[PDF] Aircraft Drag Polar Estimation Based on a Stochastic Hierarchical

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Aircraft DragPolar EstimationBased ona

Stochastic HierarchicalModel

Junzi Sun,Jacco M.Hoekstra, JoostEllerbroek

Control andSimulation, Faculty ofAerospaceEngineering

Delft UniversityofTechnology ,the Netherlands

Abstract—The aerodynamicproperties ofanaircraft determine a crucialpart ofthe aircraft performance model.Deriving accurate aerodynamiccoefcientsr equires detailedknowledge of theair craft'sdesign.Thesedesignsand parametersar ewell protectedby aircraft manufacturers.Theyrar elycanbeused in publicr esearch.Verydetailedaer odynamicmodelsare often not necessaryin airtrafc managementr elatedr esearch, asthey often usea simpliedpoint-mass aircraft performance model. In thesestudies, asimple quadraticr elationoften assumedto compute thedrag ofan aircraft basedon therequired lift. This so-calleddrag polardescribes anappr oximationof the drag coefcientbased onthe totallift coefcient.The two key parameters inthe dragpolar are thezer o-liftdragcoefcientand the factorto calculatethe lift-inducedpart ofthe dragcoefcient. Thanks tothis simplicationof theight modeltogether with accurate ightdata,we are ableto estimatetheseaerodynamic parameters basedon ightdata. Inthis paper, weestimate the drag polarbased ona nov elstochastic totalenergymodelusing Bayesiancomputi ngandMarko vchain MonteCarlosampling. The methodis basedon thestochastic hierarchical modeling approach.W ithsufcientlyaccurateight dataand somebasic knowledgeof aircraft andtheirengines,the dragpolar canbe estimated. Wealsoanalyze ther esultsand compare themtothe commonly usedBase ofAir craftData model.Themeanabsolute differenceamong 20common aircraft for zero-liftdragcoefcient and lift-induceddrag factorar e0.005 and0.003respecti vely .At the endof thispaper ,the dragpolarmodelsin different ight phases forthesecommon commerc ialair crafttypesareshared. Keywords-aircraft performance, dragpolar ,aerodynamiccoef- cient, Bayesiancomputing, MCMC

I.I NTRODUCTION

Since thein ventionofaircraft,researchersha ve beenstudy- ing theaerodynamic propertiesof airfoilsand aircraft.Exam- ples offundam entalstudiesonaerodynamic dragare giv en in [1],[2]. Fromthe start,the goalhas beento optimizethe lift overdragratiofor thecruise flight.Much effort hasbeen dedicated tocreating designsthat would reducedrag andthus increase thelift efficienc yofaircraft.Whilethezero drag coefficientcontains theparasitic dragof thewhole aircraft, the wingis mainlyresponsible forthe lift-induceddrag. Next, to thechosen airfoil,the aspectratio ofthe wingplays an important role.The wingcan beseen asa dragto liftcon verter , of whichthe alreadyhigh efficienc ycan beincreasedfurther. This isan on-goingef fort:some examplesofcurrent research are boundarylayer suction,morphing wings, plasmacontrol, and blendedwing-body aircraftshapes. Aerodynamic liftand dragforces ofan aircraftare com- plicated andcomputationally intensiv etocompute.Liftand

drag areconsidered asfunctions ofthe wingarea, dynamicairspeed, andair density, andtheremainingef fectsof theflo wfor boththe liftand dragare describedwith coefficients forboth forces.The mostcomplicated partis tomodel theselift and dragcoef ficients.Theseparametersdepend onthe Machnumber,the angleof attack,the boundarylayer andultimately on thedesign ofthe aircraftshape. For fixed-wing aircraft,these coefficientsarepresented asfunctions ofthe angleof attack, i.e.,the anglebetween theai rcraftbody axisand theairspeed vector.Inairtraf ficmanagement (ATM) research,

however,simplifiedpoint-massaircraftperformance models are mostlyused. Thesepoint-mass modelsconsider anaircraft as adimensionless point,where theangle ofattack, aswell as theside-slip angle,and theef fectof theangular ratesare not explicitlyconsidered.Hence, thestep ofcalculating these is oftenskipped andthe dragpolaris usedinstead. The relationshipbetween dragcoef ficient andliftcoeffi- cient isthe mainf actordetermining theaircraftperformance. Knowledgeof thedrag polaris thereforeessential formost ATMresearch suchas trajectoryprediction, fueloptimization, parameter estimation. Manymethods exist toexplorethe aircraftperformance during thepreliminary designphase, oftenwith afocus on the modelingof theaerodynamics. Hence,one sourceof open informationre gardingdragpolarmodelscomes from textbooks[3], [4],[5], [6].Ho wev er, onlyolderaircraftmodels are availableintheliterature. In[7], anempirical model for estimatingzero-lift dragcoef ficientsw asproposedusing existingliterature databased onse veral aircraftmodels. In general, opendata ondrag polarmodel israre, especiallyfor modern commercialaircraft. The aircraftmanuf acturerswhodesignthe aircraftdo hav e accurate aerodynamicdata. Howe ver,thesedataarerarely publicly availableduetocommercial competition.The most comprehensivecollectionof dragpolar datais theBase of AircraftData (BAD A)developedbyEurocontrol [8].It contains thedrag polarsfor nearlyall commonaircraft types. BADAisthe default "go-to"aircraft performancemodelfor current ATMresearchers.Ho wev er,itimposesastrictlicense in termsof sharingand redistribution ofthe modeldata.The project-based licensefor newer versionsmakes iteven harder for thesame researcherto reusethe model,which willalso apply forne wusersofolder versions. The goalof thispaper isto proposean alternativ epath to estimate thedrag polarmodels formodern fixed-wing com- mercial aircraft,as wellas sharethe dragpolar mod els thatwe

haveobtained.W eapproach thisestimationproblemusing aEighth SESAR Innovation Days, 3rd ± 7th December 2018

novelstochastictotal energy (STE)model. TheSTEapproach treats theparameters ofthe standardtotal energy modelas random variables.Then,we tryto solve theparameter esti- mation usingBayesian computing,specifically ,Mark ovchain Monte Carlo(MCMC) approximations.Finally ,a databaseof drag polarmodels fordif ferentcommon commercialaircraft, which wereproduced usingthis method,is provided. The structureof thispaper isas follows. Insection two, the fundamentalsof thepoint-mas sdrag polarmodelare introduced. Insection three, wefocusonthe hierarchical model approach.In sectionfour ,e xperimentsareconductedto examineand obtaindrag polardata ofmultiple aircrafttypes based onthis method.Section fiv eand sixarededicatedto the discussionsand conclusionsof thisresearch.

II.THEORYOFAE RODY NAMICMODELING

A. Dragpolarinpoint-mass models

While anaircraft flies,the dragforce isproduced bythe airflowinteracting withthe aircraftbody .The liftforce is produced dueto thepressure difference betweenthe upper and lowersurface oftheliftingde vices(wings). With thesame airspeed andalt itudeconditions,controlof liftis performed by re-configuringthe aircraftangle ofattack and/ormodifying the surfaceshapeof liftingde vices.By changingthe elevator settings, thepitch angleand the angleof attackcanbe controlled. Onthe otherhand, achange ofthe liftingde vice surfaceis primarilyperformed byre-configuring flaps. In general,the liftand dragforces ofan aircraftthat is travelinginthe freestream canbe computedas:

L=CL12

V2S

D=CD12

V2S(1)

whereCLandCDare liftand dragcoef ficients,respecti vely., V, andSare airdensity ,trueairspeed,and thelifting surface area ofthe aircraft.In practice,CLandCDcan bemodeled as functionsof theangle ofattack ( ), Machnumber( M) and flap deflection( f): C

L=fcl(;M;f)

C

D=fcd(;M;f)(2)

In aerodynamicmodels, multi-dimensional tableinterpolation or higherorder polynomialsare used.Ho wev er, inmanyATM studies, thesix-de greeoffreedomof aircraftflight dynamicis simplified tothe three-degree offreedompoint-massmodel. Leavingout theaerodynamic anglesand pitch,ya w, androll rates meansthe aerodynamicmodels alsoneed tobe adapted. In point-massmodels, therelation betweenthe aerodynamic coefficientsCDandCLis simplifiedto thedrag polar. Itis commonly representedusing aquadrati cfunction inoneof the twoformsbelo w: C

D=CD0+C2LAe

(3)whereAis theaspect ratioof thewing (spandi videdby the averagechord)and whereeis theOsw aldfactor, whichlies typically inthe range0.70-0.90. Thisequation isoften written as: C

D=CD0+kC2L(4)

with: k=1Ae (5)

These twoparameters,CD0andkare thezero-liftdr ag

coefcientandlift-induced dragcoefcientfactor respectively. The valuesofboth parametersare consideredas constants under aspecific aerodynamicconfiguration ofthe aircraft.Fig.

1 illustratesan example ofdragpolarby usinga computational

fluid dynamics(CFD)simulation foran aircraftwith aclean (no flapsor extended landinggear)configuration.0.20.00.20.40.60.81.01.2 C

L0.010.020.030.040.050.060.070.080.09CDC

D

0 = 0.0184

K = 0.0484Fig1.Dragpolar(B747-400simulated,cleanconfiguration,<10 )B. Aircraftaerodynamiccongur ations Besides theangle ofattack thataf fectsthe values ofthelift and dragcoef ficients,thechangein theshape ofthe aircraft can alterthese values. Themostnotablechange inaircraft is flaps (andslats), speedbrak esand landinggear.Each structural setting alsohas itso wncorresponding dragpolarmodel. Flaps arecommon aircraftsurf acesdeplo yedinorderto providean increasein themaximum liftcoef ficient.The yare deployedt obeableto flyat lower speeds,typically atlo w altitudes (fore xample,duringtakeof f,initial climb,andap- proach). Differentaircrafttypes hav edif ferentconfigurations of flapsand flapsettings. Anincrease inflap angleleads toan increase int heliftcoefficientunder thesame angleofattack,at the expenseofa higherdrag. Slatsare similarto flaps,b uton the leadingedge ofthe wingand they increasethe maximum lift coefficientbyincreasing thestall angleof attack.Slats are automatically extendedwhenselecting aflap settingand are considered aspart ofthis configuration. Different flapdesigns havebeenadopted byaircraft manufacturers. InT ableI, alist of commonflap optionson airfoilsand theirapproximated maximum liftcoef ficientsarelisted.These values areproduced by [9,p.107]. Itis worth notingthat theCL;maxvaluesof an airfoil arelar gerthanthev aluesof theaircraft withthesame shaped wing,especially forswept wings[3, p.263].2

TABLEI

EXAMPLEFLAPSE TTINGS (AIRFOILS)Flap typesCL;maxIllustrationairfoil only1.5 leading-edge slat2.4 plain flap2.5 split flap2.6

Fowlersingle-slottedflap 2.9

Fowlermulti-slottedflap 3.0

with leading-edgeslat 3.3 with boundarylayer suction3.9 In thispaper ,basedonthe datafrom [6,p.2 53],the increase in liftcoef ficientduetoflap deployment ismodeled. These valuesare shown inTableII, whereT OandLDrepresent the take-offandlanding configurationrespecti vely .Extended flaps also increasethe drag.The increaseof dragcoef ficientdue to flaps canbe computedusing themodel proposedby [10] C

D;flap= 0:9cfpc

1:38SfpS

sin 2(6) wherecfp=candSf=Sare flapto wingchord ratioand surfaceratio. is theflap deflectionangl e.When theseaircraft characteristics arenot av ailable,simplifiedempiricalvalues from [6,p.253] canbe used,which arelisted inT ableIII.

TABLEII

INCREASEOFLIFT COEFFI CIENTWIT HFLAPSTrailingLeading TOLDCTOL;maxCLDL;maxplain flap20

601.60 2.00

single-slotted flap20 401.70 2.20

Flowersingle-slotted flap15 402.20 2.90

Flowerdouble-slotted flap20 501.95 2.70

Flowerdouble-slotted flapw/ slat20 502.60 3.20

Flowertriple-slotted flapw/ slat20 402.70 3.50TABLEIII INCREASEOFDRAG COEFFI CIENTWIT HFLAPSCD;flapTypicalflap angleFlight phase0.021020take-off

0.042030take-off

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