsenceofafreestream,Neuendorf&Wygnanski1999observed fullsimilarityof U innerandouterregionsintheconstantpressureregionbutonlyouter regionsimilarityof U intheadversepressuregradientregion. Prolesofmeanvelocity U and V ,turbulenceintensity u 0 2 1 = 2 and v 0 2 1 = 2 ,and Reynoldsstress u 0 v 0 fromPIVmeasurementsalongtheroundedtrailingedgeareplotted inFigure3-20atavarietyofdownstreamdistancesfromtheslotfor Re c =6.5 10 5 h=c =0 : 0019,and C =0.0039,0.015,and0.057.Prolesarenotincludedfor C = 0.0039after r=c =0.040becauseseparationoccursupstreamofthatposition.The distancefromthesurfaceisnormalizedusingtheslotheight h ,andtheowquantities arenormalizedusingthejetvelocity U jet .Uncertaintyboundsareremovedfromtheplots forclarity.Themeantangentialvelocityprolesrevealhowthejetdecaysandspreads asittravelsawayfromtheslot.As U decays,themeannormalvelocity V increasesand separationisapproached.TheturbulenceintensitiesandReynoldsstressaresignicant intheimmediatevicinityofthebluntslotlipandthehigh-shearregionthatexists there.Asthejetspreadsandentrainsuidfromthefreestream,Reynoldsstressbecomes non-negligibleoveragrowingregionofthetrailingedge.Thesimilarityofthemeasured curvedwalljetowisassessedinthefollowingsection. 3.3.4FlowSimilarity Novak&Cornelius1986andNovak etal. 1987showedthattheouterregionof U alongthetrailingedgeoftheircirculationcontrolairfoilexhibitedsimilaritywhen normalizedusingthescalessuggestedbyLaunder&Rodi1983refertoFigure3-18. Thesesamescalesareappliedtothecurrentdatasetinordertodetermineifsimilarityis achieved.Inaddition,thedataisnormalizedusingthescalesfoundbyZhou&Wygnanski 1993tocollapsetheouterregionof U foraplanewalljetinanexternalfreestream. 108

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Thosescalesincludelocal U max U 1 y max ,and y m= 2 ,whichisthenormaldistancewhere U = 1 2 U max .Inthecurrentsetofexperiments, U 1 isreplacedwith U min ,thevelocity wheretheouterregionmeanshearapproacheszero,and y m= 2 isreplacedwith y m; 1 = 2 ,the normaldistancewhere U = 1 2 U max + U min ,sincethevelocitydoesnotalwaysdecayto valuesof 1 2 U max .ThesescalesareillustratedinFigure3-21. Tonormalizetheowproles,thevariouslengthandvelocityscalesdescribedmust rstbedetermined. U max isfoundbydetectingthepointofmaximumvelocityineach prole,ttingaGaussiancurvetothispointanditsadjacentfourneighbors,andthen ndingthemaximumoftheGaussiancurve.With U max known, y max isdetermined.To nd y min ,themeanshear d U=dy iscomputedstartingat y max andmovingoutwards fromthesurfaceuntilthesignoftheshearips,andthenathird-orderpolynomialis ttothetwopointsoneithersideofthezero-crossing,andtheexactzero-crossingis determinedandsetas y min .Forafewproles,theshearneverchangessignintheouter region.Inthosesituations, y min isspeciedastherstinstancewhere h=U jet d U=dy < 4 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(6 or U / U jet changesbylessthan6 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 .Thisparticularshearthresholdis determinedbyinspectionofthedata.With y min specied, U min isfoundbyevaluating athird-orderpolynomialtofthevelocitydataat y min U m; 1 = 2 iscomputedonce U max and U min areknown,andthen y m; 1 = 2 isdeterminedbyapiecewisecubicinterpolation alongallthepointsbetween y max and y min .Todetermine y e ,rst, u 0 v 0 max isdetermined insimilarfashionto U max usingaGaussiancurvet.Thentherstinstancewhere u 0 v 0 = u 0 v 0 max < 0 : 05thisvalueisalsochosenbyinspection;slightlyhigherthresholds donotsignicantlyaecttheresultsabovethepositionofmaximumReynoldsstressis deemedthenegligibleReynoldsstresslocation.Athird-orderpolynomialtofthevelocity dataisthenevaluatedatthislocationtodetermine U e U e; 1 = 2 followsthesameapproach usedtond U m; 1 = 2 Thelengthscales y max y min y e y m; 1 = 2 ,and y e; 1 = 2 areplottedwiththemean tangentialvelocityandReynoldsstressprolesfor C =0.015inFigure3-22.Proles 109

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areprovidedfrom r=c =0 : 0027untiljustpastseparationat r=c =0.047.AsFigure3-22 attests,thereislittledierencebetween y min and y e and y m; 1 = 2 and y e; 1 = 2 Thescalesareusedtonormalizetheouterregionsofthemeantangentialvelocity prolesinFigure3-23,whereitisseenthat,usingeitherthescalingsuggestedbyLaunder &Rodi1983orthemodicationtotheapproachbyZhou&Wygnanski1993,the outerstreamwiseprolesexhibitsimilarity.Notethatthevelocitydataat r=c =0 : 0027 arenotincluded,astheydonotcollapsewiththeotherproles.Infact,onlyprolesin theconstantpressureoradversepressuregradientregionalongthetrailingedgeexhibit similarity,agreeingwiththeobservationsofNeuendorf&Wygnanski1999,Novak& Cornelius1986,andNovak etal. 1987.ThesolidlineinFigure3-23representsa hyperbolictangentoftheform U =0 : 5[1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 315tanh y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1] ; {34 where U and y arethenormalizedvelocityanddistancefromthesurface,respectively. Nowthattheouterregionof U hasbeenfoundtoexhibitsimilarity,theinnerregion of U isstudied.ItisdiculttomeasureextremelyclosetothesurfacewithPIVrecall, theslotheightistypically1mm,soinnerregiondataarelimitedcomparedtothe outerregionoftheow.Nevertheless,therearesucientdatatodeterminewhether ornotsimilarityisachieved.Innerregion U dataarenormalizedusing y max and U max andplottedinFigure3-24.Unliketheouterregion,theinnerregiondoesnotexhibit similarity.TheseresultsagreewithNeuendorf&Wygnanski1999,whofoundthatthe innerregionof U foracurvedwalljetinquiescentsurroundingsdoesnotexhibitsimilarity intheadversepressuregradientregion. Theremainingowproles, V u 0 2 1 = 2 v 0 2 1 = 2 ,and u 0 v 0 ,alsodonotexhibit similarityusinganycombinationofmeanvelocityandlengthscales.However,the similarityof U intheouterregionalonemaypermitthepredictionofthatportionofthe owandsubsequentlythemeantangentialvelocityscales,whicharerequiredtoassess 110

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Howe'smodelHowe2002.Inaddition,knowledgeof y max maybeusefulinprovidingat leastarudimentarypredictionofthelocaldisplacementthickness.Inthefollowingsection, thelengthandvelocityscalesfoundtocollapsetheouterregionof U areanalyzedforthe sixcaseslistedinTable3-3. 3.3.5LengthandVelocityScales Thesetoflengthandvelocityscalesbasedonthemeanshearfoundtosuciently collapsetheouterstreamwiseprolesdonotrequireknowledgeoftheowturbulenceand arethusmorepracticalinapplication.Assuch,theyarethefocusoftheremainderofthe analysis.Thelengthandvelocityscalesassociatedwiththisapproach, y max y m; 1 = 2 U max and U min ,areplottedinincrementsof r=c =0 : 002for C =0.015inFigure3-25.The dataareplottedfromjustdownstreamoftheslotexittoseparation,detectedbytherst signofowreversal.Figure3-25furtherillustratesthedevelopmentandspreadofthe jet.Thevelocityscalesinitiallyincrease,reachamaximumvaluenear r=c =0.015,then decayatsimilarrates.Thelengthscales y max and y m; 1 = 2 arenearlyconstantinitially,then increasegradually,andnallygrowsharplyasseparationisapproached.Ontheother hand, y min increasesimmediatelyandthenfollowsaparabolicform,similartotheother lengthscales. ForalltestcaseslistedinTable3-3,thetrendsof y max y m; 1 = 2 U max ,and U min are analyzedinanattempttocollapseeachsetofcurves.Onlydatawheretheouterregion exhibitssimilarityisincluded.Neuendorf&Wygnanski1999foundthat,foracurved walljetintheabsenceofafreestream,thedecayof U max andtherateofspreadofthejet describedbythelocationwhere U = 1 2 U max couldbescaledusingthelocalkinematicjet momentum, J = U 2 max y m= 2 Z 1 y=y m= 2 =0 U U max 2 d y y m= 2 =0 : 78 U 2 max y m= 2 ; {35 andthewallradiusintheconstantpressureregion.However,unlikethecurvedwalljet ofNeuendorfandWygnanski,theentirevelocityproleofthecirculationcontrolcurved 111

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walljetdoesnotexhibitsimilarity.Hence,thelocalkinematicjetmomentumcannotbe expectedtocollapsethescales.Likewise,NeuendorfandWygnanskiobservedthatthe rateofspreadcouldbeexpressedsolelyasafunctionofthewallradiusanddownstream angularpositionfromtheslot.Thisisclearlynotobservedforthecirculationcontrol airfoil,asshowninFigures3-26and3-27.Foraplanarwalljetinanexternalfreestream, Zhou&Wygnanski1993foundthatthelengthandvelocityscalesdescribingtheplanar walljet'sspreadanddecaycouldbeexpressedasafunctionofanormalizeddownstream distancefromtheslotdenedby XJ= 2 ,where X isthedimensionaldistancefromthe slot, isthekinematicviscosity,and J = h U jet )]TJ/F22 11.9552 Tf 12.691 0 Td [(U 1 U jet istheexcessofkinematic momentumuxnearthenozzle.Adimensionlessvelocityratio U jet )]TJ/F22 11.9552 Tf 12.341 0 Td [(U 1 = U jet + U 1 wasalsoused.Theseparametersdonotcollapsethecurrentdataseteither.Instead, thedataarebesttusing Re c C ,and h=c ,asshowninFigures3-26through3-29.A powerlawcurveisttoeachdatasetbyminimizingthesquareoftheerror.Thescales, asafunctionof r=c ,arefoundtocollapsewiththeproductof Re c and C ,referred toastheReynoldscorrectedmomentumcoecient,whichhasrecentlybeenfoundto scaleliftincrementsforactivecontrolofairfoilowseparationStalnov&Seifert2010. Thisscalingparametertakesintoconsiderationthefreestreamboundarylayerandits developmentasafunctionof Re c .RecallFigures3-15and3-16,whereathighervalues of C ,themeantangentialvelocityproleofthisupstreamboundarylayer"doesnot actuallyresembleaconventionalboundarylayer.TheReynoldscorrectedmomentum coecientprovidessomebasistoaccountforthiseect.Best-tequationsforthelength 112

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andvelocityscalesareprovidedinEquations3{36to3{39. y max h = 1 C Re c 0 : 60 9 : 0 10 10 r c 6 : 1 +110 # {36 y m; 1 = 2 h =2 : 6 10 10 1 C Re c 0 : 11 r c 5 : 7 +1 : 0{37 U jet U max 2 = h c )]TJ/F21 7.9701 Tf 6.586 0 Td [(0 : 80 7 : 0 10 6 1 C Re c 0 : 10 r c 4 : 9 +0 : 005 {38 U jet U min 2 = h c )]TJ/F21 7.9701 Tf 6.587 0 Td [(0 : 65 3 : 0 10 11 1 C Re c 0 : 18 r c 5 : 9 +0 : 04 {39 Equations3{36and3{37revealapproximatelya6thpowerdependenceonthearclength r=c .Contrastthattotheresultsofthegeneralsimilaritysolution,whichindicates that,fortheself-similarowoveralogspiralsurface,thesescalesshouldhavealinear relationshipwiththearclength. TheproductofthechordReynoldsnumberandmomentumcoecientcanalsobe rearrangedastheproductofthejetReynoldsnumberandjet-to-freestreamvelocityratio. Re c C = U 1 c 2 h c U jet U 1 2 # =2 U jet h U jet U 1 =2 Re jet U jet U 1 {40 Thus,inthecaseofthecirculationcontrolairfoil,itappearsthattherateofdecayand spreadofthejetaredependentonthejetReynoldsnumber.Thisisinstarkcontrastto thendingsofZhou&Wygnanski1993,whodeterminedthat,aslongas U max =U 1 > 2, thelengthandvelocityscalesdescribingtheplanarwalljetinanexternalfreestreamare independentofthejetReynoldsnumber.Notethat U max =U 1 > 2forsomeprolesin cases2through5listedinTable3-3. Equations3{36to3{39provideapredictivecapabilityforthelengthandvelocity scalesofthemeanouterregionow.Predictedandmeasuredscalesarecomparedinthe nextsection. 113

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3.3.6FlowPrediction Acomparisonofthepredictedandmeasuredlengthandvelocityscalesispresented inFigure3-30for Re c =6 : 5 10 5 C =0.015,and h=c =0.0019.Overall,theagreement betweenthemeasuredandpredictedscalesisgood. U min isinitiallyover-predictedinthe favorablepressuregradientregion,butrecall,onlydataintheadversepressuregradient region,wheretheouterregionof U issimilar,isusedtodetermineEquations3{36to 3{39.Despitethatfact,theotherscalesarepredictedreasonablywellinthefavorable pressuregradientregion. WhiletheagreementbetweenthepredictedandmeasuredoweldsinFigure 3-30ispromising,itisaratherexpectedresultsince,afterall,thedataforthosetest conditionsareusedindeterminingthepredictionequations.Totrulytesttheprediction's capabilities,itshouldbecomparedtomeasurementsforacasenotincludedintheprior analysis.Thus,thepredictedandmeasuredscalesarecomparedinFigure3-31for Re c =1 : 3 10 6 C =0.014,and h=c =0 : 0029.Althoughtherearelargerdierences betweenthepredictedandmeasuredscalesincomparisonwithFigure3-30,theoverall agreementisfair. Tothispoint,thecurvedwalljetowhasonlybeenconsideredpriortoseparation.In thenextsection,owseparationisdiscussed. 3.3.7SeparationandStability Itiswellknownthatthecirculationcontrolcurvedwalljetisresponsiblefor entrainingfreestreamuid,delayingseparationandincreasingcirculation.Thecause ofseparationisperhapslessclear,andcouldbetheresultofasevereadversepressure gradientseeFigure3-19,meanderingstreamwisevortices,observedincurvedwalljets absentofafreestreambyLikhachev etal. 2001,Neuendorf etal. 2004,andHan etal. 2006,oracombinationofinuences. SeparationisrststudiedinthepresentinvestigationbyusingPIVtomeasure thetrailingedgeoweldandpartofthewakeregion.Meanspeedcontoursandmean 114

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velocityvectorsarepresentedfor C =0,0.014,and0.057inFigure3-32.When C =0, alargewakeisformedaftofthetrailingedge,andthefreestreamowappearssymmetric aboutthechordline.Withblowing,theowalongtheuppersurfaceoftheairfoilremains attachedandseparatesmuchfartherdownstream,asshowninFigure3-32B.Thevectors revealtheextentofowturningduetoentrainmentfromthejet.As C furtherincreases to0.057inFigure3-32C,thefreestreamisdeectedevenmore,andtheseparationpointis shiftedfartherawayfromtheblowingslot. Figure3-32onlyprovidesaqualitativedescriptionofowseparation.Foramore quantitativeanalysis,thecurvedwalljetPIVdataareusedtoestimatetheseparation locationbydetectingtheonsetofowreversalnearthesurface.Dataalongthetrailing edgesurfaceareanalyzedinincrementsof0.13mm,whichcorrespondstothevector resolutionofthedataset.Also,sinceitisnotpossibletoresolvetheoweldvery nearthesurfacewithPIV,theseparationlocationspresentedcanonlyberegardedas estimates,andthetrueseparationlocationlikelyoccursjustupstreamoftheestimated location.Theseparationlocationsareplottedasafunctionof C Re c inFigure3-33,which indicatesthattheseparationlocationmovesfartherdownstreamfromtheslotasthe productofthechordReynoldsnumberandmomentumcoecientincreases.Inparticular, as C Re c initiallyincreases,theseparationdistanceincreasesatahighrate,butatlarger valuesof C Re c ,asignicantincreaseintheproductofthetwoparametersyieldsonly asmalldelayinseparation.Acurve,whoseequationisgivenbyEquation3{41,ist tothisdatasotheseparationdistancecanbeestimatedforagivenReynoldscorrected momentumcoecient.Equation3{41accuratelypredictstheseparationlocationsforboth casesconsideredinSection3.3.6withanerroroflessthan2%. r sep c =0 : 0085 C Re c 0 : 184 {41 ItshouldbenotedthatEquation3{41appearstobeonlyapplicabletotheairfoil geometryofthepresentinvestigation.Attemptstopredicttheseparationlocations 115

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reportedbyNovak&Cornelius1986andNovak etal. 1987resultinerrorsofjust under50%or =65 .However,thetrailingedgegeometryofthecirculationcontrol airfoilstudiedbyNovakandhiscolleaguesissignicantlydierentfromthetrailingedge ofthecirculationcontrolairfoilinthepresentinvestigation.Furthermore,Equation3{41 isonlyaccuratefor Re c > 0.Withnofreestream,thecurvedwalljetowseparateswell downstreamoftheblowingslot. Themechanismcausingseparationhasgarneredmuchattentionoverthelastdecade. RecentworkbyLikhachev etal. 2001,Neuendorf etal. 2004,andHan etal. 2006has revealedthatmeanderingstreamwisevorticesarelikelytheculpritsofcurvedwalljetow separation.Floryan1986determinedtheinviscidstabilitycriterionforboundarylayer andwalljetowsoverconvexsurfaces,givenby d U 2 dy > 0 : {42 Sincethewalljetproleisnon-monotonic,whetherornotafreestreamispresent,it automaticallyviolatestheinviscidstabilitycriterion.Inparticular, d U 2 =dy< 0inthe outerregionoftheprolewherethevelocitydecays.Therefore,theouterregionofthe walljetowoveraconvexsurfaceispotentiallyunstableandsusceptibletotheformation ofGortlervortices"Floryan1986;Saric1994.Floryanalsoperformedaviscousstability analysisandprovidedaneutralstabilitycurvefortheGortlernumber,whichhefoundto beafunctionofthemaximumvelocity,boundarylayerthickness,kinematicviscosity,and curvedwallradius. G = U max r 1 = 2 {43 Floryan1986denedtheboundarylayerthicknessby = r U max 1 = 2 : {44 TheGortlernumberisevaluatedasafunctionofdownstreamdistancefromtheslotfor Re c =6 : 5 10 5 C =0 : 015,and h=c =0 : 0019.Theresultsareplottedagainstthe 116

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neutralstabilitycurvecomputedbyFloryaninFigure3-34andindicatethatbeyond r=c =0.0047,theowisunstabletodisturbancesofcriticalwavelengthsindicatedbythe regionofthegurelabeledunstable,"where =2 = isadimensionlesswavenumber and isthedisturbancewavelength.Forreference,theGortlernumberattheseparation locationisalsoincluded.TheseGortler"vorticesareexpectedtobecenteredaboutthe outerregionofthevelocityprole,oralong y m; 1 = 2 Floryan1986. CrossowPIVisusedtodetectthepresenceofstreamwisevorticesbyilluminatinga cross-owplanenormaltothetrailingedgesurface13mmdownstreamfromtheblowing slot,or r=c =0.025.Initially,measurementsareattemptedatthesametestconditions usedtoevaluatetheGortlernumberinFigure3-34.However,toboostthejetseeddensity andimagequality,thechordReynoldsnumberisreducedto Re c =5 : 6 10 5 while maintaining C =0.014and h=c =0.0019.ThedropinReynoldsnumberreducesthe massowraterequiredtomaintainthedesired C andhenceincreasestheseeddensity, butsincethemomentumcoecientremainsconstant,thesamejet-to-freestreamvelocity ratioismaintained U jet =U 1 =2.Counter-rotatingpairsofvortices,muchlikethose describedbyNeuendorf etal. 2004,arereadilyvisibleintheacquiredimagepairs andtypicallylocatedbetweenone-halfandone-and-a-halfslotheightsfromthesurface. Aninstantaneoussnapshotofthespanwisevorticitycomputedfromanimagepairis presentedinFigure3-35andshowsmultipleregionsofpositiveandnegativevorticity, oftenlocatedinpairs.AsmallerregionoftheowisextractedinFigure3-36,wherea counter-rotatingpairofvorticesisclearlypresent.Theaxesofthevorticesarelocated between0 : 9
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pull"lower-momentumuidfromthefreestreamtowardsthesurfaceandpush"the higher-momentumuidlocatednear y max awayfromthesurface.Incontrast,vortex generatorsarecommonlyplacedinboundarylayerowstoproducestreamwisevorticity thatpulls"thehigher-momentumfreestreamuidtowardsthesurfaceandpushes" thelower-momentumnear-boundaryuidaway,delayingseparation.Thestreamwise vorticesproducedintheouterregionofthecurvedwalljetdojusttheoppositeand actuallyhaveimplicationstopossiblenoise-abatingtreatments.RecallSlomski2009 numericallyevaluatedtheeectofserratingtheslotliptoreduceliptonesbybreakingup thecoherentspanwisevorticesshedfromthelip.Itispossiblethatthesaw-toothserration patterncouldintroducethesevorticesupstreamfromwheretheynaturallyoccur,causing theowtoseparateearlier.SincethecomputationaldomainevaluatedbySlomskididnot includetheseparationregion,theinuenceoftheserrationsonowseparationcouldnot bedenitivelyassessed. 3.4Summary Fluiddynamicmeasurementsrevealagreatextentaboutboththefreestreamand circulationcontrolcurvedwalljetows. C p measurementsofthecirculationcontrolairfoil intheopenjettestsectionoftheUFAFFlackaleadingedgesuctionpeaktypically observedinclosedtestsectionmeasurements.Enclosingthetestsectioncausestheleading edgepeaktoemergebutdoesnotchangethefreestreamboundarylayerowpassingover theslotlip.Theinuenceofdierentwallboundaryconditionsonthe C p distributionis alsoexaminedusingdataandpotentialowtheory.Thecurvedwalljetowismeasured usingPIVandfoundtoexhibitsimilarityintheouterregionof U only.However,this alonepermitsthedevelopmentofequationstopredictthelengthandvelocityscales requiredforsimilarityasafunctionof Re c C ,and h=c .Finally,owseparationisfound tobedependentontheproduct C Re c ,andstreamwisevorticestheorizedtopromote separationareobservedincrossowPIVmeasurements. 118

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Theuiddynamicmeasurementsalsoprovideinsightwithregardstoacoustics.Since theturbulentboundarylayerpassingovertheslotlipisindependentofthetestsection conguration,thesoundproducedbytheinteractionoftheowwiththeairfoiltrailing edgeshouldbeindependentofthetestsectioncongurationaswell.Thesimilarity ofthecurvedwalljetowmakespredictionofsomeofthescalesrequiredforHowe's modelpossible,particularlyintheassessmentofcurvaturenoiseHowe2002.Finally, serratingtheslotlipfornoisereductionmaycauseearlierowseparationbyinstigating theproductionofstreamwisevorticesintheouterregionoftheow. Figure3-1.Lipdisplacement,normalizedbyinitialslotheight h=c =0.0019,asplenum pressureincreases. 119

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Figure3-2.Airfoilsurfacepressure C p Re c =6 : 5 10 5 h=c =0.0019.| | C =0 c l =0.020,--C =0.015 c l =0.58,4 C =0.057 c l =1.5.Theleft verticaldashedlinerepresentstheboundarylayertrip,andtherightvertical dashedlinecorrespondstotheslot. Figure3-3.Liftcoecient c l asafunctionofmomentumcoecient, Re c =6 : 5 10 5 h=c =0.0019. presentinvestigation, Abramson197520%ellipticcirculation controlairfoil, Re c =3 : 4 10 5 h=c =0.0013. 120

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Figure3-4.Photographofclosedtestsection. Figure3-5.Airfoilsurfacepressure C p C =0, h=c =0.0019.| |opentestsection, Re c =6 : 5 10 5 ;---closedtestsection, Re c;c =6 : 7 10 5 .Theleftvertical dashedlinerepresentstheboundarylayertrip,andtherightverticaldashed linecorrespondstotheslot. 121

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Figure3-6.Airfoilsurfacepressure C p h=c =0.0019.| |opentestsection, C = 0.057, Re c =6 : 5 10 5 c l =1.5;---closedtestsection, C ;c =0.057, Re c;c =6 : 7 10 5 c l =2.7.Theleftverticaldashedlinerepresentsthe boundarylayertrip,andtherightverticaldashedlinecorrespondstotheslot. Figure3-7.Liftcoecient c l asafunctionofmomentumcoecient, Re c =6 : 5 10 5 h=c =0.0019. presentinvestigation,opentestsection; 4 presentinvestigation, closedtestsection; Abramson197520%ellipticcirculationcontrolairfoil, Re c =3 : 4 10 5 h=c =0.0013. 122

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Figure3-8.PhotographofclosedtestsectionforPIVmeasurements.Thefoamceilingand onefoamwallarereplacedwithclearpolycarbonatepanels. Figure3-9.Airfoilsurfacepressure C p inclosedtestsection, Re c =6 : 7 10 5 h=c = 0.0019.| |foamwalls,upperslotblowing, C =0.057 C l =2.7;--foam/polycarbonatewalls,upperslotblowing, C =0.052 C l =3.1;4 foam/polycarbonatewalls,lowerslotblowing, C =0.053 C l =2.8.Theleft verticaldashedlinerepresentstheboundarylayertrip,andtherightvertical dashedlinecorrespondstotheslot. 123

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Figure3-10.Airfoilsurfacepressure C p inopentestsection, Re c =6 : 5 10 5 h=c = 0.0019.| |upperslotblowing, C =0.058 c l =1.5;---lowerslot blowing, C =0.059 c l =1.7.Theleftverticaldashedlinerepresentsthe boundarylayertrip,andtherightverticaldashedlinecorrespondstothe slot. 124

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Figure3-11.Potentialow C p ona20%ellipse, U 1 =20m/s.|{freestream c l =1.90; ----ceilingplane c l =2.29;-groundplane c l =1.81; ceilingand groundplanes c l =2.26. Figure3-12.Airfoilsurfacepressure C p inclosedtestsectioncomparedwithpotential owtheory. foam/polycarbonatewalls,upperslotblowing Cc l =3.1; 4 foam/polycarbonatewalls,lowerslotblowing c l =2.8;----potentialow, ceilingplane c l =2.29;-potentialow,groundplane c l =1.81. 125

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A B C D E F Figure3-13.LeadingedgemeanowstreamlinesinA-Copentestsection, Re c =6 : 5 10 5 andD-Fclosedtestsection, Re c;c =6 : 7 10 5 .Flowisfromlefttoright,and theoriginislocatedattheleadingedge.Lowerslotblowingisusedforall cases, h=c =0.0019.A C =0.B C =0.014.C C =0.057.D C ;c =0. E C ;c =0.013.F C ;c =0.053. 126

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Figure3-14.Comparisonofprolesatthelipedge, C =0. opentestsection, Re c =6 : 5 10 5 ; closedtestsection, Re c;c =6 : 7 10 6 .Theslotheightis h=c =0.0019forbothcases.Forclarity,onlyeveryfourthdatumpointis displayed.Errorboundsarerepresentedbylinesofmatchingcolorforeach dataset. 127

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Figure3-15.Comparisonofprolesatthelipedge. opentestsection, Re c =6 : 5 10 5 C =0.014; closedtestsection, Re c;c =6 : 7 10 6 C =0.013.Theslot heightis h=c =0.0019forbothcases.Forclarity,onlyeveryfourthdatum pointisdisplayed.Errorboundsarerepresentedbylinesofmatchingcolor foreachdataset. 128

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Figure3-16.Comparisonofprolesatthelipedge. opentestsection Re c =6 : 5 10 5 C =0.057; closedtestsection, Re c;c =6 : 7 10 6 C =0.053.Theslot heightis h=c =0.0019forbothcases.Forclarity,onlyeveryfourthdatum pointisdisplayed.Errorboundsarerepresentedbylinesofmatchingcolor foreachdataset. 129

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Figure3-17.Logarithmicspiralsurface. Figure3-18.LengthandvelocityscalessuggestedbyLaunder&Rodi1983. Figure3-19.Coandasurface C p .PleaserefertoTable3-3fortestconditionsforeachcase. 130

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Figure3-20.Circulationcontrolcurvedwalljetproles, Re c =6 : 5 10 5 h=c =0 : 0019. r=c =a0.0036,b0.011,c0.018,d0.026,e0.033f0.040,andg 0.048. C =0 : 015, C =0 : 0039, C =0 : 057. 131

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Figure3-21.Lengthandvelocityscalesbasedonzeroshear. Figure3-22.Circulationcontrolcurvedwalljetproles, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019. r=c =a0.0027,b0.010,c0.017,d0.025,e0.032f 0.040,andg0.047.Lengthscalesareindicatedby y max y min y e 4 y m; 1 = 2 ,and y e; 1 = 2 132

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Figure3-23.Outerregionsimilarityof U usingameanshearscalesandbscales suggestedbyLaunder&Rodi1983, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019. r=c = 0.010, 4 0.017, 0.025, 0.032 + 0.040,and 0.047. Thelinerepresents U =0 : 5[ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 315tanh y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1]. Figure3-24.Lackofinnerregionsimilarityof U Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019. r=c = 0.010, 4 0.017, 0.025, 0.032 + 0.040,and 0.047. 133

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Figure3-25.Velocityandlengthscalesdescribingdecayandspreadofjet, Re c =6 : 5 10 5 C =0 : 015,and h=c =0 : 0019. Figure3-26.Rateofspreadof y max =h andcollapseddatawithbesttline, R 2 =0.95.See Table3-3fortestconditions: case1, case2, 4 case3, case4, case5, case6. 134

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Figure3-27.Rateofspreadof y m; 1 = 2 =h andcollapseddatawithbesttline, R 2 =0.97. SeeTable3-3fortestconditions: case1, case2, 4 case3, case4, case 5, case6. Figure3-28.Rateofdecayof U max =U jet andcollapseddatawithbesttline, R 2 =0.95. SeeTable3-3fortestconditions: case1, case2, 4 case3, case4, case 5, case6. 135

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Figure3-29.Rateofdecayof U min =U jet forallcaseslistedintable3-3,andcollapseddata withbesttline, R 2 =0.95.SeeTable3-3fortestconditions: case1, case2, 4 case3, case4, case5, case6. Figure3-30.Comparisonofmeasuredandpredictedavelocityscalesandblength scales, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019.Measuredscalesare representedbydatumpoints,andpredictedscalesarerepresentedbylines. 136

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Figure3-31.Comparisonofmeasuredandpredictedavelocityscalesandblength scales, Re c =1 : 3 10 6 C =0 : 014, h=c =0 : 0029.Measuredscalesare representedbydatumpoints,andpredictedscalesarerepresentedbylines. A B C Figure3-32.Meanspeed j V j =U 1 contoursandmeanvelocityvectors, Re c =6 : 5 10 5 and h=c =0.0019.Theairfoiltrailingedgeissketchedforreference.A C = 0.BUpperslotblowing, C =0.014.CUpperslotblowing, C =0.057. 137

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Figure3-33.Approximateseparationlocationasafunctionof C Re c measured,---t R 2 =0 : 996. Figure3-34.Instabilityrange,indicatedbyGortlernumber, Re c =6 : 5 10 5 C =0 : 015, h=c =0 : 0019.TheneutralstabilitycurveiscomputedbyFloryan1986. Figure3-35.Instantaneousspanwisevorticity x h=U jet distribution, Re c =5 : 6 10 5 C =0 : 014, h=c =0 : 0019.Theoriginof z=h islocatedatmidspan. 138

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Figure3-36.Instantaneousspanwisevorticity x h=U jet distributionandinstantaneous velocityvectorsrevealpresenceofcounter-rotatingpairofvortices, Re c =5 : 6 10 5 C =0 : 014, h=c =0 : 0019.Theoriginof z=h islocatedat midspan. Table3-1.ScalingoflengthandvelocityscalesfromsimilaritysolutionofGuitton& Newman1977; x isthearclengthand a isaconstant Parameter Scaling R x y 1 = 2 m x U max x a Table3-2.Scalingoflengthandvelocityscalesfromgeneralsimilaritysolutionofacurved walljetinafreestream; x isthearclengthand a isaconstant Parameter Scaling R x y 1 x y 2 x U 1 x a U 2 x a U 2 =U 1 constant 139

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Table3-3.Testcasespresented Case h=c 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(3 Re c 10 5 U 1 m=s U jet m=s U jet =U 1 C 1 1.96.5202010.0039 2 1.96.5204020.015 3 1.96.5208040.057 4 1.96.520381.90.014 5 1.91340782.00.014 6 2.96.520321.60.014 140

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CHAPTER4 ACOUSTICS Asthetitleofthischaptersuggests,thefocusofthischapterisndingsfrom acousticsmeasurements.First,thepresenceandscalingoftonesisdiscussed.Then, resultsfromphasedarraymeasurementsarepresentedinordertoidentifytheprimary noisesourcesunderdierenttestconditions.Dierentmeasurementandprocessing techniquesareusedtoestimatethebroadbandnoisespectrum.Finally,Howe'smodelof circulationcontrolacousticsiscomparedwithameasurementHowe2002. 4.1Tones Acoustictonesareundesirableforunderwatervehicleapplicationsandapotential hurdlefortheapplicationofcirculationcontroltounderwatervehicles.Thepresenceof tonesisevaluatedusingmicrophoneM1"fromthetestsetupillustratedinFigure2-9. Figure4-1includesspectraforavarietyofmomentumcoecientsat Re c =6 : 5 10 5 and h=c =0.0019.Thespectrarevealbothlowandhighfrequencytonesundercertain conditions.Thesetonesareevaluatedfurtherinthefollowingsections. 4.1.1LowFrequencyTones Spectrafromalltestcaseswherealowfrequencytoneismeasuredareplotted togetherinFigure4-2.Thetestsincludedcovervariationsin C Re c ,and h=c .The frequencyaxisisnormalizedusingtheStrouhalnumber, St = 2 y TE U 1 ; {1 whichhasanominalvalueof0.21forvortexsheddingfromabluntedge.Thelength scale y TE =4.6cm,thethicknessofthetrailingedgeattheslotexitplane,isusedasthe referencelengthscale.ThetonesplottedinFigure4-2collapseataStrouhalnumberjust above0.21,indicatingthattheselowfrequencytonesareproducedbyvortexshedding fromtheroundairfoiltrailingedge. 141

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Withsucientblowing,thesetonesarenotmeasured.Usingjustasingleblowing slot,thetonesareeliminatedif C 0 : 002. 4.1.2HighFrequencyTones Previousresearchcitedthepresenceofhighfrequencytonesthoughttobeattributed tovortexsheddingfromtheslotlipSlomski2009.MicrophonesplacedintheUFAFF measurehighfrequencytonesatavarietyoffrequenciesandamplitudes.Asamplingof thesetonesisprovidedinFigure4-3for C =0.014and h=c =0.0029.Thefrequencyaxis isagainnormalizedbytheStrouhalnumber,thistimedenedby St = fl U jet ; {2 where l isthelipthickness.TheStrouhalnumberofthetonesareincloseagreementnear 0.21,anditisconcludedthatthesetonesareproducedbyvortexsheddingfromtheslot lip. Theselip"tonesareonly,butnonnecessarily,producedwhenbothfreestreamand jetowsexist.However,exactconditionscorrespondingtothegenerationofliptones couldnotbedetermined.Nocleartrendsbetweentheappearanceormagnitudeofthe tonesandowparameterssuchas Re c C h=c ,and Re jet arefound.Thedirectionality ofthetonesisassessedtosomeextentusingthemicrophonesplacedontheoppositeside fromtheblowingslot.Thelevelsofthetonesmeasuredbythesemicrophonesareatmost onlyslightlyabovebroadbandnoiselevels,asshowninFigure4-4. Withbothvortexsheddingtonesexperimentallyveried,focusshiftstowards broadbandnoiseandnoisesourceidentication. 4.2AssessmentofNoiseSources AsdiscussedinSection2.6,asinglemicrophoneisunabletodiscernbetweenmultiple acousticsources.Furthermore,pairsortriadsofmicrophonesareunabletodistinguish betweenmultiplecorrelatednoisesources.Phasedacousticarraysare,ontheotherhand, distinctlyusefulforidentifyingnoisesources.Resultsfromphasedarraymeasurements 142

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ofthecirculationcontrolairfoilarepresentedinthissection.Caseswithonlyfreestream ow,onlyjetow,andbothowsareexamined. 4.2.1IntheAbsenceofaFreestream Thecirculationcontroljetisrstconsideredintheabsenceofafreestream.Recall Figure3-10,whichrevealsthatthe C p distributionsforupperandlowerslotblowingare nearlyidenticalinthetunnelopentestsectionconguration.Itisreasonabletoassume that,sincethe C p distributionsaresimilar,thetrailingedgeblowingslotowsandthe soundtheyproducearealsocomparable.Thus,beforebeammapsarediscussed,consider thespectrashowninFigure4-5,presentedassoundpressurelevelSPLindB,where SPL =10log 10 ^ G yy f f p 2 ref : {3 f isthebinwidthoftheprocessedspectrum.Thesespectraaremeasuredbythecenter microphoneplacedabovetheairfoilforcaseswheretheblowingslotonthesameside asthemicrophoneisused,andthisisreferredtofromthispointforwardsassameside blowingSSB.Furthermore,thecasespresentedinFigure4-5aresimilartoSSBcases analyzedwiththephasedarray.Uppermicrophonespectraareconsideredinplaceof thespectrafromthearraycentermicrophonesincethearraymicrophonesuersfrom signicantinter-microphonescatteringathigherfrequencies.TheverticallinesinFigure 4-5signifytheoctavefrequencieswherebeammapsaregenerated,notably1kHzto64 kHz.Noticethatwhen Re jet =1380,thesoundgeneratedishardlydistinguishablefrom thenoiseoor.Hot-wiredatausedtocharacterizeslotowuniformityindicatethatthe jetisjustbecomingturbulentat Re jet = hU jet = =2600.AccordingtoHowe2002, slot-jetinteractionnoiseiseliminatedifthejetislaminar. Beammaps,liketheexampledisplayedinFigure4-6,arepresentedforavarietyof jetReynoldsnumbersinFigures4-7through4-13.Eachbeammapincludesoverlayswith theairfoil,sidewalls,inlet,anddiuser,asillustratedinFigure4-6.Allmeasurements representcaseswheretheblowingslotonthesamesideoftheairfoilasthearrayisused. 143

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ThearraypointspreadfunctionPSFisalsoshownineachgureasreferenceforthe arrayresolution.CSMdiagonalremovalisappliedduringbeamformingtoeliminate microphoneself-noiseandchannelnoiseDougherty2002.Propagationtimesarenot correctedforshearlayerrefractionbecauseoftheunknowneectoftheblowingjetonthe tunnelshearlayerAmiet1978.BeammapsarepresentedinabsolutedBref.20 Pa forcomparisonbetweendierenttestconditions.Thescanningplaneislocatedalongthe chordline,1.12mabovethearray. Byandlarge,intheabsenceofafreestream,thetrailingedgeisthedominantnoise sourceatallfrequencies,providedthejetReynoldsnumberissucientlylarge.When Re jet =1340,thetrailingedgenoiseisnotuniformornosinglesourceisapparent. However,thelevelsarealsoextremelylowincomparisonwiththehigher Re jet cases, agreeingwiththehot-wiredatathatindicatesthejetisjustbecomingturbulentwhen Re jet =2600.When Re jet =2680,thetrailingedgeisthelonedominantsource,although thesoundpowerisnotdistributeduniformlyalongthetrailingedgeatfrequenciesof4 kHzand8kHz.Inaddition,thelevelsarealsoextremelylowabove16kHz.When Re jet =3980andhigher,thetrailingedgeistheprincipalnoisesource,thepowerisgenerally distributeduniformlyalongthetrailingedge,andlevelsincreasewith Re jet Havingidentiedthetrailingedgeastheprimarynoisesourceintheabsenceofa freestream,thearrayisusedtoidentifynoisesourceswiththeadditionofthetunnel freestream. 4.2.2WithaFreestream Theadditionofafreestreammayproducecontaminatingnoisesources.Potentially harmfulnoisesourcesincludesidewallscrubbingnoiseanddiuserowimpingement noise,whichwassevereenoughinpriorexperimentsintheUFAFF,thatitmotivatedthe removalofameter-longsectionofthediusertofurtherseparateitfrommicrophonesand arraysBahr2010.Ifthesenoisesourcesarepresent,thenthearrayprovidesinsightinto 144

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theirstrengthsrelativetothesourcesofinterestandthefrequenciesatwhichtheymaybe problematic. Followingthepatternoftheprevioussection,beforebeammapsareprovided, considertheuppermicrophonespectrameasuredforupperslotblowingtestconditions plottedinFigure4-14foravarietyofmomentumcoecientsat Re c =6 : 5 10 5 .Like thejet-onlyspectra,thespectrumfor Re jet =0 C =0isnearlyidenticaltothe Re jet =1330 C =0 : 004spectrum.When C =0 : 015 Re jet =2660,thereis someincreasein SPL atfrequenciesbelow1kHzandabove10kHz,butinbetween thedierenceismarginal.As C increasesfurther,thesoundlevelsincreaseoverall frequencies. BeammapsareprovidedinFigures4-15to4-21foroctavesbetween1kHzand64 kHz.ThecasesagainrepresentSSB.Diagonalremovalisused,butashearlayercorrection isnot.Thescanningregionislocated1.12mabovethearray,alongtheairfoilchordline. ArrayPSFsarealsoshownforreferencewitheachgure. InFigure4-15Hz,itisimmediatelyobviousthatcirculationcontrolisnota signicantnoisesourcewhentheslotjetisnotturbulent.MapsBthroughDindicatethat noiseiscomingfromthevicinityoftheairfoil,butitisdiculttodiscernexactlywhere thenoiseisoriginating.Thereisevidenceofowimpingementnoiseonthetunneldiuser orneighboringceilingwedgesinmapD,whichcorrespondsto C =0 : 017.Atthehighest momentumcoecientstested,thereisnoevidenceofowimpingementnoise.Inmaps EthroughG,thetrailingedgeregionisthedenitivedominantnoisesource,although thepowerappearsskewedtoonesideofthetrailingedge.Thepowerlevelsincrease signicantlywithincreasing C beginningwith C =0.017. Mapsproducedat2kHzarepresentedinFigure4-16.InmapsBthroughD,sidewall scrubbingnoise,causedbytheturbulentwallboundarylayer,appearstobesignicant, andthelevelsforthesemapsarenearlyidentical,despitethefactthat C isincreasing considerably.FlowimpingementnoiseisalsoevidentinmapsDandEandisdominantin 145

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mapDwhere C =0.017.BeginningwithmapE C =0.037,theprimarynoisesource islocatedinthevicinityofthetrailingedge,buttherearelargedeviationsinsoundpower acrossthespanofthemodelinmapsEthroughG. Thebeammapsat4kHzshowninFigure4-17havesimilarcharacteristicstothe2 kHzmaps.SidewallnoisedominatesinmapsBthroughDandisalsoreadilyapparentin mapE.FlowimpingementnoiseisobservedinmapsDandEaswell.InmapsFandG, thestrongestsourceappearsalongthetrailingedgeat37%span,anditslevelincreases with C .Other,lessdominantsourcesarevisibleatthejunctionsbetweenthesidewalls andmodeltrailingedge. InFigure4-18,whichshowsmapsat8kHz,mapsBthroughDagainrevealsidewall noise,andtheprimarysourcesarelocatedattheintersectionsofthesidewallsandthe airfoilleadingedge.Thissoundcouldbecausedbyhorseshoevorticesformedwhenthe tunnelwallboundarylayerpassesaroundtheleadingedge.Comparedwiththehigher C cases,theirlevelsarerelativelylow.InmapsEthroughG,theprimarysourcesappearat thetrailingedge-sidewallinterfacesandonceagainalongthetrailingedgeat37%span. Mapscreatedbytheinnerarrayat16kHzareshowninFigure4-19.Sourcesin mapsBthroughDincludethesidewalls,leadingedgehorseshoevortices,anddiuserow impingement.However,thelevelsofthesesourcesareverylowrelativetothesourcesat momentumcoecientsof C =0.037andhigher.Thetrailingedge-sidewallinterfacestill dominatesinmapE,butthesoundappearstodistributemoreuniformlyoverthetrailing edgeas C increasesinmapsFandG. Beammapsfor32kHzarepresentedinFigure4-20.Withnoorminimalblowing mapsBandC,theleadingedgehorseshoevorticesandtheboundarylayertripappear assources,althoughtheirlevelsareextremelylow.Thedominantsourcesarethetrailing edge-sidewallinterfacesinmapDthroughG,andthe37%spanlocationinmapsE throughG. 146

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Finally,at64kHzFigure4-21,mapsAthroughDareextremelynoisy,butspectra measuredbythecentermicrophoneinFigure4-14indicatethelevelsatthisfrequency areatornearthenoiseoor.AmongthesidelobesapparentinmapsEthroughG,the dominantsourcesstillappeartobelocatedatthejunctionsofthesidewallsandthe trailingedgeandalongthetrailingedgeat37%span. Figures4-15through4-21revealanassortmentofnoisesources,mostofwhichare undesired.Itisclearthatfor Rec =6 : 5 10 5 ,contaminatingnoisesourcesdominateat allfrequenciesconsideredwhen C < 0.017.Thesesourcesincludesidewallnoise,ow impingementnoise,andleadingedgenoise,possiblyfromhorseshoevortices.At C = 0.037,someofthesesourcesarestillpresentwiththeadditionofnoiseatthetrailing edgethatappearstooriginateatthetrailingedge-sidewalljunctionsandat37%span. Thesesametrailingedgesourcesappeartodominateforthehighermomentumcoecients considered.Nearthesidewall,afewmechanismscouldbegeneratingsound.First,vortices similartowingtipvorticeshavebeenobservedinpriorcirculationcontrolexperiments withsidewalls,andtheseveryvorticeswerethesubjectofarecentnumericalinvestigation Englar&Williams1972;Nishino&Shari2010.Second,thereisadiscontinuityat theedgeoftheslotformedbythesteelairfoilendplatejustafewmillimetersfromthe foamwall.Thisedgecouldradiatesound.Whilethesearepotentialmechanismsforsound generatednearthesidewall,thesourceappearingat37%spanissurprising,especially consideringthat,intheabsenceofafreestream,thesourceisuniformlydistributedacross theentiretrailingedge.Perhapsthesoundisonlygeneratedwithafreestreambecause itiscausedbytheturbulentboundarylayerpassingoverthemodelsurfaceorlipedge. However,whilethereisaseamoftwolippiecesnear37%span,inspectionrevealsno distinctdiscontinuityornon-uniformitycomparedtotheotherlipseams,andpriortothe tunnelentry,thisseamissealedandcheckedforleaks. Togathermoreinsightregardingthesetrailingedgesources,beammapsfrom oppositesideblowingmeasurementsOSBarecomparedwithSSBmeasurementsin 147

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Figures4-22to4-28forthethreehighestmomentumcoecientsconsidered.Theleft columnofbeammapsA,C,EcorrespondstoSSB,andtherightcolumnofbeammaps B,D,FcorrespondstoOSB.Toreiterate,thearrayislocatedonthesamesideasthe blowingslotfortheSSBcases. Thesameandoppositeslotblowingbeammapsat992Hz,showninFigure4-22, arenearlyundistinguishable,butitisinterestingthatthelevelsareslightlyhigherfor OSB.At2kHzFigure4-23,thereareslightdierencesinthesourcesidentied.With OSB,thearraydoesnotmeasureowimpingementnoisewhen C =0 : 040,likely becausetheowimpingementnoiseisbelowthearrayplane.Atthehighermomentum coecients,themapsarefairlysimilar,butthelevelsareasmuchas5dBhigherfor OSBcomparedtoSSB.At4kHzFigure4-24,therearesubstantialdierencesinthe apparentnoisesources.WithOSB,thenoiseismuchmoreevenlydistributedthanSSB, andthelevelsarebetween3-4dBhigherforOSB.At8kHzFigure4-25,however,the trailingedge-sidewallinterfacesbegintodominate,alongwitharegionnear25%span. LevelsarejustslightlyhigherfortheOSBcases.ItisdiculttodistinguishtheSSBand OSBbeammapsshowninFigure4-26for16kHz,andforallbutthehighest C shown, themaximumlevelsforSSBarehigherthanOSB.At32kHzand64kHz,presentedin Figures4-27and4-28,respectively,thebeammapsforOSBrevealjustonesourcelocated atoneofthesidewall-trailingedgejunctions.ThehighestlevelsforOSBareagain3-4dB higherthanSSBfor32kHz,butthemaximumlevelsarehigherforSSBat64kHz.Itis interestingthatsourcesarenotobservedalongthetrailingedgeat37%spanandatthe oppositesidewallforOSB. Thearraymeasurementsprovideusefulinformationaboutthetruesoundsourcesin thisexperiment.Atlowmomentumcoecients,tunnelsidewallnoise,owimpingement noise,leadingedgehorseshoevortices,andsidewall-trailingedgejunctionnoisedominate. Atthehighestmomentumcoecients,sidewallscrubbingnoiseandowimpingement noiseareinsignicant,butthesoundmaylargelybeproducedatthesidewall-trailingedge 148

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interfacesandalongthetrailingedgeat37%spanbyanunknownsource.Undoubtedly, asinglemicrophoneiscertainlynotmeasuringonlytwo-dimensionalcirculationcontrol noisesources.Furthermore,sincethetrailingedgealoneisneverthedominantsource, theapplicabilityofthecoherentoutputpowerandthree-microphonemethodsisalso questionable. 4.3BroadbandNoise Thebeamformingresultsshowthatmultiplebroadbandacousticsourcesexistover allfrequenciesregardlessofthemomentumcoecienttested.Sincesoundproducedby thetwo-dimensionalcirculationcontrolsourcesofinterestneverappearstodominate,the COPandthree-microphonemethodsarelikelynotsuitableforapplicationtowardsthis investigation.However,theywillstillbeassessedandcomparedwithasinglemicrophone autospectrum. 4.3.1Free-StandingMicrophones Thedierentfree-standingmicrophoneprocessingtechniquesdescribedinSection 2.6.1arecomparedfor Re c =6 : 5 10 5 h=c =0.0019,and C 0 : 10,whichrepresents thehighest C testedand,asindicatedbythebeammaps,yieldsinsignicantsidewall scrubbingandowimpingementnoise.DataarepresentedforSSBandOSBcases. ThesinglemicrophonespectrumrepresentstheautospectrumofthecenterG.R.A.S. microphone.TheCOPde-noisedautospectrumestimateiscomputedusingthecenter G.R.A.S.andthedownstreamB&Kmicrophone,althoughtheresultantspectrumis similariftheupstreammicrophoneisused.BoththeCOPandthree-microphonespectra representestimatesforthesignalmeasuredbythecenterG.R.A.S.microphone. Figure4-29comparesthethreemethodsforSSB.TheCOPandthree-microphone spectraareplottedasalledregionthatincludestheiruncertaintybounds.TheCOP spectrumislowerinmagnitudethanthesingleandthree-microphonespectra,particularly athigherfrequencies.RecallthediscussioninSection2.6.1.2,wheretheCOPmethodis foundtounder-predictlevelsifthesignal-to-noiseratiooftheadditionalmicrophoneis 149

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notsucientlyhigh.Thethree-microphonespectrumisincloseagreementwiththesingle microphonespectrumathigherfrequencies. SimilarresultsareshowninFigure4-30foroppositesideblowing.TheCOP spectrumlevelsarelowincomparisonwiththeothermethods,andthethree-microphone spectrumagreeswellwiththesinglemicrophonespectrum,evenathigherfrequencies. ThegeneralagreementofthespectrainFigures4-29and4-30indicatesthatthere islittleuncorrelatednoiseinthemeasurements.Sincebeamformingresultsindicate theexistenceofmultipledominantsources,thesemethodsarenotappropriatefor measuringthenoiseofinterestproducedbythecirculationcontroltrailingedge.The three-microphoneandsingle-microphonespectraareingoodagreement,suggestingthat thethree-microphonemethodprovidesasatisfactoryrepresentationoftheoverallacoustic eldcomprisedofcirculationcontrolnoiseandinstallationeects.Usingthephasedarray, itmaybepossibletoextractaspectrumofthisdesiredcirculationcontrolnoiseonly. 4.3.2Array Analternativeapproachtothemethodsdiscussedintheprevioussectionisto obtainaspectrumfromthearrayviaintegration,asdescribedinSection2.6.2.1.This processiscertainlynotwithoutitslimitations,particularlyathigherfrequencies,sincethe integratedlevelsarehighly-dependentontheaccuracyofthearraycalibration,including themeasuredsensorlocations,andtheregionofintegration. Anintegratedspectrumiscomparedtothefree-standingcenterG.R.A.S.microphone autospectruminFigure4-31for Re c =6 : 5 10 5 h=c =0.0019,andSSB C 0 : 065. Theouterarrayisusedtocomputethespectrumforfrequenciesbelow10kHz,while theinnerarrayisusedforfrequencies10kHzandhigher.Theintegrationregionis frequencydependentandaimstoexcludenoisefromthesidewall-trailingedgeinterfaces. 150

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Inparticular,theintegrationregionisdenedby )]TJ/F22 11.9552 Tf 9.298 0 Td [(BW= 2 x BW= 2 )]TJ/F22 11.9552 Tf 9.298 0 Td [(L= 2+ BW= 2 y L= 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(BW= 2 where BW referstothecomputedarray3dBbeamwidthrefertoFigure2-19, L = 1.12mistheairfoilspan,andtheoriginofthecoordinatesystemisatthemidspanpoint where x=c =1.TheshadedoverlayinFigure4-31representstheestimateduncertainty inthespectrallevelscomputedusing10,000iterationsoftheMonte-Carlosimulation describedbyYardibi etal. 2010 a .BothspectrainFigure4-31areplottedusingthe dimensionlessspectrumformsuggestedbyHowe2002, =10 log 10 U=a O ~x;! 0 U 2 2 L s a= j ~x j 2 M sin 2 = 2sin ; {4 where U isthefreestreamvelocity, O isthesingle-sidedpower-spectraldensity, 0 is thefreestreamdensity, L s isthesourcelength, j ~x j isthedistancefromthetrailingedge atmidspantotheobserver, M isthefreestreammachnumber, istheanglebetween thechordwisedirectionandtheobserver,and istheanglebetweenthespanwise directionandtheobserver.Forthesinglemicrophonespectrum, L s = L =1.12m,but forthearrayspectrum, L s isbasedonthespanwiselengthofthefrequency-dependent integrationregion.Asexpected,thearrayspectrumandsinglemicrophonespectrum deviatesignicantly,particularlyathigherfrequencieswherethedominantsourcesappear atthesidewall-trailingedgeinterfaces. Theintegratedarrayspectrumisusedinthefollowingsectionforcomparisonwith theacousticmodelderivedbyHowe2002. 4.4Howe'sModelofCirculationControlAcoustics Howe2002theorizedthreesignicantbroadbandnoisetypes:curvaturenoise, passiveslotnoise,andslot-jetinteractionnoise.Curvaturenoiseisproducedbyboundary layerturbulencescatteringotheroundedtrailingedge.Passiveslotnoiseisgeneratedby 151

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freestreamboundarylayerturbulencescatteringotheslotlip.Finally,slot-jetinteraction noiseiscausedbytheinteractionofturbulenceinthetrailingjetwiththeslotlipand trailingedgesurfaceneartheslot.Howederivedmathematicalmodelsforthesethree noisesourcesthatarefunctionsoflocalowscales,includingthedisplacementthickness, meanvelocity,andfrictionvelocity.InordertoaccuratelycompareHowe'smodelwiththe experimentalresults,thesescalesneedtobedetermined.First,theprimaryassumptions behindHowe'smodelareintroduced. 4.4.1Assumptions Howe'smodelfollowsthreeprimaryassumptions.First,thefreestreamMachnumber, M 1.Forthecasesconsidered, M =0 : 06,sothisassumptionissatised.Second,the freestreamReynoldsnumbermustbelarge.Since Re c =6 : 5 10 5 ,thisassumptionis alsosatised.Finally,Howeassumedcompactnessofthescatteringedgethickness.For curvaturenoise,Howeconsideredthetrailingedgeradiusasthelengthscaleofinterest, andhence, R .Forthecirculationcontrolairfoilofthepresentstudy, R = when f =15kHz,soprovided f< 1kHz,thecompactnessassumptionismet.Howeconsidered thelipthicknessasthelengthscaleofinterestwhendeterminingcompactnessforpassive slotnoiseandslot-jetinteractionnoise.Theairfoilunderinvestigationhasalipthickness of0.28mm.Anacousticwavelengthof0.28mmcorrespondstoafrequencyofover1200 kHz,sothiscompactnessassumptionisalsovalidforthehighestfrequencyofinterest,80 kHz. 4.4.2EstimatesofFlowScales Howe'smodelisassessedusingowmeasurementsfor Re c =6 : 5 10 5 C =0.057 Re jet =5000,and h=c =0.0019.Thiscaserepresentsthehighestmomentumcoecient studiedwithPIV.Theowscalesmeanvelocity,displacementthickness,andfriction velocityrequiredtoevaluateHowe'smodelareestimatedfromthisdata. 152

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4.4.2.1Curvaturenoise Howe2002denescurvaturenoiseasthesoundproducedbyturbulenceinthe boundarylayerpassingovertheroundedtrailingedge.Thisisperhapsanunclear denition,since,asshowninChapter3,theowovertheroundedtrailingedgeis thatofawalljetandnotaconventionalboundarylayer.Inaddition,exactlywhere toestimatethescalesalongthetrailingedgeisvague.Inhisanalysis,Howeusedthepoint ofmaximumcurvatureonanellipse.Sincetheairfoiltrailingedgeunderinvestigation hasaconstantradius,curvatureisconstant.Instead,sinceitiswell-knownthatturbulent soundproductionincreaseswithvelocity,scalesforcurvaturenoiseareevaluatedatthe pointofmaximumvelocityalongthetrailingedge. Themeantangentialvelocityproleatthepositionofmaximumvelocity,foundto be r=c =0.0129,isplottedinFigure4-32.Thethreescalesofinterestincludethemean velocityoutsidetheboundarylayer,thedisplacementthickness,andthefrictionvelocity. Sincetheowrepresentsawalljet,theinnerregionprovidestheclosestresemblanceto aconventionalboundarylayerandwillhenceforthbeusedtodeterminetheseowscales. Themeanvelocityoutsidetheboundarylayerischosentobethelocalmaximumvelocity, whichismeasuredtobe U o = U max =91.3m/s.Thedisplacementthicknessandfriction velocityaremuchmorediculttocalculatebecauseofinsucientdatanearthesurface. Furthermore,themaximumvelocityisfoundtobetheclosestdatumpointtothesurface. Asanapproximation,aone-seventhpowerlawisusedtoestimatethedisplacement thicknessandmomentumthicknessPrandtl1961. U U max = y y max 1 = 7 {5 Thedisplacementthickness, = Z y max 0 1 )]TJ/F15 11.9552 Tf 23.151 11.109 Td [( U U max dy; {6 153

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andmomentumthickness, = Z y max 0 U U max 1 )]TJ/F15 11.9552 Tf 23.151 11.11 Td [( U U max dy; {7 aredeterminedbyevaluatingthepowerlaw.Theskinfrictioncoecientisthenestimated usingtheKarmanintegralrelationWhite2006, C f 0 : 3 e )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 : 33 H log 10 Re 1 : 74+0 : 31 H ; {8 where H = = istheshapefactor,and Re = U max = istheReynoldsnumberbasedon themomentumthicknessandfreestream"velocity.Aftertheskinfrictionisobtained,the wallshearstressandfrictionvelocityareestimated. w = 1 2 0 U 2 max C f {9 v = r w 0 {10 Thedisplacementthicknessandfrictionvelocityareapproximately0.010mmand9.57 m/s,respectively. 4.4.2.2Passiveslotnoise Howe2002considerspassiveslotnoisetobethesoundproducedbyturbulencein theexteriorfreestreamboundarylayerscatteringotheslotlip.Likecurvaturenoise,the assumptionofaconventionalboundarylayerowpassingoverthelipisnotentirelytrue. Figure3-16showsthattheowpassingoverthelipedgetakestheformofawalljetmore sothanaboundarylayer.Likecurvaturenoise,themeanvelocityoutsidetheboundary layeristakenasthelocalmaximumvelocity,andthedisplacementthicknessiscomputed usingEquation4{6. ThemeantangentialvelocityprolemeasuredusingPIVisshowninFigure4-33. Themaximumvelocityisfoundtobe U s = U max =41.4m/s.Becausethereissucient dataintheprole,thedisplacementandmomentumthicknessesarecomputedusingthe 154

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dataalone.ThefrictionvelocityisestimatedusingtheKarmanintegralrelation.The displacementthicknessis0.140mm,andthefrictionvelocityis2.78m/s. 4.4.2.3Slot-jetinteractionnoise Slot-jetinteractionnoiseistheorizedbyHowe2002tobegeneratedbythe interactionofturbulenceinthejetwiththeslotlipandlowerslotsurface,i.e.thetrailing edgeinthevicinityoftheslot.Toestimatetheowscalesforthisnoisesource,themean tangentialvelocityprolejustdownstreamoftheslotexitplaneisevaluated.Themean velocityisonceagaintakentobethelocalmaximumwalljetvelocity.Becausetheupper portionoftheproleisinuencedbythelipwake,thedisplacementandmomentum thicknessesareanalyzedatthelowerCoandasurfaceusingtheone-seventhpowerlawof Equation4{5.TheKarmanintegralrelationisagainusedtoestimatethefrictionvelocity. ThemeantangentialvelocityproleattheslotexitisshowninFigure4-34.The maximumvelocityisfoundtobe U j = U max =88.8m/s,thedisplacementthicknessis estimatedas0.025mm,andthefrictionvelocityisapproximatedtobe6.84m/s. 4.4.3ComparisonwithMeasurement NowthatscaleshavebeenestimatedfromthePIVdata,theywillusedtobecompare Howe'smodeltoameasuredspectrum.Forcomparisonpurposes,rst,Howe'smodelis assessedusingthescalesitestimatesgivenjustthemodelgeometry,freestreamvelocity, andjetvelocity.Specically,thescalesarethefreestreamvelocity, U ,thejetvelocity U jet thesemi-majoraxisoftheairfoilellipticprole, a ,thetrailingedgeradius, r ,theslot height, h ,andthelipthickness, l U =18 : 9m/s ;U jet =80 : 6m/s ;a =0 : 2606m r =0 : 0222m ;h =1 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(3 m ;l =2 : 7 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(4 m ThespectrumproducedbyHowe'smodelfortheseinputsisshowninFigure4-35along withthespectrumobtainedfromthearraymeasurement.Thespectraarepresentedin theformgivenbyEquation4{4.Forthepresentcase,valuesfortheseconstantsaregiven 155

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below. 0 =1 : 21kg/m 3 ; j ~x j =1 : 12m ;M = U=c 0 =0 : 055 = = 2rad ; = = 2rad Recall,whenappliedtothearrayspectrum, L s ,thesourcelength,isfrequency-dependent andbasedonthearray's3dBbeamwidth.ForthepredictedspectruminFigure4-35, L s =1.12m. AsevidentinFigure4-35,therearesignicantdierencesbetweenthemeasured spectraandHowe'smodel.However,recallHowe'sspectrumiscomputedusingonly estimatesofthelengthandvelocityscales.ThesescalesarelistedinTable4-1and comparedwiththosedeterminedfromthePIVmeasurements.Notsurprisingly,thereare largedierencesbetweenthescales,especiallythoseforcurvaturenoise.Recall,Howe 2002considerscurvaturenoisetobeproducedbyboundarylayerturbulence,whereasin theactualow,thereisawalljetinsteadofaconventionalboundarylayerpassingover thetrailingedge.Iftheinnerregionofthewalljetistakentobetheboundarylayer,then themeanandfrictionvelocitieswillcertainlybehigher,andthedisplacementthickness willbesignicantlysmaller. WhenthescalesestimatedfromthePIVmeasurementsaresubstitutedintoHowe's model,thepredictedspectrumisquitedierent,asshowninFigure4-36.Thedierences betweenthemeasuredandpredictedspectra,though,arestillsignicant.Theshaded regionspresentedwitheachpredictedspectrumrepresentuncertaintybounds,estimated using10,000iterationsofaMonte-Carlosimulation.Thefrictionvelocities,displacement thicknesses,meanvelocities,slotheight,lipthickness,andfreestreamvelocityareall perturbedwitheachiteration.Theperturbationstothefrictionvelocities,displacement thicknesses,andmeanvelocitiesarethemselvesdeterminedfrom10,000iterationsof Monte-CarlosimulationsbasedonthePIVmeasurementuncertainty.Evenconsidering alluncertainties,thedierencesbetweenthepredictedandmeasuredspectraarelarge. 156

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However,someusefulinformationmaystillbeextractedfromHowe'smodel.According tothemodel,passive-slotnoiseisthedominantnoisesourceforfrequenciesbelow20kHz, andslot-jetinteractionnoiseistheprimarysourceathigherfrequencies. Finally,anattemptismadetodeterminevaluesfortheowscalesinorderforHowe's modelspectrumtoapproachthemeasuredarrayspectrum.Howe'smodelisttothe measuredspectrumusingaleastsquaresapproach,andtheowscalesthatprovidethe besttaredetermined.Theresultsfromtheleastsquarestaredependentontheinitial guessandtheboundssetforeachparameter.Theinitialguessesfortheparametersare themeasuredvaluesofthescaleslistedinTable4-1.Thelowerboundsforallparameters aresetto0.Theupperboundsforthemeanvelocitiesaresetto100m/s,theupper boundsforthefrictionvelocitiesaresetto20m/s,andthedisplacementthicknessbounds forthewalljetandlipboundarylayerprolesaresetto0.5mmand1mm,respectively. ThebesttspectrumiscomparedtothemeasuredspectruminFigure4-37,wherethe twoareshowntobeincloseagreement.Noticethereisnocontributionfromslot-jet interactionnoise.Toreiterate,theresultantbesttparametersaredependentonthe initialguessesandbounds,sothissolutionisprovidedsimplyforillustrativepurposes. ThescalesdeterminedfromthebesttarelistedinTable4-2.Therearegenerallylarge variationsbetweenthescalesobtainedfromthePIVmeasurementsandthescaleslistedin Table4-2.Inaddition,someofthescalesareunreasonable,including v =19m/sforthe owpassingoverthelip, =0.5mmwithacorresponding v =20m/sforthewalljet owresponsibleforcurvaturenoise,and =1.0 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(5 mmforthedisplacementthickness attheslotexit. 4.5Summary Thesoundproducedbyacirculationcontrolairfoilinanopenjetanechoicwind tunnelismeasuredandcharacterizedusingfree-standingmicrophonesandanested phasedacousticarray.Tonesproducedbyvortexsheddingfromtheroundtrailingedge andbluntslotliparedetected.Beamformingmapsindicatethatthetrailingedgeis 157

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thedominantsourceintheabsenceofafreestream,butwithafreestream,therearea multitudeofsources.Atlowmomentumcoecientscorrespondingtolaminarslotjets, contaminatingnoisesources,includingowimpingementnoiseandsidewallscrubbing noise,areobservedtodominate.Athighermomentumcoecients,noisefromthetrailing edgeistheprimarysource,butthesoundpowerisunevenlydistributedalongthetrailing edge.Thebeammapsindicatethattheinterfacesofthesidewallsandtrailingedgearethe primarysources,particularlyathigherfrequencies.SpectracomputedusingtheCOPand three-microphonemethodsarecomparedwithsinglemicrophoneautospectrumandfound tobeincloseagreement,indicatinglittleuncorrelatednoiseinthemeasurements.All threemethodsare,bytheirnature,unabletodistinguishthetwo-dimensionalcirculation controlnoisegeneratedatthetrailingedgewithnoiseproducedbytheaforementioned contaminatingnoisesources.Aspectrumisalsocomputedusingthephasedarrayby integratingoverasmall,frequency-dependentregionofthetrailingedgetominimizethe sidewallendeects.Thisintegratedspectrumiscomparedwithamodelofcirculation controlacousticspresentedbyHowe2002andfoundtodieronaverageby30dB. Howe'smodelsuggeststhat,forthetestconditionsconsidered,passiveslotnoiseisthe dominantnoisesourceforfrequenciesbelow20kHz,andslot-jetinteractionnoiseisthe principalsourceathigherfrequencies. 158

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Figure4-1.PowerspectraldensitymeasuredbymicrophoneM1abovethetrailingedge, Re c =6 : 5 10 5 h=c =0 : 0019. Figure4-2.Allcaseswithtrailingedgevortexsheddingtonesmeasuredbymicrophone M1.Theverticallinecorrespondsto St =0.21. 159

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Figure4-3.SlotlipvortexsheddingtonesmeasuredbymicrophoneM1, C =0.014, h=c = 0.0029. Figure4-4.SlotlipvortexsheddingtonesmeasuredbymicrophonesM1andM4see Figure2-9, Re c =1 : 3 10 6 C =0.014, h=c =0.0029. 160

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Figure4-5.SpectraHzbinwidthmeasuredbythecentermicrophoneabovethe trailingedge,SSB, U 1 =0, h=c =0.0019.Theverticallinesrepresentthe frequencieswherebeammapsareproduced. Figure4-6.Samplebeammapwithairfoilandtunnelcomponentslabeled. 161

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A B C D E F Figure4-7.OuterarraybeammapsdBat992Hz,SSB, U 1 =0, h=c =0.0019.A ArrayPSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet = 5300.F Re jet =6600. 162

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A B C D E F Figure4-8.OuterarraybeammapsdBat2kHz,SSB, U 1 =0, h=c =0.0019.AArray PSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet =5300. F Re jet =6600. 163

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A B C D E F Figure4-9.OuterarraybeammapsdBat4kHz,SSB, U 1 =0, h=c =0.0019.AArray PSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet =5300. F Re jet =6600. 164

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A B C D E F Figure4-10.OuterarraybeammapsdBat8kHz,SSB, U 1 =0, h=c =0.0019.A ArrayPSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet = 5300.F Re jet =6600. 165

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A B C D E F Figure4-11.InnerarraybeammapsdBat16kHz,SSB, U 1 =0, h=c =0.0019.A ArrayPSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet = 5300.F Re jet =6600. 166

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A B C D E F Figure4-12.InnerarraybeammapsdBat32kHz,SSB, U 1 =0, h=c =0.0019.A ArrayPSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet = 5300.F Re jet =6600. 167

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A B C D E F Figure4-13.InnerarraybeammapsdBat64kHz,SSB, U 1 =0, h=c =0.0019.A ArrayPSF.B Re jet =1340.C Re jet =2680.D Re jet =3980.E Re jet = 5300.F Re jet =6600. 168

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Figure4-14.SpectraHzbinwidthmeasuredbythecentermicrophoneabovethe trailingedge,SSB, Re c =6 : 5 10 5 h=c =0.0019.Theverticallinesrepresent thefrequencieswherebeammapsareproduced. 169

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A B C D E F G Figure4-15.OuterarraybeammapsdBat992Hz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 170

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A B C D E F G Figure4-16.OuterarraybeammapsdBat2kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 171

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A B C D E F G Figure4-17.OuterarraybeammapsdBat4kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 172

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A B C D E F G Figure4-18.OuterarraybeammapsdBat8kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 173

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A B C D E F G Figure4-19.InnerarraybeammapsdBat16kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 174

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A B C D E F G Figure4-20.InnerarraybeammapsdBat32kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 175

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A B C D E F G Figure4-21.InnerarraybeammapsdBat64kHz,SSB, Re c =6 : 5 10 5 h=c =0.0019. AArrayPSF.B C =0.C C =0 : 004 Re jet =1330.D C =0 : 017 Re jet =2660.E C =0 : 037 Re jet =4000.F C =0 : 065 Re jet =5300. G C =0 : 10 Re jet =6640. 176

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A B C D E F Figure4-22.OuterarraybeammapsdBat992Hz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 177

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A B C D E F Figure4-23.OuterarraybeammapsdBat2kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 178

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A B C D E F Figure4-24.OuterarraybeammapsdBat4kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 179

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A B C D E F Figure4-25.OuterarraybeammapsdBat8kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BSSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 180

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A B C D E F Figure4-26.InnerarraybeammapsdBat16kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 181

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A B C D E F Figure4-27.InnerarraybeammapsdBat32kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 182

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A B C D E F Figure4-28.InnerarraybeammapsdBat64kHz, Re c =6 : 5 10 5 h=c =0.0019.A SSB, C =0 : 037.BOSB, C =0 : 040.CSSB, C =0 : 065.DOSB, C =0 : 071.ESSB, C =0 : 10.FOSB, C =0 : 11. 183

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Figure4-29.Comparisonoffree-standingmicrophoneprocessingtechniquesforSSB, Re c =6 : 5 10 5 C =0.11,and h=c =0.0019Hzbinwidth. Figure4-30.Comparisonoffree-standingmicrophoneprocessingtechniquesforOSB, C = 0.10, Re c =6 : 5 10 5 ,and h=c =0.0019Hzbinwidth. 184

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Figure4-31.SinglemicrophoneandintegratedarrayspectraHzbinwidthforSSB, C 0 : 065, Re c =6 : 5 10 5 ,and h=c =0.0019. Figure4-32.Meantangentialvelocityprolealongtrailingedgeat r=c =0.0129, Re c =6 : 5 10 5 C =0.057, h=c =0.0019withone-seventhpowercurve insetPrandtl1961. 185

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Figure4-33.Meantangentialvelocityproleatslotlipedge, Re c =6 : 5 10 5 C =0.057, h=c =0.0019.Noticethelocalmaximumvelocity.Onlyeveryotherpointis displayedforclarity. Figure4-34.Meantangentialvelocityproleatslotexitwithone-seventhpowercurve, Re c =6 : 5 10 5 C =0.057, h=c =0.0019Prandtl1961. 186

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Figure4-35.Howe'smodel,evaluatedfor Re c =6 : 5 10 5 C =0.057, h=c =0.0019, providingonlyairfoilgeometrydetailsandtestconditions.Thearray spectrumrepresentsSSB, Re c =6 : 5 10 5 C =0.065, h=c =0.0019. Figure4-36.Howe'smodel,evaluatedfor Re c =6 : 5 10 5 C =0.057, h=c =0.0019using lengthandvelocityscalesestimatedfromPIVdata.Thearrayspectrum representsSSB, Re c =6 : 5 10 5 C =0.065, h=c =0.0019. 187

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Figure4-37.BesttofHowe'smodeltomeasuredspectrum. Table4-1.LengthandvelocityscalesusedtoevaluateHowe'smodel PredictedHowe2002 PIVMeasurement Noisetype U m/s mm v m/s U m/s mm v m/s Curvature 56.31.041.97 91.30.0109.57 Passiveslot 32.01.041.12 41.40.1402.78 Slot-jet 79.90.0192.80 88.80.0256.84 Table4-2.ExampleofscalesrequiredforHowe'smodeltomatchmeasurement Noisetype U m/s mm v m/s Curvature 63.10.520.0 Passiveslot 59.80.1719.0 Slot-jet 88.81.0 10 )]TJ/F21 7.9701 Tf 6.586 0 Td [(5 7.60 188

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CHAPTER5 CONCLUSIONSANDFUTUREWORK Theclosingchapterofthisdissertationsummarizesthekeyndingsofthisinvestigation andprovidessuggestionsforfuturework.Keyndingsarecategorizedbasedontheir primaryrelevancetoeitheruiddynamicsoracoustics. 5.1KeyFindings 5.1.1FluidDynamics Midspansurfacepressuremeasurementsofanellipticcirculationcontrolairfoil intheUFAFFrevealtheabsenceofaleadingedgesuctionpeakregularlyobservedin circulationcontrolexperimentsandnumericalstudies.Thelackofaleadingedgesuction peakcontributestoadecitintheliftproducedcomparedwithpriorexperimentsof thesamegeometryinclosedtestsectionwindtunnels.Theleadingedgesuctionpeak isrecoveredbyenclosingthewindtunneltestsection,andtheliftdecitiseliminated. However,thesurfacepressuredistributionishighly-dependentontheboundaryconditions enforcedbythesuction-sidetunnelboundary.Measuredsurfacepressuredistributions arecomparedwithpotentialowtheoryforowaroundanellipseusingtheconformal mappingtechnique.Thetheoryprovidesinsightintothebehavioroftheowandexplains theelevatedsuction-sideandreducedpressure-sidesurfacepressuremagnitudesmeasured witharigidversusporoussuction-sidetunnelboundary.Inaddition,thetheoryalso supportstheobservationsthatthepressure-sidetunnelboundaryisnearlyirrelevantin determiningthesurfacepressuredistribution. PIVmeasurementsrevealtheextenttowhichenclosingthetestsectionsignicantly modiestheleadingedgeoweld.Inanopenjettestsection,leadingedgestagnation pointmovementisminimalwithincreasingmomentumcoecient.Inaclosedtestsection, leadingedgestagnationpointmovementisconsiderablymoresignicantasthemomentum coecientincreases.Contrarytotheleadingedgeobservations,PIVmeasurementsreveal negligibledierencesintheboundarylayerowpassingovertheslotlipwhenthetest 189

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sectionisopenedorclosed.Thescatteringofturbulentpressureuctuationsinthis boundarylayerotheslotlipisoneofthenoisesourcestheorizedbyHowe2002.Hence, thisnoisesourcemechanismshouldremainlargelyunchangedregardlessofwhetherthe testsectionisenclosedornot.Thisndingcanprovidedirectiontofutureresearchers consideringcirculationcontrolexperimentsinopenorclosedwindtunnelswhenacoustic measurementsmaybeofinterest. AdditionalPIVmeasurementsfocusonthecurvedwalljetowanditssimilarity. Althoughasimilaritysolutionindicatesthatfullowsimilarityisonlyachievableifthe curvedsurfacetakestheshapeofalogarithmicspiral,theouterregionsofthemean tangentialvelocityprolesdoindeedexhibitsimilarityusingscalesbasedonthemaximum velocityandtheReynoldsstressormeanshearLaunder&Rodi1983.Thelengthand velocityscalesrequiredforsimilarityaremeasuredforacollectionoftestcaseswherethe chordReynoldsnumber,momentumcoecient,andslotheightarevaried.Thedatafor thesescalesisassembledandfoundtocollapse,andtheresultantbest-tequationsof thescalesareafunctionoftheproductofthechordReynoldsnumberandmomentum coecient.Thisso-calledReynoldscorrectedmomentumcoecientcanalsoberecastas aproductofthejetReynoldsnumberandjet-to-freestreamvelocityratioseeEquation 3{40indicatingthat,unlikethecaseoftheplanarwalljetinafreestream,thelength andvelocityscalesdescribingthecurvedwalljetowinafreestreamaredependenton thejetReynoldsnumberZhou&Wygnanski1993.Theequationsforpredictingthese lengthandvelocityscales,givenbyEquations3{36through3{39,comparefavorablywith measurements. FlowseparationisalsoassessedusingPIV,and,likethelengthandvelocityscales oftheow,theseparationlocationisafunctionoftheReynoldscorrectedmomentum coecient.Anequationforpredictingtheseparationlocationisprovidedandfoundto matchthepresentmeasurementswithexcellentaccuracy.However,theequationmayonly bevalidfortheairfoilgeometryinvestigated,asusingittopredicttheseparationlocation 190

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foradierentgeometrystudiedbyNovak&Cornelius1986andNovak etal. 1987 leadstolargeerrors.Regardingtheseparationmechanism,crossowPIVmeasurements uncoveredstreamwisepairsofcounter-rotatingvorticesonlypreviouslymeasuredina curvedwalljetintheabsenceofafreestreamLikhachev etal. 2001;Neuendorf etal. 2004;Han etal. 2006.Thesevortices,whichpull"high-momentumuidawayfromthe surface,arethoughttobeultimatelyresponsibleforowseparationNeuendorf etal. 2004;Han etal. 2006.Ifthatisthecase,thenthesevorticesmayprecludetheuseof serratedliptreatmentstoreducelipvortexsheddingtones,sincetheserrationsmay stimulateandadvancetheearlierdevelopmentofthesevorticesintheow. 5.1.2Acoustics Acousticmeasurementsrevealthepresenceoflowandhighfrequencytonesassociated withvortexsheddingfromtheroundtrailingedgeandbluntslotlip,respectively.The lowfrequencytonecanbeeliminatedwithsingleslotblowingprovidedthemomentum coecientisatleast0.002.Theemergenceandbehaviorofthehighfrequencytonesis somewhatsporadic,asnocleartrendsbasedonowparameterslikethechordReynolds number,momentumcoecient,jetReynoldsnumber,andslotheight-to-chordratioare found.However,theliptonesareonly,butnotnecessarily,producedwhenboththechord andjetReynoldsnumbersarenonzero.Also,thetonesarehardlydiscerniblefromthe broadbandnoisewhenmeasuredontheoppositesidefromtheblowingslot,indicating thatthesehighfrequencytonesarehighlydirective. Broadbandnoisesourcesareidentiedusinganestedphasedmicrophonearray. Withoutafreestream,thetrailingedgeisthedominantsoundsourceprovidedthe jetReynoldsnumberissucientlyhightocreateaturbulentjet.Withafreestream, however,manysourcesaremeasured.AtthechordReynoldsnumbertested, Re c = 6 : 5 10 5 ,contaminatingnoisefromsidewallscrubbingandowimpingementarefound todominateatlowmomentumcoecientsorjetReynoldsnumbers.Inaddition,leading edgehorseshoevorticesformedatthesidewall-modeljunctionsdominateatfrequencies 191

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below8kHzfor C < 0.017,althoughtheirlevelsarelowrelativetotheothersources identied.Athighermomentumcoecients,thetrailingedgeisthedominantsource. However,thesoundisnotevenly-distributedalongthetrailingedge.Instead,sources aremeasuredatthetrailingedge-sidewalljunctions,likelycausedbytheinteraction ofinducedstreamwisevorticeswiththesidewalls,andnear37%span,theoriginof whichisunknownbutmaybeduetotheseambetweentwoairfoilcomponents.Array measurementsoftestsutilizingtheoppositeblowingslotfromthearrayindicatethat atfrequenciesof8kHzandhigher,thetrailingedge-sidewalljunctionsaretheprimary sources. Thepresenceofmultiplesourcesmakestheestimationofbroadbandnoiseproduced bythecirculationcontrolairfoilextremelydicult.Spectracomputedusingdierent microphoneprocessingtechniques,includingsinglemicrophoneautospectrum,coherent outputpower,andthethree-microphonemethod,arefoundtobeincloseagreement. However,sincethearraydetectsmultiplesourcesoverallfrequenciesofinterest,the resultsfromthemicrophoneprocessingtechniquesareallinvalidsincethesingle-source assumptionisviolated.Thus,thesemethodsarenotsuitableformeasurementsof two-dimensional"circulationcontrolnoise,includingthesourcestheorizedbyHowe 2002.Arrayintegrationtechniquesprovidetheonlymeanstoestimatethesoundnot attributedtotheundesirednoisesources,butmoreworkisneededtoperfectcalibration techniques,andassessandreducemicrophonescatteringeects. Finally,Howe'smodelofcirculationcontrolacousticsisevaluatedat Re c =6 : 5 10 5 C =0 : 057,and h=c =0.0019wherecirculationcontrolrelatednoiseisdominantHowe 2002.Themeanvelocity,frictionvelocity,anddisplacementthicknessscalesrequiredby Howe'smodelareestimatedfromthePIVmeasurementsofthetrailingedgeoweld. Usingthesescales,largedierences 30dBareobservedbetweenHowe'smodeland abroadbandspectrumobtainedfromarraymeasurements.Howe'smodelindicatesthat, atthetestconditionsevaluated,whichcorrespondtoalowchordReynoldsnumberand 192

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amoderatemomentumcoecient,thedominantnoisemechanismforfrequenciesbelow 20kHzistheturbulentfreestreamboundarylayerowpassingovertheslotlip.The interactionofturbulenceintheslotjetwiththeslotlipisthedominantsourceathigher frequencies.Noiseproducedbytheturbulentowpassingovertheroundtrailingedgeis negligible. 5.2ResearchImpact Theinitialgoalsofthisresearchinvestigationplacedanemphasisonidentifyingand characterizingthetwo-dimensionalnoisesourcemechanismsofacirculationcontrolairfoil. However,theratherunexpectedresultsofthisstudyindicatethatthree-dimensionalnoise sourcesmaybeofmoreinterestand,fromanacousticsperspective,aprimarydeterrentto theapplicationofcirculationcontroltounderwatervehicles.Thenoisesourcesidentied atthejunctionsbetweenthetrailingedgeandsidewallsaresignicantatfrequenciesofat least4kHzandhigher.Thesenoisesources,nottheoriginaltwo-dimensionalnoisesources considered,warrantfurtherinvestigation. Thendingsofthisinvestigationalsohighlightarecurringthemeinexperimental aeroacoustics-quantifyingthesourceofinterestisoftenextremelydicult.Thenoise producedbythebaselineorlowslotblowingowsisheavilycontaminatedbysidewall scrubbingandowimpingementnoise.Theseundesiredsourcesmustbeeliminatedifthe noiseproducedbycirculationcontrolistobeassessedatsuchtestconditions.Ofcourse, thatisnotastraightforwardtask,andoptionsarelimitedbasedontheowfacility andthemodelbeingtested.Forexample,ifsidewallsareeliminated,otherundesired sourcesorthree-dimensionaleectswouldbeintroduced.Perhapsasidewallsuction system,ifimplementedinaquietfashion,couldeliminatethethree-dimensionaleects withoutintroducingadditionalnoise.Asformeasurementtechniques,themultiplesources identiedbythephasedacousticarrayprohibittheuseofmultiplemicrophone-based methodsifthetwo-dimensionalsoundsourcesareofprimaryinterest.However, thesemethodsmightbesuitableinthestudyofthethree-dimensionalnoisesources. 193

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Otherwise,array-basedmethodsprovidetheonlypossiblemannerbywhichtostudythe two-dimensionalsourcesofinterest. 5.3RecommendationsforFutureWork Sincethisinvestigationhasidentiedsubstantialthree-dimensionalnoisesource mechanisms,futurewindtunnelmeasurementsshouldfocusonthree-dimensionalmodels similartoanunderwatervehiclecontrolsurface.Ascaledsemi-spancirculationcontrol wingshouldbemountedtoasectionofamodelvehiclehullandtestedinananechoic windtunnel.Furthermore,byremovingtheporoustunnelwalls,sidewallscrubbingnoise willbeeliminated.Instead,therelativestrengthofthesoundproducedbythehull-trailing edgenoisesourcecanbeevaluated,andotherthree-dimensionaleectsrepresentative ofanactualunderwatervehiclecanbegauged.Inaddition,futuremodelsshouldbe constructedinasfewcomponentsaspossible,eliminatingthelikelihoodthatmodelseams orjointscaninuencethemeasurements. Additionalworkisneededtofullycharacterizetheemergenceandbehaviorofthe high-frequencyliptones.Thesetonescouldbeasignicantimpedimenttotheapplication ofcirculationcontroltounderwatervehicles.Sincethetoneshavebeenexperimentally veried,lipmodicationsaimedatreducingtheirlevels,likethosesuggestedbySlomski 2009,shouldbeexperimentallyevaluated.Furthermore,theeectofsuchmodications onowseparationmustbeadequatelyassessed. ThepresentinvestigationutilizesPIVtomeasurethelengthandvelocityscales requiredforHowe'smodel,butPIVisunabletomeasureextremelyclosetosurfaces, whichisessentialforhighly-accuratedisplacementthicknessandfrictionvelocityestimates ofthecurvedwalljetHowe2002.Additionalmeasurementsarerecommendedon larger-scalemodelsinlargertestfacilities,sothatspatialresolutionissubsequently improved.Hot-wireanemometryshouldalsobeconsideredinplaceofPIVformeasuring near-wallvelocityproles,providedtheowisnominallyunidirectional.Microelectromechanical systemsMEMSshearstresssensors,liketheonedevelopedbyChandrasekharan2009, 194

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shouldbeconsideredforinstallationalongtheCoandasurface.Althoughpackaging suchdevicesmaybedicultonahighly-contouredsurface,theycouldprovideaprecise measurementofthefrictionvelocity.Alternatively,globalinterferometricskin-friction measurementsoftheCoandasurfaceandlipcouldalsoprovideaccuratefrictionvelocity informationNaughton&Sheplak2002.Highly-accurateowmeasurementsatavariety oftestconditionsareessentialtoconrmunderwhatconditionspassiveslotnoiseis indeedtheprimarytwo-dimensionalnoisesource. Finally,furtherworkisneededtosupporttheuseofarray-basedmethodsin aeroacousticmeasurements.Thesemethodsareparticularlychallengingwhen,likein thepresentinvestigation,highfrequencysourcesmustbeconsidered.Higherfrequencies motivatetheneedforsmalleraperturearrays,butsmalleraperturearraysintroduce signicantmicrophonescatteringeects.Eitherthescatteringeectsneedtobe characterized,orasuitabletreatmentneedstobefoundtomitigatescattering.The lattermaynotbetooeectiveinsomecircumstances,likethepresentinvestigation, wherethedistanceseparatingmicrophonesislessthanahalf-wavelengthatthehighest frequencyofinterest.Instead,anarrayofMEMSmicrophonesshouldbeconsidered forfutureexperiments.Suchanarraywouldpermitclosersensorspacingandprovide reducedscatteringeects,bothfromneighboringsensorsandthemicrophonesthemselves Underbrink2002;Arnold etal. 2003. 195

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APPENDIXA AIRFOILNOTESANDTECHNICALDRAWINGS Thisappendixincludesalltechnicaldrawingsnecessarytoreproducethecirculation controlairfoilusedinthisstudy,picturesofthecompletedairfoil,andnotesonproperly sealingtheairfoil. A.1FabricationPicturesandNotes Asidefromtheslotlips,whicharefabricatedbyTMREngineering,themodelis constructedbyCraigJohnsonattheIllinoisInstituteofTechnologyMechanical,Material andAerospaceEngineeringMachineShop.Picturesofavarietyofnishedpartsare providedinFiguresA-1toA-7. Theairfeedcomponents,showninFigureA-1,areprinted"inaselectivelaser sinteringthree-dimensionalprinterusinganylon-basedpowder.Thetopandbottom leadingedgepiecesareshowninFigureA-2.Themachinedchannelisprovidedforrouting pressuretubingoutofaholeinthesideoftheairfoil.Thetopandbottomleadingedges, surfaceplates,dividerplate,andtrailingedgeassemblyarepiecedtogetherinFigure A-3.AcollectionoftrailingedgeinstrumentringsaredisplayedinFigureA-4.Figure A-5Aillustrateshowthesupportrodissecuredtothemodel.A2.54cmthicksteelblock isboredforapresstwiththesupportrod,anda9.53mmdiameterdowelispressed throughtherodandtheblock.Foradditionalairfeedassemblysupport,abracket,shown inFigureA-5B,surroundsthesupportrodandcouplestotheairfeedconnectorplate. Long,exible,annealed,stainlesssteelpressuretubulations.711mminnerdiameter areusedforthepressuretapsinstalledinthesurfaceplates x=c =0.22,0.51,and0.75. Theseexibletubesareguidedtowardsasmallholeintheleadingedgealongtheplenum wallsofthesurfaceplates.Oncethesteeltubespassthroughthehole,exibletubing isattachedtotheendofthetubesandtheholeissealedwithsilicone.Thelocationsof thetapsandthesteelpressuretubingpathsaremarkedbyblackarrowsandredlines, respectively,inFigureA-6A.Picturesoftheinstalledpressuretubes,includingtheexible 196

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stainlesssteeltubulations,areprovidedinFigureA-6.Thefully-assembledairfoil,shown initsshippingcrateinFigureA-7,wasdeliveredinDecember2007. A.2AirfoilSealing Afterdelivery,theairfoiliscompletelydisassembled,inspected,andthencarefully reassembledusingsealantsandgasketstopreventairleakagefromtheplenum.Thesteps takentoensurepropersealingaredescribedinthissection. First,allboltsscrewedinholesthatprovideaconnectionbetweenthetwoplenums ortheinsideofthemodelanditssurroundingsarewrappedinTeontapeorcoated withTeonpipethreadcompound.Second,abeadofahardeningsilicone-basedsealant forexample,anygenericsiliconeaquariumsealantisappliedattheinterfaceofthe leadingedgeandsurfaceplatepiecesasshowninFigureA-8.Foranon-permanent, metal-to-metalsealalongthecontactingsurfacesofthetopandbottomleadingedges, theinterfacebetweentheleadingedgesanddividerplate,andtheinterfacesbetweenthe trailingedgeassemblycomponentse.g.dividerplateextensions,anon-hardeningrubber gasketsealantisused.FigureA-9showstheapplicationofPermatexUltraRubberGasket Sealant&Dressingtotheinterfacebetweenthedividerplateandleadingedge.The contactsurfacesofthetrailingedgeinstrumentringsarecleanedandlubricatedwitha siliconeandPTFElubricant-sealantforeasyrotation.Finally,rubbergasketsarecreated from1.59mmthickrubbersheetsforeachairfeedassemblycomponentandtheelliptical sideplates.TheairfeedassemblycomponentsandtheirgasketsaredisplayedinFigure A-10.AthinlayerofthePermatexgasketdressingisappliedtoeachsideofthegaskets, asshowninFigureA-11. A.3TechnicalDrawings Technicaldrawingsforallcirculationcontrolairfoilcomponentsareattachedatthe endofthisappendix. 197

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FigureA-1.Airfeedassemblycomponentsprintedinaselectivelasersinteringmachine. FigureA-2.Topandbottomleadingedgepieces. 198

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FigureA-3.Airfoiltakingshapeaftercompletionofleadingedge,surfaceplates,divider plate,andtrailingedgeassembly. FigureA-4.Blanktrailingedgeinstrumentrings. 199

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A B FigureA-5.Supportrodandairfeedmountingschemes.ASupportrodsecuredto mountingblockviapresst.BAirfeedassembledandsecuredtosupport rodbracket. 200

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A B C FigureA-6.Pressuretapsandtubing.ARedlinesindicatepathsforexiblesteel pressuretubing.BFlexiblesteelpressuretubesinstalled.CCloseupview ofpressuretubing. 201

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FigureA-7.PictureofnishedairfoilupondeliverytoUF. FigureA-8.Pictureofabeadofahardeningsilicone-basedsealantappliedalongtheseam betweentheleadingedgeandsurfaceplate. 202

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FigureA-9.Pictureofanon-hardeningrubbergasketsealantappliedalongmetal-to-metal contactsurfacesforanon-permanentseal. FigureA-10.Airfeedcomponentsandcorrespondinggaskets. 203

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FigureA-11.Arubbergasketsealantisappliedtoeachsurfaceofthegaskets. 204

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FigureA-12.Circulationcontrolairfoilexteriorassemblyview. 205

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FigureA-13.Circulationcontrolairfoilinteriorassemblyview. 206

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FigureA-14.Midspanpressuretaps. 207

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FigureA-15.Topleadingedge. 208

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FigureA-16.Bottomleadingedge. 209

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FigureA-17.Topsurfaceplate. 210

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FigureA-18.Bottomsurfaceplate. 211

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FigureA-19.Topsideplatelipassembly. 212

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FigureA-20.Topleftsideplatelip. 213

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FigureA-21.Topcentersideplatelip. 214

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FigureA-22.Toprightsideplatelip. 215

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FigureA-23.Bottomsideplatelipassembly. 216

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FigureA-24.Bottomleftsideplatelip. 217

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FigureA-25.Bottomcentersideplatelip. 218

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FigureA-26.Bottomrightsideplatelip. 219

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FigureA-27.Dividerplate. 220

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FigureA-28.Trailingedgeassembly. 221

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FigureA-29.Dividerplateextensionassembly. 222

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FigureA-30.Dividerplateextension. 223

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FigureA-31.Dividerplateextension. 224

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FigureA-32.Dividerplateextension. 225

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FigureA-33.Dividerplateextension. 226

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FigureA-34.Longtrailingedge. 227

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FigureA-35.Shorttrailingedge. 228

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FigureA-36.Trailingedgeinstrumentplug. 229

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FigureA-37.Trailingedgeinstrumentplugwithpressuretaps. 230

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FigureA-38.Sideplateassembly. 231

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FigureA-39.Leftsideplate. 232

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FigureA-40.Leftsideplate. 233

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FigureA-41.Rightsideplate. 234

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FigureA-42.Rightsideplate. 235

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FigureA-43.Airfeed. 236

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FigureA-44.Airfeedtransition. 237

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FigureA-45.Airfeedconnectorplate. 238

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FigureA-46.Supportrod. 239

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FigureA-47.Supportrodextension. 240

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FigureA-48.Supportrodmountingbracket. 241

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FigureA-49.Supportrodconnectingplate. 242

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APPENDIXB IDEALFLOWOVERANELLIPSE Thissectionpresentsananalysisofidealincompressible,irrotational,inviscid liftingowoveraninnitecylinderusingcomplexvariabletheoryandthemappingof itssolutiontoanellipse.First,areviewofthestreamfunction,velocitypotential,and complexvariabletheoryispresented.Next,theanalyticalsolutionforliftingowovera cylinderisderived.Followingabriefintroductiontoconformalmapping,resultsfroma MATLABprogramdesignedtoanalyzeowoveranellipsearepresented.Thesourcecode isattachedattheendofthissection. B.1StreamFunctionandVelocityPotential Inatwo-dimensional,incompressibleow,thestreamfunction isdenedasfollows Panton2005. u = @ @y B{1 v = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(@ @x B{2 Thestreamfunctionsatisescontinuity.Iftheowisalsoirrotational,thestreamfunction satisesLaplace'sequation, r 2 =0. Assumingirrotationalow,avelocitypotential canbedened. u = @ @x B{3 v = @ @y B{4 ThevelocitypotentialalsosatisesLaplace'sequation, r 2 =0,iftheowisincompressible. Thus,fortwo-dimensionalidealow,boththestreamfunctionandvelocitypotential satisfyLaplace'sequation. B.2ComplexVariableTheory Complexvariabletheorycanbeusedtosolvethetwo-dimensionalLaplace'sequation. Consideraowinthe z -plane,where z = x + iy = re i .Thecomplexpotential F z is 243

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denedas F z = x;y + i x;y : B{5 Thecomplexvelocity W z canthusbefound. W z = dF dz = @ @x + i @ @x = @ @y )]TJ/F22 11.9552 Tf 11.955 0 Td [(i @ @y B{6 W z = u )]TJ/F22 11.9552 Tf 11.955 0 Td [(iv = qe )]TJ/F23 7.9701 Tf 6.587 0 Td [(i B{7 Essentially,the z -planeismappedtothe w -planeusing W z B.3LiftingFlowOveraCylinder Liftingowoveracylinderisrepresentedbythesuperpositionofauniformow, doublet,andlinevortex.Thecomplexpotentialsforauniformowatangle ,adoublet ofstrength k ,andalinevortexwithclockwisecirculation)-327(aregiveninTableB-1.Thus, thecomplexpotentialforliftingowoveracylinderis F z = U 1 ze )]TJ/F23 7.9701 Tf 6.586 0 Td [(i + k z + i )-167(ln z 2 : B{8 Assumingaparallelfreestream, =0,thecomplexpotentialreducesto F z = U 1 z + k z + i )-167(ln z 2 : B{9 B.3.1VelocityPotentialandStreamFunction Thevelocitypotentialandstreamfunctionarefoundbysubstituting z = re i into EquationB{9. F r; = U 1 re i + k re i + i )-167(ln re i 2 = U 1 r cos + i sin + k r cos )]TJ/F22 11.9552 Tf 11.955 0 Td [(i sin + i \050ln r + i 2 = U 1 r + k r cos )]TJ/F15 11.9552 Tf 13.151 8.088 Td [()]TJ/F22 11.9552 Tf 7.314 0 Td [( 2 | {z } + i U 1 r )]TJ/F22 11.9552 Tf 16.241 8.088 Td [(k r sin + )-167(ln r 2 | {z } 244

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Hence,thevelocitypotentialandstreamfunctionforliftingowoveracylinderare = U 1 r + k r cos )]TJ/F15 11.9552 Tf 13.151 8.088 Td [()]TJ/F22 11.9552 Tf 7.314 0 Td [( 2 ; B{10 = U 1 r )]TJ/F22 11.9552 Tf 16.241 8.088 Td [(k r sin + )-167(ln r 2 : B{11 B.3.2VelocityComponents Theradialvelocity v r andtangentialvelocity v are v r = @ @r = U 1 cos 1 )]TJ/F22 11.9552 Tf 27.065 8.088 Td [(k U 1 r 2 ; B{12 v = )]TJ/F22 11.9552 Tf 10.494 8.087 Td [(@ @r = )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 1 sin 1+ k U 1 r 2 )]TJ/F15 11.9552 Tf 18.755 8.087 Td [()]TJETq1 0 0 1 400.079 507.456 cm[]0 d 0 J 0.478 w 0 0 m 18.523 0 l SQBT/F15 11.9552 Tf 400.079 496.267 Td [(2 r : B{13 Onthesurfaceofthecylinder,theradialvelocityiszero.Solvingtheradialvelocityfor r yields v r =0= U 1 cos 1 )]TJ/F22 11.9552 Tf 27.065 8.088 Td [(k U 1 r 2 ; r = r k U 1 = R; B{14 where R istheradiusofthecylinder.Thevelocitycomponentsarerewritteninamore convenientformas v r = U 1 cos 1 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( R r 2 # ; B{15 v = )]TJ/F22 11.9552 Tf 9.298 0 Td [(U 1 sin 1+ R r 2 # )]TJ/F15 11.9552 Tf 18.755 8.088 Td [()]TJETq1 0 0 1 375.612 241.851 cm[]0 d 0 J 0.478 w 0 0 m 18.523 0 l SQBT/F15 11.9552 Tf 375.612 230.662 Td [(2 r : B{16 Recallingthat u = v r cos )]TJ/F22 11.9552 Tf 12.442 0 Td [(v sin and v = v r sin + v cos ,itcanbeshownthatas r !1 u = U 1 and v =0,asexpected.Finally,thevelocityonthesurfaceofthecylinder is v r =0 ; B{17 v = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 U sin )]TJ/F15 11.9552 Tf 20.459 8.088 Td [()]TJETq1 0 0 1 338.003 95.413 cm[]0 d 0 J 0.478 w 0 0 m 21.931 0 l SQBT/F15 11.9552 Tf 338.003 84.223 Td [(2 R : B{18 245

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B.3.3StagnationPoints Thestagnationpointsarelocatedonthesurfaceofthecylinder,sotheirazimuthal positionsarefoundbyevaluatingthetangentialvelocityequationat r = R andsolvingfor v =0= )]TJ/F22 11.9552 Tf 9.298 0 Td [(U 1 sin 1+ R r 2 # )]TJ/F15 11.9552 Tf 18.755 8.087 Td [()]TJETq1 0 0 1 398.465 602.932 cm[]0 d 0 J 0.478 w 0 0 m 18.523 0 l SQBT/F15 11.9552 Tf 398.465 591.743 Td [(2 r 0= )]TJ/F22 11.9552 Tf 9.298 0 Td [(U 1 sin 1+ R R 2 # )]TJ/F15 11.9552 Tf 20.459 8.087 Td [()]TJETq1 0 0 1 398.465 563.081 cm[]0 d 0 J 0.478 w 0 0 m 21.931 0 l SQBT/F15 11.9552 Tf 398.465 551.892 Td [(2 R )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 U 1 sin = )]TJETq1 0 0 1 263.556 527.417 cm[]0 d 0 J 0.478 w 0 0 m 21.931 0 l SQBT/F15 11.9552 Tf 263.556 516.227 Td [(2 R =sin )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 26.261 8.088 Td [()]TJETq1 0 0 1 309.339 497.896 cm[]0 d 0 J 0.478 w 0 0 m 38.848 0 l SQBT/F15 11.9552 Tf 309.339 486.707 Td [(4 U 1 R B{19 Therearethreepossiblesolutionsetsfor illustratedbythestreamlineplotsinFigure B-1.If)]TJ/F22 11.9552 Tf 52.896 0 Td [(< 4 U 1 R FigureB-1A,then hastwosolutionsandtherearetwo stagnationpoints.If)-436(=4 U 1 R FigureB-1B,thenthereisonlyonesolutionfor andconsequentlyonestagnationpointlocatedat =3 = 2.Finally,if)]TJ/F22 11.9552 Tf 77.826 0 Td [(> 4 U 1 R Figure B-1C,then isundened,andonestagnationpointliesinsidethecylinder,whilethe otherisplacedinthefreestream.Whiletheminimumvelocity v r = v =0occursatthe stagnationpoints,themaximumvelocityoccursat = = 2. v ;max = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 U 1 )]TJ/F15 11.9552 Tf 20.459 8.088 Td [()]TJETq1 0 0 1 340.973 282.723 cm[]0 d 0 J 0.478 w 0 0 m 21.931 0 l SQBT/F15 11.9552 Tf 340.973 271.534 Td [(2 R B{20 FromtheEuler-sequation,themaximumvelocityandminimumpressureoccuratthe samelocation.Hence,apressuredierentialexistsbetweentheupperandlowersurfaces ofthecylinder,producingaliftforce. B.3.4PressureCoecient Thepressurecoecient C p atanypointonthecylindersurfaceiscalculatedby applyingBernoulli'sequationbetweenthesurfaceofthecylinder, r = R ,andapointfar 246

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awayfromthecylinder, r !1 p s + 1 2 U 2 s = p 1 + 1 2 U 2 1 B{21 p s + 1 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 U 1 sin )]TJ/F15 11.9552 Tf 20.459 8.087 Td [()]TJETq1 0 0 1 310.466 643.668 cm[]0 d 0 J 0.478 w 0 0 m 21.931 0 l SQBT/F15 11.9552 Tf 310.466 632.479 Td [(2 R 2 = p 1 + 1 2 U 2 1 p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 = 1 2 U 2 1 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4sin 2 )]TJ/F15 11.9552 Tf 23.065 8.088 Td [(2)]TJETq1 0 0 1 317.582 607.404 cm[]0 d 0 J 0.478 w 0 0 m 32.995 0 l SQBT/F22 11.9552 Tf 317.582 596.215 Td [(RU 1 sin )]TJ/F27 11.9552 Tf 11.956 16.857 Td [( )]TJETq1 0 0 1 400.517 607.404 cm[]0 d 0 J 0.478 w 0 0 m 38.848 0 l SQBT/F15 11.9552 Tf 400.517 596.215 Td [(2 RU 1 2 # C p = p s )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 1 1 2 U 2 1 =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4sin 2 )]TJ/F15 11.9552 Tf 23.065 8.088 Td [(2)]TJETq1 0 0 1 316.623 569.158 cm[]0 d 0 J 0.478 w 0 0 m 32.995 0 l SQBT/F22 11.9552 Tf 316.623 557.969 Td [(RU 1 sin )]TJ/F27 11.9552 Tf 11.956 16.857 Td [( )]TJETq1 0 0 1 399.559 569.158 cm[]0 d 0 J 0.478 w 0 0 m 38.848 0 l SQBT/F15 11.9552 Tf 399.559 557.969 Td [(2 RU 1 2 B{22 B.3.5Drag Thedragperunitspanofthecylinderisfoundbyintegratingthesurfacepressure. D 0 = D l = )]TJ/F22 11.9552 Tf 9.298 0 Td [(R 2 Z 0 p s cos d B{23 Afterintegration,thedragisfoundtobe D 0 =0 : B{24 ThisresultisknownasD'Alembert'sParadox:thedragofasymmetricalobjectin anidealowisalwayszero.Sinceviscouseectshavebeenignored,thereisnoow separationandhencenoformorfrictiondrag. B.3.6Lift Theliftperunitspaniscalculatedfrom L 0 = L l = )]TJ/F22 11.9552 Tf 9.299 0 Td [(R 2 Z 0 p s sin d: B{25 TheresultofintegratingEquationB{25isthefamiliarKutta-Joukowskitheorem:inan idealow,theliftperunitspanisequaltotheproductoffreestreamdensity,freestream velocity,andcirculation. L 0 = U 1 )-14275(B{26 247

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B.4ConformalMapping Potentialowoveracylinderisconvenientlytransformedtoowoveranotherbody, suchasaatplate,ellipse,orairfoil,usingconformalmapping.Assumeasliceofthe cylinderisplottedinthe z -plane,where z = x + iy .Thecylinderistransformedinto anothershapeinthe -plane,where = + i ,usingthefollowingmappingfunction, where a isatransformationconstantPanton2005. = z + a 2 z B{27 If a = R ,thecylinderradius,thenthecylinderistransformedtoaatplateof innitesimalthickness.If R>a ,thenthecylinderistransformedtoanellipsewith semi-axes A and B ,where A = R + a 2 R : B{28 B = R )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(a 2 R : B{29 Iftheshapeoftheellipseisknown,thentheseequationsarerewrittentocomputethe transformationconstant a andcorrespondingcylinderradius R R = 1 2 A + B B{30 a = r 1 2 R A )]TJ/F22 11.9552 Tf 11.955 0 Td [(B B{31 Mappingpotentialowoverothersurfaces,suchasJoukowskiairfoilsorcurvedplates,is achievedbyplacingthecenterofthecylinderawayfromtheoriginofthe z -planePanton 2005. SolvingEquationB{14for k andsubstitutingtheresultintoEquationB{9yieldsthe followingformofthecomplexpotentialforliftingowoveracylinder. F z = U 1 z + R 2 z + i )-167(ln z 2 : B{32 248

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Thestreamlinesaroundanellipsearesketchedbyplottingtheimaginarycomponentof F z inthe -plane.Tocomputethepressuredistributiononthesurfaceoftheellipse,the complexvelocity, W z mustbetransformedto W W = dF d = dF dz dz d = W z 1 d dz B{33 Thus,thecomplexvelocityforliftingowoveranellipseis W = U 1 1 )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(R 2 z 2 + i )]TJETq1 0 0 1 361.929 562.467 cm[]0 d 0 J 0.478 w 0 0 m 13.659 0 l SQBT/F21 7.9701 Tf 361.929 555.356 Td [(2 z a 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F21 7.9701 Tf 15.247 4.708 Td [(1 z 2 : B{34 Substituting z = Re i onthesurfaceofthecylinder,thecomplexvelocitybecomes W = U 1 )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + i )]TJETq1 0 0 1 354.825 487.155 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 354.825 480.043 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.115 4.707 Td [(a 2 R 2 : B{35 B.5ResultsandAnalysis AMATLABprogramiswrittentocomputeandplotthestreamlinesandthesurface pressurecoecientdistributionforauser-denedellipticalbody,liftcoecient,and freestreamvelocity.Resultsfromafewsamplecasesarepresentedanddiscussedinthis section.TheparametersinTableB-2arethesameforeachcasediscussed.Notethatthe user-speciedchordisthatoftheUFcirculationcontrolairfoil.Thecorrespondingellipse chordisactually c e =1 : 018405 c TherstexampleisthecaseofzeroliftshowninFigureB-2.Asexpected,the streamlinesindicateleadingandtrailingedgestagnationpointsat =180 and 0 ,respectively.Theupperandlowersurface C p distributionsoverlapsincenoliftis generated.Whenasmallamountofliftisgenerated,asshowninFigureB-3,theleading andtrailingedgestagnationpointsmoveslightlytowardseachother,andsmallsuction peaksappearneartheleadingandtrailingedges.Asliftisfurtherincreasedto C l =1, asshowninFigureB-4,thestagnationpointsmoveclosertogetherandtheleadingand 249

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trailingedgesuctionpeaksbecomewelldened.Finally,when C l =3,asshowninFigure B-5,largeleadingandtrailingedgesuctionpeaksareevident. B.6InuenceofBoundaries Considerthepotentialowaroundacircularcylinderplacedadistance h from aceilingplane.Usingthemethodofimages,animage"cylinderwithcirculationof opposite-sensefromtherealcylinderisplaced"adistance h fromtheoppositesideofthe plane.Thecomplexpotentialforthisowis F z = F real + F image = U 1 z + R 2 z + i )]TJETq1 0 0 1 298.32 531.209 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 298.32 520.019 Td [(2 ln z + U 1 R 2 z )]TJ/F22 11.9552 Tf 11.956 0 Td [(i 2 h )]TJ/F22 11.9552 Tf 13.958 8.088 Td [(i )]TJETq1 0 0 1 414.551 531.209 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 414.551 520.019 Td [(2 ln z )]TJ/F22 11.9552 Tf 11.955 0 Td [(i 2 h : B{36 TheconformalmappingfunctiongivenbyEquationB{27willonlytransformtheow aroundtherealcylinder,leavingtheowaroundtheimagecylinderunalteredand untransformed. Instead,considerthisslightlyalternativeapproachdepictedinFigureB-6.Thereal cylinderiscenteredattheoriginofthe z 1 = x 1 + iy 1 plane,andtheimagecylinderis centeredattheoriginofthe z 2 = x 2 + iy 2 planethatisosetfromthe z 1 planeby i 2 h such that z 2 = z 1 )]TJ/F22 11.9552 Tf 12.404 0 Td [(i 2 h .Thetransformationfortherealcylinderis 1 = z 1 + a 2 =z 1 ,andthe transformationfortheimagecylinderis 2 = z 2 + a 2 =z 2 Thecomplexpotentialfortherealcylinderisgivenby, F 1 z 1 = U 1 z 1 + R 2 z 1 + i )]TJETq1 0 0 1 355.448 268.22 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 355.448 257.031 Td [(2 ln z 1 : B{37 Thecomplexpotentialfortheimagecylinderwithoppositesensecirculationisgivenby, F 2 z 2 = U 1 R 2 z 2 )]TJ/F22 11.9552 Tf 13.958 8.088 Td [(i )]TJETq1 0 0 1 333.454 196.494 cm[]0 d 0 J 0.478 w 0 0 m 12.922 0 l SQBT/F15 11.9552 Tf 333.454 185.305 Td [(2 ln z 2 : B{38 Thegoalistocomeupwithanexpressionforthecomplexvelocityoftheowaround bothcylindersasafunctionof 1 .Fortherealcylinder,thisisstraightforwardandfollows 250

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thestepsoutlinedinSection1. W 1 1 = dF 1 dz 1 dz 1 d 1 = U 1 1 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(R 2 z 2 1 + i )]TJETq1 0 0 1 373.599 675.171 cm[]0 d 0 J 0.478 w 0 0 m 23.092 0 l SQBT/F15 11.9552 Tf 373.599 663.981 Td [(2 z 1 1 1 )]TJ/F23 7.9701 Tf 13.151 4.707 Td [(a 2 z 2 1 B{39 Thecomplexvelocityfortheimagecylinderis W 2 1 = dF 2 dz 2 dz 2 d 2 d 2 d 1 : B{40 Since 2 = 1 )]TJ/F15 11.9552 Tf 10.341 0 Td [(2 ih 0 notethatthedistancebetweenthecylindersandthedistancebetween theellipseswillnotbethesame, d 2 =d 1 =1.Hence,thecomplexvelocityfortheimage cylinderis W 2 1 = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( U 1 R 2 z 2 2 + i )]TJETq1 0 0 1 329.834 494.087 cm[]0 d 0 J 0.478 w 0 0 m 23.092 0 l SQBT/F15 11.9552 Tf 329.834 482.897 Td [(2 z 2 1 1 )]TJ/F23 7.9701 Tf 13.15 4.707 Td [(a 2 z 2 2 : B{41 Next,substitute z 2 = z 1 )]TJ/F15 11.9552 Tf 13.091 0 Td [(2 ih ,andthenwritebothcomplexvelocitiesintermsof z 1 = Re i W 1 1 = U 1 )]TJ/F22 11.9552 Tf 5.479 -9.683 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + i )]TJETq1 0 0 1 358.159 411.007 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 358.159 403.896 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.115 4.707 Td [(a 2 R 2 B{42 W 2 1 = )]TJ/F15 11.9552 Tf 42.276 8.088 Td [(1 1 )]TJ/F23 7.9701 Tf 32.107 4.707 Td [(a 2 Re i )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 ih 2 U 1 R 2 Re i )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 ih 2 + i )]TJETq1 0 0 1 377.43 366.977 cm[]0 d 0 J 0.478 w 0 0 m 75.225 0 l SQBT/F15 11.9552 Tf 377.43 355.787 Td [(2 Re i )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 ih B{43 Finally,thecomplexvelocityonthesurfaceoftherealellipseis W 1 = W 1 1 + W 2 1 : B{44 Ifagroundplaneispresentinsteadofaceilingplane,thentheonlychangetothe previousanalysisisthat z 2 = z 1 +2 ih and 2 = 1 +2 ih 0 .Thepresenceofbothground andceilingplanesisaccountedforbyaninnitenumberofimageellipses,sinceeach imageitselfisreectedbytheadditionalplaneKatz&Plotkin2001.Thus,thecomplex velocityonthesurfaceoftherealellipsebetweentwoplanesisgivenbythefollowing 251

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summation. W 1 = U 1 )]TJ/F22 11.9552 Tf 5.48 -9.683 Td [(e i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + i )]TJETq1 0 0 1 242.605 684.017 cm[]0 d 0 J 0.478 w 0 0 m 15.81 0 l SQBT/F21 7.9701 Tf 242.605 676.906 Td [(2 R e i e i 2 )]TJ/F23 7.9701 Tf 14.115 4.707 Td [(a 2 R 2 )]TJ/F25 7.9701 Tf 16.355 14.944 Td [(1 X n =1 1 1 )]TJ/F23 7.9701 Tf 36.596 4.707 Td [(a 2 Re i )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 nih c 2 U 1 R 2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 nih c 2 + )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 n +1 i )]TJETq1 0 0 1 373.351 637.978 cm[]0 d 0 J 0.478 w 0 0 m 86.378 0 l SQBT/F15 11.9552 Tf 373.351 626.789 Td [(2 Re i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 nih c )]TJ/F25 7.9701 Tf 16.355 14.944 Td [(1 X n =1 1 1 )]TJ/F23 7.9701 Tf 36.596 4.707 Td [(a 2 Re i +2 nih c 2 U 1 R 2 Re i +2 nih c 2 + )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 n +1 i )]TJETq1 0 0 1 373.157 598.767 cm[]0 d 0 J 0.478 w 0 0 m 86.184 0 l SQBT/F15 11.9552 Tf 373.157 587.578 Td [(2 Re i +2 nih c B{45 ThecomplexowMATLABcodeforowoveraellipsenearboundariesisalso attachedattheendofthisappendix. A B C FigureB-1.Streamlinesforliftingowoveracylinder.A)]TJ/F22 11.9552 Tf 314.722 0 Td [(< 4 UR .B)-277(=4 UR .C )]TJ/F22 11.9552 Tf 10.635 0 Td [(> 4 UR 252

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FigureB-2.Streamlinesand C p distribution, U 1 =20m/s, C l =0. FigureB-3.Streamlinesand C p distribution, U 1 =20m/s, C l =0.5. 253

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FigureB-4.Streamlinesand C p distribution, U 1 =20m/s, C l =1. FigureB-5.Streamlinesand C p distribution, U 1 =20m/s, C l =3. 254

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FigureB-6.Schematicofmethodofimagesappliedtoliftingowoveranellipse. TableB-1.Complexpotentialfunctionsforcomponentsofliftingowoveracylinder Component ComplexPotential UniformFlow U 1 ze )]TJ/F23 7.9701 Tf 6.587 0 Td [(i Doublet k z LineVortex i )-177(ln z 2 TableB-2.MATLABcodeparameters Parameter Value Chord, c 0.5207m Thickness-to-ChordRatio 0.2 FreestreamVelocity, U 1 40m/s 255

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MATLABCode-EllipseinFreestream %%CirculationControlIdealFlowSolver %Description:Givenauser-definedellipsegeometry,liftcoefficient, %andfreestreamvelocity,streamlinesandaCpdistributionforan %ellipticalbodyareproducedusingtheanalyticalsolutiontoideal %liftingflowoveracylinder. %Author:DrewWetzel %Date:2009 clearall closeall clc %%UserInputsandConstants cl=0;%liftcoefficientperunitspan V=20;%freestreamvelocity[m/s] c=0.5207;%CCairfoilchord[m] %c=0.2032;%Abramsonairfoilchord[m] tcr=.2;%ellipsethickness-to-chordratio rho=1.2;%freestreamdensity[kg/m^3] %%EllipseandCylinderParameters %Ellipse ce=1.018405*c;%ellipsechordlength[m] A=ce/2;%semimajoraxis[m] B=tcr*A;%semiminoraxis[m] %MappingConstants R=A+B/2;%cylinderradius a=sqrtR*A-B/2;%transformationconstant %%ComputeLiftandCirculation %ComputeLiftForceperunitspan L=cl*0.5*rho*V^2*ce;%[N/m] 256

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%ComputeCirculationUsingKutta-JoukowskiTheorem circ=L/rho*V;%circulation[m^2/s] %%Streamlines [x,y]=meshgrid-4*R:.002:4*R,-4*R:.002:4*R; z=x+1i*y; forii=1:lengthx forjj=1:lengthy ifabszii,jj<=R-9e-3 zii,jj=NaN; end end end %MappingFunction eta=z+a^2./z; %ComplexPotential F=V*z+R^2./z+1i*circ/*pi*logz; %PlotStreamlines figure setgcf,'color','white' subplot,2,1 contourrealeta,imageta,imagF,29 colormap'gray' hold %PlotEllipse th=0:.02:2*pi; xe=A*costh; ye=B*sinth; fillxe,ye,[0.50.50.5]; setgca,'FontSize',36 xlabel'xm' ylabel'ym' 257

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title'Streamlines' axisequal xlim[-1.2*ce,1.2*ce] ylim[-1.2*ce,1.2*ce] %%CpDistribution %TransformedComplexSurfaceVelocity W=V*expi*th-1+1i*circ/*pi*R*expi*th./expi*th-a^2/R^2; %SurfaceVelocityComponentsandMagnitude u=realW; v=-imagW; Vs=sqrtu.^2+v.^2; %PressureCoefficient Cp=1-Vs./V.^2; %PlotCpDistribution subplot,2,2 plotxe:floorlengthxe/2+ce/2/ce,Cp:floorlengthxe/2,'r' hold plotxeceillengthxe/2:end+ce/2/ce,Cpceillengthxe/2:end,'b' setgca,'YDir','reverse' setgca,'FontSize',36 xlim[01] xlabel'x/c' ylabel'C_p' title'C_pDistribution' MATLABCode-EllipsenearBoundaries %%CirculationControlIdealFlowSolverw/Boundaries %Description:Givenauser-definedellipsegeometry,liftcoefficient, %andfreestreamvelocity,Cpdistributionsforanellipticbodyina %freestream,nearaceilingplane,nearagroundplane,andbetween %ceilingandgroundplanesareproducedusingtheanalyticalsolutionto %idealliftingflowoveracylinder.Thedistancebetweentheellipseand %planesisspecifiedusingthe"blkg"blocakgeratioterm. 258

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%Author:DrewWetzel %Date:2011 clearall closeall clc %%UserInputsandConstants cl=1.90;%liftcoefficientperunitspan V=20;%freestreamvelocity[m/s] c=0.5207;%CCairfoilchord[m] tcr=0.2;%ellipsethickness-to-chordratio rho=1.2;%freestreamdensity[kg/m^3] blkg=0.1439;%ellipseblockageratiothickness/tunnelheight %%EllipseandCylinderParameters %Ellipse ce=1.018405*c;%ellipsechordlength[m] A=ce/2;%semimajoraxis[m] B=tcr*A;%semiminoraxis[m] %MappingConstants R=A+B/2;%cylinderradius[m] a=sqrtR*A-B/2;%transformationconstant %DistancefromEllipseOrigintoWalleta-plane[m] h=B/blkg; %DistancefromCylinderOrigntoWallz-plane symszz hc_tmp=absimagevalsolvezz+a^2/zz-1i*h,zz; ifhc_tmp
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%%ComputeLiftandCirculation %ComputeLiftForceperunitspan L=cl*0.5*rho*V^2*ce;%[N/m] %ComputeCirculationUsingKutta-JoukowskiTheorem circ=L/rho*V;%circulation[m^2/s] %%DefineEllipse %Ellipse th=0:.005:2*pi; xe=A*costh; ye=B*sinth; %Grid [x,y]=meshgrid-10*R:.02:10*R,-10*R:.02:10*R; z=x+1i*y; forii=1:lengthx forjj=1:lengthy ifabszii,jj<=R-9e-3 zii,jj=NaN; end end end %%FREESTREAM %TransformedComplexSurfaceVelocity W=V*expi*th-1... +1i*circ/*pi*R*expi*th./expi*th-a^2/R^2; %SurfaceVelocityComponentsandMagnitude u=realW; v=-imagW; Vs=sqrtu.^2+v.^2; 260

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%PressureCoefficient Cp=1-Vs./V.^2; %PlotCpDistribution figure setgcf,'color','white','units','inches','position',[224.53.5] boxon holdon plotxe:floorlengthxe/2+ce/2/ce,Cp:floorlengthxe/2,... '-k','linewidth',1 plotxeceillengthxe/2:end+ce/2/ce,Cpceillengthxe/2:end,... '-k','linewidth',1,'handlevisibility','off' setgca,'YDir','reverse' xlim[01] xlabel'$x/c$','fontname','Times','fontsize',10,'interpreter','latex' ylabel'$C_p$','fontname','Times','fontsize',10,'interpreter','latex' setgca,'fontsize',10,'fontname','Times' %ComputeLift Cle=trapzxeceillengthxe/2:end+ce/2/ce,... Cpceillengthxe/2:end... -trapzfliplrxe:floorlengthxe/2+ce/2/ce,... fliplrCp:floorlengthxe/2 %%WALLABOVE %TransformedComplexSurfaceVelocity Wa=V*exp*1i*th-1+1i*circ/*pi*R*expi*th./exp*1i*th... -a^2/R^2-1./-a^2./R*expi*th-2*1i*h.^2.*... V*R^2./R*expi*th-2*1i*h.^2+... 1i*circ./*pi*R*expi*th-2*1i*h; %SurfaceVelocityComponentsandMagnitude ua=realWa; va=-imagWa; Vsa=sqrtua.^2+va.^2; %PressureCoefficient 261

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Cpa=1-Vsa./V.^2; %PlotCpDistribution plotxe:floorlengthxe/2+ce/2/ce,Cpa:floorlengthxe/2,... '--k','linewidth',1 plotxeceillengthxe/2:end+ce/2/ce,Cpaceillengthxe/2:end,... '--k','linewidth',1,'handlevisibility','off' %ComputeLift Cla=trapzxeceillengthxe/2:end+ce/2/ce,... Cpaceillengthxe/2:end... -trapzfliplrxe:floorlengthxe/2+ce/2/ce,... fliplrCpa:floorlengthxe/2 %%WALLBELOW %TransformedComplexSurfaceVelocity Wb=V*exp*1i*th-1... +1i*circ/*pi*R*expi*th./exp*1i*th-a^2/R^2... -1./-a^2./R*expi*th+2*1i*h.^2.*... V*R^2./R*expi*th+2*1i*h.^2+... 1i*circ./*pi*R*expi*th+2*1i*h; %SurfaceVelocityComponentsandMagnitude ub=realWb; vb=-imagWb; Vsb=sqrtub.^2+vb.^2; %PressureCoefficient Cpb=1-Vsb./V.^2; %PlotCpDistribution plotxe:floorlengthxe/2+ce/2/ce,Cpb:floorlengthxe/2,... '-.k','linewidth',1 plotxeceillengthxe/2:end+ce/2/ce,Cpbceillengthxe/2:end,... '-.k','linewidth',1,'handlevisibility','off' %ComputeLift 262

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Clb=trapzxeceillengthxe/2:end+ce/2/ce,... Cpbceillengthxe/2:end... -trapzfliplrxe:floorlengthxe/2+ce/2/ce,... fliplrCpb:floorlengthxe/2 %%TWOWALLS %TransformedComplexSurfaceVelocity imgsum=0; forii=1:2:100%i.e.1,3,5,7,... imgsum=imgsum-1./-a^2./R*expi*th-2*ii*1i*h.^2.*... V*R^2./R*expi*th-2*ii*1i*h.^2+... 1i*circ./*pi*R*expi*th-2*ii*1i*h... -1./-a^2./R*expi*th+2*ii*1i*h.^2.*... V*R^2./R*expi*th+2*ii*1i*h.^2+... 1i*circ./*pi*R*expi*th+2*ii*1i*h... -1./-a^2./R*expi*th-2*ii+1*1i*h.^2.*... V*R^2./R*expi*th-2*ii+1*1i*h.^2-... 1i*circ./*pi*R*expi*th-2*ii+1*1i*h... -1./-a^2./R*expi*th+2*ii+1*1i*h.^2.*... V*R^2./R*expi*th+2*ii+1*1i*h.^2-... 1i*circ./*pi*R*expi*th+2*ii+1*1i*h; end Wc=V*exp*1i*th-1... +1i*circ/*pi*R*expi*th./exp*1i*th-a^2/R^2+imgsum; %SurfaceVelocityComponentsandMagnitude uc=realWc; vc=-imagWc; Vsc=sqrtuc.^2+vc.^2; %PressureCoefficient Cpc=1-Vsc./V.^2; %PlotCpDistribution plotxe:30:floorlengthxe/2+ce/2/ce,... Cpc:30:floorlengthxe/2,'o:k','linewidth',1,'markersize',4 plotxeceillengthxe/2:30:end+ce/2/ce,... 263

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Cpcceillengthxe/2:30:end,'o:k','linewidth',1,'markersize',4,... 'handlevisibility','off' %ComputeLift Clc=trapzxeceillengthxe/2:end+ce/2/ce,... Cpcceillengthxe/2:end... -trapzfliplrxe:floorlengthxe/2+ce/2/ce,... fliplrCpc:floorlengthxe/2 %Legend hl=legend'Freestream','Ceiling','Ground','Both','location','north'; sethl,'interpreter','latex','box','off' xlim[-0.021.02] ylim[floorminCpa1.1] 264

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APPENDIXC VENTURIMETEREQUATIONS AVenturimeter,showninFigureC-1,isanobtrusiveowmeterusedtomeasure volumetricowrate.Comparedtootherowmeters,liketheoriceandnozzlemeters, headlossisminorWhite2003.ThisAppendixexplainsthegeneraltheorybehinda Venturimeter,followedbyhowtocomputethevolumetricowratefortheLambda SquareVenturimetersusedintheexperiment. Assumingsteady,idealow,continuityandBernoulli'sequationcanbecombinedto ndthevolumetricowrate.First,continuityisrearranged. AV = const 4 D 2 1 V 1 = 4 D 2 2 V 2 D 2 1 V 1 = D 2 2 V 2 V 1 = V 2 D 2 D 1 2 C{1 TheresultfromcontinuityissubstitutedintoBernoulli'sequation. p + 1 2 V 2 = const p 1 + 1 2 V 2 1 = p 2 + 1 2 V 2 2 p 1 + 1 2 V 2 D 2 D 1 2 # 2 = p 2 + 1 2 V 2 2 V 2 = 8 > > < > > : 2 p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(p 2 1 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( D 2 D 1 4 9 > > = > > ; 1 = 2 C{2 Finally,thevolumetricowrateiscomputed. Q = D 2 2 V 2 4 C{3 265

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Inpractice,however,compressibilityandfrictioneectsareimportant.Thefollowing moredetailedstepsforcomputingthevolumetricowrateareprovidedbyLambda SquarefortheowofairthroughtheirVenturimeters.Themetersusedhavethe followingdiameterratio: = D 2 D 1 =0 : 8152 : C{4 Allpressuresandtemperaturesshouldbeexpressedinunitspsiaand F,respectively. Theexpansionfactor Y 1 ,whichaccountsfortheexpansionofthegasfromtheinlettothe throat,iscalculatedattheupstreamhighpressurelocationusing Y 1 = 8 > > < > > : )]TJ/F22 11.9552 Tf 11.955 0 Td [( 4 )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 p 2 p 1 2 = 1 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( p 2 p 1 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 1 )]TJ/F22 11.9552 Tf 11.956 0 Td [( 4 p 2 p 1 2 = 1 )]TJ/F23 7.9701 Tf 13.15 5.256 Td [(p 2 p 1 9 > > = > > ; 1 = 2 : C{5 Next,thedischargecoecient,denedastheratiooftheactualowratetothe theoreticalowrate,iscomputedusing C =1 : 005 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 471 +0 : 546 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 514 3 : C{6 Thedischargecoecientisrecastastheowcoecient, K = C p 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [( 4 : C{7 Finally,thevolumetricowrateSCFMisdeterminedwiththefollowingequation. Q = 5 : 9816 D 2 2 KY h 2 : 703 p 1 SG h=w 460+ T 1 i 1 = 2 39 : 73 SG 460+ T b C{8 FortheLambdaSquareVenturimetersusedintheexperiment, D 2 =1.3125in.Also, h=w = p 1 )]TJ/F22 11.9552 Tf 12.712 0 Td [(p 2 inunitsinH 2 O, SG =1isthespecicgravityofair, T b =60 Fisthe basetemperature,and T 1 isthemeasuredtemperature Fattheupstreamhighpressure location. 266

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Notethat T 1 isthestatictemperature,buttheRTDinstalledintheairdelivery systemmeasuresthestagnationtemperature T 01 .Foranadiabaticow,thestaticand stagnationtemperaturesexpressedinKelvinarerelatedusing T 1 T 01 = 1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 M 2 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; C{9 whichcanberewrittenas T 1 = T 01 1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 V 2 1 RT 1 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 ; C{10 where V 1 isthevelocityattheVenturimeterinlet.Solvingfor T 1 ,EquationC{10becomes T 1 = T 01 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 R V 2 1 : C{11 Forair, =1.4and R =287J/kgK,soEquationC{11becomes T 1 = T 01 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 : 98 10 )]TJ/F21 7.9701 Tf 6.587 0 Td [(4 V 2 1 : C{12 Hence,fortherelativelylowowvelocitiesthroughtheVenturimeter V 1 = O m/s, T 1 T 01 Themassowrateiscalculatedusing m = 1 Q; C{13 wherethedensityiscomputedassumingaperfectgaswithconstantspecicheatsfrom 1 = p 1 RT 1 : C{14 Inpractice, Q and 1 areconvertedtoSIunitssothenalmassowrateiscomputedin unitskg/s. 267

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FigureC-1.VenturimeterschematicadaptedfromWhite2003. 268

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APPENDIXD JETVELOCITYCALCULATION Thederivationofthecirculationcontrolairfoilslotjetvelocityispresentedinthis appendix. D.1JetVelocity:GeneralCase Considerone-dimensional,steady,isentropic,subsonicowofaperfectgasthrough theplenumofthecirculationcontrolairfoil,whichismodeledasthesimplenozzle sketchedinFigureD-1.Positions1and2inFigureD-1correspondtothelocationsof largestcross-sectionalareaintheplenumandtheslot,respectively. Giventhestatedassumptions,theenergyequationreducestoanexpressionof constantstagnationenthalpy. h 1 + 1 2 U 2 1 = h 2 + 1 2 U 2 2 = const D{1 Foraperfectgas, h = c p T ,anditfollowsthat c p T 1 + 1 2 U 2 1 = c p T 2 + 1 2 U 2 2 : D{2 DividingEquationD{2by c p T 2 andrearrangingyields T 1 T 2 =1+ U 2 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 1 2 c p T 2 : D{3 Recallingthat c p = R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; D{4 EquationD{3becomes T 1 T 2 =1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 U 2 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 1 2 RT 2 : D{5 Sincethesoundspeedforisentropicowofaperfectgasis a = p RT ,EquationD{5can beexpressedintermsoftheMachnumberatthenozzleexitplane, M 2 T 1 T 2 =1+ 1 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 M 2 2 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 2 1 a 2 2 : D{6 269

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Foranisentropicow, T 0 T = p 0 p )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 ; D{7 andthestagnationtemperaturesandpressuresremainconstant T 01 = T 02 p 01 = p 02 Thus, T 1 T 2 = T 1 T 01 T 02 T 2 = p 1 p 01 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 p 02 p 2 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 = p 1 p 2 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 ; D{8 andEquationD{6becomes p 1 p 2 )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 =1+ 1 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 M 2 2 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 2 1 a 2 2 : D{9 RearrangingEquationD{9andsolvingfor M 2 yields M 2 = v u u t 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 1 p 2 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 # + U 2 1 a 2 2 : D{10 Thejetvelocitycanbeexpressedas U 2 = a 2 M 2 = p RT 2 M 2 : D{11 IfEquationD{8isrearrangedsuchthat T 2 = T 1 p 2 p 1 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 ; D{12 thenanexpressionforthejetvelocitycanbefoundbysubstitutingEqs.D{12andD{10 intoEquationD{11. U 2 = v u u t 2 RT 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F27 11.9552 Tf 11.956 16.857 Td [( p 2 p 1 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 # + U 2 1 D{13 D.2PlenumVelocity TheeectoftheplenumvelocityonEquationD{13isassessedinthissection. Considerowexitingthenozzleatthemaximumjetvelocitystudied, U 2 =100m/s,and 270

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standardtemperature, T 2 =293K.Thecross-sectionalareasatpositions1and2are A 1 = y p s; D{14 A 2 = hs; D{15 where s isthespan, y p =40.6mmisthemaximumheightintheplenum,and h istheslot height,whichvariesbetween0.5,1.0,and1.5mm.Thus,thearearatiois A 1 A 2 = y p h = 40 : 6mm h : D{16 ThesoundspeedandMachnumberatposition2are a 2 = p RT 2 = p : 4J = kgKK=343m = sD{17 M 2 = V 2 a 2 = 100m = s 343m = s =0 : 29D{18 Sincetheowisconsideredisentropic,thestagnationtemperatureratio T=T 0 andthe arearatio A=A canbefound.Notethat A isdenedasthecross-sectionalareanecessary fortheowtoreachsonic M =1conditions. T T 0 = 1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 M 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 D{19 A A = 1 M 2 +1 1+ )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 M 2 +1 2 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 D{20 Thus,theseratiosatthenozzleexitcanbecomputed. T 2 T 02 = 1+ 1 : 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 : 29 2 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 =0 : 9835D{21 A 2 A = 1 0 : 29 2 1 : 4+1 1+ 1 : 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 : 29 2 1 : 4+1 2 : 4 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 =2 : 098D{22 Inordertondthevelocityattheinlet,thearearatio A 1 =A mustrstbe determined. A 1 A = A 2 A A 1 A 2 =2 : 098 40 : 6mm h = 85 : 2mm h D{23 271

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Then, M 1 iscomputedbyndingtherootsofEquationD{20usingNewton'smethod John&Keith2006.First,EquationD{20isrewrittenas f M = M )]TJ/F22 11.9552 Tf 11.955 0 Td [(c b 1+ M 2 2 bc b ; D{24 where = A A ; D{25 b = +1 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; D{26 c = 2 +1 : D{27 Next,thederivativeofEquationD{24iscomputed. df dM = f 0 = )]TJ/F22 11.9552 Tf 11.955 0 Td [(c b )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 M 1+ M 2 2 bc b )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 D{28 Newton'smethodisrepeateduntiltheresultconverges. M i +1 = M i )]TJ/F22 11.9552 Tf 13.496 8.088 Td [(f i f 0 i D{29 Once M 1 isknown, T 1 =T 01 isfoundusingEquationD{19,and T 1 iscalculated. T 1 = T 1 T 01 T 02 T 2 T 2 D{30 Finally,thevelocityintheplenumisfound. U 1 = p RT 1 M 1 D{31 AshortMATLABroutineisdevelopedtocompute M 1 T 1 =T 01 T 1 ,and U 1 .The resultsfromthisanalysisforallthreeslotheightstested h =0.5,1.0,and1.5mmare summarizedinTableD-1.Thelargestestimatedplenumvelocityis3.53m/swhen h =1.5 mm. 272

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D.3InuenceofNonzeroPlenumVelocityonJetVelocityEstimate EquationD{13canberewrittenas U 2 = q C + U 2 1 ; D{32 where C =2 RT 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( p 2 p 1 )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 # : D{33 Hence,solvingfor C ,EquationD{32canberewrittenas C = U 2 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 1 : D{34 Forthelargestplenumvelocity U 1 =3.53m/s, C =9988m 2 /s 2 .Iftheinuenceofthe plenumvelocityisneglected,thejetvelocity, U 2 ,canbeestimatedfrom U 2 ;est = p C =99 : 94m = s : D{35 Theerroroftheestimateis %error= U 2 ;est )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 U 2 100= 99 : 94m = s )]TJ/F15 11.9552 Tf 11.955 0 Td [(100m = s 100m = s 100= )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 06% : D{36 Thiserrorisconsiderednegligible,andhencetheplenumvelocitycanbeassumedtobe U 1 0.Hence, T 1 = T 01 = T 0 and p 1 = p 01 = p 0 .Also,recall p 2 = p 1 .Therefore,Equation D{13canberewrittenas U jet = v u u t 2 RT 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( p 1 p 0 )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 = # : D{37 273

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FigureD-1.Circulationcontrolairfoilplenummodeledasasimple,one-dimensional nozzle. TableD-1.Resultsofplenumvelocityanalysis h mm A 1 =A 2 A 1 =A M 1 T 1 =T 0 T 1 K U 1 m/s 0.5 81.21700.0034040.9999982981.18 1.0 40.685.20.0067920.9999912982.35 1.5 27.156.80.010190.9999792983.53 274

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APPENDIXE PIVUNCERTAINTYANALYSIS E.1Samplecountandconvergence FiguresE-1,E-2,andE-3showrepresentativevectorcountsforthethreedatasets whereprolesareextracted.Asnotedpreviously,between500and1200imagepairsare acquiredforeachcase,dependingontheexperiment,toensureanadequatenumberof samplesforstatisticalconvergence.Thedataarecheckedforconvergenceatparticular pointsintheow.Forexample,inFiguresE-2andE-3,apointwithalowvector countisselected.InFigureE-1,apointnearthelipedgeisselectedsincetheboundary layerpassingoverthelipisofmostinterestthere.Runningaveragesofdimensionless speedarealsocomputedandplottedineachFigurewith95%condenceintervals.The dimensionlessspeedisobservedtoconvergewithinthecondenceintervalafterfewerthan 150samplesineachcase.Forallowprolespresented,averagesarecomputedwithmore than150samplesateachpoint. E.2Biasuncertainty ThebiasuncertaintyisdeterminedusingtheapproachoutlinedbyColeman&Steele 2009andappliedtoPIVdatabyMurray&Ukeiley2007.First,Equationsforthebias uncertaintyinthevelocitymagnitudeanddirectionaredetermined.Theresultsofthese Equationsarenecessaryforcomputingthebiasuncertaintyinthetangentialandnormal velocitycomponents,turbulenceintensities,andReynoldsstress. E.2.1Velocitymagnitudeanddirection Thesystematicorbiasuncertaintyinthemagnitudeofthevelocitymeasurementis theroot-sum-squareoftheuncertaintycomponentsassociatedwiththepixeldisplacement, D ,themeasuredlengthsofthecalibrationtargetinphysicalunits, L m ,andpixels, L p andthetimebetweenexposures, t .Thus,thetotalbiasuncertaintyisgivenby B j V j = h D B D 2 + L m B L m 2 + )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [( L p B L p 2 + t B t 2 i 1 = 2 : E{1 275

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B D ,thebiasuncertaintyassociatedwiththecalculatedpixeldisplacement,is0.03px forLaVisionDaVissoftwareDaVis2010. B L m ,thebiasuncertaintyinthephysical lengthofthecalibrationtarget,istypicallybasedonthetargetgridthickness. B L p ,the biasuncertaintyinthemeasuredpixellengthofthecalibrationtarget,isalsotakento be0.03pxsinceitismeasuredinDaVis.Finally, B t ,thecombinedbiasuncertainty ofthelaserpulsetimingsynchronizerandthelaser'sresponsetothesynchronizer,is approximately1nsTSI2008.ThetermsarepartialderivativesofthecomputedPIV velocitymagnitude j V j j V j = DL m L p t E{2 withrespecttoitssubscriptedcomponents,i.e. D = L m L p t ; L m = D L p t ; L p = L m D L 2 p t ; t = L m D L p t 2 : Sincethebiasuncertaintyinthepixeldisplacementis0.03px,theuncertaintyinthe directionofthevelocityvectorcanalsobedetermined.RefertoFigureE-4,where isthe angleofthevectorrelativetothe X Y coordinatesystemdenedinsection2.5.3.3,and B isthedesiredangularuncertainty.Itfollowsthat B =sin )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 0 : 03px D ; E{3 where D isthepixeldisplacement. E.2.2Tangentialandnormalvelocitycomponents RecallthecoordinatetransformationEquationsdescribedinsection2.5.3.3.The tangentialvelocity U isafunctionof V X V Y ,and .Since V X and V Y arerelatedto j V j throughtheEquations V X = j V j cos and V Y = j V j sin ,respectively, U canberewritten as U = j V j cos cos )-222(j V j sin sin : E{4 276

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Hence,thebiasuncertaintyin U isafunctionoftherespectiveuncertaintiesin j V j ,and B U = h )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( j V j B j V j 2 + B 2 + B 2 i 1 = 2 E{5 Thepartialderivativetermsfollow. j V j =cos cos )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin sin = j V j cos sin )-222(j V j sin cos = j V j sin cos )-222(j V j cos sin B j V j and B followfromEquationsE{1andE{3,respectively. B isthebiasuncertainty inthecomputationoftheanglebetweentheproleandtheglobal X -axisandis determinedbycalculatingthestandarddeviationofaseriesof30angles computed fromindividualtrialsoftheanglecalculationalgorithm.Thevariationintheangle calculatedisduetouserinput,sincetheusermustselectthesurfacepointsforttingthe ellipse. B isfoundtobe0.003rad. Thebiasuncertaintyin V isfoundsimilarlytothebiasuncertaintyin U B V = h )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [( j V j B j V j 2 + B 2 + B 2 i 1 = 2 E{6 Thepartialderivativetermsaregivenbelow. j V j =cos sin +sin cos = j V j cos cos )-222(j V j sin sin = j V j sin cos + j V j cos cos 277

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E.2.3Turbulenceintensities Theturbulenceintensityofthetangentialvelocitycanbewrittenas u 0 2 1 = 2 = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 N X n =1 U n )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U 2 # 1 = 2 = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 U 1 )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U 2 + U 2 )]TJ/F15 11.9552 Tf 13.954 3.022 Td [( U 2 + + U N )]TJ/F15 11.9552 Tf 13.954 3.022 Td [( U 2 1 = 2 : E{7 Hence,thebiasuncertaintyin u 0 2 1 = 2 canbeexpressedas B u 0 2 1 = 2 = U 1 B U 1 2 + U 2 B U 2 2 + + U N B U N 2 + U B U 2 1 = 2 : E{8 B U n and B U arecalculatedusingEquationE{5.Thepartialderivativetermsare U n = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 u 0 2 1 = 2 U n )]TJ/F15 11.9552 Tf 13.954 3.022 Td [( U U = 1 N )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 1 u 0 2 1 = 2 N X n =1 U n )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U Similarly,thebiasuncertaintyofthenormalvelocityturbulenceintensityis B v 0 2 1 = 2 = V 1 B V 1 2 + V 2 B V 2 2 + + V N B V N 2 + V B V 2 1 = 2 ; E{9 wherethepartialderivativetermsaregivenbelow. V n = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 v 0 2 1 = 2 V n )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V V = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 v 0 2 1 = 2 N X n =1 V n )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V E.2.4Reynoldsstress Liketheturbulenceintensities,theReynoldsstresscanalsobeexpanded. u 0 v 0 = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 N X n =1 U n )]TJ/F15 11.9552 Tf 13.954 3.022 Td [( U V n )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 U 1 )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U V 1 )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V + U 2 )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U V 2 )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V + U N )]TJ/F15 11.9552 Tf 13.953 3.022 Td [( U V N )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V E{10 278

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ThebiasuncertaintyoftheReynoldsstressthenbecomes B u 0 v 0 = U 1 B U 1 2 + V 1 B V 1 2 + U 2 B U 2 2 + V 2 B V 2 2 + + U N B U N 2 + V N B V N 2 + U B U 2 + V B V 2 1 = 2 : E{11 ThebiasuncertaintiesinthevelocitycomponentsarefoundusingEquationsE{5andE{6, andthepartialderivativetermsaredenedbelow. U n = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 U n )]TJ/F15 11.9552 Tf 13.954 3.022 Td [( U V n = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 V n )]TJ/F15 11.9552 Tf 13.742 3.022 Td [( V U = 1 N )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 N X n =1 U )]TJ/F22 11.9552 Tf 11.955 0 Td [(U n V = 1 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 N X n =1 V )]TJ/F22 11.9552 Tf 11.955 0 Td [(V n E.3Randomuncertainty TherandomuncertaintyofeachquantityisexpressedbytheEquationsinBenedict &Gould1996.Theserandomuncertaintyestimatesarevalidregardlessoftheestimate distribution,butBenedict&Gould1996suggestsasmanyas1000samplesperdata point.Although,duetoseedingirregularities,1000samplesperdatapointareimpractical forthePIVdatasetinthisinvestigation,thisrandomuncertaintyanalysiswillstillbe used.The95%condenceintervalsforvariousowquantitiesarelistedinTableE-1. 279

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FigureE-1.Opentestsectiondataset, Re c =0.65M, C =0.014, h=c =0.0019.a Vectorcountcontourandbrunningaverageofmeanvelocitymagnitude |{with95%condenceintervals---evaluatedatthelocationdenoted by` 'ina. FigureE-2.Curvedwalljetdataset, Re c =0.65M, C =0.015, h=c =0.0019.aVector countcontourandbrunningaverageofmeanvelocitymagnitude|{with 95%condenceintervals---evaluatedatthelocationdenotedby` 'ina. FigureE-3.Curvedwalljetseparationdataset, Re c =0.65M, C =0.015, h=c =0.0019. aVectorcountcontourandbrunningaverageofmeanvelocitymagnitude |{with95%condenceintervals---evaluatedatthelocationdenoted by` 'ina. 280

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FigureE-4.Illustrationofvectordirectionbiaserror. TableE-1.RandomuncertaintyestimatesBenedict&Gould1996 Quantity 95%Condenceinterval U 1 : 96 u 0 2 N 1 = 2 V 1 : 96 v 0 2 N 1 = 2 u 0 2 1 = 2 1 : 96 u 0 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 0 2 2 4 u 0 2 N 1 = 2 v 0 2 1 = 2 1 : 96 v 0 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v 0 2 2 4 v 0 2 N 1 = 2 u 0 v 0 1 : 96 u 0 2 v 0 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( u 0 v 0 2 N 1 = 2 281

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APPENDIXF CURVEDWALLJETSIMILARITYSOLUTION Thisappendixprovidesdetailedstepsforthesimilaritysolutionofaturbulentcurved walljetpresentedbyGuitton&Newman1977andasubsequentextensionoftheir analysistoincludetheinuenceofafreestream.Itwillbeshownthatalogarithmicspiral istheonlysurfacecontourthatpermitsaself-similarsolutionofallmeanandturbulent quantities.Aschematicofowoveralogarithmicspiral,includingdenitionsofow parameters U m and y 1 2 m ,isprovidedinFigure3-17.Thenotationinthisappendixfollows thatusedbyGuittonandNewman. F.1GoverningEquations Beforederivinganequationthatdescribesasimilarsolution,thegoverningequations inpolarcoordinateswillbeusedtoderivetheformsofthegoverningequationsforow overasurfaceoflocalradiusofcurvature R .Inthefollowinganalysis,theowisassumed tobesteady,incompressible,two-dimensional,turbulent,andinviscidwithnobodyforces. Theinviscidassumptionimpliesthattheequationswillbevalideverywhereoutside theviscoussublayer.Inpolarcoordinates,thevelocityinthe r directionis v r ,andthe tangentialvelocityateach is v .Inthemodiedgoverningequations,thetangential velocityinthe x directionisdenoted u ,andthenormalvelocityinthe y directionis givenby v .Thetotalvelocitycomponentsandpressurearedecomposedintomeanand uctuatingcomponents,e.g. u = u + u 0 F.1.1Continuity Thesteady,two-dimensionalcontinuityequationinpolarcoordinatesisgivenby White2006. @ @r rv r + @ @ v =0F{1 282

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Thefollowingexpressionsareusedtoconverttothecoordinatesystemforowovera surfaceoflocalradius R v r = v;v = u;r = y + R; @ @r = @y @r @ @y = @ @y ; @ @ = @x @ @ @x = R @ @x Hence,continuitybecomes @ @y [ y + R v ]+ R @u @x =0 @u @x + @ @y h v 1+ y R i =0F{2 Substitutingthemeananductuatingcomponentsofvelocity, @ u + u 0 @x + @ @y h v + v 0 1+ y R i =0 ; andtakingthetimeaverage, @ u @x + @u 0 @x + @ @y h v 1+ y R i + @ @y h v 0 1+ y R i =0 ; yieldsthetime-averagedcontinuityequationfortwo-dimensional,steady,turbulentow overacurvedsurfaceoflocalradius R @ u @x + @ @y h v 1+ y R i =0F{3 F.1.2 y -Momentum Theradialmomentumequationforsteady,two-dimensional,inviscidowignoring bodyforcesisgivenbyWhite2006, ~ V r v r )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 r v 2 = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @p @r ; F{4 where ~ V r = v r @ @r + v r @ @ = v @ @y + u )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ y R @ @x : 283

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Theradialmomentumequationbecomesthe y -momentumequationinthenewcoordinate scheme. v @ @y + u )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y R @ @x v )]TJ/F15 11.9552 Tf 25.005 8.088 Td [(1 y + R u 2 = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @p @y Multiplyingby )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y R yields 1+ y R v @v @y + u @v @x )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R 1 R )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ y R u 2 = )]TJ/F27 11.9552 Tf 11.291 13.27 Td [( 1+ y R 1 @p @y u @v @x + 1+ y R v @v @y )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(u 2 R = )]TJ/F27 11.9552 Tf 11.291 13.271 Td [( 1+ y R 1 @p @y : F{5 Substitutingthemeananductuatingcomponentsofvelocityandpressure, u + u 0 @ v + v 0 @x + 1+ y R v + v 0 @ v + v 0 @y )]TJ/F15 11.9552 Tf 13.151 8.088 Td [( u + u 0 2 R = )]TJ/F27 11.9552 Tf 11.291 13.271 Td [( 1+ y R 1 @ p + p 0 @y ; simplifying,andtakingthetime-averageyields u @ v @x + )]TJ 9.963 9.962 Td [()]TJ 9.049 9.047 Td [()]TJETq1 0 0 1 160.524 431.821 cm[]0 d 0 J 0.478 w 0 0 m 25.33 0 l SQBT/F22 11.9552 Tf 160.524 413.757 Td [(u 0 @ v @x + )]TJ 9.963 9.962 Td [()]TJ 9.048 9.047 Td [()]TJETq1 0 0 1 200.273 431.821 cm[]0 d 0 J 0.478 w 0 0 m 24.767 0 l SQBT/F15 11.9552 Tf 201.003 413.757 Td [( u @v 0 @x + u 0 @v 0 @x + 1+ y R v @ v @y + v 0 @ v @y + v @v 0 @y + v 0 @v 0 @y )]TJ/F15 11.9552 Tf 14.728 8.088 Td [(1 R u 2 + 2 uu 0 + u 0 2 = )]TJ/F27 11.9552 Tf 11.291 13.271 Td [( 1+ y R 1 @ p @y )]TJ/F2 9.9626 Tf 13.076 -11.518 Td [( @p 0 @y u @ v @x + 1+ y R v @ v @y )]TJ/F15 11.9552 Tf 13.88 8.088 Td [( u 2 R = )]TJ/F27 11.9552 Tf 11.291 13.271 Td [( 1+ y R 1 @ p @y )]TJETq1 0 0 1 354.382 345.006 cm[]0 d 0 J 0.478 w 0 0 m 27.562 0 l SQBT/F22 11.9552 Tf 354.382 326.942 Td [(u 0 @v 0 @x )]TJ/F27 11.9552 Tf 11.955 13.271 Td [( 1+ y R v 0 @v 0 @y + u 0 2 R : Bynotingthefollowingtwoexpressions: u 0 @v 0 @x = @ @x u 0 v 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(v 0 @u 0 @x v 0 @v 0 @y = @ @y v 0 2 2 ; themomentumequationcanbefurthersimpliednotethat C = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y R u @ v @x + C v @ v @y )]TJ/F15 11.9552 Tf 13.881 8.088 Td [( u 2 R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(C 1 @ p @y )]TJ/F22 11.9552 Tf 16.476 8.088 Td [(@ @x u 0 v 0 + v 0 @u 0 @x )]TJ/F22 11.9552 Tf 11.955 0 Td [(C @ @y v 0 2 2 + u 0 2 R u @ v @x + C v @ v @y )]TJ/F15 11.9552 Tf 13.881 8.088 Td [( u 2 R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(C 1 @ @y p + v 0 2 )]TJ/F22 11.9552 Tf 16.476 8.088 Td [(@ @x u 0 v 0 + u 0 2 R + v 0 @u 0 @x + C @ @y v 0 2 2 284

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Fromcontinuity,itfollowsthat v 0 @u 0 @x + 1+ y R v 0 @v 0 @y = v 0 )]TJ/F22 11.9552 Tf 13.563 8.088 Td [(@ @y h v 0 1+ y R i + 1+ y R v 0 @v 0 @y = )]TJ/F22 11.9552 Tf 9.299 0 Td [(v 0 1+ y R @v 0 @y )]TJ/F22 11.9552 Tf 11.955 0 Td [(v 0 2 @ @y 1+ y R + v 0 1+ y R @v 0 @y = )]TJ/F22 11.9552 Tf 9.298 0 Td [(v 0 2 R ; andthetime-averaged y -momentumequationisgivenby u @ v @x + 1+ y R v @ v @y )]TJ/F15 11.9552 Tf 13.88 8.088 Td [( u 2 R = )]TJ/F27 11.9552 Tf 11.291 13.27 Td [( 1+ y R 1 @ @y p + v 0 2 )]TJ/F22 11.9552 Tf 16.477 8.088 Td [(@ @x u 0 v 0 + u 0 2 )]TJETq1 0 0 1 476.821 554.656 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 476.821 544.393 Td [(v 0 2 R F{6 F.1.3 x -Momentum Theazimuthalmomentumequationforsteady,two-dimensional,inviscidowignoring bodyforcesisgivenbyWhite2006. ~ V r v + v r v r = )]TJ/F15 11.9552 Tf 13.386 8.088 Td [(1 r @p @ F{7 EquationF{7becomesthe x -momentumequationinthenewcoordinatesystem. v @ @y + u )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ y R @ @x u + uv y + R = )]TJ/F22 11.9552 Tf 28.343 8.088 Td [(R y + R @p @x Multiplyingby )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y R yields u @u @x + 1+ y R v @u @y + uv R = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @p @x F{8 Substitutingthemeananductuatingcomponentsofvelocityandpressure, u + u 0 @ u + u 0 @x + 1+ y R v + v 0 @ u + u 0 @y + u + u 0 v + v 0 R = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ p + p 0 @x ; 285

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simplifying,andtakingthetime-averageyieldsthefollowingequations. u @ u @x + )]TJ 9.963 9.963 Td [()]TJ 9.049 9.046 Td [()]TJETq1 0 0 1 158.85 690.245 cm[]0 d 0 J 0.478 w 0 0 m 25.341 0 l SQBT/F22 11.9552 Tf 158.85 672.182 Td [(u 0 @ u @x + )]TJ 9.963 9.963 Td [()]TJ 9.049 9.046 Td [()]TJETq1 0 0 1 198.609 690.245 cm[]0 d 0 J 0.478 w 0 0 m 25.341 0 l SQBT/F15 11.9552 Tf 199.339 672.182 Td [( u @u 0 @x + u 0 @u 0 @x + 1+ y R v @ u @y + v 0 @ u @y + v @u 0 @y + v 0 @u 0 @y + 1 R u v + uv 0 + u 0 v + u 0 v 0 = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @ p @x )]TJ/F2 9.9626 Tf 10.035 -9.196 Td [( @p 0 @x u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @ p @x )]TJETq1 0 0 1 329.4 591.763 cm[]0 d 0 J 0.478 w 0 0 m 28.136 0 l SQBT/F22 11.9552 Tf 329.4 573.699 Td [(u 0 @u 0 @x )]TJ/F27 11.9552 Tf 11.955 13.271 Td [( 1+ y R v 0 @u 0 @y )]TJETq1 0 0 1 463.46 591.342 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 463.46 581.787 Td [(u 0 v 0 R u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.586 8.087 Td [(1 @ p @x )]TJETq1 0 0 1 294.383 537.55 cm[]0 d 0 J 0.478 w 0 0 m 28.136 0 l SQBT/F22 11.9552 Tf 294.383 519.487 Td [(u 0 @u 0 @x )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R @ @y u 0 v 0 )]TJETq1 0 0 1 448.798 537.55 cm[]0 d 0 J 0.478 w 0 0 m 27.562 0 l SQBT/F22 11.9552 Tf 448.798 519.487 Td [(u 0 @v 0 @y )]TJETq1 0 0 1 498.477 537.13 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 498.477 527.574 Td [(u 0 v 0 R Usingcontinuity,itcanbeshownthat 1+ y R u 0 @v 0 @y = u 0 @ @y h v 0 1+ y R i )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(u 0 v 0 R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(u 0 @u 0 @x )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(u 0 v 0 R ; andthe x -momentumequationcanbefurthersimplied. u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.586 8.087 Td [(1 @ p @x )]TJETq1 0 0 1 282.734 394.098 cm[]0 d 0 J 0.478 w 0 0 m 28.136 0 l SQBT/F22 11.9552 Tf 282.734 376.035 Td [(u 0 @u 0 @x )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R @ @y u 0 v 0 )]TJETq1 0 0 1 430.839 394.098 cm[]0 d 0 J 0.478 w 0 0 m 28.136 0 l SQBT/F22 11.9552 Tf 430.839 376.035 Td [(u 0 @u 0 @x )]TJETq1 0 0 1 474.782 393.678 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 474.782 384.122 Td [(u 0 v 0 R )]TJETq1 0 0 1 510.126 393.678 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 510.126 384.122 Td [(u 0 v 0 R u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @ p @x )]TJ/F22 11.9552 Tf 16.476 8.088 Td [(@ @x u 0 2 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R @ @y u 0 v 0 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 u 0 v 0 R Finally,the x -momentumequationiscastinitsnalform. u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @x p + u 0 2 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R @ @y u 0 v 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 u 0 v 0 R F{9 F.1.4AdditionalAssumptions Guitton&Newman1977considered L 0 tobethewidth"ofthejet,suchthat L 0 =x 1.Since R O x ,then L 0 =R 1,andGuittonandNewmanassumedthe boundarylayerassumptionwasvalid.Fromcontinuity,therelativemagnitudesofthe 286

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velocitycomponentscanbedetermined. @u @x + @ @y h v 1+ y R i =0 u x + 1 L 0 v 1+ L 0 R 0 v L 0 x u Hence, v u .Assuming u O ,thedominanttermsofthe y -momentumequationare foundtobe u 2 R = 1 @ @y p + v 0 2 : F{10 EquationF{10illustratesthebalancethatexistsinthisowbetweenthecentripetal accelerationofthemeantangentialvelocityandthenormalgradientofmeanpressureand turbulentnormalstressinthe y -direction v 0 2 F.2Self-Preservation Inthesimilarityanalysis,onlytheportionofowoutsidetheviscoussublayeris consideredrecalltheinviscidassumption.Thesurfaceisalsoassumedtobeconvex R> 0.GuittonandNewmanassumedthefollowingsimilarityfunctions. u = U m f 0 ; )]TJETq1 0 0 1 218.353 323.927 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 218.353 314.371 Td [(u 0 v 0 = U 2 m g 12 ; u 0 2 )]TJETq1 0 0 1 344.012 324.635 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 344.012 314.371 Td [(v 0 2 = U 2 m g ; = y y 1 2 m GuittonandNewmanalsoassumedthattheReynoldsnumberwaslargeenoughsuch thattheskinfrictioncoecientwasindependentof x .Theyassumed U m / U r ,where U r = p w = isthefrictionvelocity.Theskinfrictioncoecientisgivenby C f = w 1 2 U 2 m = U 2 r 1 2 U 2 m =2 U r U m 2 : Thenormalmeanvelocity v canbefoundfromcontinuity,EquationF{3, v 1+ y R = )]TJ/F27 11.9552 Tf 11.291 16.272 Td [(Z @ u @x dy; 287

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where @ u=@x isgivenby @ u @x = @ @x U m f 0 = f 0 dU m dx + U m @f 0 @x = f 0 dU m dx + U m @f 0 @ @ @y 1 2 m @y 1 2 m @x = f 0 dU m dx )]TJ/F22 11.9552 Tf 11.955 0 Td [(U m f 00 y 1 2 m dy 1 2 m dx = f 0 dU m dx )]TJ/F22 11.9552 Tf 15.051 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 00 : Since dy = y 1 2 m d ,thenormalvelocitybecomes v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 2 m Z 0 f 0 dU m dx )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 00 d v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 2 m dU m dx Z 0 f 0 d + U m dy 1 2 m dx Z 0 f 00 d: Thesecondintegralisevaluatedusingintegrationbypartsnote f =0. Z 0 f 00 d =[ f 0 ] 0 )]TJ/F27 11.9552 Tf 11.955 16.273 Td [(Z 0 f 0 d = f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f Therefore,thenormalvelocityisgivenby v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 2 m dU m dx f )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f # : F{11 The y -momentumequationEquationF{10canalsobeintegratedfromlarge y Z 1 y u 2 R dy = 1 Z 1 y @ @y p + v 0 2 dy Z 1 U m f 0 2 R y 1 2 m d = 1 Z 1 1 y 1 2 m @ @ p + v 0 2 y 1 2 m d p + v 0 2 1 = y 1 2 m R U 2 m Z 1 f 0 2 d F{12 EquationsF{11andF{12aresubstitutedintothe x -momentumequationEquation F{9andasimilaritydierentialequationisestablished.Forconvenience,EquationF{9is 288

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restatedbelow. u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @x p + u 0 2 )]TJ/F27 11.9552 Tf 11.955 13.27 Td [( 1+ y R @ @y u 0 v 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 u 0 v 0 R Eachtermofthe x -momentumequationwillbeconsideredindividually.First,considerthe rstconvectiveaccelerationcomponent. u @ u @x = U m f 0 f 0 dU m dx )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 00 = U m dU m dx f 0 2 )]TJ/F22 11.9552 Tf 15.051 8.088 Td [(U 2 m y 1 2 m dy 1 2 m dx f 00 f 0 Next,considerthesecondconvectiveaccelerationcomponent. 1+ y R v @ u @y = )]TJ/F22 11.9552 Tf 9.299 0 Td [(y 1 2 m dU m dx f )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f # @ @y @ @ U m f 0 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(y 1 2 m dU m dx f )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f # 1 y 1 2 m U m f 00 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(U m dU m dx f 00 f + U 2 m y 1 2 m dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f Thecentripetalaccelerationtermbecomes u v R = 1 R U m f 0 )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 2 m )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R dU m dx f )]TJ/F22 11.9552 Tf 15.051 8.087 Td [(U m y 1 2 m dy 1 2 m dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f # = )]TJ/F22 11.9552 Tf 9.298 0 Td [(U m )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R y 1 2 m R dU m dx f 0 f )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U m y 1 2 m dy 1 2 m dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f # = )]TJ/F22 11.9552 Tf 9.298 0 Td [(U m )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R y 1 2 m R dU m dx f 0 f + U 2 m R )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R dy 1 2 m dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f : Thersttermontherighthandside,themeanpressuregradientandgradientofthe normalReynoldsstress u 0 2 ,becomes )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @x p + u 0 2 = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @ @x p + u 0 2 + v 0 2 + @ @x v 0 2 = )]TJ/F22 11.9552 Tf 13.82 8.088 Td [(@ @x p + v 0 2 )]TJ/F22 11.9552 Tf 16.477 8.088 Td [(@ @x u 0 2 )]TJETq1 0 0 1 341.346 122.142 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 341.346 111.878 Td [(v 0 2 289

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= )]TJ/F22 11.9552 Tf 13.82 8.088 Td [(@ @x p 1 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( p + v 0 2 )]TJ/F22 11.9552 Tf 16.477 8.088 Td [(@ @x u 0 2 )]TJETq1 0 0 1 397.153 709.342 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 397.153 699.078 Td [(v 0 2 = )]TJ/F22 11.9552 Tf 13.82 8.088 Td [(@ @x )]TJ/F22 11.9552 Tf 10.494 10.781 Td [(y 1 2 m R U 2 m Z 1 f 0 2 d )]TJ/F22 11.9552 Tf 16.477 8.088 Td [(@ @x U 2 m g = 1 R U 2 m Z 1 f 0 2 d dy 1 2 m dx + y 1 2 m Z 1 f 0 2 d d dx U 2 m + y 1 2 m U 2 m @ @x Z 1 f 0 2 d # )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 gU m dU m dx + U 2 m y 1 2 m dy 1 2 m dx g 0 = 1 R U 2 m dy 1 2 m dx Z 1 f 0 2 d +2 y 1 2 m U m dU m dx Z 1 f 0 2 d )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 2 m dy 1 2 m dx Z 1 @ @ f 0 2 d # )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 U m dU m dx g + U 2 m y 1 2 m dy 1 2 m dx g 0 = 1 R U 2 m dy 1 2 m dx Z 1 f 0 2 d +2 y 1 2 m U m dU m dx Z 1 f 0 2 d + U 2 m dy 1 2 m dx f 0 2 # )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 U m dU m dx g + U 2 m y 1 2 m dy 1 2 m dx g 0 : Next,considertheReynoldsstressgradientterm. )]TJ/F27 11.9552 Tf 11.291 13.27 Td [( 1+ y R @ @y u 0 v 0 = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( 1+ y 1 2 m R @ @y @ @ )]TJ/F22 11.9552 Tf 9.298 0 Td [(U 2 m g 12 = 1+ y 1 2 m R U 2 m y 1 2 m g 0 12 Finally,theReynoldsstresstermis )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 u 0 v 0 R = )]TJ/F15 11.9552 Tf 12.072 8.087 Td [(2 R )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 2 m g 12 =2 U 2 m R g 12 : 290

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Combiningallofthe x -momentumequationtermsresultsin U m dU m dx f 0 2 )]TJ/F22 11.9552 Tf 15.052 8.088 Td [(U 2 m y 1 2 m dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U m dU m dx f 00 f + U 2 m y 1 2 m dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f )]TJ/F22 11.9552 Tf 35.433 8.087 Td [(U m )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R y 1 2 m R dU m dx f 0 f + U 2 m R )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 = 2 m R dy 1 2 m dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f = 1 R U 2 m dy 1 2 m dx Z 1 f 0 2 d +2 y 1 2 m U m dU m dx Z 1 f 0 2 d + U 2 m dy 1 2 m dx f 0 2 # )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 U m dU m dx g + U 2 m y 1 2 m dy 1 2 m dx g 0 + 1+ y 1 2 m R U 2 m y 1 2 m g 0 12 +2 U 2 m R g 12 : Next,multiplyeachtermby y 1 2 m =U 2 m y 1 2 m U m dU m dx f 0 2 )]TJ/F22 11.9552 Tf 13.151 10.781 Td [(dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 13.151 10.781 Td [(y 1 2 m U 2 m dU m dx f 00 f + dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f 00 f )]TJ/F15 11.9552 Tf 40.476 8.088 Td [(1 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ y 1 = 2 m R y 1 2 m R y 1 2 m U m dU m dx f 0 f + 1 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ y 1 = 2 m R y 1 2 m R dy 1 2 m dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f = y 1 2 m R dy 1 2 m dx Z 1 f 0 2 d +2 y 1 2 m U m dU m dx Z 1 f 0 2 d + dy 1 2 m dx f 0 2 # )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 2 m U m dU m dx g + dy 1 2 m dx g 0 + 1+ y 1 2 m R g 0 12 +2 y 1 2 m R g 12 GuittonandNewmanwrotetheresultingsimilarityequationtoorder y 1 = 2 m =R .Hence, 1 1+ y 1 = 2 m R y 1 2 m R = 1 )]TJ/F22 11.9552 Tf 13.151 10.781 Td [(y 1 2 m R + y 1 2 m R 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(::: # y 1 2 m R y 1 2 m R ; andthesimilarityequationbecomes y 1 2 m U m dU m dx f 0 2 )]TJ/F2 9.9626 Tf 10.959 -9.279 Td [( dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 13.15 10.781 Td [(y 1 2 m U 2 m dU m dx f 00 f + dy 1 2 m dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f )]TJ/F22 11.9552 Tf 13.151 10.781 Td [(y 1 2 m R y 1 2 m U m dU m dx f 0 f + y 1 2 m R dy 1 2 m dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f = y 1 2 m R dy 1 2 m dx Z 1 f 0 2 d +2 y 1 2 m U m dU m dx Z 1 f 0 2 d + dy 1 2 m dx f 0 2 # )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 y 1 2 m U m dU m dx g + dy 1 2 m dx g 0 + 1+ y 1 2 m R g 0 12 +2 y 1 2 m R g 12 : 291

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Finally,rearrangingandsimplifyingyieldsGuittonandNewman'ssimilaritysolution. y 1 2 m U m dU m dx )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f +2 g )]TJ/F22 11.9552 Tf 13.15 10.781 Td [(dy 1 2 m dx f 00 f + g 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(g 0 12 )]TJ/F22 11.9552 Tf 13.151 10.781 Td [(y 1 2 m R y 1 2 m U m dU m dx f 0 f +2 Z 1 f 0 2 d + dy 1 2 m dx f 0 f + Z 1 f 0 2 d + g 0 12 +2 g 12 # =0 F{13 AlltermsoutsidetheparenthesesinEquationF{13mustbeconstant.Since dy 1 2 m dx =constant ; itfollowsthat y 1 2 m / x .Similarly,since y 1 2 m R =constant ; R / y 1 2 m / x .Therefore,inorderforsimilaritytobeachieved,thesurfacemustbea logarithmicspiral,i.e. r / e R=x F.3Self-Preservation:ModiedSimilarityFunctions Thepreviousanalysisrevealedthatinorderforsimilaritytobeachievedintheform assumedbyGuittonandNewman,thenthesurfacemustbealogarithmicspiral.Inthis section,anarbitrarysimilarityfunctionbasedontheformknowntoproducesimilarityin theouterregionofthemeantangentialvelocityproleisappliedLaunder&Rodi1983; Novak&Cornelius1986;Novak etal. 1987.ThescalingsuggestedbyLaunder&Rodi 1983isgivenbelow. u )]TJ/F22 11.9552 Tf 11.955 0 Td [(U e U m )]TJ/F22 11.9552 Tf 11.955 0 Td [(U e vs. y )]TJ/F22 11.9552 Tf 11.955 0 Td [(y m y 1 = 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(y m Thisscalingcanberewrittenintheformofsimilarityfunctions. u = U m )]TJ/F22 11.9552 Tf 11.956 0 Td [(U e f 0 + U e ; = y )]TJ/F22 11.9552 Tf 11.955 0 Td [(y m y 1 = 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(y m 292

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Arbitraryvelocityandlengthscaleswillbeused,suchthatthesimilarityfunctionstake thefollowingnalform. u = U 1 f 0 + U 2 ; )]TJETq1 0 0 1 232.411 657.829 cm[]0 d 0 J 0.478 w 0 0 m 18.341 0 l SQBT/F22 11.9552 Tf 232.411 648.274 Td [(u 0 v 0 = U 2 1 g 12 ; u 0 2 )]TJETq1 0 0 1 356.063 658.538 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 356.063 648.274 Td [(v 0 2 = U 2 1 g ; = y )]TJ/F22 11.9552 Tf 11.955 0 Td [(y 2 y 1 IfthescalingsuggestedbyLaunderandRodiisapplied,then U 1 = U m )]TJ/F22 11.9552 Tf 12.503 0 Td [(U e U 2 = U e y 1 = y 1 = 2 )]TJ/F22 11.9552 Tf 11.967 0 Td [(y m ,and y 2 = y m .Theremainderoftheanalysisfollowsthesamestepsoutlined intheprevioussection. Thenormalmeanvelocity v canbefoundfromcontinuity,EquationF{3, v 1+ y R = )]TJ/F27 11.9552 Tf 11.291 16.272 Td [(Z @ u @x dy; where @ u=@x isgivenby @ u @x = @ @x U 1 f 0 + U 2 = f 0 dU 1 dx + U 1 @f 0 @x + dU 2 dx = f 0 dU 1 dx + U 1 @f 0 @ @ @y @y @x + dU 2 dx = f 0 dU 1 dx + U 1 @f 0 @ @ @y 1 @y 1 @x + @ @y 2 @y 2 @x + dU 2 dx = f 0 dU 1 dx + U 1 f 00 1 y 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [( dy 1 dx )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 dy 2 dx + dU 2 dx = f 0 dU 1 dx )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 1 dx f 00 )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 2 dx f 00 + dU 2 dx : Since dy = y 1 d ,thenormalvelocitybecomes v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 Z 0 f 0 dU 1 dx )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 1 dx f 00 )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 2 dx f 00 + dU 2 dx d v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 dU 1 dx Z 0 f 0 d + U 1 dy 1 dx Z 0 f 00 d + U 1 dy 2 dx Z 0 f 00 d )]TJ/F22 11.9552 Tf 11.955 0 Td [(y 1 dU 2 dx Z 0 d: Thesecondintegralisevaluatedusingintegrationbypartsnote f =0. Z 0 f 00 d =[ f 0 ] 0 )]TJ/F27 11.9552 Tf 11.955 16.272 Td [(Z 0 f 0 d = f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 293

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Therefore,thenormalvelocityisgivenby v 1+ y R = )]TJ/F22 11.9552 Tf 9.298 0 Td [(y 1 dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx : F{14 The y -momentumequationEquationF{10canalsobeintegratedfromlarge y Z 1 y u 2 R dy = 1 Z 1 y @ @y p + v 0 2 dy Z 1 U 1 f 0 + U 2 2 R y 1 d = 1 Z 1 1 y 1 @ @ p + v 0 2 y 1 d p + v 0 2 1 = y 1 R U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d F{15 EquationsF{14andF{15aresubstitutedintothe x -momentumequationEquation F{9andasimilaritydierentialequationisestablished.Forconvenience,EquationF{9is restatedbelow. u @ u @x + 1+ y R v @ u @y + u v R = )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @x p + u 0 2 )]TJ/F27 11.9552 Tf 11.955 13.271 Td [( 1+ y R @ @y u 0 v 0 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 u 0 v 0 R Eachtermofthe x -momentumequationwillbeconsideredindividually.First,considerthe rstconvectiveaccelerationcomponent. u @ u @x = U 1 f 0 + U 2 f 0 dU 1 dx )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 1 dx f 00 )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 2 dx f 00 + dU 2 dx = dU 1 dx U 1 f 0 2 + U 2 f 0 )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 1 dx U 1 f 00 f 0 + U 2 f 00 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx U 1 f 00 f 0 + U 2 f 00 + dU 2 dx U 1 f 0 + U 2 Next,considerthesecondconvectiveaccelerationcomponent. 1+ y R v @ u @y = )]TJ/F22 11.9552 Tf 9.299 0 Td [(y 1 dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx @ @y @ @ U 1 f 0 + U 2 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(y 1 dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx 1 y 1 U 1 f 00 = )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 1 dU 1 dx f 00 f + U 2 1 y 1 dy 1 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f + U 2 1 y 1 dy 2 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 dU 2 dx f 00 294

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Thecentripetalaccelerationtermbecomes u v R = 1 R U 1 f 0 + U 2 )]TJ/F22 11.9552 Tf 9.299 0 Td [(y 1 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ y 1 + y 2 R dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx = )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 1 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ y 1 + y 2 R y 1 R dU 1 dx f 0 f )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 2 dx f 0 2 + dU 2 dx f 0 )]TJ/F22 11.9552 Tf 37.518 8.088 Td [(U 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ y 1 + y 2 R y 1 R dU 1 dx f )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx Thersttermontherighthandside,themeanpressuregradientandgradientofthe normalReynoldsstress u 0 2 ,becomes )]TJ/F15 11.9552 Tf 10.587 8.088 Td [(1 @ @x p + u 0 2 = )]TJ/F15 11.9552 Tf 10.586 8.088 Td [(1 @ @x p + u 0 2 + v 0 2 + @ @x v 0 2 = )]TJ/F22 11.9552 Tf 13.82 8.088 Td [(@ @x p + v 0 2 )]TJ/F22 11.9552 Tf 16.476 8.088 Td [(@ @x u 0 2 )]TJETq1 0 0 1 355.336 484.351 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 355.336 474.087 Td [(v 0 2 = )]TJ/F22 11.9552 Tf 13.82 8.087 Td [(@ @x p 1 )]TJ/F27 11.9552 Tf 11.955 16.856 Td [( p + v 0 2 )]TJ/F22 11.9552 Tf 16.476 8.087 Td [(@ @x u 0 2 )]TJETq1 0 0 1 411.143 448.086 cm[]0 d 0 J 0.478 w 0 0 m 13.616 0 l SQBT/F22 11.9552 Tf 411.143 437.823 Td [(v 0 2 = )]TJ/F22 11.9552 Tf 13.82 8.087 Td [(@ @x )]TJ/F22 11.9552 Tf 10.494 8.087 Td [(y 1 R U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F22 11.9552 Tf 16.477 8.087 Td [(@ @x U 2 1 g = 1 R @y 1 @x U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R @ @x U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R U 2 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 gU 1 dU 1 dx )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 1 @g @ @ @y 1 @y 1 @x + @ @y 2 @y 2 @x = 1 R dy 1 dx U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d +2 y 1 R U 1 dU 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R U 2 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 gU 1 dU 1 dx )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 2 1 g 0 1 y 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [( dy 1 dx )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(dy 2 dx 295

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= 1 R dy 1 dx U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d +2 y 1 R U 1 dU 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R U 2 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 gU 1 dU 1 dx + U 2 1 y 1 g 0 dy 1 dx + dy 2 dx Next,considertheReynoldsstressgradientterm. )]TJ/F27 11.9552 Tf 11.291 13.27 Td [( 1+ y R @ @y u 0 v 0 = )]TJ/F27 11.9552 Tf 11.291 16.857 Td [( 1+ y 1 + y 2 R @ @y @ @ )]TJ/F22 11.9552 Tf 9.299 0 Td [(U 2 1 g 12 = 1+ y 1 + y 2 R U 2 1 y 1 g 0 12 Finally,theReynoldsstresstermis )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 u 0 v 0 R = )]TJ/F15 11.9552 Tf 12.071 8.087 Td [(2 R )]TJ/F22 11.9552 Tf 9.298 0 Td [(U 2 1 g 12 =2 U 2 1 R g 12 : 296

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Combiningallofthe x -momentumequationtermsresultsin dU 1 dx U 1 f 0 2 + U 2 f 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 1 y 1 dy 1 dx U 1 f 00 f 0 + U 2 f 00 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx U 1 f 00 f 0 + U 2 f 00 + dU 2 dx U 1 f 0 + U 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 dU 1 dx f 00 f + U 2 1 y 1 dy 1 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f + U 2 1 y 1 dy 2 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(U 1 dU 2 dx f 00 )]TJ/F22 11.9552 Tf 37.518 8.087 Td [(U 1 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 + y 2 R y 1 R dU 1 dx f 0 f )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(U 1 y 1 dy 2 dx f 0 2 + dU 2 dx f 0 )]TJ/F22 11.9552 Tf 37.518 8.088 Td [(U 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 1 + y 2 R y 1 R dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx = 1 R dy 1 dx U 2 1 Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d +2 y 1 R U 1 dU 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R U 2 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 U 1 dU 1 dx g + U 2 1 y 1 g 0 dy 1 dx + dy 2 dx + 1+ y 1 + y 2 R U 2 1 y 1 g 0 12 +2 U 2 1 R g 12 Next,multiplyeachtermby y 1 =U 2 1 y 1 U 2 1 dU 1 dx U 1 f 0 2 + U 2 f 0 )]TJ/F15 11.9552 Tf 16.566 8.087 Td [(1 U 1 dy 1 dx U 1 f 00 f 0 + U 2 f 00 )]TJ/F15 11.9552 Tf 16.565 8.087 Td [(1 U 1 dy 2 dx U 1 f 00 f 0 + U 2 f 00 + y 1 U 2 1 dU 2 dx U 1 f 0 + U 2 )]TJ/F22 11.9552 Tf 14.272 8.088 Td [(y 1 U 1 dU 1 dx f 00 f + dy 1 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f + dy 2 dx f 00 f 0 )]TJ/F22 11.9552 Tf 14.272 8.088 Td [(y 1 U 1 dU 2 dx f 00 )]TJ/F15 11.9552 Tf 40.932 8.088 Td [(1 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1+ y 1 + y 2 R y 1 U 1 y 1 R dU 1 dx f 0 f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 1 y 1 dy 2 dx f 0 2 + dU 2 dx f 0 )]TJ/F15 11.9552 Tf 40.932 8.088 Td [(1 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1+ y 1 + y 2 R U 2 U 1 y 1 U 1 y 1 R dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx = y 1 R dy 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d +2 y 1 R y 1 U 1 dU 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R y 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 U 1 dU 1 dx g + g 0 dy 1 dx + dy 2 dx + 1+ y 1 + y 2 R g 0 12 +2 y 1 R g 12 Thesimilarityequationisexpressedtoorder y 1 =R 1 1+ y 1 + y 2 R y 1 R = 1 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 1 + y 2 R + y 1 + y 2 R 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(::: # y 1 R y 1 R ; 297

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Thus,thesimilarityequationbecomes y 1 U 2 1 dU 1 dx U 1 f 0 2 + U 2 f 0 )]TJ/F15 11.9552 Tf 16.566 8.088 Td [(1 U 1 dy 1 dx U 1 f 00 f 0 + U 2 f 00 )]TJ/F15 11.9552 Tf 16.565 8.088 Td [(1 U 1 dy 2 dx U 1 f 00 f 0 + U 2 f 00 + y 1 U 2 1 dU 2 dx U 1 f 0 + U 2 )]TJ/F22 11.9552 Tf 14.272 8.087 Td [(y 1 U 1 dU 1 dx f 00 f + dy 1 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f + dy 2 dx f 00 f 0 )]TJ/F22 11.9552 Tf 14.272 8.087 Td [(y 1 U 1 dU 2 dx f 00 )]TJ/F22 11.9552 Tf 14.272 8.087 Td [(y 1 U 1 y 1 R dU 1 dx f 0 f )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 0 f )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 1 y 1 dy 2 dx f 0 2 + dU 2 dx f 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 2 U 1 y 1 U 1 y 1 R dU 1 dx f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 1 y 1 dy 2 dx f 0 + dU 2 dx = y 1 R dy 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d +2 y 1 R y 1 U 1 dU 1 dx Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d + y 1 R y 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 U 1 dU 1 dx g + dy 1 dx g 0 + dy 2 dx g 0 + 1+ y 1 + y 2 R g 0 12 +2 y 1 R g 12 : Notethat y 1 R y 1 @ @x Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d = y 1 R y 1 @ @y 1 @y 1 @x + @ @y 2 @y 2 @x @ @ Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d = y 1 R y 1 1 y 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [( dy 1 dx )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(dy 2 dx @ @ Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d = )]TJ/F22 11.9552 Tf 10.494 8.088 Td [(y 1 R dy 1 dx @ @ Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(y 1 R dy 2 dx @ @ Z 1 f 0 2 +2 U 2 U 1 f 0 + U 2 U 1 2 # d = y 1 R dy 1 dx f 0 2 +2 U 2 U 1 f 0 )]TJ/F27 11.9552 Tf 11.956 16.857 Td [( U 2 U 1 2 # + y 1 R dy 2 dx f 0 2 +2 U 2 U 1 f 0 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( U 2 U 1 2 # 298

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Finally,rearrangingandsimplifyingyieldsanewsimilaritysolution. y 1 U 1 dU 1 dx f 0 2 )]TJ/F2 9.9626 Tf 10.959 -8.302 Td [( dy 1 dx f 00 f 0 + dy 2 dx f 00 f 0 + y 1 U 1 dU 2 dx f 0 + y 1 U 1 U 2 U 1 dU 1 dx f 0 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(U 2 U 1 dy 1 dx f 00 + U 2 U 1 dy 2 dx f 00 + y 1 U 1 U 2 U 1 dU 2 dx )]TJ/F22 11.9552 Tf 14.272 8.088 Td [(y 1 U 1 dU 1 dx f 00 f + dy 1 dx f 00 f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f + dy 2 dx f 00 f 0 )]TJ/F22 11.9552 Tf 14.272 8.088 Td [(y 1 U 1 dU 2 dx f 00 )]TJ/F22 11.9552 Tf 14.271 8.088 Td [(y 1 U 1 y 1 R dU 1 dx f 0 f + y 1 R dy 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.956 0 Td [(f 0 f + y 1 R dy 2 dx f 0 2 )]TJ/F22 11.9552 Tf 14.272 8.088 Td [(y 1 U 1 y 1 R dU 2 dx f 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 2 U 1 y 1 U 1 y 1 R dU 1 dx f + U 2 U 1 y 1 R dy 1 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f + U 2 U 1 y 1 R dy 2 dx f 0 )]TJ/F22 11.9552 Tf 13.151 8.087 Td [(U 2 U 1 y 1 U 1 y 1 R dU 2 dx )]TJ/F22 11.9552 Tf 13.15 8.087 Td [(y 1 R dy 1 dx Z 1 f 0 2 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 R dy 1 dx U 2 U 1 Z 1 f 0 d )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 1 R dy 1 dx Z 1 U 2 U 1 2 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 R y 1 U 1 dU 1 dx Z 1 f 0 2 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 y 1 R y 1 U 1 dU 1 dx U 2 U 1 Z 1 f 0 d )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 R y 1 U 1 dU 1 dx Z 1 U 2 U 1 2 d )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(y 1 R dy 1 dx )]TJ 6.604 6.595 Td [()]TJ/F22 11.9552 Tf -5.607 -2.19 Td [(f 0 2 +2 U 2 U 1 f 0 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( U 2 U 1 2 # )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 1 R dy 2 dx )]TJ 6.604 6.595 Td [()]TJ/F22 11.9552 Tf -5.608 -2.19 Td [(f 0 2 +2 U 2 U 1 f 0 )]TJ/F27 11.9552 Tf 11.955 16.857 Td [( U 2 U 1 2 # +2 y 1 U 1 dU 1 dx g )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(dy 1 dx g 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(dy 2 dx g 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 12 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 1 R g 0 12 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 2 R g 0 12 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 y 1 R g 12 =0 Thenalsimilarityequationis y 1 U 1 dU 1 dx f 0 2 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 f +2 g )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(dy 1 dx f 00 f + g 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(g 0 12 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 1 R y 1 U 1 dU 1 dx [ A ]+ dy 1 dx [ B ]+ g 0 12 +2 g 12 + y 1 U 1 dU 2 dx [ C ]+ dy 2 dx [ D ] + y 1 U 1 dU 2 dx f 0 )]TJ/F22 11.9552 Tf 11.955 0 Td [(f 00 )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(dy 1 dx U 2 U 1 f 00 + dy 2 dx U 2 U 1 f 00 )]TJ/F15 11.9552 Tf 11.955 0 Td [( g 0 + y 1 U 1 U 2 U 1 dU 1 dx f 0 + dU 2 dx )]TJ/F22 11.9552 Tf 13.151 8.088 Td [(y 2 R g 0 12 =0F{16 where A = f 0 f +2 Z 1 f 0 2 d + U 2 U 1 f +4 Z 1 f 0 d +2 Z 1 U 2 U 1 2 d B = f 0 f + Z 1 f 0 2 d + U 2 U 1 f + f 0 +2 Z 1 f 0 d )]TJ/F27 11.9552 Tf 11.955 20.443 Td [( U 2 U 1 2 + Z 1 U 2 U 1 2 d C = f 0 + U 2 U 1 D = U 2 U 1 f 0 )]TJ/F22 11.9552 Tf 13.15 8.088 Td [(U 2 2 U 2 1 299

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AlltermsoutsidetheparenthesesinEquationF{16mustbeconstant.Notetheterm, dy 1 dx f 00 f 0 + g 0 : Since dy 1 =dx =constant,itfollowsthat y 1 / x .Similarly,notetheterm, y 1 R g 0 12 +2 g 12 : Since y 1 =R =constant,then R / y 1 / x .Therefore,inorderforsimilaritytobeachieved, thesurfacemustbealogarithmicspiral,i.e. r / e R=x 300

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BIOGRAPHICALSKETCH DrewWetzelwasborninAllentown,Pennsylvaniain1984andraisedinthenearby boroughofMacungiepronouncedma-k U n-je;LenapeIndianforbearswamp".Drew graduatedfromEmmausHighSchoolin2002andwentontoreceiveaBachelorofScience withDistinctioninmechanicalengineeringfromthePennsylvaniaStateUniversityin 2006.Lessthanamonthlater,DrewbegangraduatestudiesattheUniversityofFlorida undertheguidanceofDr.LouisCattafestaandsubsequentlyreceivedhisMasterof ScienceinmechanicalengineeringfromUFin2008.Uponcompletionofhisdegree,Drew willbeginworkingintheAcousticsandFluidMechanicsTechnologygroupatBoeing CommercialAirplanesinEverett,Washington. 308