NASA Technical Reports Server (NTRS) 20170001315: DPW-VI Results Using FUN3D with Focus on k-kL-MEAH2015 (k-kL) Turbulence Model

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by  NASA Technical Reports Server (NTRS)

DPW-VI Results Using FUN3D with Focus on
k-kL-MEAH2015 (k-kL) Turbulence Model
K.S.Abdol-Hamid,∗Jan-Renee´ Carlson,†ChristopherL.Rumsey,‡
ElizabethM.Lee-Rausch,§andMichaelA.Park¶
NASALangleyResearchCenter,Hampton,VA
TheCommonResearchModelwing-bodyconfigurationisinvestigatedwiththek-kL-MEAH2015turbu-
lencemodelimplementedinFUN3D.ThisincludesresultspresentedattheSixthDragPredictionWorkshop
andadditionalresultsgeneratedaftertheworkshopwithanonlinearQuadraticConstitutiveRelation(QCR)
variantofthesameturbulencemodel. Theworkshopprovidedgridsareused,andauniformgridrefinement
studyisperformedatthedesigncondition. Alargevariationbetweenresultswithandwithoutareconstruc-
tionlimiterisexhibitedon“medium”gridsizes,indicatingthatthemediumgridsizeistoocoarsefordrawing
conclusionsincomparisonwithexperiment.Thisvariationisreducedwithgridrefinement.Atafixedangleof
attackneardesignconditions,theQCRvariantyieldeddecreasedliftanddragcomparedwiththelineareddy-
viscositymodelbyanamountthatwasapproximatelyconstantwithgridrefinement. Thek-kL-MEAH2015
turbulencemodelproducedwingrootjunctionflowbehaviorconsistentwithwindtunnelobservations.
Nomenclature
area halfwingreferencearea Re Reynoldsnumber
ref
b wingsemispan
ref Conventions
c localchord
CFD ComputationalFluidDynamics
C dragcoefficient
D
CFL Courant-Friedrichs-Lewy
C liftcoefficient
L NAS NASAAdvancedSupercomputing
C pitchingmomentcoefficient
m NTF NationalTransonicFacility
C x-componentofskinfrictioncoefficient
fx QCR QuadraticConstitutiveRelation
c meanaerodynamicchord
ref RANS Reynolds-averagedNavier-Stokes
h characteristicgridspacing
C surfacepressurecoefficient Symbols
p
M∞ freestreamMachnumber α angleofattack,degree
N numberofnodesingrid η wingspanlocation
∗SeniorResearchScientist,ConfigurationAerodynamicsBranch,MS499,AssociateFellow.
†ResearchScientist,ComputationalAeroSciencesBranch,MS128,SeniorMember.
‡SeniorResearchScientist,ComputationalAeroSciencesBranch,MS128,Fellow.
§AssistantBranchHead,ComputationalAeroSciencesBranch,MS128,AssociateFellow.
¶ResearchScientist,ComputationalAeroSciencesBranch,MS128,SeniorMember.
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I. Introduction
THEAIAAAppliedAerodynamicsTechnicalCommittee(APATC)conductedtheir6thDragPredictionWorkshop
(DPW-VI)ainthesummerof2016tocontinuetheevaluationofCFDtransoniccruisedragpredictionsforsubsonic
transports.Thestatedobjectivesoftheworkshopwereasfollows:1)tobuildonthesuccessofthepastfiveAIAADrag
Prediction Workshops (DPW-I–V); 2) to assess the state-of-the-art computational methods as practical aerodynamic
tools for aircraft force and moment prediction of industry relevant geometries; 3) to provide an impartial forum for
evaluatingtheeffectivenessofexistingcomputercodesandmodelingtechniquesusingNavier-Stokessolvers;and4)
toidentifyareasneedingadditionalresearchanddevelopment. ThefocusofthisworkshopwastheNASACommon
ResearchModel(CRM)withwind-tunnelmeasuredwingtwist;bothwing-body(WB)andwing-body-nacelle-pylon
(WBNP) configurations were considered. CFD predictions of absolute and incremental force and moment values
were examined and compared. The workshop included grid convergence and code verification studies as well as an
angle-of-attack sweep with static aeroelastic deformations. As with prior workshops, grids were made available for
allrequiredcases.
ResultsfortheDPW-VIrequiredcasestudiesweresubmittedtotheworkshopfortheFUN3D1–4unstructured-grid
Reynolds-averagedNavier-Stokes(RANS)solveronasetofworkshop-suppliednode-basedmixed-elementmeshes.
FUN3Dhasbeenusedinpreviousworkshopstudies,includingDPW-IV5 andDPW-V.6 ThepriorFUN3Dworkshop
studiesfocusedontheuseofthestandardSpalart-Allmaras(SA)turbulencemodel.FortheDPW-IVgridconvergence
studyatthedesignliftcondition,FUN3Dtotalforce/momentpredictionswiththeSAmodelwerewithinonestandard
deviationoftheworkshopcoresolutionmedians. TheDPW-IVdownwashstudyandReynoldsnumberstudyresults
also compared well with the range of results shown in the workshop presentations. Similarly, the FUN3D results
fromtheDPW-VgridconvergencestudycomparedcloselywithresultsfromothercodeswhenusingtheSAmodel.
However,resultsfromtheDPW-Vbuffetstudyproducedalargervariationthanthedesigncaseprimarilyduetothe
large differences in the predicted side-of-body separation. Park et al.6 summarize the DPW-IV methods and size of
the simulated side-of-body separations. They also studied the impact of modeling differences and grid effects on
separationextentinthecontextofDPW-V.Alargewing-rootseparationbubblewasnotobservedinthewind-tunnel
tests.
FUN3D results for DPW-VI were submitted for the SA turbulence model and for a recently developed k-kL-
MEAH2015turbulencemodel.7 Thek-kL-MEAH2015modelhasbeenimplementedinFUN3Dinalooselycoupled
manner. Forbrevity,wewillrefertok-kL-MEAH2015ask-kL,herein. Theimplementationofk-kLinbothCFL3D
and FUN3D was verified in Ref. 7. The results were compared with theory and experimental data, as well as with
results that employ the SST turbulence model.8 They demonstrated that the k-kL model has the ability to produce
resultssimilarorbetterthantheSSTmodelincomparisonwithexperiment.Forexample,foraseparatedaxisymmetric
transonicbumpvalidationcase,thesizeoftheseparationbubble(separationandreattachmentlocations)predictedby
thek-kLmodelisclosertoexperimentalmeasurements.
As in prior workshops, the FUN3D SA results from the DPW-VI test cases compared closely with results from
othercodes whenusing theSA modelb, and sothe currentpaperwill focusonly ona subsetofDPW-VI caseswith
the k-kL model. The current study will focus on the effects of limiter and turbulence model formulation on the
computationalresultsoftheCRMWBconfiguration. Theresultswillincludeaconstant-liftgridconvergencestudy
fromtheworkshopaswellasanadditionalconstantangle-of-attackgridconvergencestudy.Anangle-of-attacksweep
withstaticaeroelasticdeformationswillbeconsideredandcomparisonswillbemadewiththeexperimentaldata.
II. CommonResearchModel
The CRMc is a full-span wing body configuration with optional horizontal tail and optional nacelle/pylon. It
is designed to be representative of a contemporary high performance transonic transport.9 The derived reference
quantitiesofthefull-scalevehiclearesummarizedinTable1,whichcorrespondtothegeometryandgridsprovidedby
theDPWcommittee. TheCRMWB(nonacelle/pylon)wasanalyzedwithandwithoutahorizontaltailinDPW-IV.10
TheCRMWBwithoutboththenacelle/pylonandhorizontaltailwasthefocusofDPW-V.11ThefocusofDPW-VIis
ontheWB(notail)withandwithoutthenacelle/pylon.
AnexperimentalaerodynamicinvestigationoftheNASACRMhasbeenconductedintheNASALangleyNational
TransonicFacility12andintheNASAAmes11-ftTransonicWindTunnel.13 Classicalwallcorrectionsaccountingfor
ahttps://aiaa-dpw.larc.nasa.gov[retrieved9/12/2016].
bhttps://aiaa-dpw.larc.nasa.gov[retrieved9/12/2016].
chttp://commonresearchmodel.larc.nasa.gov[retrieved8/12/2016].
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modelblockage, wakeblockage, tunnelbuoyancy, andliftinterferencehavebeenappliedtotheexperimentaldata.d
Alargeoffsetinpitchingmomentbetweentheexperimentaldataandthefree-aircomputationalresultsfromDPW-IV
hasbeennoted.10 Subsequentcomputationalassessmentsofthemodelsupportsysteminterferenceeffectsindicated
thattheCRMpitchingmomentissensitivetothepresenceofthemountinghardware.14,15 Themodelsupportsystem
wasnotincludedintheDPW-IV,DPW-VorDPW-VIgridsystems.Additionally,theinvestigationsofRef.15alsoled
tothediscoveryofalargediscrepancybetweentheas-builtandtestedwindtunnelmodelwing-twistandtheDPW-IV
computational wing-twist. Hue16 showed that including the experimentally measured twist distribution reduced lift
andimprovedthecomparisonwithwind-tunnelmeasurements. Keyeetal.17 alsoappliedfluid-structurecouplingand
confirmedtheshiftinpredictedforcesandmomentduetowingtwist. Duetothedifferencesbetweenthewindtunnel
wing-twistandtheDPW-IVandDPW-Vgeometries,one-to-onecomparisonsbetweentheseworkshopcasesandthe
experimental data have been problematic. The CRM geometry for DPW-VI includes the static aeroelastic twist and
deformationexperiencedbythemodelatdifferentanglesofattack, butomitsthemodelsupportfeaturesandtunnel
walls.
Table1. ReferencegeometryfortheCRM.
Parameter Value
c ,meanaerodynamicchord 275.80inch
ref
area ,one-halfwingreferencearea 297,360inch2
ref
b ,semispan 1159.75inch
ref
X-MomentCenter 1325.90inch
Z-MomentCenter 177.95inch
AR,aspectratio 9.0
III. MethodDescription
FUN3D1–4isafinite-volumeRANSsolverinwhichtheflowvariablesarestoredatthenodesofthemesh.FUN3D
solvestheflowequationsonmixedelementgrids,i.e.,tetrahedra,pyramids,prisms,andhexahedra. Atinterfacesde-
limitingneighboringcontrolvolumes,theinviscidfluxesarecomputedwithanapproximateRiemannsolverbasedon
the values on either side of the interface. The flux difference splitting method of Roe18 is used in the current study.
For second-order accuracy, interface values are obtained by a U-MUSCL scheme19,20 with gradients of the mean
flowequationscomputedatthemeshverticesusinganunweightedleast-squarestechnique. TheU-MUSCLscheme
coefficient is set to 0.0 for purely tetrahedral grids and 0.5 for grids with mixed element types. Several reconstruc-
tionlimitersareavailableinFUN3D,twoofwhichareusedinthecurrentstudy. ThedimensionalVenkatakrishnan
limiter21 is scaled to the mean aerodynamic chord to have the same behavior as the airfoil example with unit chord
inVenkatakrishnan.21 Inthepresentthree-dimensionalanalysis, asmoothlimitercoefficientisusedbasedon3/c
ref
wherec isthemeanaerodynamicchordoftheconfiguration. ThislimiterisreferredtoastheVenkatlimiter. Addi-
ref
tionally,thecurrentstudyusesastencil-basedmin-modlimiter22 augmentedwithaheuristicpressurelimiter.23 This
limiterisreferredtoastheh-minmodlimiter. Computationsarealsoperformedwithnolimiter.
For tetrahedral meshes, the full viscous fluxes are discretized with a finite-volume formulation in which the re-
quiredvelocitygradientsonthedualfacesarecomputedwiththeGreen-Gausstheorem. Ontetrahedralmeshesthisis
equivalenttoaGalerkintypeapproximation. Fornontetrahedralmeshes,thesameGreen-Gaussapproachcanleadto
odd-evendecoupling.Apureedge-basedapproachcanbeusedtocircumventtheodd-evendecouplingissuebutyields
onlyapproximateviscousterms.Fornontetrahedralmeshes,theedge-basedgradientsarecombinedwithGreen-Gauss
gradients;thisimprovestheh-ellipticityoftheoperatorandallowsthecompleteviscousstressestobeevaluated.2,24
ThisformulationresultsinadiscretizationofthefullNavier-Stokesterms.
The solution at each time step is updated with a backwards Euler time-integration scheme. At each time step,
thelinearsystemofequationsisapproximatelysolvedwithamulticolorpoint-implicitprocedure.25 Localtime-step
scalingisemployedtoaccelerateconvergencetosteadystate. Forturbulentflows,avarietyofturbulencemodelsare
availablewithinFUN3D.Thek-kLmodel7usedinthisstudyisalineareddyviscositytwo-equationturbulencemodel,
which is solved in a loosely coupled approach with the mean-flow equations. The nonlinear Quadratic Constitutive
dhttp://commonresearchmodel.larc.nasa.gov[retrieved10/13/2016].
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Relation(QCR)developedbySpalart26 willalsobeusedtocomputetheReynoldsstressterms. Thiswillbereferred
toas+QCRnonlinearoptionusedbyk-kL.7 Thediscretizationoftheturbulencemodeldiffusiontermsishandledin
thesamefashionasthemean-flowviscousterms.
IV. FocusCasesfromDPW-VIandResults
In the present paper, we will cover the effect of limiter and turbulence model formulations in the computational
resultsofCRMWBconfigurationforanumberoftestcasesatM =0.85andRe=5million:
∞
1. CRMgridconvergencestudy(C =0.5): CRMwing-body(WB)configurationscomputedwithk-kLincluding
L
reconstructionlimitereffects;
2. CRM grid convergence study (α = 2.75 degree): CRM wing-body (WB) configurations computed with k-kL
includingreconstructionlimitereffects;
3. CRMWBstaticaeroelasticeffect: Angle-of-attacksweepcomputedwithk-kL,bothlineareddy-viscosityand
QCRvariations.
ThefirstandthirdcasesarebasedontherequiredcasesfromDPW-VI.Thesecondcaseisatraditionalgridconver-
gencestudywhereallflowconditionsarefixed,includingangleofattack.
The current FUN3D solutions were computed on mixed-element unstructured grids provided by the DPW com-
mittee. These grids were generated using the gridding guidelines specified for the workshop.e Table 2 lists the grid
sizesforthefamilyofgridsusedforthegridconvergencestudy.Thesegridshaveprismaticcellsnearthesolidsurface
of the wing/body and tetrahedral cells away from the wall with a small number of pyramids in the interface region
betweentheprismsandtetrahedralcells. Table3liststhegridsusedfortheangle-ofattacksweeps. Thesegridsareat
themediumgridlevelandhaveasimilartopologyasthosecreatedforthegridconvergencestudy. Recallthatthese
grid geometries include the static aeroelastic twist and deformation experienced by the model at different angles of
attack,butomitthemodelsupportfeaturesandtunnelwalls. Alltheforceandmomentcoefficientspresentedherein
are the total value (combined pressure and viscous components) for the wing and body. Computed results will be
comparedwithexperimentaldataforC ,C C andC run197R44f fromtheNTFtestatNASALangleyResearch
L D m P
Center. WewillalsocompareCFDwithexperimentaldata27 thatweregeneratedwithapressure-sensitivepaintmea-
suringtechniquedoneattheNASAAmes11-fttransonictunnel.Asmentionedearlier,theCFDmodeldidnotinclude
supportsystemortunnelwalls.
Table2. SummaryofDPW-VIWBgridsforgridconvergencestudies.
Level N
Tiny 20,472,098
Course 29,916,005
Medium 44,249,828
Fine 66,228,067
X-Fine 100,781,934
U-Fine 151,316,926
ehttps://aiaa-dpw.larc.nasa.gov[retrieved9/12/2016].
fhttp://commonresearchmodel.larc.nasa.gov[retrieved8/12/2016].
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Table3. SummaryofDPW-VIWBgridsforangle-of-attacksweep.
α,degrees N
2.50 44,154,687
2.75 44,249,828
3.00 44,174,034
3.25 44,180,700
3.50 44,242,617
3.75 44,217,067
4.00 44,238,097
A. ConstantLiftGridConvergenceStudy
One of the required cases computed for DPW-VI is a grid convergence study performed at the design condition of
M =0.85,Re=5millionandC =0.5fortheWBconfiguration. Eachcaseisstartedatα=2.75◦togetinitially
∞ L
converged solutions. Then, the angle of attack is automatically adjusted to reach the target lift coefficient within
±0.0001. Figure 1 shows a typical total lift and drag coefficient convergence history through the two-step solution
process. FUN3Dtakesapproximately6000iterationstocompletetheentireprocess,targetingC =0.5.
L
C
L
C
D
0.7 0.05
0.6 0.04
L D
C C
0.5 0.03
0.4 0.02
0 2000 4000 6000
Iteration
Figure1. Typicalliftanddragconvergence,M =0.85,Re=5million,C =0.5,k-kLturbulencemodel.
∞ L
DragcoefficientC andpitchingmomentcoefficientC areplottedasafunctionofacharacteristicgridspacing
D m
squared, h2, inFigs.2(a)and(b), respectively. Theh2 iscomputedasthe(numberofnodes=N)raisedtothe−2/3
power. This exponent is based on two assumptions: (1) the characteristic length of the grid spacing varies with the
cuberootofthecellvolume,and(2)thesolutionlieswithinthe“asymptoticrange”ofgridconvergenceanditsspatial
errordecreaseswith2nd-orderaccuracy. Whentheseassumptionsaremet,thecomputedoutputsshouldvarylinearly
withh2.Onthefinergrids,thedifferencebetweennolimiterandh-minmodgenerallydiminishesforthevaluesshown
inFig.2. TheC valuesofthefinestgridwithnolimiterandh-minmodarewithinonecount,0.0001. TheC values
D m
ofthefinestgridwithnolimiterandh-minmodarewithin0.001. TheangleofattackrequiredforC = 0.5isalso
L
plottedasafunctionofh2inFig.2(c). Thistrimangleofattackdecreasesforbothmethodsasthegridisrefined. This
decreaseinangleofattackwithgridrefinementatconstantcoefficientofliftisanalogoustoanincreaseincoefficient
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ofliftwithgridrefinementatconstantα. Theh-minmodlimiterreducesthedifferencesbetweentinyandultragrids
bymorethan50%(comparedtonolimiter)forallthevaluesplottedinFig.2.
No limiter
No limiter
0.0265 0.07 h minmod limiter
h minmod limiter
0.0260 0.08
D m
C C
0.0255 0.09
0.0250 0.10
0.000000 0.000005 0.000010 0.000015 0.000000 0.000005 0.000010 0.000015
h2 h2
(a)Coefficientofdrag. (b)Coefficientofpitchingmoment.
2.80
No limiter
h minmod limiter
2.75
2.70
)
2.65
(
2.60
2.55
2.50
0.000000 0.000005 0.000010 0.000015
h2
(c)Angleofattack,αdegrees.
Figure2. Theeffectsoflimiterasafunctionofcharacteristicgridspacing,C =0.5,k-kLturbulencemodel.
L
B. ConstantAngle-of-AttackGridConvergenceStudy
An alternate grid convergence study fixes geometry and all flow conditions (independent variables) to evaluate the
effectofgridinpredictingflowquantitiessuchassurfacepressurecoefficientandtotalforcecoefficients(dependent
variables). ThegridsprovidedbytheDPW-VIcommitteearebasedonthegeometryatα = 2.75◦,whichiscloseto
designconditionfortheCRMWBconfiguration.
The first set of results evaluates the effect of limiters when using the basic linear k-kL turbulence model. Drag,
lift, and moment coefficients as well as surface pressure coefficients at different locations on the wing are assessed.
Thebasicno-limiterresultsarecomparedtotheVenkatandtheh-minmodlimiterresults. Thelift,dragandmoment
coefficientsareplottedasafunctionofh2 inFigs.3(a), (b)and(c), respectively. Onthefinergrids, the differences
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betweennolimiter,Venkatlimiter,andh-minmodlimiterdiminish. TheC valuesonthefinestgridarewithinabout
D
five counts, 0.0005, for the various methods. The C values of the finest grid are within about 0.008, and the C
L m
values of the finest grid are within about 0.005. As the grid is refined, C and C increase for either no limiter or
D L
h-minmodanddecreasefortheVenkatlimiter. TheoppositeoccursforC . Inotherwords,theVenkatlimiterhasa
m
differenttrendthantheotherapproaches. Thisdifferenttrendbetweenthelimiterswithgridrefinementcauses10%
differencesbetweentheresultsatthemediumgridlevel(h2 ≈ 0.000008). Themediumgridlevelisusuallybuiltto
computetherequiredcasesclosetodesigncondition(α=2.50◦−4.00◦)withan“industry-standard”acceptablelevel
ofaccuracy. However,thelargeinfluenceofthelimiteronthemediumgridlevel(over44millionnodes)indicatesthe
likelihoodthatthisgridisstillnotyetfineenoughformanyengineeringpurposes.
No limiter No limiter
0.030 h minmod limiter 0.60 h minmod limiter
Venkat limiter Venkat limiter
0.58
0.029
0.56
0.028
D L 0.54
C C
0.027
0.52
0.026
0.50
0.025 0.48
0.000000 0.000005 0.000010 0.000015 0.000000 0.000005 0.000010 0.000015
h2 h2
(a)Dragcoefficient. (b)Liftcoefficient.
No limiter
h minmod limiter
Venkat limiter
0.08
m
C
0.10
0.12
0.000000 0.000005 0.000010 0.000015
h2
(c)Pitchingmomentcoefficient.
Figure3. Theeffectsoflimiterasafunctionofcharacteristicgridspacing,α=2.75◦,k-kLturbulencemodel.
Next, chordwisesurfacepressurescoefficientsfromthemediumandultragridsolutionsareexamined. Figure4
showssurfacepressurecoefficientsatthreespanlocations: η =0.131,0.502,and0.95. Figures4(a),(c)and(e)show
results from the medium grid solutions, whereas (b), (d) and (f) show results from the ultra grid solutions. At the
mediumgridlevel,thenolimiterandh-minmodlimitergivesimilarresultsatallstations,whicharedifferentthanthe
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VenkatlimiterresultsasshowninFig.4(a),(c)and(e).Overall,bothnolimiterandh-minmodlimiterresultsaremuch
closertotheexperimentalpressuredatag ascomparedwithVenkatlimiterresults. However,thisdoesnotnecessarily
meanthattheyaremoreaccurate,becausenumericalerrorsduetoinsufficientgridrefinementmaystillbeplayinga
largerole. Infact,allthreeapproachesaremuchclosertoeachotherattheultragridlevelasshowninFigs.4(b),(d)
and(f). Thedifferenceisslightlywideratη =0.95towardthewingtipbutdiminishedfromthemediumgridlevels.
ThesurfacepressureresultsareconsistentwithwhatwasdiscussedforC andC inFig.3. Attheultragridlevel,
D L
C andC arewithinlessthan1%betweenalltheapproaches.
D L
The basic (linear) k-kL results are compared with (nonlinear) k-kL+QCR turbulence model results in Fig. 5.
Thesesolutionswerecomputedwithoutalimiter. ThedragcoefficientC ,liftcoefficientC ,andmomentcoefficient
D L
C are plotted as a function of h2 in Figs. 5(a), (b), and (c), respectively. The variation in C , C , and C with
m L D m
gridrefinementshowssimilartrendsforbothk-kLandk-kL+QCR.Thedifferencesareveryconsistentacrossthegrid
levelswithk-kL+QCRresultinginasmallerC andC andhigher(lessnegative)C thanthebasick-kLmodel.
L D m
ThecorrespondingsurfacepressurecoefficientsfromthemediumandultragridsolutionsareshowninFig.6at
η =0.131,0.502and0.95. Figures6(a),(c)and(e)aretheresultsfromthemediumgridsolutions. Figures6(b),(d)
and(f)aretheresultsontheultragridlevel. Overall,resultsfromthelinearandnonlinearturbulencemodelsarevery
similarforη =0.131and0.502,withaslightdifferenceattheη =0.95station.
ghttp://commonresearchmodel.larc.nasa.gov[retrieved8/12/2016].
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No limiter No limiter
1.5 h minmod limiter 1.5 h minmod limiter
Venkat limiter Venkat limiter
= 0.131, Experimental Data 0.131, Experimental Data
1.0 1.0
0.5 0.5
p p
C C
0.0 0.0
0.5 0.5
1.0 1.0
0 0.5 1 0 0.5 1
x / c x / c
(a)η=0.131,mediumgrid. (b)η=0.131,ultragrid.
No limiter No limiter
1.5 h minmod limiter 1.5 h minmod limiter
Venkat limiter Venkat limiter
= 0.502, Experimental Data 0.502, Experimental Data
1.0 1.0
0.5 0.5
p p
C C
0.0 0.0
0.5 0.5
1.0 1.0
0 0.5 1 0 0.5 1
x / c x / c
(c)η=0.502,mediumgrid. (d)η=0.502,ultragrid.
No limiter No limiter
1.5 h minmod limiter 1.5 h minmod limiter
Venkat limiter Venkat limiter
= 0.95, Experimental Data 0.95, Experimental Data
1.0 1.0
0.5 0.5
p p
C C
0.0 0.0
0.5 0.5
1.0 1.0
0 0.5 1 0 0.5 1
x / c x / c
(e)η=0.95,mediumgrid. (f)η=0.95,ultragrid.
Figure4. Comparisonofchordwisesurfacepressurecoefficientdistributions,α=2.75◦,k-kLturbulencemodel.
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k kL k kL
0.030 k kL+QCR 0.60 k kL+QCR
0.58
0.029
0.56
0.028
D L0.54
C C
0.027
0.52
0.026
0.50
0.025 0.48
0.000000 0.000005 0.000010 0.000015 0.000000 0.000005 0.000010 0.000015
h2 h2
(a)Dragcoefficient. (b)Liftcoefficient.
0.07
0.08
0.09
m
C
0.10
0.11 k kL
k kL+QCR
0.12
0.000000 0.000005 0.000010 0.000015
h2
(c)Pitchingmomentcoefficient.
Figure5.TheeffectofQCRasafunctionofcharacteristicgridspacing,α=2.75◦,nolimiter,k-kLturbulencemodel.
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