DSpace at VNU: Aerodynamic design optimization of helicopter rotor blades including airfoil shape for forward flight

DSpace at VNU: Aerodynamic design optimization of helicopter rotor blades including airfoil shape for forward flight tài...

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N.A Vua,1, J.W Leeb,2

aHo Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam

bKonkuk University, Seoul 143-701, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:

Received 19 September 2013

Received in revised form 19 May 2014

Accepted 25 October 2014

Available online xxxx

Keywords:

Rotor blades design

Airfoil

Design optimization

Thisstudyproposesaprocesstoobtainanoptimalhelicopterrotorbladeshapeincludingbothplanform

representationalgorithmwhichusestheClassFunction/ShapeFunctionTransformation(CST)isemployed

to generate airfoil coordinates With this approach, airfoil shape was considered in terms of design variables.The optimizationprocess wasconstructedbyintegratingseveralprogramsdevelopedbythe author.Airfoilcharacteristicsareautomaticallygeneratedbyananalysistoolwherelift,drag,andmoment coefficientsofairfoilarepredictedforsubsonictotransonicflowandawiderangeofattackangles.The designvariablesincludetwist,taperratio,pointoftaperinitiation,bladerootchord,andcoefficientsof theairfoildistributionfunction.Aerodynamicconstraintsconsistoflimitsonpoweravailableinhoverand forwardflight,aerodynamicrequirements(lift,dragandmomentcoefficients)forcriticalflowcondition occurring on rotor blades The trim condition must be attainable in any flight condition Objective functionischosenasacombinationexpressionofnon-dimensionalrequiredpowerinhoverandforward flight

©2015PublishedbyElsevierMassonSAS

1 Introduction

In contrast to fixed wingdesign, most rotorcraft research

fo-cuseson the design of the rotor blade to optimize performance,

vibration, noise, andso on because the rotor blade performance

playsanessential roleinmostofthedisciplinesinhelicopter

de-sign The aerodynamics of helicopter rotor blades is a complex

discipline Diverse regimes of flow occur on blades, such as

re-verseflow,subsonicflow,transonicflow,andevensupersonicflow

In forward flight, a component of the free stream adds to the

rotational velocity at the advancing side and subtracts from the

rotational velocity at the retreating side The blade pitch angle

andblade flapping aswell asthe distribution ofinduced inflow

through the rotor will all affect the blade section angle of

at-tack(AoA) [16].Thenon-uniformity ofAoAovertherotor disk in

conjunctionwiththeinconstant distributionofvelocityalong the

helicopterrotorblademakesaerodynamicanalysisdifficult

Thereare twocommonapproachestoblade aerodynamic

per-formancedesign.First,someresearchersnowfocusonbladeshape

E-mail addresses:vna2006@hotmail.com (N.A Vu), jwlee@konkuk.ac.kr

(J.W Lee).

1 Lecturer, Department of Aerospace Engineering.

2 Professor, Department of Aerospace Information Engineering, Member AIAA.

designby selectingthepoint oftaperinitiation,rootchord, taper ratio,and maximumtwist which minimize hover power without degrading forwardflight performance [31].This approachusually deals with integration ofseveral programs to build an optimiza-tionprocess.MichaelandFrancisinvestigatedtheinfluenceoftip shape,chord,bladenumber,andairfoilonrotorperformance.Their wind tunneltest demonstratessignificant improvementsthat can

be gained from planform tailoring and further development of airfoils, specifically for high speed rotor operation [19] Second, someworkstriedtosolvethisproblemusingnumericalmethods Joncheray used the vortex method, which schematizes the blade and rotational flow areas on the basis of a distribution of vor-tices, tocalculatethe airflow arounda rotor inhover [13].Pape andBeaunier createdan aerodynamic optimization forhelicopter rotor blade shape in hover based on the coupling of an opti-mizerwitha three-dimensionalNavier–Stokes solver[22].Morris andAllendevelopedagenericcomputationalfluiddynamics(CFD) based aerodynamic optimization tool for helicopter rotor blades

in hover [21] Gunther Wilke performed a methodological setup

of variable fidelity framework for the aerodynamic optimization

of helicopterrotor blades and demonstratedits capabilitiesfora single and multi-objectivetest case[32].M Imiela andG Wilke investigated an optimizationusinga multi-fidelityapproach with multiple designparameters on twist,chord, sweep, andanhedral http://dx.doi.org/10.1016/j.ast.2014.10.020

1270-9638/©2015 Published by Elsevier Masson SAS.

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problems in hover starting withthe easy task ofoptimizing the

twistrate forthe7A modelrotor The last optimizationin hover

involved all design parameters, namely twist, chord, sweep,

an-hedral, transition point of two different airfoil, starting point of

the blade tip showing its superiority over simpler optimization

problems with respect to the achieved improvement [11] These

CFDmethodsarereasonableforthehovercasebutverytime

con-suming.Moreover,applicationoftheCFDmethodtotheflowfield

passingthebladeinforwardflightisverycomplex.Therefore,the

CFDmethodisnotsuitableforthepreliminarydesignphasewhere

theneedforquickestimationandconsideringofallfactors

includ-ingairfoilarerequired

Theairfoilshapewhichsignificantlyaffectstheperformance of

helicopterrotorbladesisusuallyconsideredasaseparateproblem

Hassanetal.developeda procedurebased onthecoupled

three-dimensional direct solutions to the full potential equation and

two-dimensionalinversesolution toan auxiliary equationforthe

designof airfoilsectionsforhelicopterrotor blades[9].Bousman

examinedtherelationshipbetweenglobalperformanceofatypical

helicopterandtheairfoilenvironment[4].McCroskeyattemptedto

extract asmuch useful quantitative informationas possible from

criticalexaminationandcorrelationsofexistingdataobtainedfrom

over40windtunneltests[18].Therefore,thismethodisnot

appli-cabletoalargenumberofnewgenerationsofairfoilshapes

Mar-ilyn J Smith [24] evaluated computational fluid dynamics (CFD)

codes suchas OVERFLOW[6],FUN2D[1],CFL3D [23],Cobalt LLC

[25],andTURNS[27] todetermine2Dairfoilcharacteristics.With

theadvancementofcomputertechnology,E.A.MaydaandC.P.van

DamdevelopedaCFD-basedmethodologythatautomatesthe

gen-erationof2Dairfoilperformancetables[17].Themethodemploys

ARC2Dcode,whichcontrolsa2DReynolds-AveragedNavier–Stokes

(RANS)flowsolver.Themethodwasshowntoperformwellforthe

largely“hands-off”generationofC81tables,forusemainlyin

com-prehensiverotorcraftanalysiscodes.Nevertheless,thestate ofthe

artofrotorcraftstudiesisnotonlyforanalysisbutalsofordesign

The method is a very expensive approach for rotorcraft analysis

anddesignpurposeswheredesignersaimtocompromiseonmany

factors(designvariables)toconstructacertainobjective

The lack of less expensive analysis methods has been

block-ingmulti-variableconsiderationofrotorbladedesignoptimization

Therefore,rotorbladeairfoilshapesandplanformsareusually

ex-aminedinisolateddesignoptimizations.Aneffectivelyautomated

approach that is less expensive could contribute greatly to the

rapidgeneration of C81tables, to providethe ability toconsider

allaerodynamic aspects inrotor bladedesignoptimization Vu et

al.havedevelopedatool thatcanrapidlyandaccuratelycompute

airfoildatathatareneededforrotorcraftdesignandanalysis

pur-poses[29]

Withthe aimof allowing quick estimationin thepreliminary

designphase, thisstudyproposesa process toobtain an optimal

helicopter rotor blade shape including both planform and airfoil

shape for helicopter aerodynamic performance In this study, a

newgeometryrepresentationalgorithmwhichusestheClass

Func-tion/Shape FunctionTransformation (CST)methodwas applied to

acteristicstablesisemployedinthedesignprocess.Theprocess as-sociatesanumberofcommercialsoftwarepackagesandin-house codes that employ diverse methodologies including the Navier– Stokesequation-solvingmethod,thehigh-orderpanelmethodand Eulerequationssolvedwiththefullycoupledviscous–inviscid in-teraction(VII)method

ThedesignprocessisrepresentedinFig 1.Thisprocessalso in-cludesasizingmodule.Aftersettingthesizeofthehelicopter,the helicopterrotorblade shapeoptimizationprocessisperformedas thenextstepofthedesignprocess.Followingthisprocess,asetof initialvaluesfordesignvariablesischosenfromthesizingmodule Theairfoilbaseline,whichisairfoilNACA0012,waschosenforthe firststepofthedesignprocess.Then,bladeshapevariablessuchas chorddistribution,twistdistribution,andairfoilpoint coordinates are generated.The requiredpowerforhoverandforwardflightis computedby theKonkukHelicopter Design Program(KHDP),and thetrimconditionischecked.Airfoilanalysisisperformedbythe automated process program The airfoilaerodynamic characteris-tics are represented in C81 table format Some other additional codestogenerateairfoilcoordinates,chorddistribution,andtwist distribution are implemented in order to build a full framework forthe optimizationprocess inModelCentersoftware ModelCen-terisapowerfultoolforautomatingandintegratingdesigncodes Onceamodelhasbeenconstructed,tradestudiessuchas paramet-ric studies,optimizationstudies,andDesign ofExperiment(DOE) studiesmaybeperformed[20]

2 Design process

2.1 Design considerations

The powerrequired todrive themain rotor isformed by two components: induced powerandprofile power(toovercome vis-couslossesattherotor).Theinducedpowerandtheprofilepower primarily influence the blade aerodynamics performance design [16].Helicopterhoverperformanceisexpressedintermsofpower loading or figure of merit (FM) A helicopter having good hover performance mayhaveinferiorperformance inforwardflight.The compromisebetweenhoverandforwardflightleadsustoexpress the target design value in terms ofthe required power in hover andforwardflight

Theconventionalapproachtobladeaerodynamicsperformance design fixed the airfoilshape Ingeneral, the choice ofairfoils is controlled bythe needtoavoidexceedingthesection drag diver-gence Machnumber onthe advancingside ofthe rotor disk, the maximumsectionliftcoefficientsontheretreatingsideofthe ro-tordisk andthezero-liftpitchingmoments

Thepresentworkconsiderstheeffectofbladeairfoilshapeon required power Therefore, a baseline airfoilNACA0012 was cho-sen as a unique airfoil for the blade to simplify the process of optimum design.Theairfoilshape isrepresentedbyCST function coefficients Thesecoefficientsare alsothedesignvariablesofthe examinedoptimizationproblem

Theabovediscussionshowsthattheinducedandprofilepower canberepresentedasfunctionsoftwist,taperratio,pointoftaper

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Fig 1 Design synthesis process.

initiation, bladerootchord, and coefficientsofairfoildistribution

function Aerodynamics performance is defined by the following

requirements:

+ Therequiredpowermustbelessthanthepoweravailable

+ Thehelicoptermustbeabletotrimathoverandforwardflight

condition

+ Theairfoilshouldhavethefollowingcharacteristics:low

zero-lift pitching moment at low speed M=0.3 approximately,

highmaximumlift betweenM=0.3 and M=0.5,highdrag

divergenceMachnumberatzerolift

2.2 Design synthesis process

ThedesignsynthesisprocessisshowninFig 1.Thedashed-line

rectangle represents a module which is integrated in

ModelCen-tersoftware Each module is connected with the other modules

bydatainput/outputflows,whicharethemutualpart.Four

mod-ulesare implemented inthisoptimizationframework: the chord,

twist,andradiusdistribution generationmodule;theairfoilpoint

coordinates generation module; the airfoil characteristics library

withC81 format module; and the sizing, trim, and performance

analysismodule.Thechord,twist,andradiusdistributionsare

gen-erated by a code in which the geometry representation can be

changed;forexample,itcan bea linearornonlinearfunction.In

thisstudy,chorddistributionisgeneratedbasedontherootchord,

thepointoftaperinitiation,andthetaperratio.Twistdistribution

is assumed to vary as a linear function along the blade Radius

distribution was divided by the equal annulus area of the rotor

disk.These distributions are the input data forthe trim code in

thetrimmingprocess

Tencoefficientsoftheairfoildistributionfunctionweredefined

asthe initialinput dataof thedesignprocess afterobtaining the

fittingcurveoftheairfoilbaselineNACA0012.Then,airfoil

coordi-natepoints were generated by usingthe CSTfunction The

auto-matedprocessgeneratesanairfoilcharacteristicslibrarywithC81

format comprising the airfoil lift, drag, and moment coefficients

with respect to the angle of attack for different Mach numbers

(from0.05to1.0)

TheairfoilcharacteristicsinC81formatandrotorblades plan-formconfigurationarethenusedforperformanceandtrim analy-sis.Itshouldbenotedthatthebaselinerotorbladesconfiguration canbeobtainedfromthesizingprocess.Itisassumedthatthe siz-ingprocess generatesrotorblades configurationsimilartothat of theBo105helicopter.Thisassumptionisforcomparisonpurposes

ofdesignoptimization

TheKHDPprogramwiththeperformanceanalysismodule pro-videsmanyoptionsfortheobjectivefunction.Theobjective func-tion ofthisstudyis chosen asa combinationexpression of non-dimensionalrequiredpowerinhoverandforwardflight.Helicopter data areanalyzed by the performance codeobtained fromeither thesizingmoduleoruserinputs

Afterachievingthetrimcondition,meaningthatthetrim con-dition is attainable, the required power is evaluated in order to proceed to the next loop ofthe optimization process So, a new set ofinitial data (root chord, thepoint of taper initiation, taper ratio,pre-twist,andA0 toA4 coefficientsoftheairfoildistribution function)aregenerateddepending ontheoptimizationalgorithm Thisloopcontinuesuntiltheconvergenceconditionissatisfied

2.2.1 Geometry representation CST method [2]

TheCSTmethodisbasedonanalyticalexpressionstorepresent andmodifythevariousshapes[15].Thecomponentsofthis func-tionare“shapefunction”and“classfunction”

UsingtheCSTmethod,thecurvecoordinatesaredistributedby thefollowingequation:

Fortheformulationof theCSTmethod,Bernstein polynomials areusedasashapefunction

Fig 2 shows the airfoil geometry represented using the CST method andnon-uniformrational basis B-spline (NURBS).In this case,thecontrolvariablesarethecoordinatesofthecontrolpoints (five variables forthe upper curve andfive forthe lower curve)

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Fig 2 RAE 2822 airfoil representation[2]

Fig 3 Absolute errors in airfoil generation[2]

TheCSTmethodwithfourcontrolvariablesfitstheexistingairfoil

betterthanNURBS,whichusestencontrolvariables[2]

Fig 3showstheabsoluteerrorsofairfoilgenerationusingCST

andNURBS(fivecontrolpointsforeachcurve,fourthorder

blend-ingfunctions).GenerationbyNURBSgivesbiggererrorsatthetail

partoftheairfoil

The advantage of the CST method in comparison with other

methods such as Spline, B-Splines, or NURBS is that it can

rep-resentcurvesandshapesveryaccurately usingfewscalarcontrol

parameters

Inthisstudy,theairfoilbaselinewaschosenasNACA0012.With

the givendatacoordinate points in Cartesian coordinatespace, a

curve fitting was generated usingfourth orderBernstein

polyno-mials

Theclassfunctionfortheairfoilwas:

The airfoil distribution functions defined as upper and lower

curvesarepresentedsequentiallyasbelow

y l(x) =C(x) 

A l0(1−x)4+A l1 4x(1−x)3+A l2 6x2(1−x)2

+A l3 4x3(1−x) +A l4 x4

y u(x) =C(x) 

A u0(1−x)4+A u1 4x(1−x)3+A u2 6x2(1−x)2

+A u3 4x3(1−x) +A u4 x4

(4)

where A u0=0.1718; A u1=0.15; A u2=0.1624; A u3=0.1211;

A u4=0.1671; A l0= −0.1718; A l1= −0.15; A l2= −0.1624; A l3=

−0.1211; A l4= −0.1671

Changes in the coefficients A0 and A4 in the CST method

aresufficientforairfoilshapemodification[31].Thesecoefficients

werealsothedesignvariablesoftheexaminedoptimization

prob-lem

Fivecoefficientsoftheairfoildistributionfunctionweredefined

astheinitial inputdata ofthedesign processafter obtainingthe

fittingcurveoftheairfoilbaselineNACA0012.Then,airfoil

coordi-natepointsweregeneratedbyusingtheCSTfunction

Fig 4 Automated process of 2D airfoil characteristics estimation[29]

2.2.2 An effective tool for the automated generation of airfoil characteristics tables [29]

The aerodynamicsof helicopterrotor bladesis a complex dis-cipline Diverse regimesof flow occur onblades, such as reverse flow, subsonic flow, transonic flow, andeven supersonicflow An effectively automated approach that is lessexpensive could con-tributegreatlytotherapidgenerationofC81tables,toprovidethe ability to consider all aerodynamic aspects inrotor blade design optimization

Thissection describesthedevelopment ofamethodologythat integrates anumber ofcommercialsoftware componentsand in-housecodes thatemploydiversemethodsincludingthe2DRANS equation-solving method, a high-order panel method, and Euler equations solved withthe fully coupled viscous–inviscid interac-tionmethod

Thesequentapplicationsofeachmethodareasfollows:

•Ahigh-orderpanelwiththefullycoupledviscous–inviscid in-teractionmethodforM∞≤0.4

•The Euler equations solved with the fully-coupled viscous– inviscidinteractionmethodfor0.4 <M∞≤0.7

•The2DRANSequation-solvingmethodforM∞>0.7

The2DRANSmethodisonlyusedforM∞>0.7 wherethetwo lessexpensivemethods(Eulerequationsandthehigh-orderPanel solvedwiththefullycoupledviscous–inviscidinteractionmethod) arelesssuitable

Byintegratingcommercialsoftwareandin-housecodes,afully automated process has been developed for generating C81 ta-bles quickly andaccurately forarbitraryairfoil shapes Moreover, thecommercialsoftwareincludingGridgenV15andFluent6.3.26, used for mesh generation and CFD modeling, are very common

in the CFDresearch community Therefore,the proposed method couldbe applicabletoanyautomationprocess employingGridgen andFluentinparticular,aswellasCFDtoolsingeneral

The SC1095 that is used inthe UH-60A main rotor was cho-senforvalidationpurposesbecauseofthewealthofdataavailable from theUH-60A Airloads flight test program [5],aswell asthe currentevaluationofthe UH-60Arotor loadsby a numberof re-searchers

Fig 4showsthetotalautomatedprocessforairfoil characteris-ticestimation

Anairfoilanalysisprogram,2KFoil,was developedforsubsonic isolatedairfoils.ThecodewasadaptedfromthewellknownXFOIL codesoastobesuitableforthepresentstudy.Thecodeemploys

asimplifiedenvelopeversionofthee nmethodforpredicting tran-sition locations Theuser-specifiedparameter “Ncrit”isset to9.0

1 67

Fig 5 The automatic process of MSES execution[29]

(theambientdisturbance levelofan averagewind tunnel) forall

ofthepredictions[8]

MSES,acoupledviscous/inviscidEulermethodforasingle

air-foil section and multiple sections design and analysis, was

em-ployedto predict airfoilcharacteristics from M∞=0.4 to M∞=

0.7

The in-housecode shownin Fig 5was developed to manage

theMSESrun

Fluent6.3.26, comprehensive software for CFD modeling,was

employedtoanalyze2D airfoilcharacteristics inthetransonic

re-gion.The softwareis widely utilizedby CFD research and

indus-tries, thereby ensuring that thedevelopment isapplicable to the

community.Moreover, itwouldbe straightforwardto supportfor

othersolvers

Anin-housecodeshowninFig 6hasbeendevelopedto

man-agethe Fluentrun A libraryofjournal filesthat are utilizedfor

therun ofthecasesettingAoA = 0 deg iscreated.Forinstance,

thejournalfilesarecreatedforthefollowingM∞ andAoA pairs:

M∞=0.75,AoA= 0 deg;M∞=0.80,AoA= 0 deg;M∞=0.85,

AoA=0 deg;etc.AjournalfilecontainsasequenceofFluent

com-mands,arrangedastheywouldbetypedinteractivelyintothe

pro-gramorentered througha GUI TheGUIcommands arerecorded

asschemecodelinesinjournalfiles

Figs 7 and8showthevalidationoftheautomatedprocessfor

airfoilcharacteristicstablesatM=0.4 andM=0.8.Thelift,drag

andpitchingmomentcoefficientsoftheautomated process

calcu-lationatM∞=0.4 forAoAfrom−20 degto20 degareshownin

Fig 7.Theautomated processresultsareveryclosetotheARC2D

results

Stall behavior still remains difficult for CFD researchers The

currentstudyandMayda’s studyhavethesameproblemforthis

Fig 6 Automatic process of Fluent execution[29]

region.Forotherregions,theautomatedprocessresultsand exist-ingC81tabledataareingoodagreement

Thedragcoefficientcalculatedbytheautomatedprocessagrees verywellwiththeC81dataasARC2D

TheexistingC81dataandthemomentcoefficientcalculatedby theautomatedprocessarealsoingoodagreement

The lift, drag and pitching moment coefficients of the auto-matedprocess calculationat M∞=0.8 for AoAfrom−20 deg to

20degareshowninFig 8.AtthisM∞,Fluentisemployedto cal-culatethe2Dairfoilcharacteristics

Ingeneral, theARC2Dandautomated process resultshavethe samedatatrendduetousingthesameSAturbulencemodel.The pitchingmomentvariesnon-linearlynearAoA=0 degbecauseof theshockcommencingontheairfoil

The zero-lift drag coefficient data of the experiment and au-tomated process are shownin Fig 9.There isfairly good agree-mentbetweentheexperimentaldataandthecalculateddata.Itis seen that the calculated results representthe lower boundary of theexperimental data.DifferentRe andboundary layertransition locations cause scatter in the experimental data The automated process resultsshowgoodagreement withthe experimentinthe drag–divergencezonewherethedragcoefficientsharplyincreases

2.2.3 Konkuk helicopter design program (KHDP)

KHDP is a helicopter sizing, performance analysis, and trim analysisprogramthat was developedatKonkukUniversity.These codesweredevelopedforuseintheconceptualdesignphaseand

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Fig 7 Lift, drag and moment coefficients at M∞=0.4 for the SC1095 airfoil [29]

hence they used empirical formulas to reduce computing times

[14]

Blade element theory was implemented to calculate the

re-quired power in different helicopter operations, namely hover,

climb,cruise,descent,andautorotation[26,10]

Helicopterdataareanalyzed bytheperformancecodeobtained

fromeitherthesizingmoduleoruserinputs

Thedifferencesbetweenthecalculatedresultsandexistingdata

arewithin5%ingeneral,henceacceptableforthepreliminary

de-signphase[28]

3 Optimization formulation and method

3.1 Design variables

The blade shape including maximum pre-twist, taper ratio,

pointoftaperinitiation,bladerootchordaredesignvariables

Ad-ditionally,the A0to A4coefficientsoftheairfoildistribution

func-tionaredesignvariablesforairfoilshape.Thebladeisassumedto

Fig 8 Lift, drag and moment coefficients at M∞=0.8 for the SC1095 airfoil [29]

Fig 9 Drag coefficients at zero lift as a function of M for the SC1095 aerofoil [29]

1 67

berectangularuntilthestationofthepointoftaperinitiationand

thentaperedlinearlytothetip.Thetwistvarieslinearlyfromthe

roottothetip.NACA0012waschosenasthebaselineairfoil

3.2 Constraints

Therequirementsareasfollows:theairfoilsectionsshouldnot

stallinforwardflight;theMach numberattheblades tipshould

avoidthedragdivergenceMachnumber

Thedrag–divergenceMachnumberatzeroliftisameasureof

theusefulnessofasectionnearthetipofahelicopterrotorblade

inforwardflight.Itisaparametertoquantifythedragpenalty

as-sociatedwithstrongcompressibilityeffects[7].ThedesirableMach

numberin this case is M DD0 ≥0.81 However, estimation of the

dragdivergenceMach number(M DD)isnot available inthis

pro-cess.Thepurposeoftheseconstraintsistoavoidaveryhighdrag

atblades tip on the advancing side Therefore, the requirements

arechangedtoconstraintsontheairfoilsectiondragcoefficient

Thetransonic dataareestimatedby solving theNavier–Stokes

equationusingFluentsoftware.Therefore,thesectionaldrag

coef-ficientconstraintcanbedefinedasbelow:

Thisconstraintisconstructedbecauseaportionofthe

advanc-ing blade generally operates beyond M DD Low drag rise beyond

dragdivergenceisdesirable

ThehighM DDproperty requiresathinandlesscambered

air-foil,whilethehighC lmax requiresathickandmorecambered

air-foil.Theseconstraintsareconflictinganddifficulttoachieveinone

design.Therefore,theseconstraintsarecompromisedandbuiltup

inTable 1

Themaximumlift(0.3≤M≤0.5)iscriticalindelaying

retreat-ingblade stall.Separationathighliftlevelsdependsonboth the

freestreamMachnumberandairfoilshape.Forthetypical airfoil

employedonhelicopterrotorblades,themaximumliftrequiredis

greaterthan1.5

Bensonetal.indicatedthat smallnose-uppitchingmomentis

necessarytominimizerotorloadsinforwardflight[3].The

pitch-ingmomentatzeroliftshouldsatisfythecriteriabelow

Thetrimconstraintinhoverandforwardflightisimplemented

byexpressingtheconstraintintermsofthe numberoftrim

iter-ationsITER, andthemaximumnumberoftrimiterations allowed

ITERmax

Anotherconstraintusedtoensurethatthebladetipchorddoes

notbecometoosmall

Allconstraintsarenormalized.Thenormalizingfactorsare

cho-senasapossiblemaximumvalue basedontheexperienceofthe

designers.ThisstudyperformsoptimizationofthebladeoftheBO

105helicopter

3.3 Objective function and optimization tool

The performance module allows forthe objective function of

theoptimizationproblemtobeveryvaried

Inthisstudy,a linearcombinationofrequiredpowerinhover

andforwardflightwasperformedastheobjectivefunction

Table 1

Constraints of optimization at 120 kts forward speed flight.

C d0, M=MDD0+ 0.02 0.0 0.04 0.03

ModelCenter [20]

F=0.75 P h

P h ref +0.25 P f

P f ref

(9)

Weightfactors are0.75and0.25chosen by thedesigner’s ex-perience Reference values P h ref, P f ref are used to normalize the objectivefunctioncomponents

AllmoduleswerewrappedintheModelCenterprogram,which

is a powerful tool for automating and integrating design codes Genetic algorithm is widely used to perform a global optimiza-tion problem However, this method requires a large number of runs Therefore, the Design Explorer tool was used to perform theoptimization searchusing ModelCenter.Design Explorer’s key technologiesare the systematicandefficientsamplingof the de-sign space using Design of Experiments(DOE) methods and the intelligentuseof“surrogate”modelsforproblemanalysisand op-timization The smooth surrogate models serve assubstitutes for potentiallyexpensiveand“noisy”computersimulationsandmake globalanalysisandoptimizationofcomplexsystemspractical

ThesurrogatemodelsusedbyDesignExplorerareKriging inter-polationmodels[23].Tocreateasurrogatemodel,DesignExplorer executes the analysis code (ModelCenter model) multiple times and storesthe results of each run in a table The input variable valuesfor thisseriesof runsare chosen to efficientlycanvas the designspace(usinganorthogonalarray).Initialonehundredforty samples(tentimesofthenumberofdesignvariables)areusedto generatesurrogatemodel

The aim of Kriging interpolation is to estimate the value of

an unknown function, f , at a point x∗ using weighted linear

combinations ofthe valuesofthe function atsome other points,

x1, x2, , x n.Thepredictedvalue ˆf(x∗)isexpressedas:

ˆfx∗

=

n



i=1

w i

x∗

Theweights w iaresolutionsofasystemlinearequationwhich

isobtainedby calculatingthepartial firstderivativesofthe error variance and setting the results to zero The error of prediction

ε (x)isexpressedas:

ε (x) = f(x) −

n



i=1

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Where:

TAPR: Taper ratio; POTAP: Position of taper initiation; CHOR: Chord length; POWER_HOVER: required power in hover flight; AU0, AU4, AL0, and AL4: Coefficients of airfoil shape distribution function: TWIST: Twist of the rotor blades.

Fig 11 Sensitivity analysis of design variables[30]

1 67

Table 2

Design variables of optimization at 120 kts forward speed flight.

TheprocessofusingasurrogatemodelintheDesignExplorer

toolisshowninFig 10.Thesurrogatemodels areselectively

up-dated and refined asthe optimization process progresses Global search mechanismsare implementedto avoidlocalminima.A fi-nalpatternsearchguaranteesthatthebestdesignfoundisatleast

alocalminimum

4 Results

Inthisstudy,theconvergencehistoryoftheobjectivefunction shows that the objective function is reduced to 0.956, so it re-ducesby 4.4% afterthe optimizationprocess Thefigure ofmerit increasesby 4.3%(from0.7to0.73).From Eqs.(9),we caneasily obtain 5.3% reduction on the required powerin 120 kts forward flight The study assumed that the drag divergence Mach num-beris0.83.Aportionoftheadvancingblademayoperatebeyond

M DD in higher forward speed or maneuver flight In these flight conditions,thezero-lift drag could riseto0.03 However, the ob-jective functionwas considered for120kts forwardspeed where the Machnumber atthetip ofrotor blades could approach0.81

Fig 12 Optimal rotor blade shape and airfoil for 120 kts forward speed.

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