Aerodynamic Design of Propeller and Parametric Considerations：螺旋桨的气动设计和参数化的思考

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AerodynamicPerformancesofPropellerswithParametricConsiderationsontheOptimalDesign

S.D’Angelo,F.Berardi,E.Minisci

DepartmentofAeronauticalandSpaceEngineering

PolitecnicodiTorino,Turin-Italy

ABSTRACT

Inthispapertwonumericalproceduresarepresented:thefirstalgorithmallowsforthedeterminationofthegeometriccharacteristicsofthemaximumefficiencypropellerforagivenoperativeconditionandprofiledistributionalongtheblade;theoutputofthisnumericalprocedureisthechorddistributionandtwistangleoftheblade,togetherwithitsefficiencyanditstorqueandthrustcoefficientsfortheprescribedoperativecondition.Theaerodynamiccharacteristicsoftheoptimumpropellerwhenoperatinginaconditiondifferentfromthedesignoneareobtainedbyasecondalgorithmthatallowsfortheevaluationoftheefficiency,thethrustandtorquecoefficientsofapropellerofknowngeometry,whenthebladepitchandoperativeconditionarevaried.

Inthepapertheformulationusedforderivingthegeometryoftheoptimumpropelleranddeterminingitsperformanceswhenoperatingoff-designisdescribedindetail.Theresultsobtainedfromtheproposedpropellermodelhavebeenvalidatedbycomparisonwithexperimentaldata.

Nomenclature

b

nondimensionalbladesectionchord,b=l/R

c

soundspeed

cd

airfoildragcoefficient

cl

airfoilliftcoefficient

D

drag

E

aerodynamicefficiency

FA

totalaerodynamicforce

kP

Prandtlcorrectionfactor

K

Lagrangefactor(constant)

l

bladesectionchord

L

lift

M

enginetorque

Ma

freestreamMachnumber,Ma=V/c

Ma

localMachnumberatstation,Ma=VE/c

n

bladenumber

P

shaftpower

r

coordinatealongbladespan

R

propellerradius

Re

freestreamReynoldsnumber,Re=R*V*/

Re

localReynoldsnumberatstation,Re=l*VE*/

T

thrust

uD

inducedvelocityintheplaneofthepropeller

V

forwardspeed

VA

apparentvelocity,VA=(V2+(r)2)1/2

VE

actualvelocityofflowapproachingtheairfoil

nondimensionalactualvelocity,=VE/V

actualairfoilangleofattack

A

apparentairfoilangleofattack

Emax

angleofattackformaximumairfoilefficiency

i

inducedincidence

tw

twistangle

advanceratio

circulation

anglebetweenactualvelocityandpropellerplane

A

anglebetweenapparentvelocityandpropellerplane

efficiency

bladesectionpitchanglerelativetopropellerplane,()=tw()+0

0

collectivepitch,0=(=0.75)

airviscosity

nondimensionalcoordinatealongtheblade

airdensity

thrustcoefficient

torquecoefficient

angularcomponentofinducedvelocityinthepropellerplane

propellerangularvelocity

INTRODUCTION

Thenumericalprocedureproposedinthepresentpaperrequirestheaprioriknowledgeoftheaerodynamiccharacteristicsoftheairfoilsusedforthepropeller.Thiscanbeobtainedfromanexperimentalorcomputationaldata-base.

Itisnotalwayspossibletoobtaindetaileddata-basesthattakeintoconsiderationbothReynoldsandMachnumbereffectsinsuchawiderangeasthatencounteredatdifferentsectionsofanaeronauticalpropellerblade.LinearinterpolationandextrapolationbetweenexperimentaldataisusedinordertotakeintoconsiderationReynoldsnumbervariation,whilesemiempiricalcorrectionsareimplementedfortakingintoconsiderationcompressibilityeffects.

Thepropellerperformancedeterminedbyasimplifiednumericalprocedurecomparewellwiththeexperimentalresultsobtainedfromwind-tunneltests,asfarasthesectionat0.75ofbladespanisoperatingbelowstallangleandbelowdrag-divergenceMachnumberintheconsideredcondition.

Inwhatfollows,theclassicalaerodynamicpropellertheoryisdiscussed.Theformulationisorientedtowardstheimplementationofanumericalalgorithmforthedeterminationofthepropellerofmaximumefficiencyforagivenoperatingcondition.

Afteradetaileddiscussionoftheaerodynamictheoryofpropellers,theexperimentaldatabaseofairfoilcharacteristicsusedintheevaluationoftheoptimumbladeispresentedandacomparisonbetweenexperimentaldataandnumericalresultsforthegivenairfoilisdiscussedforvalidatingtheproposedalgorithm.

Aparametricstudyoftheoptimumbladeshapeatdifferentoperatingpointsispresented,soastoderivesomegeneralconsiderationsonthegeometryofthemaximumefficiencypropeller.Thesegeneralprinciplescanbeusefulindefiningproperinitialsolutionswhenmoresophisticatedoptimizationtoolsareused,withmeritfunctionsdifferentfromtheaerodynamicefficiencyoreveninthecaseofmulti-objectiveoptimization.

Thesameformulationusedfortheoptimizationroutineisimplementedinasecondnumericalalgorithmfortheevaluationoftheaerodynamiccharacteristicsofagivenpropellerinanyoperatingcondition,intermsofbladepitch,,andadvanceratio,.Thevaluesofthrustcoefficient,torquecoefficientandefficiencyareobtained,i.e.(,),(,),and(,).

Thisroutineisvalidatedcomparingitsresultswithavailableexperimentaldata.Theaccuracyofthenumericalpredictionseemstobesatisfactoryfromanengineeringstandpoint.Moreoverthecomputationaleffortrequiredbytheproposedalgorithmisverylimitedandthismakeitsuitableforitsimplementationindifferentoptimizationprocedures,singleobjectiveormulti-objective,deterministicornon-deterministic.

Fromaerodynamicpropellertheorytopropellerdesign.

Inthispaperwewillconsiderpropellergeneratingthrustinanaxialflow.Thebladehasvariablechordandtwistanglebutthefeatheringaxisisassumedrectilinearandlyinginaplaneduringtherevolution.

Theaerodynamictheoryadoptedisbasedonclassicalresults[1],[2],obtainedfromtheintegrationofvortextheory,wingtheoryandmomentumtheory.

Whenvortextheoryisadopted,propellerthrustandtorqueareexpressedasafunctionofcirculationalongtheblade.Thrustdistributionofminimumenergydissipationisobtainedbyavariationalapproach,adoptingPrandtlsimplifiedapproach[3].

Ifwingtheoryischosenforrepresentingtheaerodynamicbehavioroftheblade,thrustandtorqueareobtainedfromintegrationofelementaryliftanddragcontributionactingonaninfinitesimalbladeelement.Theinducedvelocityisevaluatedlocallycombiningresultsfromwingtheoryandmomentumconservationprinciple.

Vortextheory

Figure1representagenericbladeelement.ThemeaningofthesymbolsusedinthefollowingpagescanbefoundintheNomenclatureandisrepresentedinthesamefigure.

Theactualvelocityoftheflowpastagivensection,VE,isgivenbythevectorsumoftheapparentvelocity,VA,andtheinducedvelocityincrementattheconsideredsectionuD.VAisgivenbythesumofforwardspeedVandrotationalspeedofthesectionr.

Whenabladeelementdratpositionralongthebladespanisconsidered,thecirculation(r)canbeexpressedaccordingtoPrandtlapproximationas:

(1)

(2)

Thrust,torque,andpowerdissipatedbythepropellerareobtainedintegratingtheelementarycomponentsalongasinglebladeandmultiplyingtheresultbythenumberofblades.

Onagivenbladeelement,theaerodynamicforcedFA,thrustdT,torquedManddissipatedenergyperseconddPdaregivenrespectivelyby:

(3)

(4)

(5)

(6)

Calculusofvariationsallowsforthedeterminationoftheoptimalcirculationdistribution(r)thatminimizestheenergylossforagiventhrustT.

Letting-KVbetheLagrangefactor,thederivativeofthelinearcombinationofthrustandpowerlossisequatedtozero.Theresultingequationissolvedwithrespecttothecirculation,andtheconditionforminimumpowerlossisobtained:

(7)

Thecirculationdistribution,theactualvelocityandinducedvelocityforthesectionatcoordinaterfromthepropelleraxisarethenexpressedby:

(8)

(9)

(10)

Givenn,T,V,,andR,theLagrangemultipliercanbeobtainedintwodifferentways:

bynumericalsolutionoftheimplicitfunction:

(11)

where

(12)

byasimplifiedformulationobtainedneglectingK2withrespecttothefirstordertermasfarasK1,sothat:

(13)

where

(14)

Wingtheory

Asstatedintheintroduction,thewingtheoryallowsfortheevaluationofthrustandtorquegeneratedbyapropellerinagivenoperatingconditionbyintegratingtheelementaryaerodynamicactionsactingonaninfinitesimalportiondroftheblade.

Inthiscaseelementarythrustandtorqueareexpressedas:

(15)

(16)

Theactualvelocityandtheinducedincidenceareobtainedequatingtheexpressionforthepropellerthrustderivedfromvortextheoryandthatobtainedbywingtheory.Theresultingexpressionsare:

(17)

(18)

Non-dimensionalcoefficientandindependentparameters

Theadvanceratio,thethrustcoefficient,thetorquecoefficient,andthepropellerefficiencyaredefinedasfollows:

(19)

(20)

(21)

(22)

Thrustandtorquecoefficientscanbeexpressedasafunctionof6independentparameters,i.e.:

(23)

(24)

IfweapplythetheoremofBuckingam,weobtainthefollowingrelationsintermsofnon-dimensionalparameters:

(25)

(26)

whereMaistheMachnumberoftheundisturbedflowupstreamthepropeller,whileReistheReynoldsnumber,thereferencelengthbeingthepropellerradiusR.

WewillassumethattheinfluenceofReandMaoneisdueonlytotheireffectontheaerodynamiccoefficientsclecdofeachbladesection,i.e.ontheaerodynamiccharacteristicsofeachairfoil.

Finallywewillintroducealsothefollowingnon-dimensionalquantities:

(27)

(28)

(29)

(30)

(31)

whereisanon-dimensionalcoordinatealongthebladeradius,bisthebladesectionnon-dimensionalchord,isthenon-dimensionalactualvelocitypasttheconsideredbladesection,ReandMaarethelocalReynoldsandMachnumber,respectively.

Aerodynamicdatabase

Theevaluationoftheaerodynamicperformanceofapropellerrequiresthedetailedknowledgeofcharacteristicoftheairfoilusedfortheblades.TheaerodynamicdatabaseshouldprovidethevaluesofairfoildragandliftcoefficientsasafunctionofangleofattackandReynoldsnumber.

Usuallydifferentairfoilsareusedalongtheblade,butaerodynamiccharacteristicsareavailableforonlyfewofthem.Inthiscaseitisassumedthatintermediatesectionsbetweenknownprofilesarecharacterizedbyan“intermediate”aerodynamicbehaviour,i.e.liftanddragcoefficientsareobtainedbyaproperlinearcombinationthatweightstheairfoilcharacteristicsasafunctionoftherelativedistancefromtheprofileswithknowncharacteristics.Ifxisalocalnondimensionalcoordinatebetweentheknownsections,liftanddragcoefficientfornon-compressibleflowwillbeexpressedas:

(32)

(33)

Inordertoobtainsignificantresultsfromtheinterpolation,itisnecessarytoconsidertheeffectsofReynoldsnumberontheaerodynamiccoefficients,asfarasRexexperiencesasignificantvariationwhendifferentsectionsareconsidered.

Compressibilitycorrection

Compressibilityeffectscanhaveasizeableinfluenceonthevaluesofliftanddragcoefficients.

ThelocalMachnumberalonganaeronauticalpropellerbladecanvaryinsuchawiderangethatthiseffectscannotbeneglectedwithoutanunacceptablelossofaccuracy.IfaerodynamicdataexhaustivelyincludecompressibilityeffectsitispossibletointerpolateaerodynamicdatawithrespecttoMachnumberinthesamewayasitisdoneforReynoldsnumber.WhentheavailabledataarelimitedtolowMachnumber,asemi-empiricalfactorisderivedsoastoprovidethenecessarycorrectionforclandcdduetocompressibilityeffectsfromtheirvaluesforthecaseofnoncompressibleflow

Thecorrectionfactorusedinthepresentpaperwasoriginallyderivedforsymmetricalairfoilswithrelativethicknesst=0.21,forMa<0,9and-25°25°,butitisreasonabletoextendtheuseofthiscorrectiontoairfoilswithmoderatecamberandhighervaluesofrelativethickness.

Thedeterminationofthecorrectionfactorisbasedon:

thedeterminationofthelocalcriticalMachnumber

thedeterminationofthelocaldrag-riseMachnumber

thecorrectionoftheaerodynamiccoefficients

ThecriticalMachnumberisrelatedtotheminimumpressurecoefficientactingontheairfoilinanon-compressibleflowby([4],[5]):

(34)

Reference6providesthevalueoftheminimumpressurecoefficientatzeroliftincidence,cpi,min0,forseveralairfoil.Fromthisdataitispossibletoexpresscpi,min0forthe4digitNACAprofilesasafunctionoftherelativethicknessasfollows:

(35)

Anotherrelationbetweenminimumpressurecoefficientandliftcoefficientinincompressibleflowandrelativethicknesst,isdiscussedinRef.7:

(36)

Theresultsobtainedfromthislatterapproachshowapooragreementwithexperimentaldata,asitisclearlyvisibleinFig.2.

Abetterrepresentationisobtainedbyaddingtocpi,min0acontributionproportionaltocli2/t,withacoefficientasmallerthanunity.Comparisonwiththesameexperimentaldataprovidedforaavalueof0.75,sothatwecanexpresscpi,minas:

(37)

ThevalueofthecriticalMachnumberisobtainedsolvingthefollowingequation:

(38)

InmostcasesMacrindicatesathresholdbeyondwhichcompressibilityeffectsonaerodynamicperformancecannolongerbeneglected,eveniftherearenoimmediateconsequences.ButwhentheratioMa/Macrisgreaterthanavaluebetween1.04and1.20,dragincreasesdramatically.

ThevalueofMaforwhichdcd/dMa=0.1isthedragriseMachnumber,MaDR,andindicatesalimitbeyondwhichairfoilaerodynamicperformancedegradesseriously.

FromtheexperimentaldatareportedinRef.[8]and[6],arelationbetweenMaDRandMacrisderived,wheretheliftcoefficientinincompressibleflowappearsasaparameter:

(39)

WhenMa<MaDR,theliftcoefficientincreaseswithMachnumber,forthesameincidence.

Theliftcoefficientincompressibleflow,clc,isobtainedaccordingtoKaplanrelation(thatisamodifiedversionofPrandtl-Glauertcorrectionfactor):

(40)

(41)

ForMa>MaDR,liftcoefficientstartstodecreasewithMachnumber,withaminimumforMa0.9.Inthiscasethecorrectionfactoris:

(42)

ThedragcoefficientcddoesnotdependonMachnumberbelowMaDR,but,forMa>MaDR,energylosscausedbytheshockwaveisresponsibleofasharpincreaseinaerodynamicdrag.Thedragcoefficientisevaluatedaddingtoitsvaluefornon-compressibleflowCdiacontributionduetowavedrag,accordingto[9],[10]:

(43)

Acomparisonbetweenexperimentaldata(inRef.[8])andresultsobtainedaccordingtothecorrectionsofnon-compressiblecoefficientsisreportedinFigs.3and4.Itisevidentthatwehavegoodagreementforvaluesoftheangleofattackbelowstall.Asafinalconsiderationonthiscorrectionprocedureforcompressibilityeffects,weunderlinethattheresultsareaccurateforsymmetricalprofiles,withthicknessratiolessthan0.15andforMachnumberbetween0.3and0.9,inthelinearrangeoftheliftcurve.Thesimplicityoftheprocedureallowsforareducedcomputationaleffort.

Characteristicsofann-bladespropellerofradiusR

Thefeatheringaxisoftheblade,representedbyanon-dimensionalcoordinateintheinterval[0,1],isdividedintomelements,thusdefiningm+1sections.Theactiveportionofthebladebeginsatminandnisthenumberofsectionswithmin.Ifnpsectionsairfoilsofknowncharacteristicsareused,thegenericairfoilatcoordinatealongthebladewillbeidentifiedbythedistancexfromtheclosestairfoiliponthehubsideandthedistance1-xfromtheclosestairfoilip+1onthetipside.

Ourproblemcanbestatedasfollows:evaluatethrust,torqueandefficiencyofapropellerwithnbladesofknowngeometry(radiusR,chorddistribution,twistangle),fromtheaerodynamiccharacteristicsoftheairfoilused,forgivenoperatingconditions(forwardspeed,V,angularvelocity,,altitudeandbladepitch,thelatterbeingdefinedasthepitchangleofthesectionat0.75·R).

Foranybladesectionthelocalaerodynamiccharacteristicsareevaluatedaccordingtowingtheory.Itisthusnecessarytocalculateactualvelocity,itsanglewithrespecttothepropellerdiskandtheaerodynamiccoefficientsofthesectionsasafunctionoftheinducedincidence.

Innon-dimensionaltermsthethrustandtorquecoefficientsandthepropellerefficiencyareevaluatedasafunctionofbladenumbern,b(),tw,airfoilcharacteristics,,Re,Ma,.

Thesoftwareinputisdividedintotwosetsofparameters,thefirstonefordescribingthepropelleroperatingpoint,thesecondforthepropelleraerodynamiccharacteristicsanddiscretizationparameters:

bladenumber,n

numberofdiscretizationintervals,n

numberofairfoilswithknownaerodynamiccharacteristics,np

aerodynamiccharacteristicsoftheknownairfoils

foreachvalueofj,j=1,2,…,n,thevaluesofb,tw,ip(indexoftheclosestknownairfoilonthehubside,i.e.airfoil1),ip+1(indexoftheclosestknownairfoilonthetipside,i.e.airfoil2),thevalueofx,non-dimensionaldistancefromairfoil1,i.e.x_=_(j–ip)/(ip+1–ip).

Foreverysectiontheevaluationoftheaerodynamiccoefficientsiscarriedoutaccordingtotheprocedureoutlinedinthepreviousparagraph.Giventhevaluesof,Re,Ma,,n,b,twandprofiledistributionalongtheblade,thevaluesof,,andiaredeterminedbyiterationfori.Startingfromtherelations:

(44)

(45)

(46)

(47)

(48)

thelocalaerodynamiccoefficientsareevaluated:

(49)

(50)

(51)

(52)

Onceconvergenceoniisreached,thelocalincrementsforthrustandtorquecoefficientsareexpressedasfollows:

(53)

(54)

Finallythevaluesofdanddareintegratedalongthebladespan,giving:

(55)

(56)

(57)

Comparisonwithexperimentaldata

AcomparisonbetweennumericalresultsandexperimentaldatareportedinRef.11ispresentedinFigs.5,6and7tovalidatetheproposedalgorithm.TheaerodynamiccharacteristicsoftheairfoilsoftheNACA44XXseriesusedforthepropelleranalyzedinRef.11wereobtainedfromRefs.12and6.

Thevaluesof,andarereportedinFigs.5,6and7,respectively,asafunctionoftheadvanceratio.

TheagreementbetweenourmethodandexperimentalresultsappearssatisfactorywhentherepresentativesectionlocalMachnumberisbelowMaDRandisangleofattackisbelowstall.Forhighervaluesoflocalvelocityorbladepitchangletheempiricalcorrectionsarenolongersufficientforobtaininganaccuraterepresentationofthecomplexphysicalphenomenaaroundaconsistentportionoftheblade.

Inparticular,theplotsshowanoveroptimisticpredictionofthepropellerperformanceforhighvaluesofthebladepitchangle.

Designofthemaximumefficiencypropeller

Inthisparagraphanumericalprocedureforthedeterminationoftwistangleandchorddistributionasafunctionofthedistancerfromthehubforann-bladepropellerofradiusRispresented.TheoptimumpropellerwillminimizestheenergylossforprovidingarequiredthrustT,giventheoperatingpointintermsofforwardspeed,propellerangularvelocityandaltitude.

Themaximumefficiencyofthepropellerwillbeobtainedifalltheairfoilsalongthebladespanareattheirmaximumefficiencyangleofattack.Foreachsection,theangleofattackformaximumefficiencyisdetermined,asafunctionoflocalReynoldsandMachnumber.Aniterativeprocedureisnecessaryfordeterminingthevalueoftheairfoilchordforagivensection.

Theresultingchordandtwistangledistributionsalongthebladewilldependon4independentparameters,thrustcoefficient,advanceratio,ReynoldsandMachnumbers,Re,Ma.Aparametricstudyoftheoptimumbladeshapeasafunctionoftheseparametersprovidesinterestingresults.

Thenumericalprocedurefordeterminingtheoptimumbladeshape,canbesummarizedasfollows.

Again,theinputdatacanbegroupedintothreesets,afirstonefortheoperatingpoint,,,ReandMa,andasecondone,withthepropellercharacteristicsalreadyassignedbythedesigner:

bladenumber,n,

airfoildistributionalongtheblade,

aerodynamiccharacteristicsoftheairfoilusedandthelastone,givenbythedisctretizationparameters.

Firstofall,theLagrangemultiplierKisdetermined:

foreverysectioniofthediscretizedblade,wherei=1,2,…,n,itis:

(58)

(59)

Theequation

(60)

isthensolvednumericallybyabisectionmethod.

Thedistributionoftwistangleandprofilechordalongthebladeisdeterminedasfollows.Theangleandthelocalvelocityaregivenby:

(61)

(62)

(63)

Foreachsectionithefollowingequationsareiterateduntilconvergenceisreached:

(64)

(65)

Thentheangleofattachofmaximumefficiencyisdetermined:

(66)

wherexand1-xaretherelativedistancefromthenearestknownairfoils(seepar.4)

(67)

Whenconvergenceisreached,theresultinglocalpitchangleisgivenby:

(68)

Thentheglobalcharacteristicsaredeterminedbynumericalintegrationof

(69)

(70)

where

(71)

(72)

Theresultingefficiencyis

(73)

Attheendoftheprocedureitisnecessarytocheckwhethertheassignedbladenumberniscompatiblewiththeprescribedoperatingpoint.Inparticular,ifbmax/R<0.15itisnecessarytoreducenforstructuralreasons,whileifbmax/R>0.24,increasingncanproducesignificantimprovementintermsofaerodynamicefficiency.

Parametricanalysisoftheoptimumblade

Thegeometryoftheoptimumbladedependsonthenon-dimensionalparameters,,ReandMa.Aparametricstudyispresented,limiting,forthesakeofsimplicity,theanalysistothestudyoftwo-bladedpropellerswithconstantairfoilNACA0012.Thecharacteristicsofthisairfoilareknownwithgreatdetailandthecompressibilitycorrectionisparticularlywellsuited.

Therangeofvariationoftheparametersrepresentingtheoperatingpointofthepropelleraredeterminedtakingintoconsideration20singleengineairplaneswithtwobladepropellers.

Theircharacteristicsintermsofcruisealtitudeandairspeed,thrust,diametera