NASA Technical Reports Server (NTRS) 20150021271: Aeroelastic Analysis of a Flexible Wing Wind Tunnel Model with Variable Camber Continuous Trailing Edge Flap Design

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by  NASA Technical Reports Server (NTRS)

Aeroelastic Analysis of a Flexible Wing Wind Tunnel Model with
Variable Camber Continuous Trailing Edge Flap Design
NhanNguyen∗
NASAAmesResearchCenter,MoffettField,CA94035
EricTing†
StingerGhaffarianTechnologiesInc.,MoffettField,CA94035
SoniaLebofsky‡
StingerGhaffarianTechnologiesInc.,MoffettField,CA94035
Thispaperpresentsdataanalysisofaflexiblewingwindtunnelmodelwithavariablecambercontinuoustrailing
edge flap (VCCTEF) design for drag minimization tested at the University of Washington Aeronautical Laboratory
(UWAL).ThewindtunneltestwasdesignedtoexploretherelativemeritoftheVCCTEFconceptforimprovedcruise
efficiencythroughtheuseoflow-costaeroelasticmodeltesttechniques. Theflexiblewingmodelisa10%-scalemodel
ofatypicaltransportwingandisconstructedofwovenfabriccompositesandfoamcore. Thewingstructuralstiffness
inbendingistailoredtobehalfofthestiffnessofaBoeing757-eratransportwingwhilethetorsionalstiffnessisabout
thesame. Thisstiffnessreductionresultsinawingtipdeflectionofabout10%ofthewingsemi-span. TheVCCTEF
is a multi-segment flap design having three chordwise camber segments and five spanwise flap sections for a total of
15individualflapelements. Thethreechordwisecambersegmentscanbepositionedappropriatelytocreateadesired
trailing edge camber. Elastomeric material is used to cover the gaps in between the spanwise flap sections, thereby
creating a continuous trailing edge. Wind tunnel data analysis conducted previously shows that the VCCTEF can
achieveadragreductionofupto6.31%andanimprovementinthelift-to-dragratio(L/D)ofupto4.85%.Amethodfor
estimatingthebendingandtorsionalstiffnessesoftheflexiblewingUWALwindtunnelmodelfromstaticloadtestdatais
presented.Theresultingestimationindicatesthatthestiffnessoftheflexiblewingissignificantlystifferintorsionthanin
bendingbyasmuchas3to1.Theliftpredictionfortheflexiblewingiscomputedbyacoupledaerodynamic-structural
model. ThecoupledmodelisdevelopedbycouplingaconceptualaerodynamictoolVorlaxwithafinite-elementmodel
oftheflexiblewingviaanautomatedgeometrydeformationtool. Basedonthecomparisonoftheliftcurveslope,the
liftpredictionfortherigidwingisingoodagreementwiththeestimatedliftcoefficientsderivedfromthewindtunnel
testdata. DuetothemovementoftheVCCTEFduringthewindtunneltest,uncertaintyintheliftpredictiondueto
theindicatedvariationsoftheVCCTEFdeflectionisstudied.Theresultsshowasignificantspreadintheliftprediction
whichcontradictstheconsistencyintheaerodynamicmeasurements, thussuggestingthattheindicatedvariationsas
measuredbytheVICONsystemmaynotbereliable. Theliftpredictionoftheflexiblewingagreesverywellwiththe
measuredliftcurveforthebaselineconfiguration. Thecomputedbendingdeflectionandwash-outtwistoftheflexible
wingalsomatchreasonablywellwiththeaeroelasticdeflectionmeasurements. Theresultsdemonstratethevalidityof
theaerodynamic-structuraltoolforusetoanalyzeaerodynamicperformanceofflexiblewings.
I. Introduction
Theaircraftindustryhasbeenrespondingtotheneedforenergy-efficientaircraftbyredesigningairframestobeaerodynam-
icallyefficient,employinglight-weightmaterialsforaircraftstructuresandincorporatingmoreenergy-efficientaircraftengines.
Reducing airframe operational empty weight (OEW) using advanced composite materials is one of the major considerations
forimprovingenergyefficiency. Modernlight-weightmaterialscanprovidelessstructuralrigiditywhilemaintainingsufficient
load-carrying capacity. As structural flexibility increases, aeroelastic interactions with aerodynamic forces and moments can
alteraircraftaerodynamicssignificantly,therebypotentiallydegradingaerodynamicefficiency.
UndertheFundamentalAeronauticsPrograminNASAAeronauticsResearchMissionDirectorate,theFixedWingproject
isconductingmultidisciplinaryresearchtoinvestigateadvancedconceptsandtechnologiesforfutureaircraftsystems.ANASA
study entitled “Elastically Shaped Future Air Vehicle Concept” was conducted in 20101,2 to examine new concepts that can
∗NASAAmesResearchCenter,ResearchScientist,AIAAAssociateFellow,[email protected]
†StingerGhaffarianTechnologiesInc.,NASAAmesResearchCenter,ResearchEngineer,[email protected]
‡StingerGhaffarianTechnologiesInc.,NASAAmesResearchCenter,ResearchEngineer,[email protected]
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enableactivecontrolofwingaeroelasticitytoachievedragreduction.Thisstudyshowedthathighlyflexiblewingaerodynamic
surfacescanbeelasticallyshapedin-flightbyactivecontrolofwingtwistandverticaldeflectioninordertooptimizethelocal
angles of attack of wing sections to improve aerodynamic efficiency through drag reduction during cruise and enhanced lift
performanceduringtake-offandlanding.
Thestudyshowsthatactiveaeroelasticwingshapingcontrolcanhaveapotentialdragreductionbenefit. Conventionalflap
andslatdevicesinherentlygeneratedragastheyincreaselift. Thestudyshowsthatconventionalflapandslatsystemsarenot
aerodynamically efficient for use in active aeroelastic wing shaping control for drag reduction. A new flap concept, referred
to as Variable Camber Continuous Trailing Edge Flap (VCCTEF) system, was conceived by NASA to address this need.1
Initial study results indicate that, for some applications, the VCCTEF system may offer a potential pay-off in drag reduction
that could provide significant fuel savings. In order to realize the potential benefit of drag reduction by active span-load and
aeroelasticwingshapingcontrolwhilemeetingallotherperformancerequirements,theapproachforhighliftdevicesneedsto
beconsideredaspartofthewingshapingcontrolstrategy.
NASAandBoeingarecurrentlyconductingajointstudytodeveloptheVCCTEFfurtherundertheresearchelementActive
AeroelasticShapeControl(AASC)withintheFixedWingproject.3,4 ThisstudyisbuiltuponthedevelopmentoftheVCCTEF
systemforNASAGenericTransportModel(GTM)whichisessentiallybasedontheBoeing757airframe,5 employinglight-
weight shaped memory alloy (SMA) technology for actuation and three separate chordwise segments shaped to provide a
variablecambertotheflap. Thiscamberedflaphaspotentialfordragreductionascomparedtoaconventionalstraight,plain
flap. Theflapisalsomadeupofindividual2-footspanwisesectionswhichenabledifferentflapsettingateachflapspanwise
position. This results in the ability to control the wing twist shape as a function of span, resulting in a change to the wing
twist to establish the best lift-to-drag ratio (L/D) at any aircraft gross weight or mission segment. Wing twist on traditional
commercialtransportdesignsisdictatedbytheaeroelasticdeflectionofafixed“jigtwist”shapeappliedatmanufacture. The
design of this jig twist is set for one cruise configuration, usually for a 50% fuel loading or mid-point on the gross weight
schedule. TheVCCTEFoffersdifferentwingtwistsettings,hencedifferentspanwiseloadings,foreachgrossweightcondition
andalsodifferentsettingsforclimb,cruiseanddescent,amajorfactorinobtainingbestL/Dconditions.
ThesecondfeatureofVCCTEFisacontinuoustrailingedgeflap.Theindividual2-footspanwiseflapsectionsareconnected
with a flexible covering, so no breaks can occur in the flap planforms, thus reducing drag by eliminating these breaks in the
flapcontinuitywhichotherwisewouldgeneratevorticitythatresultsinadragincreaseandalsocontributestoairframenoise.
Thiscontinuoustrailingedgeflapdesigncombinedwiththeflapcamberresultinlowerdragincreaseduringflapdeflections.
Inaddition,italsooffersapotentialnoisereductionbenefit.
Fig. 1-WingConfiguredwiththeVariableCamberContinuousTrailingEdgeFlap
TheVCCTEFisdividedinto14sectionsattachedtotheouterwingand3sectionsattachedtotheinnerwing,asshownin
Fig. 1.4 Each24-inchsectionhasthreecamberflapsegmentsthatcanbeindividuallycommandedtoformavariablecamber
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trailingedge,asshowninFig. 2. Thesecamberflapsarejoinedtothenextsectionbyaflexibleandsupportedmaterial(shown
inblue)installedwiththesameshapeasthecamberandthusprovidingcontinuousflapsthroughoutthewingspanwithnodrag
producinggaps.
Fig. 2-VariableCamberFlap
Usingthecamberpositioning,afull-span,low-drag,high-liftconfigurationcanbeactivatedthathasnodragproducinggaps
andalowflapnoisesignature. ThisisshowninFig. 3. Tofurtheraugmentlift,aslottedflapconfigurationisformedbyanair
passagebetweenthewingandtheinnerflapthatservestoimproveairflowovertheflapandkeeptheflowattached. Thisair
passageappearsonlywhentheflapsareextendedinthehighliftconfiguration.
Fig. 3-CruiseandHighLiftVCCTEFConfigurations
Figure 4 illustrates the GTM equipped with the VCCTEF for wing shaping control. By actively shaping the wing aero-
dynamicsurfaceusingtheVCCTEF,optimalaerodynamicperformancecouldpotentiallyberealizedatanypointintheflight
envelope. TheVCCTEFreliesontwomechanismstoimproveaerodynamicperformance: 1)wingtwistoptimizationforflexi-
blewingdesign,and2)variablecamberandcontinuoustrailingedgeforimprovedaerodynamics. Thisfixed-wingtechnology
maybereferredtoasPerformanceAdaptiveAeroelasticWing(PAAW)technology.
Fig. 4-GTMwithVCCTEF
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II. WindTunnelModelofFlexibleWingwithVCCTEF
For exploratory assessment of the aerodynamic potential of the VCCTEF concept, a 10%-scale aeroelastic model of a
softened Boeing 757-based GTM wing was constructed for a wind tunnel experimental investigation in the University of
WashingtonAeronauticalLaboratory(UWAL)inAugustof2013.6 Thesemi-spanofthemodelis5.6075ft,asshowninFig.
5. Themodelisconstructedofwovenfabriccompositesskinandextrudedpolystyrenefoamcore. Thecompositelaminates
andextrudedpolystyrenefoamcorearestructurallytailoredtoattainhalfofthebendingstiffnessofthescaledbaselineGTM
wingstiffnesswhilekeepingtorsionalstiffnessaboutthesame. Thistailoredstiffnessistoachievea10%wingtipdeflection.
The VCCTEF parts are fabricated by 3D printing. The flap segments are mechanically interlocking aerodynamic surfaces in
thechordwisedirectionandmatewithsiliconeelastomermaterialbetweenspanwiseflapsections,asshowninFig. 6. Theflap
segmentsarehingedatthreechordwiselocationsandaredesignedtobefullyadjustable. ThedimensionsoftheVCCTEFare
showninFig. 7.
Fig. 5-UWALWindTunnelModelwithVCCTEF(CourtesyofUniversityofWashingtonAeronauticalLaboratory)
Fig. 6-VCCTEFConstruction(CourtesyofUniversityofWashingtonAeronauticalLaboratory)
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Fig. 7-VCCTEFDimension(CourtesyofUniversityofWashingtonAeronauticalLaboratory)
Figure 8 shows an exploded view of the UWAL wind tunnel model of the flexible wing mated to a center body fairing
attachedtoanexternalfloor-mountedbalance.
Fig. 8-ExplodedViewofWindTunnelModel(CourtesyofUniversityofWashingtonAeronauticalLaboratory)
TheUWALaeroelasticwindtunnelmodelwasbuiltwithdifferentstiffnessthanthescaledGTMwinganditsjigshapewas
notoptimizedforbestaerodynamicperformanceattestconditions. ThesamejigshapeasthatoftheGTMwingwasused. The
wash-outtwististhereforenon-optimalforthemodelwhenitoperatesatthedesignliftcoefficientof0.51.ACFDoptimization
wasconductedpriortothetesttoidentifyanoptimaljigtwist. However,thisoptimizedjigtwistwasnotincorporatedintothe
finalmodelfabricationduetoprogrammaticissues.
TherelevantmodelscalinginformationisgiveninTable1.
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Full-Scale Semi-SpanModel
M 0.797 0.1162
∞
C 0.51 0.51
L
h,ft 36,000 0
q ,psf 211.09 20.00
∞
S/2,ft2 975.5 9.638
c¯,ft 16.6417 1.5963
b/2,ft 62.4167 6.1262
Table1-ModelParameters
Thewindtunneltestisdesignedtobeanexploratory,proof-of-conceptstudy. Theobjectiveofthewindtunnelexperiment
istoexploretherelativemeritoftheVCCTEFdesignasadragreductioncontroldevice,andtheabilitytosimulatetheproblem
inarelativelylow-costtest. Lift,drag,sideforce,pitchingmoment,yawingmoment,androllingmomentwererecordedfrom
the external floor-mounted balance. In addition, aeroelastic deflections of the flexible wing model were also measured by a
VICONmotiontrackingsystem. TheVICONsystemmeasuredthe3Ddisplacementofthewindtunnelmodelat54pointson
themodel.
To ensure that the wind tunnel model has correct aeroelastic properties, static load tests and frequency measurements
were conducted. A detail 3D NASTRAN model was constructed by UWAL for comparison with the measurements.6 The
3D NASTRAN results demonstrate an excellent agreement with the static load test data and measured frequencies. A stick
NASTRAN model was also constructed by UWAL to match the deflection information from the 3D NASTRAN model. In
addition,aNASTRANDoubletLatticeflutteranalysiswasperformedbyUWALtoensurethatthewindtunnelisflutter-freein
thewindtunnel.6 Flutterspeedsweredeterminedtobewellabovethetestsectionairspeedof39.54m/sec.
III. WindTunnelTest
Thetestwasconductedatanominaldynamicpressureof20psf. Off-conditiondataandadditionalrunsat10,15,25and
30psfwerealsocollectedforsomecases. ThenominaltestsectionairspeedwasMach0.1162. Figure9isaphotographofthe
flexiblewingwindtunnelmodelintheUWALtestsection.
Fig. 9-FlexibleWingWindTunnelModelinUWALTestSection(CourtesyofUniversityofWashingtonAeronautical
Laboratory)
Thewindtunnelmodelwastestedwithatotalof13VCCTEFconfigurationsrangingfromzerotofulldeflection. These
VCCTEFconfigurationsaredesignatedas:
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• FLAP0-baselinezerodeflection
• FLAP1-fulldeflectionforallflapsections
• FLAP2-varyingfromamaximumdeflectionattheinboardandoutboardflapstoaminimumdeflectionatthemid-span
flap
• FLAP3-varyingfromaminimumdeflectionattheinboardandoutboardflapstoamaximumdeflectionatthemid-span
flap
• FLAP4-varyingmonotonicallyfromamaximumdeflectionattheinboardflaptozerodeflectionattheoutboardflap
• FLAP5-varyingmonotonicallyfromzerodeflectionattheinboardflaptoamaximumdeflectionattheoutboardflap
• FLAP6-similartoFLAP4configurationbutwithasmallerdeflection
• FLAP7 - varying monotonically from a maximum positive deflection at the inboard flap to a negative deflection at the
outboardflap
• FLAP8-rigid-bodydeflectionwiththetwooutercambersegmentsatzerorelativedeflection
• FLAP9-deflectionofthetrailingedgecambersegments
• FLAP10-intermediatedeflection
• FLAP11-fullnegativedeflection
• FLAP12-FLAP6configurationplusagurneyflap
Figure10illustratessomeoftheseflapconfigurations.
Fig. 10-VCCTEFFlapConfigurations(CourtesyofUniversityofWashingtonAeronauticalLaboratory)
ThetestrunmatrixfortheVCCTEFisshowninTable2.
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FLAP 0 1 2 3 4 5 6 7 8 9 10 11 12
q =0−20psf,α =0o 18 25 32 42 55 73 79 86 93
∞
87
q =0−20psf,α =1o 19 26 33 38 43 49 56 64 74 80 94
∞
88
q =0−20psf,α =1.5o 27 34
∞
q =0−20psf,α =2o 20 28 35 39 44 50 57 65 75 81 89 95
∞
q =0−20psf,α =2.5o 29 36
∞
q =0−20psf,α =3o 30 40 45 51 58 66 76 82 90 96
∞
q =0−20psf,α =4o 22 41 46 52 59 67 77 83 91 97
∞
q =0−20psf,α =4.5o 47 53
∞
q =0−20psf,α =5o 23 60,61 68 78 84 92 98
∞
q =0−20psf,α =5.5o 62
∞
q =0−20psf,α =6o 99
∞
q =0−20psf,α =6.5o 100
∞
q =10psf,α =−2o−10.5o 117 121
∞
q =15psf,α =−2o−8.5o 118 122
∞
104 106 109
q =20psf,α =−2o−7o 111 102 113
∞
116 107 123
q =25psf,α =−2o−5.5o 119 124
∞
q =30psf,α =−2o−4.5o 120 125
∞
Table2-TestUW2087RunMatrix
TheVCCTEFsegmentpositionsfortheseconfigurationsareshowninTable3. Thefollowingconventionisused. Aflap
sectionisaportionoftheVCCTEFalongthespanwisedirection. Thereare5flapsectionsnumberedfrom1attheinboardto
5 at the outboard. Each flap section is comprised of three camber segments labeled as A for the innermost camber segment,
Bforthemiddlecambersegment, andCforthetrailingedgecambersegment. ThisisshowninFig. 11. Thepositionangle
isdenotedby(a/b/c)wherea,b,andcareflappositionsindegreesrelativetotheforwardflapsegment. ForsegmentA,the
positionangleiswithrespecttothefixedwingportion.
Fig. 11-VCCTEFFlapNotation
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FLAP Run Section1 Section2 Section3 Section4 Section5
1 25 (5/4/9) (6/7/9) (9/7/10) (7/8/10) (6/9/9)
1* 106 (1/7/11) (3/9/7) (5/7/8) (5/9/7) (5/10/10)
1** 107 (1/7/11) (3/9/7) (5/7/8) (5/9/7) (5/10/10)
32 (3/2/10) (4/5/4) (5/3/4) (4/6/5) (5/10/10)
2
36 (2/3/9) (4/5/4) (5/3/5) (4/6/4) (4/10/10)
38 (2/3/5) (5/6/5) (6/6/11) (5/6/5) (4/6/5)
3
41 (2/3/5) (4/6/5) (4/6/11) (4/6/5) (3/6/5)
42 (2/5/9) (3/6/3) (4/1/6) (2/5/2) (0/0/0)
4
47 (1/4/9) (3/6/2) (4/1/6) (2/5/1) (0/0/0)
49 (0/0/0) (2/5/1) (4/2/7) (3/5/6) (3/8/9)
5 51 (0/0/0) (2/5/1) (4/2/7) (3/5/6) (3/8/9)
53 (0/0/0) (2/5/1) (4/2/7) (3/5/5) (3/8/9)
55 (1/1/1) (0/2/0) (0/0/2) (0/0/0.5) (0/0/0)
61 (1/1/1) (0/2/0) (0/0/2) (0/0/0.5) (0/0/0)
6
62 (1/1/1) (0/2/0) (0/0/2) (0/0/0.5) (0/0/0)
111 (1/1/1) (0/1/1) (0/0/2) (0/0/0) (0/0/0)
7 64 (3/3/4) (2/3/2) (0/0/3) (0/0/0) (0/0/−2)
7* 109 (3/4/3) (2/5/1) (2/0/2) (0/−1/0) (0/0/−2)
73 (4/0/0) (4/0/0) (5/0/0) (5/0/0) (4/0/0)
8
78 (3/0/0) (4/0/0) (5/0/0) (4/0/0) (4/0/0)
79 (0/0/6) (0/0/6) (0/0/8) (0/0/6) (0/0/6)
9 81 (0/0/6) (0/0/6) (0/0/8) (0/0/6) (0/0/6)
84 (0/0/6) (0/0/6) (0/0/8) (0/0/6) (0/0/6)
86 (2/3/2) (3/3/1) (4/2/5) (2/5/2) (2/5/3)
10 91 (2/3/2) (3/3/1) (4/2/5) (2/5/2) (2/5/3)
92 (1/3/2) (3/3/1) (4/2/5) (2/5/2) (2/5/2)
93 (−1/0/−3) (0/0/−6) (−1/−4/0) (−2/0/−5) (−4/0/−4)
11
100 (−1/0/−4) (0/0/−5) (−1/−3/0) (−2/0/−5) (−3/0/−4)
Table3-VCCTEFDeflections
IV. AeroelasticEffectsonLiftandDrag
Theaeroelasticdeflectionsofaflexiblesweptbackwindtunnelmodelcontributesignificantlytotheaerodynamicperfor-
manceduetothewash-outeffectresultingfromwingbendingandtwist. Additionally,theVCCTEFdeflectionsalsocontribute
toangleofattackchanges. Theaeroelasticangleofattackofawingsectioncanbeexpressedas7
dW(y¯) N ∂α
α (y)=α−α (y)−γ(y¯)cosΛ−Θ(y¯)cosΛ− sinΛ+∑ cδ (y )cosΛ (1)
c i i h h
dy¯ ∂δ
i=1 i
whereα isthegeometricangleofattackofthewingsectionaboutthepitchaxisy,α istheinducedangleofattackduetothe
i
downwashaboutthepitchaxisy,γ isthewingpre-twistangleabouttheelasticaxisy¯=y/cosΛ(positivenosedown),Θisthe
wing torsional twist about the elastic axis y¯(positive nose down),W is wing vertical bending along the elastic axis (positive
upward),Λisthesweepangleoftheelasticaxis,δ istheabsolutedeflectionofthei-thflapsegmentoftheVCCTEFaboutthe
i
hingeaxisy whichhasasweepangleofΛ ,and∂α /∂δ istheangleofattacksensitivityorcambercontrolderivativedueto
h h c i
theVCCTEFflapdeflection.
Thefollowingconventionisused.Aquantityexpressedwiththeindependentvariableyisonewithrespecttothestreamwise
direction,e.g.,c(y)isthesectionalchordinthestreamwisedirection. Aquantityexpressedwiththeindependentvariabley¯is
onewithrespecttotheelasticaxis;e.g.,Θ(y¯)isthewingtwistabouttheelasticaxis. Thequantityδ (y )istheflapdeflection
i h
aboutthehingeaxis.
In general, the aeroelastic deflections are functions of the angle of attack and the flap deflection. Therefore, they can be
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expressedas
∂Θ(y¯) N ∂Θ(y¯)
Θ(y¯)=Θ (y¯)+ α+∑ δ (2)
0 i
∂α ∂δ
i=1 i
∂W(y¯) N ∂W(y¯)
W(y¯)=W (y¯)+ α+∑ δ (3)
0 i
∂α ∂δ
i=1 i
where Θ andW are the aeroelastic deflections at zero angle of attack, and the partial derivatives are the sensitivities of the
0 0
aeroelasticdeflectionswithrespecttotheangleofattackandtheflapdeflection.
Therefore,theexpressionfortheaeroelasticangleofattackofawingsectioncanbewrittenas
(cid:20) (cid:18) (cid:19) (cid:21)
∂Θ(y¯) ∂ dW(y¯) dW (y¯)
α (y)=α 1− cosΛ− sinΛ −α (y)−γ(y¯)cosΛ−Θ (y¯)cosΛ− 0 sinΛ
c i 0
∂α ∂α dy¯ dy¯
N (cid:20)∂α ∂Θ(y¯) ∂ (cid:18)dW(y¯)(cid:19) (cid:21)
+∑ ccosΛ − cosΛ− sinΛ δ(y ) (4)
h h
∂δ ∂δ ∂δ dy¯
i=1 i i i
Itcanbeseenthattheaeroelasticdeflectionscancausethedesiredsectionalangleofattacktobenon-optimal. Theeffect
oftheadaptiveaeroelasticwingshapingcontrolbytheVCCTEFiscapturedinthelastterm. Theterm ∂αc istherigidcamber
∂δ
(cid:16) (cid:17)
controltocompensateforthenon-optimalsectionalangleofattack. Thetwoterms ∂Θ and ∂ dW aretheaeroelasticwing
∂δ ∂δ dy¯
shapingcontrolbyleveragingwingflexibilitytochangethewash-outtwistofawinginordertoachieveimprovedaerodynamic
performance. Thus,theeffectofadaptiveaeroelasticwingshapingcontrolistooptimizethespanloadatanyoperatingpoint
insideagivenflightenvelope.
Thecambercontrolderivative ∂αc canbedirectlyestimatedfromthinairfoilpotentialflowtheorybyevaluatingthefollow-
∂δi
ingintegraltransformwiththepotentialkernelfunction f(θ)=cosθ−1as8
ˆ ˆ
∂α 1 θi+1 1 θi+1
=− f(θ)dθ =− (cosθ−1)dθ (5)
∂δ π π
i θi θi
where
(cid:48)
c
(cid:48)
x = (1−cosθ) (6)
2
(cid:48)
c
(cid:48) (cid:48)
c −x = (1−cosθ) (7)
i 2 i
(cid:48) (cid:48)
andc istheairfoilchordandx istheflaphingepositionofthei-thflapsegmentmeasurednormaltothehingeaxisfromthe
i
trailingedgeandisgivenby
(cid:48) (cid:48)
x =(n+1−i)c (8)
i f
(cid:48)
wherec istheflapchordofacambersegmentmeasurednormaltothehingeaxis.
f
(cid:48) (cid:48) (cid:48) (cid:48) (cid:48)
Sothefirsthingepositionisatx =Nc ,thelasthingepositionisatx =c ,andthetrailingedgepositionisatx =0.
1 f N f N+1
Thisintegralisevaluatedas
√ (cid:12)c∗
∂α cos−1(−c∗)− 1−c∗2(cid:12) i+1
c = (cid:12) (9)
∂δ π (cid:12)
i (cid:12)c∗
i
where
(cid:48)
x
c∗=1−2 i (10)
c(cid:48)
Thesectionalliftcoefficientisexpressedas
(cid:34) (cid:35)
dW(y¯) N ∂α
c (y)=c (y)+c (y) α(y)−γ(y¯)cosΛ−Θ(y¯)cosΛ− sinΛ+∑ cδ (y )cosΛ (11)
L L0 Lα dy¯ ∂δ i h h
i=1 i
wherec isthesectionalliftcoefficientatzeroangleofattackandc isthesectionalliftcurveslope,bothofwhichaccount
L0 Lα
fortheinducedangleofattack.
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