On vortex–airfoil interaction noise including span-end effects, with application to open-rotor ...

24 Pages·2013·4.28 MB·English

by  Michel Roger

JournalofSoundandVibration333(2014)283–306
ContentslistsavailableatScienceDirect
Journal of Sound and Vibration
journal homepage: www.elsevier.com/locate/jsvi
On vortex–airfoil interaction noise including span-end effects,
with application to open-rotor aeroacoustics
Michel Rogera,n, Christophe Schramb, Stéphane Moreauc
aLaboratoiredeMécaniquedesFluidesetAcoustique,ÉcoleCentraledeLyon,69134Écully,France
bVonKármánInstitute,1640RhodeSt-Genèse,Belgium
cDépartementdeGénieMécanique,UniversitédeSherbrooke,Québec,CanadaJ1K2K1
a r t i c l e i n f o a b s t r a c t
Articlehistory: Alinearanalyticalmodelisdevelopedforthechoppingofacylindricalvortexbyaflat-
Received19March2013 plateairfoil,withorwithoutaspan-endeffect.Themajorinterestisthecontributionof
Receivedinrevisedform the tip–vortex produced by an upstream rotating blade in the rotor–rotor interaction
6September2013 noisemechanismofcounter-rotatingopenrotors.Thereforetheinteractionisprimarily
Accepted10September2013
addressed inanannularstripoflimitedspanwiseextentbounding theimpingedblade
HandlingEditor:R.E.Musafir
segment,andtheunwrappedstripisdescribedinCartesiancoordinates.Thestudyalso
Availableonline10October2013
addresses the interaction of a propeller wakewith a downstreamwingor empennage.
Cylindrical vortices are considered, for which the velocity field is expanded in two-
dimensionalgustsinthereferenceframeoftheairfoil.Foreachgusttheresponseofthe
airfoilisderived,firstignoringtheeffectofthespanend,assimilatingtheairfoiltoarigid
flatplate,withorwithoutsweep.Thecorrespondingunsteadyliftactsasadistributionof
acoustic dipoles, and the radiated sound is obtained from a radiation integral over the
actualextentoftheairfoil.Inthecaseoftip–vortexinteractionnoiseinCRORstheacoustic
signatureisdeterminedforvortextrajectoriespassingbeyond,exactlyatandbelowthe
tipradiusoftheimpingedbladesegment,inareferenceframeattachedtothesegment.
Inasecondstepthesameproblemisreaddressedaccountingfortheeffectofspanend
ontheaerodynamicresponseofabladetip.Thisisachievedthroughacompositetwo-
directionalSchwarzschild'stechnique.Themodificationsofthedistributedunsteadylift
andoftheradiatedsoundarediscussed.Thechainedsourceandradiationmodelsprovide
physicalinsightintothemechanismofvortexchoppingbyabladetipinfreefield.They
allowassessingtheacousticbenefitofclippingtherearrotorinacounter-rotatingopen-
rotorarchitecture.
&2013ElsevierLtd.Allrightsreserved.
1. Introduction
Thispaperpresentsananalyticalinvestigationontheimpingementofacylindricalvortexonathinrigidairfoilandthe
associatedsoundradiation.Thevortexaxisisassumedinaplaneperpendiculartotheairfoilplane.Thisgenericvortex–airfoil
interaction mimics a class of mechanisms encountered in rotating blade technology, for which the associated acoustic
signatureisacrucialconcern.Inparticularthemainmotivationoftheworkistheunderstandingofblade–tipeffectsinthe
rotor–rotorinteractiontonalnoisegeneratedbyadvancedcounter-rotatingopenrotors(CRORs).CRORsareidentifiedasan
nCorrespondingauthor.Tel.:þ33472186011;fax:þ33472189143.
E-mailaddress:[e-mail](M.Roger).
0022-460X/$-seefrontmatter&2013ElsevierLtd.Allrightsreserved.
http://dx.doi.org/10.1016/j.jsv.2013.09.012
284 M.Rogeretal./JournalofSoundandVibration333(2014)283–306
Nomenclature ðx ;x ;x Þ observerCartesiancoordinates
1 2 3
ðy ;y ;y Þ sourceCartesiancoordinateswithoutsweep
1 2 3
A0 amplitudefactoroftheexponentialvortex ðy′1;y′2;y3Þ sourceCartesiancoordinateswithsweep
a;a ;a ;bcombinednon-dimensionalwavenumbers ðX;YÞ axialandtangentialcoordinates
1 2
c airfoilchord α skewnessangleofagust
c speedofsound γ staggerangleofablade
0
e blade–tiptovortex–radiusdistance Γ qffiffiffiffiffiffiffiffivffiffiffioffiffiffiffirtexstrength
E;En Fresnelintegrals β¼ 1(cid:2)M2 compressibilityparameter
G~ two-dimensionalwavenumberspectrum qffiffiffiffiffiffiffi0ffiffiffiffiffiffiffiffiffiffi
I¼I1þI2 chordwiseradiationintegrals β1;2¼ 1(cid:2)M21;2 partialcompressibilityparameter
K totalnon-dimensionalaerodynamic κ;κ′ supercriticalandsub-criticalHelmholtz
wavenumber parameters
k¼ω=c0 acousticwavenumber λ acousticwavelength
ðk1;k2Þ chordwiseandspanwiseaerodynamic ϕ;ϕ′;Φ velocitydisturbancepotentials
wavenumbers φ vortexinclinationangle
ðk′1;k′2Þ sameasðk1;k2Þwithsweep Φð0Þ complexerrorfunctionwithcomplexarguments
L¼2d spanlength ψ sweepangle
ℓ¼ℓ1þℓ2 unsteadyliftonabladesegment ρ0 meanfluiddensity
M0;1;2¼U0;1;2=c0 Machnumbers ω angularfrequency
p acousticpressure Ω fanrotatingspeed
ðr;θÞ cylindricalcoordinatesaroundvortexaxis
r0 vortexcoreradius Subscripts/superscripts
R radiusofrotorannulus
0
S0 convection-correcteddistance ð(cid:3)Þn madenon-dimensionalbyc/2,orcomplex
T relativerotor–rotorinteractionperiod
conjugate
U0 incidentflowspeed ð(cid:3)~Þ;ð(cid:3)^Þ Fouriertransforms
U chordwiseconvectionspeed
c ð(cid:3)Þ incident-flowrelatedquantity
V¼ΩR tangentialspeed 0
0 ð(cid:3)Þ chordwisequantity
w upwashvelocity(normaltoabladesurface) 1
ð(cid:3)Þ spanwisequantity
w gustamplitude 2
0
alternativetotheturbofan-enginetechnologyforfutureaircraftpropulsion.Theyarecharacterizedbylowerfuelconsumption
andhigherpropulsiveefficiencyassociatedwithequivalentveryhighby-passratios,upto25or30.Butthemainconcernis
aboutthenoiseexposurearoundairportsduringtake-offandlandingbecausethebladesoperateinfreefield.Furthermore
evenifaCRORsystemismountedattherearofafuselageinordertominimizecabinnoise,thecomfortofpassengersremains
anotherchallengingissue.Thisiswhyeffortsarepresentlymadeintheaeroacousticcommunitytowardsacousticprediction
strategiesthatcanbeusedattheearlydesignstage,inordertodefinelow-noiseconfigurationswiththehelpofoptimization
algorithms.
The most annoying acoustic signature of CRORs at low and moderate frequencies is tonal noise associated with all
periodic aerodynamic fluctuations experienced by the rotating blades. This includes various mechanisms. Typically the
mounting on the airplane is responsible for stationary inflow distortions due to the vicinity of the fuselage, the flow
deviation associated with the mean lift of the wing and the wake of the pylon. This contribution is not intrinsic to the
technologybutisratherqualifiedasaninstallationeffect.Itradiatesatharmonicsofthebladepassingfrequenciesofboth
rotorsandinvolvesbothrotorsnearlyindependently.Othersourcesareintrinsictothecounter-rotatingpropulsivesystem
inthesensethattheydonotessentiallydependontheinstallation.Firstly,thewakesshedfromthefrontrotorimpingeon
thebladesoftherearrotorandproducewhatiscalledwake–interactionnoise.Secondly,thepotentialfieldoftherearrotor
caninduceliftfluctuationsonthebladesofthefrontrotor,generatingpotential-interactionnoise.Finally,thetipvortices
shed fromthefront rotorcan alsobechopped bytherearrotordependingon thecompared tipradiiand/oroperational
conditions.Bothwakeimpingementandblade/tip–vortexinteractionsinvolvesourcesdistributedontherear-rotorblades,
whereas potential-interaction noise is radiated from the front rotor. All these interactions radiate at linear combinations
of multiples of the blade passing frequencies of both rotors, for this reason the acoustic signature is referred to as
combinationtones.Thepresentfocusisonthefundamentalaspectsofblade/tip–vortexinteraction.Itisworthnotingthat
themechanismdiffersfromthewell-knownblade–vortexinteraction(BVI)thatoccursonthemainrotorsofhelicopters.In
typicalBVIthevortexaxisandthebladespanlieinalmostparallelplanes.Forthemostincriminatedoperationalconditions
the excitation by the oncoming vortex extends over a wide portion of the span. For the blade/tip–vortex interaction on
CRORs, the plane tangent to the vortex helical path and the blade span are roughly perpendicular to each other. The
impingementconcentratesonthebladearoundthespanwise/radiallocationofthevortexpath.Thisiswhydesigningrear
rotorbladeswithshortertipradiusthanthefrontrotorblades,calledclippingorcropping,isapossiblewayofavoidingor
M.Rogeretal./JournalofSoundandVibration333(2014)283–306 285
minimizingthisinteractionandtheresultingnoise.Neverthelesstheinteractioncanhavearesiduallevel,eitherbecauseof
theextensionofthevortexouterpartorbecauseofthecontractionofthefront-rotorstreamtube.Furthermoreside-winds
or flight incidence possibly deflect tip vortices and make them interact with blade tips even in the presence of clipping.
This justifies a specific address of the mechanism. The present contribution is a mathematical analysis dealing with two
complementaryaspectsofvortex–airfoil interactions. The firstaspect isthemodelingof thetip–vorticesshed fromrotor
bladesforthesakeofdescribingtheirsubsequentimpingementondownstreamsurfaces.Thesecondone,specifictoCRORs,
istheeffectofbladeterminationattheouterradiusofarear-rotorbladeontheresponseofthatbladetoanoncomingfront-
rotortip–vortex.Initsfinalandcompleteformtheaddressedmechanismcanthenbereferredtoasblade–tip/tip–vortex
interaction(BTTV).
Apart from CROR aeroacoustics, oblique and more general vortex–rotor interactions are encountered in rotorcraft
technology,typicallyasthetipvorticesgeneratedbythemainrotorofahelicopterarechoppedbythetailrotor[1].The
genericproblemofvorteximpingementonanextendedairfoilalsopossiblymimicsthemechanismofnoiseproductionby
interaction of blade–tip vortices with any downstream lifting surface of an airplane. This includes the interaction of the
vorticesshedbyconventionalpullingturbopropswiththeleadingedgeofawing[2],ortheinteractionofthewakefroma
highlyloadedCRORwithatailempennage.Becausetheassociatedtonalnoiseofinstalledpropellersisseldomaddressedin
theliterature,someinteresthasbeenfoundinprovidingphysicalinsightintoit;thisisasecondaryobjectiveofthepaper.
2. Methodologyandproblemstatement
ForthinbladesoperatinginsignificantlydisturbedflowsandatthesubsonicMachnumberscharacteristicoftake-offand
landingregimesofCRORs,FfowcsWilliamsandHawkings'formulationof theacousticanalogy[3,4]statesthatthenoise
is essentially produced by unsteady lift forces distributed on the blades. These forces act as equivalent acoustic dipoles
radiating in a homogeneous propagation medium. Once they are known the sound field is simply obtained from the
background of linear acoustics, therefore the critical task is to determine the forces first. When dealing with open-rotor
tonalnoisetheforcesareperiodicandcanbesimulatedaccuratelybyComputationalFluidDynamics,forinstanceUnsteady
Reynolds-AveragedNavier–Stokesmethods[5–8].Thisrequiresprohibitivecomputationalresources.Fast-runninganalytical
techniquesaremuchmorecompatiblewiththeneedforrepeatedcalculationsinthecontextofoptimizationalgorithms.
Thecounterpartisthatcrudesimplificationsareoftenmadeofboththeflowfeaturesandthegeometryoftheblades.
NowtheliftdipolesofCRORtonalnoisearecoherentsourcesdistributedoverhighlynon-compactbladescharacterized
by sweep, twist and variable chord. They interfere in a way that must be reproduced with minimum inaccuracy so that
thenoiseestimatesmakesense.Thisaccuracyrequirementisbeyondtheperformancesofmostexistinganalyticaltheories
which state about the aeroacoustic response of rectangular flat-plates free of tip effects. The sound-generating flow
disturbancessuchaswakesortipvorticesalsoneedbeingproperlydescribedinathree-dimensionalcontext.Itappearsthat
gettingaphysicallyconsistentevaluationofthetonal-noisesourceswithanalyticalmethodsismuchmorechallengingthan
describingthesourcesofbroadbandnoise.Indeedthestatisticaldescriptionofrandomlydistributedsourcesislesssubject
tointerferenceissues[9].Inatonal-noisecontextitisessentialtoextendtheanalyticalapproachsothatkeyfeaturesofthe
realdesignareaccountedforintheblademodel.
Thepresentdevelopmentsstartfromthelinearizedtheoryofthinflat-plateresponsetooncomingfrozendisturbances.
As an alternative to the approach proposed by Howe [10], they are basically similar to Amiet's theory of vortex–airfoil
interaction[11],inwhichtheimpulsivenoiseproducedbyasinglevortexchoppingisderivedinthetimedomainfroma
responsefunctionoftheairfoilviainverseFouriertransform.Incontrastthewholeanalysisisheremadeinthefrequency
domain because it is dedicated toperiodic interactions. Vortexchopping is interpretedas pure variations of the angle of
attack around zero mean loading. As a consequence vortex stretching at the leading-edge stagnation point and vortex
deformationbyinductionofthebladecirculationareignored(see[12,13]).Theyarebelievedtobeofminorimportancefor
thinCRORbladesbecausetheradiusofcurvatureoftheleadingedgeisverysmall.Incontrastassumingfrozenoncoming
vorticeswouldproducewrongacousticestimatesforthicksurfacessuchasaircraftwingsorhelicopterblades[14].Other
problemstatementstakingintoaccounttheactualairfoildesignareproposedforinstancebyLeppingtonandSisson[15].
Fromanotherstandpointtheinterestofthepresentstudyisthatitaddressestheeffectofairfoilspanendontheunsteady
aerodynamicresponse.
Despitetheiridealizedcharacterthedevelopmentsareafirstattempttoincludetipvorticesinanefficientmethodology
of CROR tonal noise prediction currently developed elsewhere for wake–interaction noise [16,17,7]. A similar approach
basedonamodelofhelicalvorticesisdescribedbyKinganandSelf[18].Theproblemoftheeffectofbladeterminationon
theresponsetoincidentdisturbanceswasfirstconsideredbyRogerandCarazo[16]anddeclinedtothecaseofanincident
Oseen'svortexbyRogerandSchram[19]andbyRogeretal.[20].Thepresentworkextendsandcontinuespreviousstudies
byincludingbotharbitraryvortexangleandbladesweepangle.Forthesakeofcheckingthesensitivitytothedescriptionof
theoncomingdisturbancesaso-calledexponential-vortexmodelisalsoconsideredasalternativetoOseen'smodel.Finallya
parametricstudyoftheeffectofradialdistancebetweenthevortexpathandthebladetipisperformed,inordertoassess
thebenefitofclipping.
Theanalyticaldevelopmentsresorttoastrip-theoryapproach,accordingtowhichapropellerbladeanditssurrounding
are split into thin annuli. Each annulus is unwrapped for the sake of locally describing the interaction in Cartesian
coordinates. The unsteady aerodynamic response of a blade segment is determined as if it was in a translating motion
286 M.Rogeretal./JournalofSoundandVibration333(2014)283–306
tangenttotherealhelicalmotion.Thisapproachallowsapplyingairfoilresponsefunctionsbasedontheassumptionoflocally
homogeneous flow conditions in both streamwise and spanwise directions, and at the same time introducing spanwise
variations bychanging the geometrical and aerodynamic parameters from one annular strip to another. Conventionally the
analysisisrestrictedtoasinglestripcenteredontheradiusofthetip–vortexpath.Boththeonsetofunsteadyliftandthefar-
fieldradiationareconsideredinareferenceframeattachedtoanisolatedbladesegment,inordertoprovideaclearinsightinto
theintrinsiccontributionsofvortexcirculation,combinedvortexandblade-chordangles,radialtip/vortexdistanceandother
parameters tothe radiationefficiency.Therotational motionof ablade segmentand thedescriptionof thesound field ina
stationaryreferenceframeareoutofthescopeofthepresentpaper.
TheconfigurationofaCRORsystemwithrotorsofequalradiiisshowninFig.1a.Foranalyticalpurpose,theimpinged
bladesegmentisapproximatedbyasweptparallelogramofconstantchordcdefinedperpendiculartotheleadingedge.Itis
featured by the gray surface in Fig.1b and can be shifted along the spanwise direction y′ in order to generate different
2
configurations relative to the incident vortex. Two reference frames are introduced, with axes along the chordwise and
spanwisedirectionsy′ andy′,andalongthestreamwiseandnormaldirectionsy andy ,respectively.Notethattherelative
1 2 1 2
streamisassumedalongthey directionforconsistency.Coordinatesmadedimensionlessbythehalfchordlengthc/2and
1
denotedbystarswillalsobedefinedforfurtherderivationsinSection4.Thecomplementarydescriptionintheunwrapped
mid-planeofthestripisdefinedinFig.2forthecaseofzerosweepψ¼0.Sweepisintroducedafterwardsbymakingthe
airfoilrotateintheclockwisedirectionaroundthey -axisfromthezero-sweepconfiguration.
3
Becausethenumberofbladesismoderateintheapplicationsandthesolidityissmallatthetip,theresponseofabladeis
notsignificantlyinfluencedbythepresenceofadjacentblades;thereforeisolated-airfoilresponsefunctionsareconsidered.
The latter are also best suited in the case of an impinged wing or empennage. The excitation of the blade segment, its
aerodynamicresponseandtheassociatedsoundradiationareaddressedinwhatfollowsasthreedistinctmodelingsteps.
FirsttheanalyticalvortexmodelsandtheirgustexpansionsaredetailedinSection3below.Theaerodynamicresponseof
a blade segment is discussed in Section 4 as if it had an infinite span. Yet the radiated sound is further calculated by
Fig.1. (a)Schematicviewoftip–vortexinteractioninaCROR,featuringthetipannulusaddressedinthemodel,caseofequalradii.(b)Setsofcoordinates
attachedtoasweptblade–tipsegment.NotethatsweepisdefinedbackwardinthedesignofCRORs,intheclockwisedirectionintheaxesðy ;y Þ.
1 2
M.Rogeretal./JournalofSoundandVibration333(2014)283–306 287
Fig.2. Setsofcoordinatesforthestatementofvortex–airfoilinteractionwithnosweep,from[19].Non-zerosweepisintroducedthroughadditionalairfoil
rotationaroundthey3-axis,accordingtoFig.1b.
integrating the sources over the actual span for various vortex trajectories. The effect of the termination of the blade
segmentisaccountedforbymeansofatipcorrectionderivedinSection5,wherethesoundradiationisre-considered.This
allowspointingatnotonlythecombinedeffectsofvortexandbladefeatures,butalsotheeffectofthetipresponse,onthe
soundpredictions.
3. Tip–vortexmodel
3.1. Gustexpansionofmodelvortices—zerosweep
The blade/tip–vortex interaction is described first assuming zero sweep in this section, using the coordinate systems
illustratedinFig.2.Theselectedstripfortheanalysishasthesamemeanradiusasthehelicalpathoftheincidentvortex.
Thisradiuswillbetakenastheoriginofthespanwisecoordinatey .Forsubsequentanalyses,theblade–tipsectionwillbe
2
consideredateitherpositiveornegativevaluesofthatcoordinate.Accordingtothelinearizedairfoiltheory,theunsteady
loadsonabladedependessentiallyonthenormalvelocityperturbationinducedbytheincomingvortex,expressedinthe
ðy ;y ;y Þcoordinatesystemattachedtotheblade.Itisassumedthatthevortexdiffusiontakesplaceoverdistanceswhich
1 2 3
remainlargewhencomparedtothebladechord,sothatthetraceofacylindricalline-vortexdescriptionisrelevantinthe
unwrapped plane of coordinates (X, Y) with angle φ between the vortex axis and the tangential direction Y (Fig. 2). Any
consistentvortexmodelcombinesasolid-bodyrotationcoreandsomeradialdecreaseofthetangentialvelocityprofilein
theouterregion.Thefree-vortexOseenmodelVθðrÞ¼ðΓ=rÞ½1(cid:2)e(cid:2)ðr=r0Þ2(cid:4),where2πΓandr0arethecirculationandthecore-
radiusparameter,respectively,isusuallyselectedforitsanalyticaltractability.Onceprojectedinthereferenceframeofthe
consideredbladesegment,thecorrespondingupwashiswrittenas
w(cid:3)y ;y ;t(cid:4)¼(cid:2)Γ cosðγþφÞy 1(cid:2)e(cid:2)ðy22þsin2ðγþφÞ½y1(cid:2)Uct(cid:4)2Þ=r20 (1)
1 2 2 y2þsin2ðγþφÞ½y (cid:2)U t(cid:4)2
2 1 c
introducingthephasespeedU ¼ sinφ V=sinðγþφÞalongthechordline,whereV¼ΩR denotesthetangentialspeedof
c 0
thebladesegmentrelativetothevortex(R isthemeanradiusoftheconsideredstripandΩtherelativerotationalspeed).
0
Though U does not coincide with the relative flow speed U introduced in Fig.1b, both will be assumed equal for the
c 0
preliminaryinvestigationsachievedinthepaper.γisthestaggerangleoftheblades.NotethatAmiet'sanalysis[11]isbased
onthesamevortexmodelwrittendifferentlyas
(cid:5) (cid:6) h i
VθðrÞ¼ 1þ21a rr0v0 1(cid:2)e(cid:2)aðr=r0Þ2
wherethevaluea¼1.25643ensuresthatthemaximumspeedisexactlyobtainedforr¼r .
0
Elementary blade/tip–vortex impingement is an impulsive mechanism, but its declination in the context of CRORs is
periodicwiththerelativeblade-passingperiodofthefrontrotor,notedT,asseenfromtheaforementionedreferenceframe.
Assuchitcanbeanalyzedinthefrequencydomaininthesamewayaswhatiscurrentlydoneforwake–interactionnoisein
turbomachines.Thischoiceisinherenttotheunsteadyaerodynamicmodeldescribedlateroninthepaper.ThetimeFourier
transformofthesignalwðy ;y ;tÞaccordingtotheconventione(cid:2)iωt formonochromaticwavesandforpositivefrequencies
1 2
288 M.Rogeretal./JournalofSoundandVibration333(2014)283–306
isfirstderivedas[21]
Z
w~ðωÞ¼ 1 1 w(cid:3)y ;y ;t(cid:4)eiωtdt¼(cid:2)ΓcotðγþφÞeik1y1(cid:7)h(cid:3)(cid:2)y (cid:4)(cid:2)h(cid:3)y (cid:4)(cid:8) (2)
2π 1 2 4U 2 2
(cid:2)1 c
with
(cid:9) (cid:5) (cid:6)(cid:10)
h(cid:3)y2(cid:4)¼eik1y2=sinðγþφÞ 1(cid:2)erf 2sink1ðrγ0þφÞþyr2
0
andk ¼ω=U ,erfdenotingtheerrorfunction.Thisisanoddfunctionofy ,asexpected.
1 c 2
Another model vortex with a much faster exponential decay of the tangential velocity profile as VθðrÞ¼A0re(cid:2)r=r0, A0
beinganotherconstant,canbeproposedasanalternativeifthe1/rdecayisconsideredabusivelyslow,eventhoughitis
incompatiblewiththedefinitionofanon-zerocirculation.Itisworthnotingthatstill anotherexpression isproposedby
GeorgeandChou[1]as
Γ r2
VθðrÞ¼
rr2þr2
0
whoalsostressedthatverysimilaracousticsignaturesareproducedbythismodelandtheOseenvortexvelocitylaw.The
three model expressions are compared in Fig. 3 where it is found that they produce similar velocity profiles around the
maximum, thus similar descriptions of the vortex core, provided that the parameters are properly tuned. Helical vortex
modelsdedicatedtoCRORshavebeenproposedbyKinganandSelf[18];theyarenotassessedherebecausetheemphasisis
onthelocalimpactaroundthetipofanimpingedblade.OnlytheOseenandexponentialmodelsareretainedlateron.The
lackofconsistencyoftheexponentialvortexisnotanissueherebecausetheanalysisresortstothevelocityfieldandnotto
vorticity.Verydifferentmodelsarejusttheopportunityofassessingthesensitivityofthefinaltrendswithrespecttothe
descriptionofthevortices.
Theupwashoftheexponentialvortexiswrittenas
wðy1;y2;tÞ¼(cid:2)cosðγþφÞA0y2e(cid:2)fy22þsin2ðγþφÞ½y1(cid:2)Uct(cid:4)2g1=2=r0 (3)
andthetimeFouriertransformreads[21]
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
(cid:5) (cid:6) (cid:5) (cid:6)
w~ðωÞ¼ q(cid:2)ffi2ffiffiffiAffiffiffi0ffiffiffifficffiffioffiffiffisffiffiffiðffiffiγffiffiþffiffiffiffiffiφffiffiÞffiffiffieffiffiffiiffikffi1ffiffiyffiffi1ffiffiffiffiffiffiffiy2K 4y 1 2þ k1 25 (4)
r ω2þðsinðγþφÞU =r Þ2 2 1 2 r0 sinðγþφÞ
0 c 0
for y 40 , where K is the modified Bessel function. Since the cut through a vortex features antisymmetric signals and
2 1
becausenegativeargumentsofK leadtomathematicaldifficulties,theresultfory o0mustbetakenastheoppositeofthe
1 2
onefory 40.
2
Thespectralcontentoftheimpingementdependsonthespanwisecoordinatey .Forthesakeofderivingtheinduced
2
unsteadyloadsfromSchwarzschild'stechniquelateroninthepaper,theexpressionoftheupwashmustbenowexpanded
insinusoidalgustsinthespanwisedirection,resortingtoaspaceFouriertransformeventhoughsuchatransformseems
Fig.3. ModeltangentialvelocityprofilesoftheOseenvortex(—),GeorgeandChou'svortex(––)andtheexponentialvortex(——).Modelparameters
indicatedontheplots.
M.Rogeretal./JournalofSoundandVibration333(2014)283–306 289
somewhatinappropriateforconcentratedpatterns:
Z h i
G~ðk1;k2Þ¼21π 1 w~ðωÞe(cid:2)ik1y1 e(cid:2)ik2y2 dy2:
(cid:2)1
Thisfinallyprovidesthecomplexamplitudeofthegustofwavenumbersðk ;k Þas[21]
1 2
G~ðk ;k Þ¼iΓk2cotðγþφÞe(cid:2)r20ðk22þ½k1=sinðγþφÞ(cid:4)2Þ=4 (5)
1 2 2πUc k22þ½k1=sinðγþφÞ(cid:4)2
fortheOseenvortex.Similarlytheresultfortheexponentialvortexisderivedintroducingthegenericfunctionx2K ðxÞand
1
itsFourierintegral
Z
F~ðkÞ¼ 1 1 x2K ðxÞe(cid:2)ikxdx
2π 1
(cid:2)1
withK ðxÞ¼K ðxÞforx40andK ðxÞ¼(cid:2)K ð(cid:2)xÞforxo0.Theresultisfoundas[21]
1 1 1 1
8 (cid:5) (cid:6) (cid:5) (cid:6)9
>><F 4;3;7;(cid:2)1(cid:2)ik F 4;3;7;(cid:2)1þik >>=
F~ðkÞ¼pΓffiðffiffi4ÞΓð2Þ 2 2 1þik (cid:2) 2 2 1(cid:2)ik
πΓð7=2Þ>>: ð1þikÞ4 ð1(cid:2)ikÞ4 >>;
~
where F stands for the general hypergeometric functions. Note that here F is a pure imaginary and odd function of its
argument.Thecomplexamplitudeofthegustofwavenumbersðk ;k Þfinallyreads
1 2
(cid:5) (cid:6)
G~ðk ;k Þ¼(cid:2)2A0cotðγþφÞ1F~ k2 (6)
1 2 r0Uc Q4 Q
with
"(cid:5)1(cid:6)2 (cid:5) k (cid:6)2#1=2
Q¼ þ 1
r sinðγþφÞ
0
fork 40.
2
ThealgorithmforthehypergeometricfunctionsisbasedonGauss'seriesofGammafunctions[22]
ΓðcÞ 1 ΓðaþnÞΓðbþnÞZn
Fða;b;c;ZÞ¼ ∑ :
ΓðaÞΓðbÞ ΓðcþnÞ n!
n¼0
Theseriesconvergesonlyforc4bþaandmayfeaturesomeoscillationswhentheconditionisapproached.Thereforethe
valuesofaandbarereducedbymakinguseofpropertiesofcontiguoushypergeometricfunctions.Thepresentcalculations
relyonthefollowingrelationshipstogetFð4;3=2;7=2;ZÞ[22]:
(cid:3) (cid:4) (cid:2)ð3(cid:2)ZÞ (cid:3) (cid:4) 5(cid:2)3Z (cid:3) (cid:4)
F 4;3=2;7=2;Z ¼ F 2;3=2;7=2;Z þ F 1;3=2;7=2;Z ;
8ðZ(cid:2)1Þ 8ðZ(cid:2)1Þ2
(cid:3) (cid:4) (cid:2)6 (cid:3) (cid:4) 5(cid:2)3Z (cid:3) (cid:4)
F 2;3=2;7=2;Z ¼ F 2;(cid:2)1=2;7=2;Z þ F 2;1=2;7=2;Z :
Z(cid:2)1 Z(cid:2)1
~
ThewavenumberspectrumGoftheOseenvortexisillustratedinFig.4.Aqualitativelysimilarplotwouldbeobtainedfor
theexponentialvortex.Dependingontherelativevaluesofthewavenumbersk andk ,twosetsofgustsaredefined.When
1 2
k 4M k =β2,thegustissaidtobesub-critical;otherwiseitissaidsupercritical.Themeaningofthisnotionisdiscussedin
2 0 1
Section4.1.Thetransitionbetweentherangesofsub-criticalandsupercriticalgustsisfeaturedbytheverticalgrid-planein
thefigure.Itisremarkablethatmoreenergyisdistributedinthesub-criticalrange.
3.2. Periodicinteractioninopen-rotorarchitecture
Theperiodicimpingementofsuccessivetipvorticesissimplyreproducedbyduplicatingthepreviouslyderivedsolutions
viaaninfiniteseries,accordingtotheconvolutionproduct
1 1
f ðtÞ¼ ∑ wðy ;y ;t(cid:2)nTÞ¼wðy ;y ;tÞn ∑ δðt(cid:2)nTÞ
W 1 2 1 2
n¼(cid:2)1 n¼(cid:2)1
with T¼2πR =ðBVÞ, where B is the number of blades. The resulting signals for the two investigated model vortices are
0
shown in Fig. 5. Despite the difference in the description of a single vortex, combining a series of vortices to simulate
theperiodicinteractionprovidesverysimilarvelocityprofilesinthepresentcase.Thissuggeststhatselectingonemodelor
theotheronemakesnegligibledifferences.Neverthelessbotharedetailedinthisstudyforcompleteness.Thefigurealso
compares the model tangentialvelocityprofiles tonumerical predictions based on Unsteady RANS simulations, achieved
on a full open-rotor architecture [23]. The radialvelocityis extracted at two different axial locations fromthe numerical
mesh,as themostrepresentative quantity forthe tipvortex.The negativemean-valueof the profilesis attributedtothe
290 M.Rogeretal./JournalofSoundandVibration333(2014)283–306
Fig.4. Two-dimensionalgust-wavenumberspectrumoftheincidentupwashoftheOseenvorteximpingingonarectangularunsweptblade.SeeSection4.1
for definitions. Parameters: γ¼251, φ¼201, V¼300m/s, r0¼5cm, Γ¼4m2/s. The value of the blade passing frequency (BPF) chosen for illustration
inFig.9isshownbythedashedarrow.
contractionofthestreamtubeofthefrontrotor.Themodelprofilesnicelyrecoverapartofthenumericalresults,atleastfor
theshortestdownstreamdistanceof0.3chordlength.Someasymmetryisfoundinthesimulatedpatterns,departingfrom
theanalyticalmodels.Thisasymmetryincreasesfartherdownstream.Itisattributedtothedynamicsofthevortexandmore
precisely its coupling with the complementary accelerations induced by the swirl of the front rotor. The comparison
confirms that the aforementioned models are physically consistent but that they should be improved to include swirl-
induced asymmetry in a future work. The velocity profiles could also be taken directly from the simulations to provide
alternative,numericalvaluesofthefunctionG~ðk ;k Þ.
1 2
TheFourierseries
(cid:5) (cid:6)
f~ ðωÞ¼ ∑1 f δ(cid:3)ω(cid:2)n2π=T(cid:4) withf ¼2πw~ 2π
W n n T T
n¼(cid:2)1
ofaperiodicvortextrainselectsdiscretefrequenciesinpracticalapplications.Thereforesimplyaddressingtheinteractionat
asinglefrequencyfromanyexpressionofw~ makessense.Withinthescopeofacompletemodelingofopenrotors,thegust-
splittingofaseriesofvorticeswouldbeusedtoderivefirsttheunsteadyloadsonablade–tipsegment.Thesegmentwould
bediscretizedasasetofpatchesandthefar-fieldradiationcalculatedapplyingHanson'sformulation[24]ortheequivalent
implementationderivedbyCarazoetal.[17].Thedeclinationofthemodelinarotating-bladeconfigurationisbeyondthe
scopeofthepaper.
4. Infinite-airfoilresponsefunctions
Theresponseofaninfinite-spanairfoiltoarbitraryincidentgustsisformulatedinthissectioninthegeneralconfigurationof
Fig.1b.Withrespecttotheaxesðy′;y′Þ,sweepisequivalenttoanon-zerospanwisecomponentU oftheincidentflowspeed.
1 2 2
FromthisgeneralframeworktheresponseofanunsweptairfoilfollowsbyjustsettingU ¼0.
2
4.1. Expressionsoftheunsteadylift
Considerthesinusoidal,normal-velocitygustwðy′;y′;tÞ¼w ðk′;k′Þexpfiðk′y′ þk′y′ (cid:2)ωtÞge ,convectedovertheairfoil
1 2 0 1 2 1 1 2 2 3
attheobliquespeedU ¼U e þU e .Theimpingementofthegustgeneratesapotentialvelocitydisturbanceu′¼∇ϕ′such
0 1 1 2 2
thatϕ′issolutionofthegeneralconvectedHelmholtzequation
(cid:5) (cid:6)
∂2ϕ′ ∂2ϕ′ ∂2ϕ′ ∂ϕ′ ∂ϕ′
β2 þβ2 þ þ2ik M þM
1∂y′2 2∂y′2 ∂y2 1∂y′ 2∂y′
1 2 3 1 2
∂2ϕ′
(cid:2)2M M þk2ϕ′¼0 (7)
1 2∂y ∂y
1 2
M.Rogeretal./JournalofSoundandVibration333(2014)283–306 291
30
20
10
0
s ) −10
m/
( θ −20
v
−30
−40
−50
−60
−10 0 10 20 30 40 50 60 70
azimuth (°)
30
20
10
0
s ) −10
m/
( θ −20
v
−30
−40
−50
−60
−10 0 10 20 30 40 50 60 70
azimuth (°)
Fig.5. Comparedperiodictip–vortexvelocityprofilesaccordingtoOseen'svortexmodel(–(cid:3)–),exponentialvortexmodel(——)andURANScomputations
ð(cid:5)Þ[23].(a)30percentchorddownstream;(b)50percentchorddownstream.Parameters:ðΓ;r0Þ¼ð1:5m2=s;2:5cmÞ,(A,r0)¼(3700s(cid:2)1,2.7cm).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
withk¼ω=c ,β ¼ 1(cid:2)M2,β ¼ 1(cid:2)M2,M andM beingthechordwiseandspanwiseMachnumbers,respectively.The
0 1 1 2 2 1 2
boundaryconditionsony ¼0arethecancellationofthepotentialupstreamoftheleadingedgeðy′ o0Þ,thecancellationof
3 1
thenormalvelocityontheairfoilsurfaceð0oy′ ocÞandtheKuttaconditionatthetrailingedgeanddownstreamðy′ 4cÞ.
1 1
Sincenoconditionisimposedalongthecoordinatey ,thedisturbancepotentialissoughtaccordingtotheincidentgust
2
obliquenessasϕ′¼ϕðy′1;y3Þeiðk′2y′2(cid:2)ωtÞ,sothatthereducedconvectedHelmholtzequationonϕisobtainedasfollows:
(cid:11) (cid:12) (cid:11) (cid:12)
∂2ϕ ∂2ϕ ∂ϕ
β2 þ þ2i kM (cid:2)M M k′ þ k2(cid:2)k′2β2(cid:2)2M kk′ ϕ¼0: (8)
1∂y′2 ∂y2 1 1 2 2 ∂y′ 2 2 2 2
1 3 1
This leads to the canonical Helmholtz equation for the transformed velocity potential Φ¼ϕðy′1n;yn3ÞeiM21k′1ny′1n=β21 once the
coordinatesandwavenumbersaremadedimensionlessbyc/2(superscriptn):
∂Φ ∂Φ
þ þκ2Φ¼0; (9)
∂y′n2 ∂yn2
1 3
where
! (cid:9) (cid:10)
k′n2 M2 1
κ2¼ 2 1 (cid:2)1 ¼μ2 1(cid:2) ; Θ¼μβ =k′n
β4 sin2α Θ2 1 2
1
292 M.Rogeretal./JournalofSoundandVibration333(2014)283–306
with μ¼k′nM =β2, k′n¼K cos α and k′n¼K sin α, introducing the gust skewness angle αand the total non-dimensional
1 1 1 1 2
aerodynamic wavenumber K¼jKj such that ωc=2¼U (cid:3)K¼U k′nþU k′n¼U kn with kn¼K cosðα(cid:2)ψÞ. Θ is known as
0 1 1 2 2 0 1 1
Graham'sparameter[25].
Schwarzschild'stheorem[26]aimedatsolvingEq.(9)withtheboundaryconditions
Φðy′n;0Þ¼fðy′nÞ; y′n40;
1 1 1
∂Φ
∂yn3jyn3¼0¼0; y′1no0;
wherefisaknownfunction,statesthatthetraceofthepotentialonthecomplementaryhalf-spacey′no0reads
Z 1
Φ(cid:3)y′1n;0(cid:4)¼1π 1ð(cid:2)y′1n=ξÞ1=2(cid:7)1=(cid:3)ξ(cid:2)y′1n(cid:4)(cid:8)eiκðξ(cid:2)y′1nÞfðξ;0Þdξ:
0
Amiet'smethod[27]isbasedontheiterativeuseofthistheoremtodeterminetheunsteadyliftduetoanincidentgust,
each iteration accounting for the contribution of one edge. The impingement of the incident gust at leadingedge is first
interpreted as a problem of wave scattering by the edge of a rigid half-plane. A zero-order disturbance potential which
wouldcancelthenormalvelocityw overtheentireplaney ¼0isintroducedasinitialsolution[28].Thenthedominant
3
leading-edge scattering is derived by adding a first-order potential ensuring the condition of no flow across the surface
ðy ¼0;y′ 40Þ,andcancelingtheinitialpotentialonðy ¼0;y′ o0Þ.ThisstepisachievedbySchwarzschild'stheorem,and
3 1 3 1
the resulting first approximation of the solution Φ ðy′n;0Þ yields the corresponding lift or pressure-jump surface density,
1 1
expressedas
ℓ~1(cid:11)k′1n;k′2n(cid:12)¼q(cid:2)ffiπ2ffiffiðρffiffikffi0ffi′ffiUnffiffiffiþ1ffiffiffiwffiβffiffi20ffiffiffiκeffiffiÞiffiπyffiffi=ffi′ffi4nffiffie(cid:2)iðμM1(cid:2)κÞy′1neik′2ny′2ne(cid:2)iωt: (10)
1 1 1
Thisquantityistwicethedisturbancepressure,sincethepressurefluctuationshaveoppositephasesonbothsides.Ina
secondstepthesolutioniscorrectedbyasecond-ordertermℓ~ whichcancelsℓ~ atthetrailing-edgeandbeyondinorderto
2 1
fulfillaperfectKuttacondition.ThecorrectionisderivedbywritingdownanotherSchwarzschild'sproblemforthepressure
jumponthehalf-planeðy ¼0;y 4cÞ,acceptingsomeapproximationintheintegral[28],as
3 1
ℓ~2(cid:11)k′1n;k′2n(cid:12)¼q2ρffi2ffi0ffiπffiUffiffiðffikffi1ffiffiw′ffinffiffiþffi0ffiffieffiffiβiffiπffi2ffi=ffiκffi4ffiffiÞffi(cid:6)(cid:7)1(cid:2)ð1þiÞEn(cid:7)2κ(cid:3)y′1n(cid:2)2(cid:4)(cid:8)(cid:8)e(cid:2)iðμM1(cid:2)κÞy′1neik′2ny′2ne(cid:2)iωt; (11)
1 1
EbeingtheFresnelintegralintroducedbyAmiet
Z
x eiξ
EðxÞ¼ pffiffiffiffiffiffiffiffidξ
0 2πξ
and the asterisk denoting the complex conjugate, with the property that Eð(cid:2)xÞ¼iEnðxÞ and Enð(cid:2)xÞ¼(cid:2)iEðxÞ. The total
aerodynamicresponseoftheairfoilisgivenbyℓ~¼ℓ~ þℓ~ .ItmustbenotedthatEqs.(10)and(11)havebeenfirstderivedby
1 2
AdamczykusingtheWiener–Hopftechnique[29].Theyholdforsupercriticalgustssuchthatκ240.Thecaseofthesub-critical
pffiffiffiffiffiffiffiffiffiffi
gustsiseasilyfoundbyusingthemodifiedvalueiκ′¼i (cid:2)κ2insteadofκ,andreplacingthesquarebracketinvolvingtheFresnel
integralEnbytheterm1(cid:2)erfð½2κ′ð1(cid:2)y′nÞ(cid:4)1=2Þinvolvingtheerrorfunction.Thesimplifiedexpressionsforobliquegustsimpinging
1
on an unswept airfoil are simply obtained by considering zero sweep ψ¼0. They are found equivalently
byMishandDevenport[30],Moreauetal.[31]andRoger[32].Thecompletesolutionisnotsensitivetotheexactbehaviorof
the correction ℓ~ far upstream from the trailing edge, where the lift is dominated by the leading-edge inverse square-root
2 qffiffiffiffiffiffi
singularity.Forabettertwo-dimensionalrepresentationoftheliftdistributionovertheairfoil,thequantityℓ~ y′n referredtoas
1
the‘regularizedlift’willbechosensubsequently.
Sub-criticalgustsarecharacterizedbyafasterdropoftheunsteadyliftdownstreamfromtheleadingedgebecauseofthe
factore(cid:2)κ′y′1n.Theyalsoradiatelessefficientlyforspanwise-distributedsourcesandtheiractualcontributioninvortex–airfoil
interactionnoiseisapointofinterest.InthetestcaseofFig.4thedominanthumpofthefunctionpreciselyenterstherange
of sub-critical gusts. Furthermore gust–airfoil interactions at relatively high frequencies take place well beyond
the maximum energy of the wavenumber spectrum. This is expected for instance for most combination tones produced
bytip–vortexinteractionsonCRORs.
4.2. Bladeresponsewithouttipcorrection–zerosweep
The unsteady lift locally induced on an airfoil of arbitrary large span and zero sweep by an incident Oseen vortex is
shown in Fig. 6 for both the supercritical and sub-critical gusts as well as their combination. Sub-critical gusts appear
to contribute substantially more than supercritical gusts. Both sets produce extended traces when separated whereas
theircombinedeffectresultsinamoreconcentratedtracefeaturinganupwash–downwashpattern,atleastaslongasthe