NASA Technical Reports Server (NTRS) 20170001393: Investigating the Transonic Flutter Boundary of the Benchmark Supercritical Wing

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by  NASA Technical Reports Server (NTRS)

Investigating the Transonic Flutter Boundary of the
Benchmark Supercritical Wing
JenniferHeeg∗&PawelChwalowski†
NASALangleyResearchCenter,Hampton,VA,23681-2199
Thispaperbuildsonthecomputationalaeroelasticresultspublishedpreviouslyandgeneratedinsupport
ofthesecondAeroelasticPredictionWorkshopfortheNASABenchmarkSupercriticalWingconfiguration.
ThecomputationalresultsareobtainedusingFUN3D,anunstructuredgridReynolds-AveragedNavier-Stokes
solverdevelopedattheNASALangleyResearchCenter. Theanalysisresultsfocusonunderstandingthedip
inthetransonicflutterboundaryatasingleMachnumber(0.74),exploringanangleofattackrangeof−1◦to
8◦ anddynamicpressuresfromwindofftobeyondflutteronset. Therigidanalysisresultsareexaminedfor
insightsintothebehavioroftheaeroelasticsystem.Bothstaticanddynamicaeroelasticsimulationresultsare
alsoexamined.
Nomenclature
α Angleofattack,degs
c Wingchord,16inches α0 Angle of attack, without any aeroelastic influ-
ences(rigid,windoff),degs
C Liftcoefficient
L
α Total angle of attack, explicitly incorporatating
C Derivative of lift coefficient due to angle of at- total
Lα aeroelasticinfluences,degs
tack
α¯ Rigidangleofattack,biasedsuchthatC =0,
Cm Pitchingmomentcoefficient m0
seeeqn7
C Derivativeofpitchingmomentcoefficientdueto
mα
angleofattack ω, f Frequency-radians/second,Hz
C Coefficientofpressure θ Aeroelasticincrementinthepitchangleorangle
p
ofattack,degs
gvel Initialperturbationofthegeneralizedvelocity
0
ζ Dampingratio
K Stiffnessofthepitchingdegreeoffreedom
θ
AePW AeroelasticPredictionWorkshop
M Machnumber
q¯ Dynamicpressure,psf BSCW BenchmarkSupercriticalWing
S Wingarea,512inches2 CFL Courant-Freidrichs-Lewynumber
SE Estimatedstandarderror FRF FrequencyResponseFunction
x¯ac Aerodynamic center location, in fraction of OTT OscillatingTurntable
wingchord,measuredpositiveaftfromthelead-
ingedge PAPA PitchandPlungeApparatus
x/c chordwiselocation,normalizedbychordlength RANS Reynolds-AveragedNavier-Stokes
x¯ Centerofpressurelocation, infractionofwing
cp TDT TransonicDynamicsTunnel
chord, measured positive aft from the leading
edge TE TemporalError
∗SeniorResearchEngineer,AeroelasticityBranch,AssociateFellowAIAA.
†SeniorAerospaceEngineer,AeroelasticityBranch,SeniorMemberAIAA.
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I. Introduction
GeometricsimplicityfirstdrewustotheBenchmarkSupercriticalWing(BSCW)asananalysisconfigurationfor
the Aeroelastic Prediction Workshop (AePW). With an outer mold line machined from a block of aluminum,
thewingitselfisoneofthemostrigidaeroelasticmodelsevertestedintheTransonicDynamicsTunnel(TDT);the
flexibility was provided principally by the TDT Pitch and Plunge Apparatus (PAPA). The PAPA is a two degree-of-
freedom(pitchandplunge)sidewallmountsystemconstructedofbarsarrangedtoproducethedesiredpitchandplunge
mode stiffness values, and masses positioned to decouple the wind off modal mass properties. This mount system
was engineered by Farmer,1 who reported experimental results for a supercritical wing showing similar transonic
aeroelasticcharacteristicstothecomputationalresultspresentedinthecurrentpaper.
ThesimplicityofthePAPAstructurehasallowedustostudytheflowfieldevolutionasMach,angleofattackand
excitation level are varied, and the resulting effects on the aeroelastic deformation (static aeroelastic deflection) and
stability (flutter). This paper and its companion publication2 endeavor to present a sufficient summary of results to
showtheinfluencesofshockdevelopmentandseparationonsetontheaeroelasticcharacteristicsoftheBSCWwing.
Thegoaloftheworkistodevelopandunderstandthebucketinthetransonicflutterboundaryforthisconfiguration.
Historically, the location (in terms of Mach number) and depth (in terms of dynamic pressure or velocity) of the
transonic dip have been shown to be functions of the angle of attack. There are many configurations where this
behavior has been observed in the technical literature, including both computational and experimental examples. A
specificexamplefortheNACA64A-010airfoilisreproducedfromreference3inFigure1.
Figure1. EffectofangleofattackonNACA64A-010flutterboundaries;Source:Reference3,Figure7.
WhilepreviouspublicationsassociatedwithAePW-2concentratedontheoverallsummaryofthecomputational
resultsgeneratedfortheworkshop,4inthispaper,wewillbereportingonthepredictionsfromasinglecode,FUN3D.
Wefocusprincipallyontheinfluencesoftheseparatedflowonset,inducedinthisstudythroughthreeeffects: angle
ofattack,dynamicpressureandinitialexcitation. OnlyoneMachnumberisexamined—Mach0.74. Anglesofattack
from-1◦to8◦areexamined,withtheprimaryinvestigationsatanglesofattackof0◦,3.15◦,5◦and6.5◦. Thedynamic
pressurerangesexaminedforeachangleofattackbeginattherigidconditionandrangetoabovethepredictedflutter
onset condition. Distributions of steady pressure coefficient for the static rigid BSCW configuration are compared
with experimental information over the angle of attack range where that information is available, which ends at 5◦.
The flutter onset prediction at 0◦ is compared with the corresponding result from experiment. Unfortunately, the
experimentalflutterboundaryatnonzeroanglesofattackisnotavailableforasimilarcomparisonatthisMachnumber.
In fact, very little quantitative information is available for any of the conditions where the simulations indicate that
separatedfloweffectsdominatethestability.
The paper is organized into five major sections. Background material on the configuration and its uses in the
AeroelasticPredictionWorkshopsarepresentedfirst. Theanalysisprocessesarethendiscussed,followedbyresults
fromeachofthethreeanalysisprocessesusedhere: rigidsteady,staticaeroelasticanddynamicaeroelasticanalyses.
The dynamic aeroelastic results include effects of matched point conditions, single degree of freedom representa-
tion, perturbationlevelandspatialresolution. Thefinaltwosectionscontaindiscussionsofcontinuinganalysesand
conclusions.
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II. Background
A. TheBSCWModelandTestCases
TheBSCWmodel,showninFigure2,hasasimple,rectangular,16-x32-inchwingplanform,withaNASASC(2)-
0414airfoil. ItwasfirsttestedintheTDTin1991.5 Forthistest,thewingwasmountedontheTDTPitchandPlunge
Apparatus(PAPA)toobtaintheflutterboundaryatvariousMachnumbersandanglesofattackforthistwo-degreeof
freedomsystem. In2000,thewingwastestedagain,thistimeonanOscillatingTurntable(OTT).6Thepurposeofthe
OTTtestwastomeasureaerodynamicresponseduringsinusoidal(forced)pitchoscillationofthewing.
ForboththeOTTandPAPAtests,themodelwasmountedtoalargesplitterplate,sufficientlyoffsetfromthewind-
tunnelwall(40inches)to(1)placethewingclosertothetunnelcenterlineand(2)beoutsidethetunnelwallboundary
layer.7 Thewingwasdesignedtoberigid,withthefollowingstructuralfrequenciesforthecombinedinstalledwing
andOTTmountingsystem: 24.1Hz(spanwisefirstbendingmode),27.0Hz(in-planefirstbendingmode),and79.9
Hz (first torsion mode). When installed on the PAPA mount, the combined system frequencies were 3.33 Hz for
the plunge mode and 5.20 Hz for the pitch mode.8 The plunge and pitch modes are the only modes included in the
aeroelasticanalysesofthePAPA-mountedconfigurationtobepresentedinthispaper.
For instrumentation, the model has pressure ports in chordwise rows at the 60% and 95% span locations. For
the BSCW/PAPA test, both rows were fully populated with unsteady in situ pressure transducers. The quantitative
information obtained consists of unsteady data at flutter points and averaged data on a rigidized apparatus at the
flutterconditions. FortheBSCW/OTTtest, onlytheinboardrowat60%spanwaspopulatedwithtransducers. The
quantitative information for the OTT test consists of unsteady pressure data and accelerometer data for forced pitch
oscillationsandfortheunforcedsystematstabilizedflowconditions.
(a) PhotographoftheBSCWmodelmountedontheOTTin (b) BSCWgeometry.
theTDT.
(c) Cross-sectionalviewoftheSC(2)-0414airfoil,withBSCWin-
strumentation.
Figure2. BSCWmodel.
B. TheFirstAeroelasticPredictionWorkshop
ThefirstAIAAAeroelasticPredictionWorkshop(AePW-1)tookplaceinconjunctionwiththeAIAAStructuralDy-
namics and Materials (SDM) conference in April 2012.9 This workshop utilized three configurations, including the
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BSCW.TherewerethreeBSCWtestcasesforthisworkshop: arigidsteadyanalysisandtwoforcedoscillationanal-
yses,allatMach0.85,5◦ angleofattack. ThesubmittedsimulationresultsfortheBSCWcontainedthemostwidely
varyingresultsofalloftheAePW-1configurations.10 Thesevariationswereobservedinthesteadyresultsaswellas
inthefrequencyresponsefunctions. Predictionsoftheuppersurfaceshocklocationspannedmorethan20percentof
thechord.11 Becauseofthesewidevariations,furtherstudieswereinitiatedtounderstandtheunderlyingcauses.
C. TheSecondAeroelasticPredictionWorkshop
ThesecondAIAAAeroelasticPredictionWorkshop(AePW-2)tookplaceinconjunctionwiththeAIAASciTech2016
Conference.4 ThecomputationalaeroelasticitycommunitywaschallengedtoanalyzetheBSCWconfigurationatthree
conditions(M=0.7at3◦angleofattackontheOTT;M=0.74at0◦angleofattackonthePAPA;andM=0.85at5◦angle
ofattackonthePAPA)andtopresenttheirresultsattheworkshop. TheexperimentaldatafromtheOTTtestindicated
thattheBSCWexhibitedastrongshockandboundary-layer-inducedseparatedflowatamoderateangleofattackat
transonicconditions.12 Thecomputationsofthetransonicflow,inconjunctionwiththeflutterboundarypredictions,
were therefore the focus of the workshop.13 The summary of flutter onset predictions from AePW-2 is shown in
Figure3. ThepredictionsatMach0.74,0◦ angleofattackhaveconsiderablylessscatterthanthepredictionatMach
0.85, 5◦ angle of attack. The lower Mach number, lower angle of attack predictions also surround the experimental
flutteronsetcondition,169psf.
Figure3. FlutterboundaryresultsfromAePW-2.
D. Grids
For the present study, two of the unstructured grids that are detailed in reference 2 were utilized: the designated
“medium” grid (9 million nodes) and the designated “Grid D” (35 million nodes). In the current paper, Grid D is
referred to as the extra fine grid. The grid distributions for both the wing surface and the plane of symmetry are
presentedinFigure4.
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(a) Mediumgrid;GridB;9millionnodes. (b) Extrafinegrid,GridD;35millionnodes.
Figure4. Mediumandextrafinegrids.
III. AnalysisProcesses
SolutionstotheReynolds-AveragedNavier-Stokes(RANS)equationswerecomputedusingtheNASALangley-
developedcomputationalfluiddynamics(CFD)softwareFUN3D.14Theflowsolutionswereproducedwithturbulence
closureobtainedusingtheSpalart-Allmaras(SA)one-equationmodel.15,16 Inviscidfluxeswerecomputedusingthe
Roe scheme,17 primarily without a flux limiter. The solutions were produced by running three distinct analyses,
differentiated by the treatment of the structure, the initial condition and the temporal parameters. The rigid, static
aeroelasticanddynamicaeroelasticsolutionprocessesaredescribedbelow.
A. RigidAnalysis
TheinitialsolutionforeachangleofattackatMach0.74wasgeneratedforarigidwing. Ingeneral,thesesolutions
wererunina“steady”mode,inwhichthesolutionscomputedwerenottime-accurateandthestructurewasrepresented
as a fixed position outer mold line. The position (angle) was different for each angle of attack and analyzed in an
individual simulation. Time integration was accomplished by an Euler implicit backwards difference scheme, with
localtimesteppingtoaccelerateconvergence. Althoughsomeofthecaseswerenotasymptoticallysteady, therigid
steady simulation was still the first step performed. The cases that were not asymptotically steady occurred at the
higher angles of attack and will be identified as a subset of the results presented. These cases did not converge to a
fixedpressuredistributionandshouldberuninatime-accuratemannerwheretherangeofthephysicalsolutioncan
beestablished. Resultsoftime-accuraterigidsimulationsarenotpresentedinthecurrentpaper.
B. UnsteadyAeroelasticAnalyses
For aeroelastic analyses, the flow equations were solved in a time-accurate mode18 coupled with a structural solver,
allowing the grid to deform. For this work, this coupling was performed internal to the FUN3D code19 using a
modalrepresentationofthestructuraldynamics. ThismodalsolverwasformulatedandimplementedinFUN3Dina
mannersimilartootherNASALangleyaeroelasticcodes(CAP-TSD20 andCFL3D21).FortheBSCWcomputations
presentedhere,thestructuralmodeswereobtainedvianormalmodesanalysis(solution103)withtheFiniteElement
Model(FEM)solverinMSCNastranTM.22 Themodesweretheninterpolatedtothesurfacemeshusingthemethod
developedbySamareh.23 TheBSCWFEMisdescribedinreference13.
Tosolvetheunsteadyflowequations, theFUN3Dsolveremploysadual-time-steppingmethod, whichiswidely
used in CFD. This method involves adding a pseudo-time derivative of the conserved variables to the physical time
derivativethatappearsintheNavier-Stokesequations. Subiterationsarerunwithineachphysical(global)timestep
to converge the flow solution. Convergence here refers to the process where the pseudo-time derivative vanishes as
thesubiterationsproceedwithineachphysicaltimestep. Attheendoftheiterativeprocess,asolutiontotheoriginal
unsteadyNavier-Stokesequationsisobtainedforthatphysicaltimestep. Foraninfinitelylargephysicaltimestep,the
dual-timesolverbecomesidenticaltothesteady-statesolver,thoughofcourse,alltimeaccuracyislost.
Aeroelasticanalysisrequiresagriddeformationcapability. ThegriddeformationinFUN3Distreatedasalinear
elasticityproblem.Inthisapproach,thegridpointsnearthebodycanmovesignificantly,whilethepointsfartheraway
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maynotmoveatall. Fortheaeroelasticsimulationsinthispaper,apredictor-correctorcouplingmethodbetweenthe
flow solution and the structural solution has been used. Within each physical time step, the flow field solution is
convergedusingthesubiterationsandsubsequentlycoupledwiththestructuralequations. Thiscouplingoccursonly
onceforeachphysicaltimestep. Thecouplingisdescribedfurtherandexploredindetailinreference2.
1. StaticAeroelasticAnalysis
The static aeroelastic simulation procedure begins from the last point in the rigid steady solution at identical Mach
numberandunloaded(rigid)angleofattack.StaticaeroelasticsimulationsareobtainedbyrestartingtheCFDanalysis
from this rigid steady solution. A high value of structural damping (0.9999) is used to allow the structure to find
the equilibrium position with respect to the mean flow. The fluid-structure coupling provides a step input to the
system,initiatingthemigrationfromtherigidsolutiontothecoupledstaticaeroelasticsolution. Ingeneral,thestatic
aeroelasticanalysisisperformedusinglargertimestepsandfewersubiterationsthanhavebeenshowntoberequired
foratime-accuratetemporally-convergedsolution. Thisissueisbrieflyexploredlaterinthispaper.
2. DynamicAeroelasticAnalysis
Dynamic (flutter) analysis is performed in a multistep process, with the dynamic solution started from the static
aeroelasticsolutionatcorrespondingMachnumber, angleofattackanddynamicpressure, andsettingthestructural
damping value to zero. An excitation is provided to the system by adding an initial condition to the generalized
(modal) velocities. For all results presented in the current paper, identical values of initial generalized velocity are
applied simultaneously to both the plunge and pitch modes. The effects of the initial excitation value on the flutter
solutionarediscussedlaterinthispaper.
Thewingresponse,intheformofthetimevaryingpitchangle,iscomputedandalogarithmicdecrementmethod
isusedtocalculatethedampingratio. Forastablesolution,thedampingratioisgreaterthanzero,andforanunstable
solution, thedampingratioislessthanzero. Todeterminetheflutteronsetdynamicpressure, linearinterpolationis
performed to find the zero damping case using the two cases on either side of the stability boundary (that is, using
the two dynamic pressures where the sign of the damping changes in between them). The flutter frequency is then
calculatedusingthesameinterpolationcoefficients.
Each simulation performed has a different number of time steps and each case has different transient behavior.
These two factors mean that the uncertainty associated with each calculated damping value will be different. In
the current paper, the uncertainty is captured by standard error values, which are presented along with the damping
valuesinsomecases. Thestandarderroroftheestimateofthedampingvalueiscomputedbasedonthelogarithmic
decrementprocessinformation.Thelogarithmicdecrementtechnique,asapplied,performsalinearfittothelogarithm
oftheextremevaluesoftheoscillations(thatis,thepeaksandvalleysofthecycles). Thedampingiscomputedasin
equation1,wheremistheslopeofthelinearestimateandω istheestimateofthefrequencyinradians.
m
ζ =− (1)
ω
The standard error estimate of the slope from a linear regression model is calculated as in equation 2, where x
i
arethevaluesoftheindependentvariable(time), andy arethecorrespondingvaluesofthedependentvariable(the
i
logarithm of the extreme values of the pitch angle). yˆ are the estimated values of the dependent variable computed
i
usingthelinearregressionmodel(thatis,usingtheslopeandinterceptcomputedfromtheoriginaldatapairs(x,y)).
i i
Nisthenumberofdegreesoffreedom(thenumberofpeaksandvalleysusedinthelogarithmicdecrementprocess).
Notethatthenumberofdegreesoffreedomisreducedby2,since2parameters(slopeandintercept)wereestimated
toformulatethesampleestimates,yˆ.
i
(cid:113)
∑Ni=1(yi−yˆi)2
SE = N−2 (2)
m (cid:112)
(x −x¯)2
i
Thestandarderrorestimateofthedampingvalueisthencomputedfromthestandarderrorestimateoftheslope,
assigningnoerrorinthecalculationofthefrequency,equation3.
SE =SE /ω (3)
ζ m
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IV. SimulationResults
Rigid, static aeroelastic and dynamic aeroelastic analyses were conducted at Mach 0.74 to investigate the influ-
ence of separation onset. The angle of attack was was varied from −1◦ to 8◦ and different excitation levels were
appliedtothesystem. Multipledynamicpressures,eachanalyzedthroughanindependentsimulation,wereanalyzed
atcombinationsofangleofattackandexcitationlevel.
A. RigidSteadySimulations
Resultsfromrigidsimulationsarepresentedinthissection. Theseresultswillshowthattheflowfieldcharacteristics
changewhentheairfoilexceeds6◦. Thischangeisshownthroughsurfaceandvolumetricplots(Figure5),integrated
liftcoefficients(Figure6),andpressurefieldslices(Figures7and 8).
In the rigid steady simulations, as the angle of attack increases, the region of flow separation increases and the
shockstructuremovesforwardasshowninFigure5. Therigidsteadysimulationsproduceconvergedresultsforthe
simulationsbelowthecriticalangleofattackof6◦;abovethisangle,thesolutionmeandersratherthanconverges.This
meanderingbehaviorisillustratedinFigure6,whichshowsthedevelopmentoftheliftcoefficientfordifferentangles
ofattackastheiterationsproceed. InFigure6,theiterationhistoriescorrespondingtothehighesttwoanglesofattack
(α =6.5◦ and 8◦) do not converge to fixed values. This nonconvergent behavior corresponds to conditions where
0
theflowfieldisextensivelyseparatedandwhichareoutsidetherangewherewethinkthatURANSanalysisproduces
credibleresults. Aspreviouslynoted,experimentalinformationdoesnotexistforthesehigheranglesofattack,sothis
latterassessmentisadmittedlymadewithoutbeingabletobenchmarkthecomputationalresults.
(a) 0◦ (b) 3.15◦ (c) 5◦
(d) 6.5◦ (e) 8.0◦
Figure 5. Skin friction surface plots from rigid analysis and Mach contour plots at splitter plate; Snapshots at end of
simulation.
Figure7showsthepressurecoefficientversusnondimensionalchordlocationatthe60%spanstationonboththe
upperandlowersurfacesforthesesameanglesofattack.Intheseplots,andallpressurecoefficientplotspresented,the
verticalaxisisinverted. Asshowninthefigure,theshockmovesaftwithincreasingangleofattackforanglesbelow
6.5◦ and then reverses direction as the angle of attack is further increased. The upper surface shock forms between
1.15◦ and2.4◦; theshockcontinuestogainstrengthandmoveaftuntil6◦. Forhigherangles,theshockthenmoves
forwardwithincreasingangleofattack,stillstrengthening.
Aftoftheshock,asmallregionofseparationisindicatedforanglesofattackof5◦andhigher,asillustratedinthe
skinfrictionplotsinFigure5. Intheseplots,skinfrictionisshownonthesurfaceofthewing,withorangeindicating
positiveskinfriction(attachedflow)andblueindicatingnegativeskinfriction(separatedflow). Ontheuppersurface,
theplotsfor3.15◦andbelowindicatenoseparatedflowregion. Forhigherangles,theblueregionsindicatetheextent
oftheseparatedflowregion. Inthesepictures,aslicethroughtheflowfieldisalsoshownattheintersectionbetween
thesplitterplateandthewingroot. ThecontoursonthissliceshowthelocalMachnumber, emphasizingtheupper
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Figure6. Rigidsimulationasiterationsproceed;Mediumgrid.
surface shock location through an abrupt color change, generally from red to purple. The shock extends across the
wing,withasmallregionofseparationbehindtheshockforcasesof5◦andabove.Astheangleofattackincreases,the
figuresshowthatnearthesplitterplate, theshockprogressestowardtheleadingedgeandisstrengthened, indicated
by the progression of the color shown in the shock from red to orange to yellow. Near the splitter plate, there is
noobservedreversalofthedirectionofshocktravel,buttheshockappearstoweakennearthesplitterplateforcases
above6.5◦.ThisisincontrasttothepressuredatashowninFigure7atthe60%spanstation.Inaddition,theseparated
flowregionshownnearthesplitterplateinFigure5isverysmallinextentinalldirections. Startingat6◦,however,
thereisalargerregionofseparatedflow, shownbythegrowthoftheblueregiononthewingastheangleofattack
increases.
TheplotsinFigure5alsoindicatetheeffectofthetipvortex. Thegreystreamersintheflowarestreaklines, as
aretheblacklinesonthesliceatthesplitterplate. Studyofthestreaklinesshowsthattheflowovertheaftportionof
thewinguppersurfaceflowsinboard,aswellasstreamwise. Onthelowersurface,thestreaklinesshowthattheflow
movesoutboardaswellasstreamwise. Althoughtheinfluenceofthetipvortexhasnotbeenexaminedindetailinthe
currentwork,limitedstudieswereconductedatα =5◦onatwo-dimensionalairfoilsection. Thisworkisincomplete
at this point in time and the preliminary results are not presented here, but it was evident that the two-dimensional
resultswerequalitativelydifferentfromthethree-dimensionalresults.
(a) Uppersurface (b) Lowersurface
Figure7. Finaltimepointsnapshotsofpressurecoefficientsat60%spanstation;Mediumgrid;Rigidsteadyanalysis.
Comparison of the computational results with experimental results is shown in Figure 8 for selected angles of
attack where experimental quantitative information is available. The computational results were taken as snapshots
at the last time point from each simulation and are shown by the lines; experimental information is represented by
the symbols. The grey shaded regions show the range of the experimental pressure data at the flutter point. The
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computations predict the upper surface shock to be stronger and further aft than indicated by the experimental data.
ThistrendisconsistentwiththetrendsobservedintheAePWdatasets.4,11 Althoughthepreviousworkhasidentified
thelowersurfaceaftregionasbeingmispredicted,thistrendisn’tsomuchinevidenceinthecurrentcalculations.
(a) Uppersurface (b) Lowersurface
(c) Uppersurface,zoomed
Figure8. Pressurecoefficientsat60%spanstation;Rigidsteadycase.Symbols:Experiment.ColoredLines:Analysis.The
greyregionshowstherangefortheflutterinformationfromtheexperiment;Symbolsshowthemean.
The lift and pitching moment coefficients are shown as functions of angle of attack in Figure 9 for the medium
grid. Boththeliftandthepitchingmomentcoefficientshowlineardependenceontheangleofattackuntil6◦,where
dramaticchangesinbothcurvesareseenandstallisindicated.
TheBSCWairfoilisasupercriticalairfoil(SC(2)-0414),soitwasdesignedwiththegoalofbeingshock-freeat
a given lift coefficient; the design lift coefficient for this airfoil is 0.4. Note that at the Mach number of interest for
this study, 0.74, a lift coefficient of 0.4 is produced by an angle of attack near 2◦. The pressure coefficient plots in
Figure7showthattheshockformsbetween1.15◦and2.4◦,correlatingwellwiththedesignedbehavioroftheairfoil.
Thecoefficientvaluespresentedarethefinaltimepointofeachsimulation, notthemeanvaluenortherangeofthe
coefficients. Statistical values should be obtained from time-accurate simulations to better capture the behavior that
themethodispredicting.
Theaerodynamiccenterandthecenterofpressurewerecomputedfromthecoefficientdata.Thecenterofpressure
isshowninFigure9c,measuredpositiveaftwiththeoriginatthewingleadingedge. Thewingchord,16inches,was
used as the reference length. Thus, the trailing edge location is at a value of 1 in the figure. The center of pressure
wascalculatedfromtherigiddata,usingequation4afterapplyingtheparallelaxistheoremtoreferencethepitching
moment coefficient to the wing leading edge. (The pitching moment coefficient is defined as positive leading edge
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(a) Liftcoefficient (b) Pitchingmomentcoefficient,referencedtothemid-chord
(c) Centerofpressurelocation,relativetowingleadingedge,normal-(d) Aerodynamiccenterlocation,relativetowingleadingedge,nor-
izedbychord malizedbychord
Figure9. Steadyrigidsimulationresults;Mediumgrid.
up. Theliftcoefficientisdefinedaspositiveupward. Thecenterofpressurelocationasdefinedinequation4isthus
positiveaftfromtheleadingedge.)
C
m
x¯ =− (4)
cp
C
L
Thecenterofpressureisaftofthetrailingedgeoftheairfoilfornegativeanglesofattack,asshowninFigure9c.At
0◦angleofattack,thecenterofpressureisaftofthemidchordandmovesforwardastheangleofattackincreases.The
forward-mostpositionofthecenterofpressureoccursat6◦ andisat34.6%chordbehindtheleadingedge. Further
increases in angle of attack result in moderate migration of the center of pressure aftward. The center of pressure
migration is similar to the movement measured and reported in the earliest of airfoil tables for asymmetric airfoils,
particularlyparallelingthosewithlowersurfacecuspregionsintheaftportionoftheairfoil.24
Theaerodynamiccenterwasalsocalculatedfromtherigidsteadydatasetsusingequation5.
C
x¯ =− mα (5)
ac
C
Lα
TheresultsareshowninFigure9d. Thepitchingmomentcoefficientswerecalculatedrelativetotheleadingedge,
thus the location values given are also with respect to the leading edge. In the range where C and C are linear
L m
functions of the angle of attack, the aerodynamic center location is constant (21.5% chord aft of the leading edge).
Intheregionwherethebehaviorsofthecoefficientsarenonlinear,theaerodynamiccenterlocationshiftserratically.
Calculationsatarefinedsetofanglesofattackinthenonlinearresponserangewouldberequiredtobettercharacterize
thismovement. Theaftshiftintheaerodynamiccenterisproducedbyseparatingflow,whichwillbeshowninalater
sectionofthispapertohaveadestabilizingeffectonthesystem.
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