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BROADBAND

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BROADBAND

    BROADBAND

    Availableonlineat^?.sciencedirectcom

    ScienceDirectSciencerect

    .1oumaIofHydl.odynainic~

    2010.22(3):387392

    DOI:10.1016/S1O016058fO916OO69X

    387

    l}{t."sciencedireclc0,

    science.'iiournM/lOtJl005g

    BRoADBANDRoToRNoISEPREDICTIoNBASEDoNANEW

    FREQUENCY.DOMAINFOUMULATION

    dou,FANGBin XIEJianbo,ZHOUQi

    DepartmentofNavalArchitectureandOceanEngineering,NavalUniversityofEngineering,Wuhan430033.

    China,Email:xiejb@126.corn

    (ReceivedOctober29,2009,RevisedMarch72010)

    Abstract:Thisarticlestudiesthebroadbandnoiseofarotorinupstreamturbulence.Anumerica1approachisproposedbasedon

    frequencydomain,forpredictingrotorbroadbandnoisewhichrequirestheaerodynamicsourcestobeintegratedovertheactual

    bladesurfaceratherthanoverthemean

    chordsurface.Thepredictionoftheradiatedrotorbroadbandnoiseduetoturbulenceismade Thismethodisvalidatedthroughacomparisonbetweennumericalpredictionsandmeasureddata,withareasonableagreement.

    Noiseleve1directivityshowsthatthemainlobeis1ocatedalongtherotoraxis,whiletheminimumnoiseoccursinthedirection

    vertica1totherotoraxis

    Keywords:acoustics,broadbandnoise,rotor,frequencydomainformulation 1.Intr0ducti0n

    Thenoisegeneratedbyinflowdisturbancescan resultintheradiationofbothbroadbandanddiscrete tonesdependinguponofthestructureofthe turbulenceenteringtherotor.Iftheturbulenteddvis

    longenoughtobecutbymorethanoneblade,discrete toneswouldbegeneratedatthebladepassing frequencyandhigherharmonicsIftheeddiesare smallcomparedwiththedilnensionofbladesan inducedunsteadybladeloadingwouldradiate broadbandnoisetothefarfield.

    Followingtheacousticanalogy.theclassical approachinaeroacousticsistorepresenttheacoustic sourcesasanequivalentdistribut

    

    ion

    1

    ofacoustic

    monopoles.dipolesandquadrupoles'.Thestrength ofthesesourcesisusuallydeterminedeither experimentallyoranalytically.Inpropellerorrotor problems.thesourceareaofintegrationismostly takentobeonthepro_iectiondiskoftheblade meanchordsurface.WithoutdueconsideratiOnof theactualpropellerbladesurfaces.significanterrors Biography:XIEJianbo(1978),Male,Ph.D.Candidate

    maybeincurredinthisapproximation.

    Theobjectiveofthepresentstudyistodevelopa frequency.domainnumericalmethodforpredicting

    broadbandnoiseofrotorsinupstreamturbulence. Withthismethod,thesourceintegrationisperformed oiltheactualbladesurfaceratherthanontheprojected diskorblademeanchordsurface.Numerica1

    predictionsofthebroadbandnoiseofarotorare comparedwithexistingexperimentaldata. 2.Theoryofairfoilbroadbandnoise

    Anairfoilwithsmallthickness.chordof2b andspanof2displacedinaturbulentincidentflow. andtheconvectivevelocityiSU.Figure1showsthe geometricalcharacteristicsoftheairfoilaswellasthe calculationreferenceflameandtheobserverposition. Theobservationpointislocatedat=(x,,,x3), andthesourcepointislocatedatY(Yl,Y2,Y3).The surfacepressurefluctuatingfieldisexpressed accordingtotheincidentturbulentfield.Zhou numericallypredictedtheradiatedbroadbandnoise withthefollowingbasicformulation

    鳢一

    388

    Spp(,)=fIH(,,K,)1.

    (K,K:)dK(1)

    whereSrepresentsthefarfieldacousticpower spectraldensity,P0isthedensityoftheambient flows.K.=o9/Uisthewavenumberinchordwise

    directionK.iSthewave.numberinspanwise direction,Hrepresentsthetransferfunction betweenthesurfacepressureandtheradiatedsound, and3(KL,)isthetwodimensionalspectrumof

    theisotropicVon.Karmanmode1.

    Fig.1Schematicdiagramofanairfoilandthereferenceflame ThoughthetheoryiSforthepredictionofairfoil broadbandnoiseinturbulentflows,itcanbeextendto rotatingbladeswithsomesimplifications. AssumingthattheacousticwavelengthIS sufficientlysmallincomparisonwiththesemispand, asimplifiedexpressionofEq.(1)iSobtainedas Spp(x,):l,(,K1,)l()

    whereS33(K1)istheonewavenumberspectrumof

    Von.Karmanmodel,whichisdiscussedindetailin Section3,andHpisthetransferfunction H(x,l,co)(3)

    whereg(YI,KI)istheaero'acoustictransfer functionbetweenturbulencedisturbancesandsurface pressure,assuggestedinRef.[10]

    g(y,K1)=(g1+g2)ei['']

    where

    (,)=(

    1+M)(1+

    g(,K)={1+(1+i)[2(1)])?

    [2~k-(1+M)]l,2

    whereMrepresentstheMachnumber,iSthe chordwisecoordinatenormalizedbythehalfchord, =

    /,=41M,k=Klbisthe

    wavenumberinthechordwisedirectionnormalized bythehalfchord,typicallycalledthereduced frequency.FistheFresnelintegration,definedby ()

    I(x,J,)inEq.(3)istheproductofunitvector

andthederivativesofGreenfunction

    

    )()f81

    whereisthenormalunitvectorpointinginwards fromthebladesurface,and(x,Y)istheFourier transf0mofGreenfunction[

    ,

    whichtakestheform

    (,)=1eE

    whereE=R+M(ylx1),:k./fl,and

    :o)/c0istheacousticwavenumberrelatedtothe

    observationfrequency.

    3.Turbulencemodel

    Inmostofthemodels,theturbulenceenteringthe rotorplaneisassumedtobeisotropic.Therefore. thepowerspectraldensityofupwashfluctuationscan beexpressedintermsoftheturbulenceintensityU andtheintegrallengthscale.Thetwo

    dimensionalturbulencespectrumofVon-Karman ,ll,

    /,?,

    \

    d

    ,,-,

    X

    ,,?,

    ,

    T,L

    ,l?,

    K

,,?,

    ,

    g

    ?

    mode11]isoftheform

    ,罴备

    where

    

    Ki

    

    '

    IntegratingEq.(1O)overspanwisewavenumber

    K2,theonewave-numberspectrumofVonKarman

    modelcanbeobtainedas

    4.Broadbandnoisepredictionbasedon

    frequency.domainformulation

    ForarotorwithBbladesoperatinginthe turbulentfield,weconsiderarotatingbladesegment locatedatradialdistanceandazimuthalangle, asshowninFig.2.TheobserveriSsupposedtobein the(,Y)planeinthepositione(o,r),where iSthedistancebetweentheobservationandtherotor center.AlocalreferenceflamefX,y,z)iSdefined whereXisinthechordwisedirection,Yinthe spanwisedirection,andrrepresentsthedistance betweentheobserverandthesource.

    ,P

    

    Fig.2Schematicdiagramofarotorandthelocalreference frame

    Supposingthatthefrequencyofthesource 389

    emissioninthereferenceflamefixedattherotor bladeisCOo,andtheobservedfrequencyis,the Dopplerfactorrelatedtothetwofrequenciesis['

    :

    l+sinsin0

    09

    whereMr

    representsthetangentialMachnumberat radialdistancer0.

    Sincethebladesegmentrotatesinacircular motionandtheinstantaneousspectrumchangeswith rotationalangle,itisnecessarytoperform integrationwithrespecttoazimuthalangleovera circleinapplyingairfoilbroadbandnoisetheorytoa rotatingbladesegment.Takingintoaccountofthe Dopplerfrequencyshift,thespectrumduetothat particularbladesegmentisthereforegivenby [()]=27C(,)

    Substitutingtheairfoilbroadbandnoise predictionexpressionEq.(2)intoEq.(13),and replacingconvectivevelocityUinEq.(2)with relativevelocityatthatparticularradialdistance r0,oneobtains

    [Spp(x,)]=1,

    2w

    2

    .

    i(x,Kl,co)1.

()a~odo9(14)

    However.theturbulenceintensityandthe turbulencelengthscale/linVonKarmanmodelof

    theonewavenumberspectrums33(K1)inEq.(11) varywiththerotorradius.Sotheturbulencemodel f)shouldbeincludedwithinthearea

    integration.Replacingtheareaintegrationwiththe azimuthaIintegration.andsummingoverallblade segmentstogether.thefinalexpressionforthe broadbandnoiseproducedinthefarfieldbvB bladestakestheform

    (,)=B?

    390

    ()d()o9'(15)

    whereK1:/U,

    isthewavenumberinthe

    chordwisedirection,anddS(y)representsthe aerodynamicsourcestobeintegratedovertheactual bladesurface.

    5.Comparisonbetweennumericalresultsand measureddata

    FedalaandKouidriexperimentallymeasuredthe bloadbandnoiseofafanrotorinupstream turbulencel"J.Theaxia1faniscomposedofsix camberedrotorsandaducttypehousingwithout

    guidevanes.Thesectionprofileofthetestrotoristhat OfNACA4509.andtheothermainparametersofthe rotorarelistedinTable1.

    TablelMainparainetersoftherotor

    MainparametersValue

TipRadius,Rr(mm)

    Ratioofhubtotip

    Rotationalvelocity(rpm)

    149.5

    0.45

    3000

    8

    ?

    

    '

    4

    O

    0.405060.70.80.9

    ?r

    Fig.3Radialvariationofaxialmeanvelocity Theexperimentwasconductedinananechoic windtunnel,andthefarfieldnoiselevelismeasured bymicrophonearrays2mawayfromtherotorcenter. Hot.wiresensorswereemployedformeasuringthe fluctuationsofturbulence.Theradialvarimionsof axialvelocity,turbulenceintensityandthe. turbulencelengthscaleareshowninFigs.3,4 and5,respectively.

    O2

    01

    O

    0.4050,6070809

    Fig.4Radialvariationofturbulenceintensity 0.40.50.6070.809

    ?

R7

    Fig.5Radialvariationofturbulenceintegrallength AnumericaIschemeisdevelopedforthe efhcientcomputationoftheradiatedbroadbandnoise. Therotorbladesurfaceismeshedintotriangle elementswithamaximumdimensionof8.10times theminimumacousticwavelength.Figure6showsa pressuresidemeshusedtoperforrnthenumerical integrationrequiredinEq.f151.Thetransferfunction Htherefore,canbeexpressedas

    H(1,(Yle,K)Ie(,Y)]

    whereNrepresentsthetotalnumberofelements, andthesubscriptPreferstothethelementonthe bladesurface.

    Substitutingtheaxialvelocity,turbulence intensityandturbulenceintegrallengthscale ?

    2

    ,I,

    X

    /』?

    rf

    I,

    K

    /,l,

    \

    g

    f0

    

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