Wind turbine aerodynamics and vorticity based methods fundamentals and recent applications 1st editi

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Turbine Aerodynamics and Vorticity Based Methods Fundamentals and Recent Applications 1st Edition Emmanuel Branlard (Auth.)

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Emmanuel Branlard

Wind Turbine Aerodynamics and Vorticity-Based Methods

Fundamentals and Recent Applications

ResearchTopicsinWindEnergy

Volume7

Serieseditors

JoachimPeinke,UniversityofOldenburg,Oldenburg,Germany

e-mail:peinke@uni-oldenburg.de

GerardvanBussel,DelftUniversityofTechnology,Delft,TheNetherlands

e-mail:g.j.w.vanbussel@tudelft.nl

AboutthisSeries

TheseriesResearchTopicsinWindEnergypublishesnewdevelopmentsand advancesinthe fi eldsofWindEnergyResearchandTechnology,rapidlyand informallybutwithahighquality.WindEnergyisanewemergingresearch field characterizedbyahighdegreeofinterdisciplinarity.Theintentistocoverallthe technicalcontents,applications,andmultidisciplinaryaspectsofWindEnergy, embeddedinthe fieldsofMechanicalandElectricalEngineering,Physics, Turbulence,EnergyTechnology,Control,MeteorologyandLong-TermWind Forecasts,WindTurbineTechnology,SystemIntegrationandEnergyEconomics, aswellasthemethodologiesbehindthem.Withinthescopeoftheseriesare monographs,lecturenotes,selectedcontributionsfromspecializedconferencesand workshops,aswellasselectedPhDtheses.Ofparticularvaluetoboththe contributorsandthereadershiparetheshortpublicationtimeframeandthe worldwidedistribution,whichenablebothwideandrapiddisseminationofresearch output.TheseriesispromotedundertheauspicesoftheEuropeanAcademyof WindEnergy.

Moreinformationaboutthisseriesathttp://www.springer.com/series/11859

EmmanuelBranlard

WindTurbineAerodynamics andVorticity-BasedMethods

FundamentalsandRecentApplications

EmmanuelBranlard DepartmentofWindEnergy,Aeroelastic Design

TechnicalUniversityofDenmark Roskilde

Denmark

ISSN2196-7806ISSN2196-7814(electronic) ResearchTopicsinWindEnergy

ISBN978-3-319-55163-0ISBN978-3-319-55164-7(eBook) DOI10.1007/978-3-319-55164-7

LibraryofCongressControlNumber:2017933865

© SpringerInternationalPublishingAG2017

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Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.

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Tolove, 2K 2K

Preface

Thestandardapproachinthestudyofwindturbineaerodynamicsconsistsinusing momentumanalyses.Themomentumtheoryofanactuatordiskisanexampleof momentumanalysis.Bladeelementmomentum(BEM)andtheconventional computation fluiddynamics(CFD)aretwonumericalmethodsalsobasedon momentumanalyses.Velocityandpressurearethemainvariablesusedin momentumanalysis.Theequationscanalsobeformulatedusingvorticityasmain variable.Thisleadstoanalternativeapproachreferredtoasvorticity-based methods.Thegreatpotentialofvorticity-basedmethodscomesfromthemultitude offormulationstheyoffer,rangingfromsimpleanalyticalmodelstoadvanced numericalmethods.Theanalyticalmodelwillbereferredtoasvortextheoriesand thenumericalmethodsasvortexmethods.

Thetermvorticityoftenintimidatesthenewcomer,butthisfearvanisheswhen onerealizesthatvelocityandvorticityoffertwodifferent,butoftenequivalent, pointsofview.Forinstance,themomentumtheoryofanactuatordiskwithconstantloadingcanbeequivalentlystudiedbyconsideringthetubularvorticitysheet thatispresentatthesurfaceofthestreamtube.Vorticityplaysanimportantrolein windturbineaerodynamicssincestrongvorticesarepresentinthewakesin particular.Vorticityandvorticity-basedmethodscannotbeomittedinabookonthe topic.MostoftheanalyticalmodelsusedinBEMmethodsarederivedfromanalyticalvortexmodels.Further,numericalvortexmethodsarenowcompetingwith conventionalCFDmethodsintermsofaccuracyandcomputationaltime,andthey arebecomingacommontoolforthestudyofwindturbineaerodynamics.

Theaimofthisbookistoshowtherelevanceofvorticity-basedmethodsforthe studyofwindturbineaerodynamicsandtopresenthistoricalandrecentdevelopmentsinthe fieldwithasufficientlevelofdetailsforthebooktobeself-contained. Thisbookisintendedforstudentsandresearcherscuriousaboutrotoraerodynamicsand/oraboutvorticity-basedmethods.Thebookintroducesthefundamentalsof fluidmechanics,momentumtheories,vortextheories,andvortex methodsnecessaryforthestudyofrotorsandwindturbinesinparticular.Rotor theoriesarepresentedinagreatlevelofdetailsatthebeginningofthebook.These theoriesincludethebladeelementtheory,theKutta–Joukowskitheory,the

momentumtheory,andtheBEMmethod.Differentmomentumtheoriesarederived from firstprinciplesusingacriticalapproach.Theremainingofthebookfocuseson vortextheoryandvortexmethodswithapplicationtowindturbineaerodynamics. Examplesofvortextheoryapplicationsthatarediscussedinthisbookareoptimal rotordesign,tip-losscorrections,yawmodels,anddynamicinflowmodels. Historicalderivationsandrecentextensionsofthemodelsarepresented.The cylindricalvortexmodelisanotherexampleofasimpleanalyticalvortexmodel usedinthisbook.Inthismodel,awindturbineanditswakearesimplifiedusinga vortexsystemofcylindricalshape.Formulationsequivalenttotheonesusedina BEMalgorithmareobtained.Themodelprovidesawake-rotationcorrectionwhich greatlyimprovestheaccuracyofBEMalgorithms.Thecylindricalmodelisalso usedtoprovidetheanalyticalvelocity fieldupstreamofaturbineorawindfarm (i.e.,theinductionzone)underalignedoryawedconditions.Suchresultsare obtainedinacoupleofsecondswithanimpressiveaccuracycomparedtonumerical resultsfromCFDmethodswhichwouldrequiredaysofcomputation.Different applicationsofnumericalvortexmethodsarepresentedinthisbook.Numerical methodsareusedforinstancetoinvestigatetheinfluenceofawindturbineonthe incomingturbulence.Shearedinflowsarealsoinvestigated.Itisshowninparticular thatmostvortexmethodsomitatermresultinginexcessiveupwarddisplacement ofthewindturbinewake.Manyanalytical flowsarederivedindetailinthisbook: vortexrings,Hill’svortex,vortexblobs,etc.Theyareusedthroughoutthebookto devisesimplerotormodelsortovalidatetheimplementationofnumericalmethods. SeveralMATLABprogramsareprovidedtoeasesomeofthemostcomplex implementations:BEMcodes,vortexcylindervelocityfunctions,Goldstein’scirculation,lifting-linecodes,Karman–Trefftzconformalmap,projectionfunctions forvortexparticlemethods,etc.

PartIintroducesthe fluidmechanicsfoundationsrelevanttothisbook.PartII introducesrotoraerodynamics,includingmomentumanalyses,vortexmodels,and theBEMmethod.PartIIIfocusesonclassicalvortextheoryresultswhichoriginatedfromthestudyofrotorswithoptimalcirculation.PartIVpresentstherecent developmentsinrotoraerodynamicsbasedonanalyticalvortex flows.PartV presentsrecentapplicationsofvortexmethods.PartVIprovidesdetailedanalytical solutionsthatarerelevantforrotoraerodynamics,eitherforthederivationofvortex modelsorfortheimplementationandvalidationofvortexmethods.PartVIIis dedicatedtovortexmethods.PartVIIIprovidesmathematicalcomplementstosome chaptersofthebook.

Roskilde,DenmarkEmmanuelBranlard January2017

Acknowledgements

Thecurrentworkwouldnothavebeenpossiblewithoutthesupportandhelpofmy PhDsupervisorMacGaunaaandthecontributionsfromSpyrosVoutsinas,Ewan Machefaux,PhilippeMercier,GregoireWinckelmans,NielsTroldborg,Giorgios Papadakis,andHenrikBrandenborgSørensen.Iwouldliketothankmycolleagues fortheirinspirationandfruitfuldiscussions:JakobMann,NielsSørensen,Curran Crawford,PhilippeChatelain,TorbenLarsen,AndersHansen,GeorgPirrung, FrederikZahle,MadsHejlesen,JuanPabloMurcia,AlexanderForsting,Christian Pavese,MichaelMcWilliams,LucasPascal,andJacobusDeVaal.

Iamgratefultothepersonswhoacceptedtoreviewsomechaptersofthisbook despitealimitedtime:DamienCastaignet,MichaelMcWilliams,MacGaunaa,Jens Gengenbach,Gil-ArnaudCoche,JulienB.,andBjörnSchmidt.

Aboveall,IamgladforthemomentsoflifeandloveIexperiencedthankstomy familyandfriends.Iwishtosharemoreofthosewithallofyou:Ewan,François, Aghiad,Mika,Dim,Heidi,Mike,K,Ozi,Bertille,Julie,Kiki,Loïc,Milou,Romain, Sofie,LucasP.,LucasM.,Philipp,Jeanne,Alessandro,Julien,Sophie,Dad,and Mom.

2.3.5VorticityEquationinParticularCases

2.3.6Pressure

2.3.7VortexForce,Image/Generalized/BoundVorticity,

2.4DifferentDimensionsofVorticity:Surface, LineandPoints

2.7VorticesinViscousandInviscidFluid-Results

2.8SurfaceRepresentations-VortexSheets

2.9IncompressibleFlowEquationsinPolarCoordinates-2D and3DFlows-AxisymmetricFlows

2.9.12DArbitraryFlow(CylindricalCoordinates)

2.9.23DArbitraryFlow(CylindricalCoordinates)

2.9.33DAxisymmetricFlowswithSwirl(Cylindrical

2.9.43DAxisymmetricFlowsWithoutSwirl (CylindricalCoordinates)

2.9.53DArbitraryFlow(SphericalCoordinates)

2.9.63DAxisymmetricFlowswithSwirl (SphericalCoordinates)

2.9.73DAxisymmetricFlowsWithoutSwirl (SphericalCoordinates)

2.11.3Joukowski’sConformalMap

2.11.4Karman-TrefftzConformalMap

2.11.5VandeVoorenConformalMap

3.1CharacteristicsofLiftingBodies

3.1.1FluidForceonaBody:Lift,Drag,Moment andCenterofPressure

3.1.2CenterofPressure,AerodynamicCenter andQuarterChordPointofanAirfoil

3.1.3VorticityAssociatedwithLiftingBodies

3.1.4KuttaCondition

3.2PolarDataofanAirfoilandRelatedEngineeringModels

3.2.1Introduction

3.2.2ModelsforLargeAngleofAttacks

3.2.3DynamicStallModels

3.2.4InviscidPerformances

3.2.5ModelofFully-SeparatedPolar fromKnownPolar

3.3VorticityBasedTheoriesofTwo-Dimensional

3.5InviscidLifting-SurfaceTheoryofaWing

3.6InviscidLifting-LineTheoryofaWing

3.6.2LiftingLineTheory-FromCirculationDistribution toLoads

3.6.3Prandtl’sLiftingLine Equation-Integro-DifferentialForm

3.6.4EllipticalLoadingandEllipticalWing UnderLiftingLineAssumptions andLinearTheory

4.4VelocityInductionFactorsUndertheLiftingLine

5VortexSystemsandModelsofaRotor-Bound, RootandWakeVorticity

5.1MainComponentsofVorticityInvolvedAboutaRotor

5.2SimplifiedVorticityModelsofRotors

5.2.1MainSimplificationsUsedbytheModels

5.2.2HelicalVortexModelsofaRotor

5.2.3CylindricalandTubularVortexModel ofaRotor

5.2.4VortexRingModelofaRotor

6ConsiderationsandChallengesSpeci

6.1YawandTilt

7.3.2ParticularCasesofFlowswithRotational

fi

9.2.1NotationsandAssumptions ....................

9.2.2DeterminationofPower,Thrust andRotorVelocity ..........................

9.2.3InductionFactorsandRotorPerformance .........

9.2.4DiscussionontheAssumptions .................

9.3GeneralMomentumTheory

9.3.1Introduction

9.3.2Derivation

9.4GeneralAxialMomentumTheory(NoWakeRotation)

9.4.1Assumptions

9.4.2ResultsoftheGeneralAxialMomentumTheory

9.5StreamtubeTheory(Simpli fiedMomentumTheory)

9.5.1Assumptions

9.5.2DerivationoftheMainStreamtube TheoryResults

9.5.3LoadsfromStreamtubeTheory .................

9.5.4MaximumPowerExtraction fromSTT- “OptimalRotor

10.1TheBEMMethodforaSteadyUniformIn

10.1.1Introduction ................................

10.1.2FirstLinkage:VelocityTriangleandInduction Factors

10.1.3SecondLinkage:ThrustandTorquefromMT andBET

10.1.4BEMEquations

10.1.5SummaryoftheBEMAlgorithm

10.2CommonCorrectionstotheSteadyBEMMethod

10.2.1DiscreteNumberofBlades,Tip-Losses andHub-Losses

10.2.2CorrectionDuetoMomentum TheoryBreakdown- a Ct Relations

10.3UnsteadyBEMMethod ..............................

10.3.1Introduction ................................

10.3.2DynamicWake/Infl ow........................

10.3.3YawandTiltModel .........................

10.3.4DynamicStall ..............................

10.3.5TowerandNacelleInterference

10.3.6SummaryoftheUnsteadyBEMAlgorithm

13.4DifferentExpressionsofPrandtl’sTip-LossFactor

13.5.1TheoreticalTip-LossCorrections................

14.3ComputationofGoldstein’sFactor

15.1Simple1DMomentumTheory/VortexCylinderModel

15.2CylinderAnalogExpansion

15.3Theodorsen’sWakeExpansion

15.4Far-WakeExpansionModels

15.5ComparisonofWakeExpansions

andVortexCylinderResults

19.3.3Step6:InductionsfortheImprovedBEM ofMadsenetal. ............................

19.3.4Step6:InductionsfortheActuatorDiskModel (AD) .....................................

20HelicalModelforTip-Losses:DevelopmentofaNovelTip-Loss FactorandAnalysisoftheEffectofWakeExpansion

21Yaw-ModellingUsingaSkewedVortexCylinder

22SimpleImplementationofaNewYaw-Model

23.3HelicalPitchfortheSuperpositionofSkewedCylinders

23.4Yaw-ModelImplementationUsingaSuperposition ofSkewedCylinders

23.5PartialApproach-FocusontheInboardPartoftheBlade

25AnalyticalModelofaWindTurbineinShearedIn

26ModelofaWindTurbinewithUnsteadyCirculation orUnsteadyIn

PartVLatestApplicationsofVortexMethodstoRotor AerodynamicsandAeroelasticity 27ExamplesofApplicationsofVortexMethods

28Representationofa(Turbulent)VelocityFieldUsingVortex Particles

29EffectofaWindTurbineontheTurbulentIn

30AeroelasticSimulationofaWindTurbineUnderTurbulent andShearedConditions

30.1Introduction

30.2RepresentationofShearinVortexMethods

30.3FullAeroelasticSimulationIncludingShear andTurbulence

30.4Conclusions

PartVIAnalyticalSolutionsforVortexMethodsandRotor Aerodynamics

31ElementaryThree-DimensionalFlows ........................

31.2.2VortexPoint(VortexParticle/Blobs)

31.3VortexFilaments

31.3.1VortexSegmentandLineofConstantStrength

31.3.2VortexSegmentofLinearlyVaryingStrength

31.4Multipoles

31.4.1Dipole-Doublet

31.4.2Multipoles

31.4.3ConstantPanels

31.4.4EquivalencesBetweenElements

32ElementaryTwo-DimensionalPotentialFlows

32.1UniformFlow

32.2PointSource,PointVortexandDistributionsofPoints

32.2.1PointSource/Sink ...........................

32.2.2PointVortex

32.2.3PeriodicPointVortices

32.2.4ContinuousDistributionof2DPoints

32.3DoubletandMultipoles

32.3.1Doublet ...................................

32.3.2Multi-poles

32.4Cylinder/EllipseFlows

32.4.1CylinderFlow-Acyclic-NoLift

32.4.2FlowArounda2DEllipse-NoLift

32.4.3CylinderFlow-Cyclic-withLift

32.4.4FlowAboutQuadrics ........................

32.5.1RigidRotation ..............................

32.5.2CornerFlow,FlatPlateandStagnationPoint

33FlowswithaSpreadDistributionofVorticity

33.1AxisymmetricVorticityPatches

33.1.1ExamplesofVorticityPatches

33.1.2CanonicalExample:TheInviscidVorticityPatch

34SphericalGeometryModels:FlowAboutaSphere

35VortexandSourceRings ..................................

35.1VortexRings-GeneralConsiderations

35.2FormulaeforthePotential,VelocityandGradient

36FlowInducedbyaRightVortexCylinder

36.1RightCylinderofTangentialVorticitywithArbitraryCross Section

36.1.1FiniteCylinder-GeneralVelocityField ..........

36.1.2FiniteCylinder-VelocityinTerms ofSolidAngle..............................

36.1.3InfiniteandSemi-in finiteCylindersofArbitrary CrossSections ..............................

36.1.4FiniteCylinderofTangentialVorticity andLinktoSourceSurfaces ...................

36.2RightVortexCylinderofTangentialVorticity-Circular CrossSection ......................................

36.2.1FiniteVortexCylinderofTangentialVorticity

36.2.2Semi-infiniteVortexCylinderofTangential Vorticity

36.3VortexCylinderofLongitudinalVorticity

36.3.1InfiniteCylinderofLongitudinalVorticity

36.3.2FiniteCylinderofLongitudinalVorticity

36.3.3Semi-infiniteCylinderofLongitudinalVorticity

37FlowInducedbyaVortexDisk

37.1Introduction

37.2IndefiniteFormoftheBiot–SavartLaw

37.3De finiteFormoftheBiot–SavartLaw

38FlowInducedbyaSkewedVortexCylinder

38.1Semi-infiniteSkewedCylinderofTangentialVorticity

38.1.1PreliminaryNoteontheIntegralsInvolved

38.1.2ExtensionoftheWorkofCastlesandDurham

38.1.3LongitudinalAxis-WorkofColemanetal.

38.2Semi-infiniteSkewedCylinderwithLongitudinalVorticity

38.3InfiniteSkewedCylinderwithLongitudinalVorticity (EllipticCylinder)

39FlowInducedbyHelicalVortexFilaments

39.1PreliminaryConsiderations

39.1.1Introduction

39.1.2Semi-infiniteHelixandRotorTerminology

39.2ExactExpressionsforInfiniteHelicalVortexFilaments

39.3ApproximateExpressionsforInfiniteHelicalFilaments

39.4ExpressionsforSemi-in finiteHelicesEvaluated ontheLiftingLine

39.5NotationsIntroducedforApproximateFormulae

39.6SummationofSeveralHelices-LinkBetweenOkulov’s RelationandWrench’sRelation

40ABriefIntroductiontoVortexMethods

40.4Classi ficationofVortexMethods

41TheDifferentAspectsofVortexMethods

41.1FundamentalEquationsandConcepts

41.2DiscretizationandInitialization

41.2.1InformationCarriedbytheVortexElements

41.2.2InitializationandReinitialization

41.2.3Initialization-InviscidVortexPatchExample

41.3Viscous-Splitting

41.3.1Viscous-SplittingAlgorithm

41.3.2RateofConvergenceoftheViscous-Splitting Algorithm

41.3.3ApplicationtotheVorticityTransportEquation

41.4ConvectionandStretchingofVortexElements ............

41.4.1Introduction ................................

41.4.2ConvectionofVortexElements

41.4.3Stretching .................................

41.4.4Applications ...............................

41.5Grid-FreeandGrid-BasedMethods

41.5.1Grid-FreeVortexMethods

41.5.2Grid-BasedVortexMethods (MixedEulerian–LagrangianFormulation) ........ 505

41.5.3CoupledLagrangianandEulerianSolvers

41.6ViscousDiffusion-SolutionoftheDiffusionEquation 506

41.6.1DiffusionEquationandVorticityTransport Equation 506

41.6.2FundamentalSolutionandLamb–OseenVortex 507

41.6.3Core-SpreadingMethod 509

41.6.4Random-WalkMethod 510

41.6.5Grid-BasedFinite-DifferencesMethod 511

41.6.6Particle-Strength-Exchange(PSE) 511

41.6.7NumericalApplication:Lamb–OseenVortex 513

41.6.8VorticityRedistributionMethod ................ 514

41.7Boundaries,BoundaryConditionsandLifting-Bodies ....... 514

41.7.1Introduction ................................ 514

41.7.2FluidBoundaryConditions:Free-Flow andPeriodicBoundaries ...................... 515

41.7.3SolidBoundariesinInviscidFlows .............. 515

41.7.4SolidBoundariesinViscousFlows-Vorticity Generation

41.7.5ViscousBoundariesUsingCoupling (Viscous-InviscidorLagrangian–Eulerian) ........

41.7.6Lifting-Bodies ..............................

41.8Regularization-KernelSmoothing-Mollifi cation .......... 517

41.8.1KernelSmoothingviaConvolution withaCut-OffFunction

41.8.2RequirementsontheCut-OffFunction

41.8.3SpecialCaseofSphericalSymmetry

41.8.4ExamplesUsedinParticleMethods

41.8.5RegularizationModelsforVortexFilaments

41.8.6ChoiceofCut-Off/SmoothParameter

41.8.7ApplicationtotheInviscidVortexPatch

41.9SpatialAdaptation-Redistribution-RezoningReinitialization

41.9.1Introduction

41.9.2Remeshing-Rezoning-RedistributionReinitialization .............................

41.9.3GainfromRemeshing-Application toInviscid-VortexPatch

41.9.4ProblemsIntroducedbyRemeshing .............

41.10Subgrid-ScaleModels-LES-Turbulence

41.11AccuracyofVortexMethods,Guidelines,Diagnostics andPossibleImprovements ...........................

41.11.1GuidelinesandDiagnosticsforGeneralVortex Methods

41.11.2BoundaryElements-GuidelinesandDiagnostics 535

41.11.3ParticleMethods-Convergence

41.11.4ApplicationtotheInviscidVortexPatch

42ParticularitiesofVortexParticleMethods

42.1ParticleApproximationandLagrangianMethods

42.1.1NotionofVortexBlob

42.1.2ParticleApproximation

42.1.3DynamicsofLagrangianMethods ...............

42.1.4IncompressibleVortexParticleMethods ..........

42.2StretchingTerm-DifferentSchemes ....................

42.3DivergenceoftheVorticityField .......................

42.3.1MinimizingtheErrorGrowth

42.3.2Corrections

42.3.3CriteriaforCorrection

43NumericalImplementationofVortexMethods .................

43.1InterpolationMethodRequiredforGrid-BasedMethods .....

43.1.1InterpolationinVortexMethods

43.1.2ConceptofInterpolation ......................

43.1.3InterpolationtoGrid(Projection,Griding, Assignment,Particle-to-Mesh) 556

43.1.4InterpolationfromGrid(Mesh-to-Particle) 557

43.2Tree-CodesandFastMultipoleMethod

43.2.1Tree-BasedMethod

43.2.2Tree-BasedMethod-CoefficientsuptoOrder2

43.3PoissonSolvers

43.4NumericalIntegrationSchemes

43.4.1ExpressionoftheDifferentSchemes

43.4.2ExampleofApplicationtotheInviscidPatch 563

43.4.3WorkPresentedbyLeishman 564

43.5VorticitySplittingandMergingSchemes .................

43.6ConversionfromSegmentstoParticles ..................

43.6.1CanonicalExamplesforValidation .............. 566

43.6.2RepresentationofOneSegmentbyOneParticle .... 567

43.6.3RepresentationUsingSeveralParticles

43.6.4TrailedandShedVorticityBehindaWing

43.7DistributionofControlPoints .........................

43.7.1TheWorkofJames-ChordwiseDistribution

43.7.2CosineSpacingandOtherReferences intheTopic

43.8The3/4ChordCollocationPoint

44OmniVor:AnExampleofVortexCodeImplementation

44.3Speci ficConfigurationsUsedinPublications

45.3LiftingSurface .....................................

45.4ThickBodies ......................................

45.5Unit-Tests .........................................

45.6FurtherValidation ..................................

Acronyms

a Axialinductionfactor

aB Axialinductionfactorlocaltotheblade

^ a Axialinductionfactorfrom2DMT

a0 Tangentialinductionfactor

c Chord

cn Normalaerodynamiccoeffi cient

ct Tangentialaerodynamiccoefficient

e Internalenergy et Totalenergy

h Enthalpy

ht Totalenthalpy

h Typicalgridspacinginvortexmethods

h Helixpitch

hB Apparentpitch h=B

h Normalizedpitch h=R

k Dimensionlesscirculation

k2 Ellipticalparameterforellipticintegrals

kt Turbulentkineticenergy

l Helixtorsionalparameter

l Normalizedtorsionalparameter l=R

m Ellipticalparameterforellipticintegrals

nrot RotationalspeedinRPM: X=ð2pÞ

p Staticpressure

pt Totalpressure p þ 1 2 qu2

p Frequencyassociatedwith X, p ¼ X=2p

q Heat flux

r Radialposition

rc Viscouscoreradius

t Time

t0 Parameterinthecore-spreadingmodel

r Dimensionlessradialposition r =R

r Dimensionlessradialposition r =R

s Sign

uh Tangentialinducedvelocity

uz Axialinducedvelocity

ux-componentofvelocity

vy-componentofvelocity

wz-componentofvelocity

w Wakerelativelongitudinalvelocity(Betz)

z0 Surfaceroughnesslength

A AngularImpulse

A Area

ARSeeAbbreviations

B Numberofblades

CC Dimensionlesscirculation

Cd Dragcoefficients

Cl Liftcoefficients

Cl;a Liftcoefficientslopeforsmallangles

Cp Powercoefficient

Cq Localtorquecoefficient

CQ Totaltorquecoeffi cient

Ct Tangentialaerodynamiccoefficient

Ct Localthrustcoefficient

CT Totalthrustcoefficient

D Dragforce

D Rotordiameter

D Deformationmatrix

E Completeellipticintegralofthe2ndkind

E Energy

E Enstrophy

F Tip-lossfactor

Fa Tip-lossfactorbasedonaxialinduction

FC Tip-lossfactorbasedoncirculation

FCl Performancetip-lossfactor

FGo Goldstein’stip-lossfactor

FGl Glauert ’stip-lossfactor

FPr Prandtl’stip-lossfactor

FSh Shen’stip-lossfactor

F Complexvelocitypotentialin2D

G Green’sfunctionassociatedwiththeoperator

H Heavisidefunction

H Bernoulliconstant,e.g., p þ 1 2 qu2

It Turbulenceintensity

I LinearImpulse

J Helicity

K Kernel(associatedwithagivenoperator )

K Completeellipticintegralofthe1stkind

L Liftforce

MaMachnumber

P Power

P Palinstrophy

Q Rotortorque

Q VorticalHelicity

R Rotorradius

ReReynoldsnumber

S Entropy

S Surface

S Energydensityspectrum

Sd Volumeoftheunitspherein Rd

T Thrustforce

T Temperature

U Longitudinalvelocityattherotorin1D

U Relativevelocityattherotor

U0 Longitudinalvelocityfarupstream

Ui Inducedvelocityin1D

Un Velocitynormaltotherotor

Uref Referencevelocityused,e.g.,forthenormalizationofloads

Ut Velocitytangenttotherotor

V Velocityvector

Vrel Relativevelocity

V Volume

W Inducedvelocityvectorattherotor

a Point/Blobvorticityintensity

a Angleofattack

a0 Angleofattackatzerolift

b Twistangle

c Surfacevorticity-Distributedcirculation

ct Vortexcylindertangentialvorticity

cl Vortexcylinderlongitudinalvorticity

cb Boundvorticity

d Diracfunction

e Pitchangleofthewakehelixscrew

e Regularizationparameter

f Regularization/cutoff/smoothingfunction

g Efficiency

h Azimuthalcoordinate

j Goldstein’sfactor

k Tipspeedratio ¼ XR=U0

k

r Localspeedratio ¼ kr =R

k FirstLamé’scoefficientforNewtonian fluid

l SecondLamé’scoefficient:dynamicviscosity

m Kinematicviscosity ¼ l=q

q Airdensity 1.225kg/m3

r Localbladesolidity ¼ Bc=2pr

r Cauchystresstensor

s Shearstress,viscousstresstensor

/ Flowangle

v Wakeskewangle,inyawconditions

w Azimuthalcoordinate

w Vectorpotential

x Rotationalspeedofthewake

x Vorticity

C Circulation

D Laplacianoperator r2

H Dilatation

P Gatefunction

P Completeellipticintegralofthe3rdkind

U VelocityPotential

W Streamfunction(2D)

W Stokes’ streamfunction(3D)

X Rotationalspeedoftherotor

X Rotationmatrix(fluidkinematics)

X Solidangle

X Volumeofthedomain

X Totalvorticity

@ X Surfaceboundaryofvolume X

tX Transpose

X T Transpose

r Deloperator, “nabla”

div Divergence, divX ¼r X

divT ¼ @j ðTij Þei

gradGradient,grad X ¼rX

gradGradientof first-ordertensor

curlRotational,curl X ¼r X

e.g. exempligratia: “forexample”

i.e. idest: “thatis”

viz. videlicet : “namely”

w.r.t. “withrespectto”

1DOnedimension

2DTwodimensions

3DThreedimensions

ACAerodynamiccenter